Inquiry Driven Systems • Part 11

Author: Jon Awbrey

• Overview • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Part 6 • Part 7 • Part 8 • Part 9 • Part 10 • Part 11 • Part 12 • Part 13 • Part 14 • Part 15 • Part 16 • Appendix A1 • Appendix A2 • References • Document History •

Reflective Interpretive Frameworks (cont.)

We continue the discussion of formalization in terms of concrete examples and detail the construction of a reflective interpretive framework (RIF).

A RIF is a special type of sign-theoretic setting, illustrated in the present case by building on the sign relations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(\text{A})\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(\text{B}),\!} but intended more generally to form a fully-developed environment of objective and interpretive resources, in the likes of which an “inquiry into inquiry” can reasonably be expected to find its home.

We begin by presenting an outline of the developments ahead, working through the motivation, construction, and application of a RIF that is broad enough to mediate the dialogue of the interpreters ${\text{A}}\!$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{B}.\!} The first fifteen Sections (§§ 1–15) deal with a selection of preliminary topics and techniques that are involved in approaching the construction of a RIF. The topics of these sections are described in greater detail below.

The first section (§ 1) takes up the phenomenology of reflection. The next three sections (§§ 2–4) are allotted to surveying the site of the planned construction, presenting it from three different points of view. An introductory discussion (§ 2) presents the main ideas that lead up to the genesis of a RIF. These ideas are treated at first acquaintance in an informal manner, located within a broader cultural context, and put in relation to the ways that intelligent agents can come to develop characteristic belief systems and communal perspectives on the world.

The next section (§ 3) points out a specialized mechanism that serves to make inobvious types of observation of a reflective character. The last section (§ 4) takes steps to formalize the concepts of a point of view (POV) and a point of development (POD). These ideas characterize the outlooks, perspectives, world views, and other systems of belief, knowledge, or opinion that are employed by agents of inquiry, with especial regard to the ways that these outlooks develop over time.

A further discussion (§ 5), in preparation for the task of reflection, identifies three styles of linguistic usage that deploy increasing grades of formalization in their approaches to any given subject matter.

In the next three sections (§§ 6–8), the features that distinguish each style of usage are taken up individually and elaborated in detail. This is done by presenting the basic ideas of three theoretical subjects that develop under the corresponding points of view and that exemplify their respective ideals. The next three sections (§§ 9–11) take up the classes of higher order sign relations that play an important role in reflexive inquiries and then apply the battery of concepts arising with higher order sign relations to an example that anticipates many features of a realistic interpreter. In the light of the experience gained with the foregoing styles and subjects, the next three sections (§§ 12–14) are able to take up important issues regarding the status of theoretical entities that are needed in this work.

Finally (§ 15), the relevance of these styles, subjects, and issues is made concrete by bringing their various considerations to bear on a single example of a formal system that serves to integrate their concerns, namely, propositional calculus.

A point by point outline follows:

§ 1. An approach to the phenomenology of reflective experience, as it bears on the conduct of reflective activity, is given its first explicit discussion.

§ 2. The main ideas leading up to the development of a RIF are presented, starting from the bare necessity of applying inquiry to itself. I introduce the idea of a point of view (POV) in an informal way, as it arises from natural considerations about the relationship of an immanent system of interpretation (SOI) to a generated text of inquiry (TOI). In this connection, I pursue the idea of a point of development (POD), that captures a POV at a particular moment of its own proper time.

§ 3. A Projective POV

§ 4. The idea of a POV, as manifested from moment to moment in a series of PODs, is taken up in greater detail.

A formalization for talking about a diversity of POVs and their development through time is introduced and its consequences explored. Finally, this formalization is applied to an issue of pressing concern for the present project, namely, the status of the distinction between dynamic and symbolic aspects of intelligent systems.

§ 5. The symbolic forms employed in the construction of a RIF are found at the nexus of several different interpretive influences. This section picks out three distinctive styles of usage that this work needs to draw on throughout its progress, usually without explicit notice, and discusses their relationships to each other in general terms. These three styles of usage, distinguished according to whether they encourage an ordinary language (OL), a formal language (FL), or a computational language (CL) approach, have their relevant properties illustrated in the next three sections (§§ 6–8), each style being exemplified by a theoretical subject that thrives under its guidance.

§ 6. For ease of reference, the basic ideas of group theory used in this project are separated out and presented in this section. Throughout this work as a whole, the subject of group theory serves in both illustrative and instrumental roles, providing, besides a rough stock of exemplary materials to work on, a ready array of precision tools to work with.

Group theory, as a methodological subject, is used to illustrate the mathematical language (ML) approach, which ordinarily takes it for granted that signs denote something, if not always the objects intended. It is therefore recognizable as a special case of the OL style of usage.

To the basic assumption of the OL approach the ML style adds only the faith that every object one desires to name has a unique proper name to do it with, and thus that all the various expressions for an object can be traded duty free and without much ado for a suitably compact name to denote it. This means that the otherwise considerable work of practical computation, that is needed to associate arbitrarily obscure expressions with their clearest possible representatives, is not taken seriously as a feature that deserves theoretical attention, and is thus ignored as a factor of theoretical concern. This is appropriate to the mathematical level, which abstracts away from pragmatic factors and is intended precisely to do so.

More instrumentally to the aims of this investigation, and not entirely accidentally, group theory is one of the most adaptable of mathematical tools that can be used to understand the relation between general forms and particular instantiations, in other words, the relationship between abstract commonalities and their concrete diversities.

§ 7. The basic notions of formal language theory are presented. Not surprisingly, formal language theory is used to illustrate the FL style of usage. Instrumentally, it is one of the most powerful tools available to clear away both the understandable confusions and the unjustifiable presuppositions of informal discourse.

§ 8. The notion of computation that makes sense in this setting is one of a process that replaces an arbitrary sign with a better sign of the same object. In other words, computation is an interpretive process that improves the indications of intentions. To deal with computational processes it is necessary to extend the pragmatic theory of signs in a couple of new but coordinated directions. To the basic conception of a sign relation is added a notion of progress, which implies a notion of process together with a notion of quality.

§ 9. This section introduces higher order sign relations, which are used to formalize the process of reflection on interpretation. The discussion is approaching a point where multiple levels of signs are becoming necessary, mainly for referring to previous levels of signs as the objects of an extended sign relation, and thereby enabling a process of reflection on interpretive conduct. To begin dealing with this issue, I take advantage of a second look at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{A}\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{B}\!} to introduce the use of raised angle brackets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\langle}~{}^{\rangle}),} also called supercilia or arches, as quotation marks. Ordinary quotation marks Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}~{}^{\prime\prime})} have the disadvantage, for formal purposes, of being used informally for many different tasks. To get around this obstacle, I use the arch operator to formalize one specific function of quotation marks in a computational context, namely, to create distinctive names for syntactic expressions, or what amounts to the same thing, to signify the generation of their gödel numbers.

§ 10. Returning to the sign relations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(\text{A})\!} and $L({\text{B}}),\!$ various kinds of higher order signs are exemplified by considering a series of higher order sign relations based on these two examples.

§ 11. In this section the tools that come with the theory of higher order sign relations are applied to an illustrative exercise, roughing out the shape of a complex form of interpreter.

The next three sections (§§ 32–34) discuss how the identified styles of usage bear on three important issues in the usage of a technical language, namely, the respective theoretical statuses of signs, sets, and variables.

§ 12. The Status of Signs

§ 13. The Status of Sets

§ 14. At this point the discussion touches on an topic, concerning the being of a so called variable, that issues in many unanswered questions. Although this worry over the nature and use of a variable may seem like a trivial matter, it is not. It needs to be remembered that the first adequate accounts of formal computation, Schönfinkel's combinator calculus and Church's lambda calculus, both developed out of programmes intended to clarify the concept of a variable, indeed, even to the point of eliminating it altogether as a primitive notion from the basis of mathematical logic (van Heijenoort, 355–366).

The pragmatic theory of sign relations has a part of its purpose in addressing these same questions about the natural utility of variables, and even though its application to computation has not enjoyed the same level of development as these other models, it promises in good time to have a broader scope. Later, I will illustrate its potential by examining a form of the combinator calculus from a sign relational point of view.

§ 15. There is an order of logical reasoning that is typically described as propositional or sentential and represented in a type of formal system that is commonly known as a propositional calculus or a sentential logic (SL). Any one of these calculi forms an interesting example of a formal language, one that can be used to illustrate all of the preceding issues of style and technique, but one that can also serve this inquiry in a more instrumental fashion. This section presents the elements of a calculus for propositional logic that I described in earlier work (Awbrey, 1989 and 1994). The imminent use of this calculus is to construct and analyze logical representations of sign relations, and the treatment here focuses on the concepts and notation that are most relevant to this task.

The next four sections (§§ 16–19) treat the theme of self-reference that is invoked in the overture to a RIF. To inspire confidence in the feasibility and the utility of well chosen reflective constructions and to allay a suspicion of self-reference in general, it is useful to survey the varieties of self-reference that arise in this work and to distinguish the forms of circular referrals that are likely to vitiate consistent reasoning from those that are relatively innocuous and even beneficial.

§ 16. Recursive Aspects

§ 17. Patterns of Self-Reference

§ 18. Practical Intuitions

§ 19. Examples of Self-Reference

The intertwined themes of logic and time will occupy center stage for the next eight sections (§§ 20–27).

§ 20. First, I discuss three distinct ways that the word system is used in this work, reflecting the variety of approaches, aspects, or perspectives that present themselves in dealing with what are often the same underlying objects in reality.

§ 21. There is a general set of situations where the task arises to “build a bridge” between significantly different types of representation. In these situations, the problem is to translate between the signs and expressions of two formal systems that have radically different levels of interpretation, and to do it in a way that makes appropriate connections between diverse descriptions of the same objects. More to the point of the present project, formal systems for mediating inquiry, if they are intended to remain viable in both empirical and theoretical uses, need the capacity to negotiate between an extensional representation (ER) and an intensional representation (IR) of the same domain of objects. It turns out that a cardinal or pivotal issue in this connection is how to convert between ERs and IRs of the same objective domain, working all the while within the practical constraints of a computational medium and preserving the equivalence of information. To illustrate the kinds of technical issues that are involved in these considerations, I bring them to bear on the topic of representing sign relations and their dyadic projections in various forms.

The next four sections (§§ 22–25) give examples of ERs and IRs, indicate the importance of forming a computational bridge between them, and discuss the conceptual and technical obstacles that will have to be faced in doing so.

§ 22. For ease of reference, this section collects previous materials that are relevant to discussing the ERs of the sign relations $L({\text{A}})\!$ and $L({\text{B}}),\!$ and explicitly details their dyadic projections.

§ 23. This section discusses a number of general issues that are associated with the IRs of sign relations. Because of the great degree of freedom there is in selecting among the potentially relevant properties of any real object, especially when the context of relevance to the selection is not known in advance, there are many different ways, perhaps an indefinite multitude of ways, to represent the sign relations $L({\text{A}})\!$ and $L({\text{B}})\!$ in terms of salient properties of their elementary constituents. In this connection, the next two sections explore a representative sample of these possibilities, and illustrate several different styles of approach that can be used in their presentation.

§ 24. A transitional case between ERs and IRs of sign relations is found in the concept of a literal intensional representation (LIR).

§ 25. A fully fledged IR is one that accomplishes some measure of analytic work, bringing to the point of salient notice a selected array of implicit and otherwise hidden features of its object. This section presents a variety of these analytic intensional representations (AIRs) for the sign relations $L({\text{A}})\!$ and $L({\text{B}}).\!$

Note for future reference. The problem so naturally encountered here, due to the embarrassment of riches that presents itself in choosing a suitable IR, and tracing its origin to the wealth of properties that any real object typically has, is a precursor to one of the deepest issues in the pragmatic theory of inquiry: the problem of abductive reasoning. This topic will be discussed at several later stages of this investigation, where it typically involves the problem of choosing, among the manifold aspects of an objective phenomenon or a problematic objective, only the features that are: (1) relevant to explaining a present fact, or (2) pertinent to achieving a current purpose.

§ 26. Differential Logic and Directed Graphs

§ 27. Differential Logic and Group Operations

§ 28. The Bridge : From Obstruction to Opportunity

§ 29. Projects of Representation

§ 30. Connected, Integrated, Reflective Symbols

The next seven sections (§§ 31–37) are designed to motivate the idea that a language as simple as propositional calculus can be used to articulate significant properties of $n\!$ -place relations. The course of the discussion will proceed as follows:

§ 31. First, I introduce concepts and notation designed to expand and generalize the orders of relations that are available to be discussed in an adequate fashion.

§ 32. Second, I elaborate a particular mode of abstraction, that is, a systematic strategy for generalizing the collections of formal objects that are initially given to discussion. This dimension of abstraction or direction of generalization will be described under the thematic heading of partiality.

§ 33. Third, I present an alternative approach to the issue of defective, degenerate, or fragmentary $n\!$ -place relations, proceeding by way of generalized objects known as $n\!$ -place relational complexes. Illustrating these ideas with respect to their bearing on sign relations the discussion arrives at a notion of sign-relational complexes, or sign complexes.

In the next three sections (§§ 34–36) I consider a collection of identification tasks for $n\!$ -place relations. Of particular interest is the extent to which the determination of an $n\!$ -place relation is constrained by a particular type of data, namely, by the specification of lower arity relations that occur as its projections. This topic is often treated as a question about a relation's reducibility or irreduciblity with respect to its projections. For instance, if the identity of an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n\!} -place relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L\!} is completely determined by the data of its Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\!} -place projections, then $L\!$ is said to be identifiable by, reducible to, or reconstructible from its $k\!$ -place components, otherwise Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L\!} is said to be irreducible with respect to its Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\!} -place projections.

§ 34. First, I consider a number of set-theoretic operations that can be utilized in discussing these identification, reducibility, or reconstruction questions. Once a level of general discussion has been surveyed enough to make a start, these tools can be specialized and applied to concrete examples in the realm of sign relations and also applied in the neighborhood of closely associated triadic relations.

§ 35. This section considers the positive case of reducibility, presenting examples of triadic relations that can be reconstructed from their dyadic projections. In fact, it happens that the sign relations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(\text{A})\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(\text{B})\!} fall into this category of dyadically reducible triadic relations.

§ 36. This section considers the negative case of reducibility, presenting examples of irreducibly triadic relations, or triadic relations that cannot be reconstructed from their lower dimensional projections or faces.

§ 37. Finally, the discussion culminates in an exposition of the so called propositions as types (PAT) analogy, outlining a formal system of type expressions or type formulas that bears a strong resemblance to propositional calculus. Properly interpreted, the resulting calculus of propositional types (COPT) can be used as a language for talking about well-formed types of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\!} -place relations.

§ 38. Considering the Source

§ 39. Prospective Indices : Pointers to Future Work

§ 40. Interlaced with the structural and reflective developments that go into the OF and the IF is a conceptual arrangement called the dynamic evaluative framework (DEF). This utility works to isolate the aspects of process and purpose that are observable on either side of the objective interpretive divide and helps to organize the graded notions of directed change that can be actualized in the RIF.

§ 41. Elective and Motive Forces

§ 42. Sign Processes : A Start

§ 43. Reflective Extensions

§ 44. Reflections on Closure

§ 45. Intelligence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Rightarrow} Critical Reflection

§ 46. Looking Ahead : The “Meta” Issue

§ 47. Mutually Intelligible Codes

§ 48. Discourse Analysis : Ways and Means

§ 49. Combinations of Sign Relations

§ 50. Revisiting the Source

The Phenomenology of Reflection

The concepts with which a theory operates are not all objectivized in the field which that theory thematizes.

Paul Ricoeur, The Conflict of Interpretations, [Ric, 166]

This part of the discussion is fair to cast as the phenomenology of reflection. It aims to amass the kinds of observations extremely simple reflective agents, as a matter of principal and with a minimum of preparation, can make on the ebb and flow of their own reflective acts. But this is not the kind of phenomenology which pretends it can bracket every assumption of a sophisticated or a theoretical nature off to one side of the observational picture, or thinks it can frame the description of reflection without the use of formal concepts, such as depend on the bracing and support of a technical language.

On the contrary, the brand of phenomenology being wielded here makes the explicit assumption that there are likely to be an untold number of implicit assumptions that contribute to and conspire in the framing of the picture to be observed, while it is precisely the job of reflective observation to detect the influence of these covertly acting assumptions. Further, this style of phenomenology is deliberately set free of prior constraints on the choice of descriptive devices, since it can appeal to any formal means or any technical language that serves to articulate the description of its subject.

Certain things need to be understood about the aims, the scope, and the self-imposed limits of this phenomenology, especially when it comes to the question of what it hopes to explain. It is not the task of this phenomenology to explain consciousness but only to describe its course. This it does by making an inventory of the “contents” that appear in consciousness and by delineating the relationships that appear among these contents. Along the way, it must take into account, of course, that each moment of taking stock and each moment of charting relations needs to have its resulting list or map, respectively, realized as the content of a particular moment of consciousness.

Already, this lone requirement of the descriptive task raises a host of questions about what it means for something to be counted as a content of consciousness, and it leads, according to my present lights and aims, to a closer examination of a critical relationship, the logical relation “content of”, taken abstractly and in general. Since it does not appear that very extensive lists or very detailed maps can be “wholly realized” as contents within a limited field of consciousness, it is necessary to recognize an extended sense of “realization”, where a list or a map can be “partially” or “effectively” realized in a content of consciousness if and when an indication, pointer, or sign of it is present in awareness.

In particular, this tack suggests that some things, that otherwise loom too large to fit within the frame of immediate awareness, can be treated as contents of consciousness, in the extended sense, if only an effective indication of them is present in awareness. For instance, an effective indication of a larger text is a sign that can be followed to the next, and this to the next, and so on, in a way that incrementally leads to a traversal of the whole. By extension, a list of contents of consciousness or a map of relations among these contents is “effectively realized” in a single content of consciousness if that content effectively points to it, and if the object to which it points has the structure of an object that pointedly reveals itself in time. Given the evidence of the sign and the effective analysis of its object, a manifest of contents can be prized for the sake of the items it enumerates or the estates it maps, with each in due proportion to their values. Both parts of this condition are needed, though, since knowing the name alone of a thing, even if it lends itself to knowing the thing, does not itself amount to knowing the thing itself.

In short, my philosophical working hypothesis is concrete reflection, i.e., the cogito as mediated by the whole universe of signs.

Paul Ricoeur, The Conflict of Interpretations, [Ric, 170]

This understanding of the task of phenomenology bears on three features of the approach to consciousness that I am charting here.

It is under the heading of description, especially as qualified by the adjective effective, that the rationale of using mathematical models and the strategy of seeking computational implementations of these models can be found to successively fall.
As a rule, I find it helps to avoid hypostatizing consciousness or self-awareness as statically constituted entities, but to use the systematic notions of dynamic agency and developing organization. However, in order to make connections with other approaches to phenomenology I need occasionally to mention concepts and even to make use of language that I would otherwise prefer to avoid.
Finally, it is under the cumulative aims of effective description and systematic dynamics that the utility of sign relations is key. Sign relations are the minimal forms of models that are capable of compassing all that goes on in thinking along with whatever it is that thinking relates to in all the domains that it orients toward. The use of sign relations as models, as mathematical descriptions, and as computational simulations of what appears in reflecting on conduct is especially well suited to including in these models a description of what transpires in the conduct of reflection itself.

The phenomenology envisioned here depends on no assured power of introspection but only on a modest power to reflect on conduct and give it a description. These descriptions, all the better inscribed in external media, can be examined with increasing degrees of detachment and have their consequences projected by deductive means. The mass of descriptions growing over time as experience continues and reflection persists constitutes a de facto model of behavior. The accumulating code of practice or catalog of procedure ranges from empirical standards of comparison through provisional regulations to tentative ideals for future conduct. But the status models of behavior and codes of practice bear in their early days matters far less than their ability to test their prescriptions, along with their deductive and pragmatic implications, against the corpus of continuing observation, reflection, and description.

Reflection and consciousness no longer coincide. …

What emerges from this reflection is a wounded cogito, which posits but does not possess itself, which understands its originary truth only in and by the confession of the inadequation, the illusion, and the lie of existing consciousness.

Paul Ricoeur, The Conflict of Interpretations, [Ric, 172, 173]

It is needful at this point to draw a distinction between the power of reflection, a capacity required for inquiry, and what is likely to be confused with it, the presumptive power of introspection. “Introspection”, in the sole part of its technical meaning which excludes it from empirical inquiry, refers to an infallible, and thus incorrigible, power of observation one is imagined to possess with respect to one's own private experiences, matters over which there is assumed to be no higher court of appeal than one's own particular and immediate awareness. But the horizon of experience plotted with regard to this static standpoint fails to reckon with the dynamic nature of an ongoing circumstance, that subsequent experience continually rides a circuit around its antecedents and ever constitutes a higher court for every proceeding and every precedent that falls within its jurisdiction.

The distinction which marks reflection and sets it apart from introspection is its own acknowledged fallibility, which involves its ability to be seen as false in subsequent reflections. Naturally, this has an import for the status of reflection in empirical inquiry. Paradoxically, its admission of fallibility is actually a virtue from the standpoint of making reflection useful in science. If reflection on conduct leads to a description that cannot be falsified by any contingency of conduct, then that description is insufficient to specify any particular conduct at all. This means one of several things about the description, either (1) it remains a condition of conduct in general, or (2) it resides as a part of a necessary logic at the bounds of all experience, or (3) it rests in a realm of metaphysics that abides, if anywhere, beyond the bounds of purely human experience and thus abscounds altogether from the sphere of empirical inquiry.

In this way the psyche is itself a technique practiced on itself, a technique of disguise and misunderstanding. The soul of this technique is the pursuit of the lost archaic object which is constantly displaced and replaced by substitute, fantastic, illusory, delirious, and idealized objects.

Paul Ricoeur, The Conflict of Interpretations, [Ric, 185]

One of the most difficult problems facing the phenomenology of reflection falls under the heading of “fallibility” in a markedly strong way. That is the problem of systematic distortion. Beyond the host of idols deliberately constructed are still more false images so thoroughly systematic in their generation that only their escaping consciousness prevents them from being called “deliberate”. All the more naive projects of enlightenment, capitalized or not, are brought down by a failure to recognize this category of human frailty.

If the phenomenology of reflection that is developed and justified from this point on is not to be naive about this brand of fallibility, then it needs to constitute safeguards, a system of checks and balances, if you will, against it. If no method of remediation can permanently arrest the perpetrator of these schemes from generating distractions in perpetuity, then at least one can hope for ways to arraign the forms of fallibility under various recognizable themes, so that their dangers can be avoided in the future. In this vein, it is necessary to institute the study of those more opaque obstructions that limit the medium of investigation and to facilitate the analysis of those more refractory resistances to clear reflection, whose names are legion, but whose characters can be diversely noted under the themes of obstruction, resistance, the shadow, the unconscious, the “dark side of the enlightenment”, or even better yet, the “underbrush of the clearing”.

In the general scheme of things, the forms of distortion that remain peculiar to particular agents of reflection need garner to themselves nothing outside the incidental degrees of interest. The best check to counter this species of distortion, to which the isolated individual is likely to fall prey, is the balance of cultural wisdom that is commonly stored up and invested in the living praxis of a reflective community.

It is only when the incidence of singular distortions is not damped out by the collective incitement of countermeasures, when the aggregation of local distortions is overlooked by the powers of a general reflection, when the flaws in the individual lights and mirrors of the scientific organon are not taken into account and duly compensated in the shape of the social “panopticon”, or when the grinding accumulation and the precipitous mounting up of infinitesimal but significant deviations from accurate reflection are not met with an adequate power of oversight, one that can maintain solely the interests of community integrity at heart, that a truly false ideal begins to hold sway over the very perceptions of every specialized agent of reflection.

When these aberrations and astigmatisms develop unchecked, and when the strain to see things clearly reaches the point of breaking all the instruments thereof, then the most circumscribed faults, the distorted reflections of individual hypocrisy, the strange lack of insight and the missing sense of mutual reciprocity that manifest themselves in the most parochial forms of self interest, then all of these defects, and ills, and shocks begin to “pass through” to the collective strata, to be inherited and propagated by the highest levels of social organization, and then a systematic and widespread falsification of the whole conduct of society begins to pervade its view of itself.

On macroscopic scales of organization, with medium sized bodies and bodies of media that extend over considerable distances, with masses of activity that successfully propagate their own forms through vastening expanses of time, the general condition of thoughtfulness cancels out and compensates for all but the most singular of disturbances, namely, those that are peculiar to the microscopic realm of observation. If the matter is regarded on this grander scale, then it is not hard to find a sufficient reason for the stubborn persistence of the cosmic order, and thus the desirable necessity of doing just this is never far from mind. In the case of whole societies, a like reason is often enough to explain their inertia, their resistance, and their overall slowness to change.

If there is felt a need to devise an object explanation, a presumptive sources of troubles that is already compact, concrete, and thus confined enough to accuse, apprehend, and hopefully imprison on account of the mass's retarded potential, then resorting to a hypostasis posed in the form of an “archaic object” is a prototypical way of controlling anxiety, and it frequently, if not infallibly, can serve just as well as any other device on which to pin the common blame. This highlights the question: What sort of archaic object would account for the general malaise in a community whose dedication to inquiry has become root-bound?

I wish to apply a determinate philosophical method to a determinate problem, that of the constitution of the symbol, which I described as an expression with a double meaning. I had already applied this method to the symbols of art and the ethics of religion. But the reason behind it is neither in the domains considered nor in the objects which are proper to them. It resides in the overdetermination of the symbol, which cannot be understood outside the dialecticity of the reflection which I propose.

Paul Ricoeur, The Conflict of Interpretations, [Ric, 175]

The archaic object of this global aimlessness, informing the course of the general drift and fixing inquiry in an orbit of constant acceleration with no immediate risk of resolution, is very likely nothing more than the whole community of interpretation itself, effectively realized as an object of its own devising.

The community of interpretation, whose currency funds the community of inquiry as a going enterprise within its fold, has sufficient reason to preserve itself in its present form as a valuable object, commodity, or resource. But the dialectical nature of the process that is currently conducted between them, due in part to the dialectrical charges of the “-ionized” terms that pass for information between them. A term of this charge splits the action from the end and shares it between the parties to an ambiguity, the active and passive objects that together comprise its full denomination. This division of denotation forces interpretation to vacillate between the two extremes of meaning in a vain and eternal effort to rejoin their senses of value to the realm of the rendered and misspent coin, in hopes of regaining the meaning what was mint in their original condition. The stowing away of one portion or the other drives the potential that drives both themselves and all the actions that they are meant to convey toward their designate and their destinate ends, but the unstable equilibrium that is their due, especially when it is permitted to be waged by uncontrolled forms of oppositional attraction, does not permit the dialogue to rest. It continues to remain in doubt and does not fail to renew its ambivalence regarding the maintenance of any fixed form it happens to take, always wondering whether its present form is literally necessary, precisely sufficient, or whether it is but transiently and contingently convenient. Accordingly and otherwise the whirl of dialogue, for all its own reasons, is always in imminent danger of wasting away into the echo of its own narcissism.

The problem arises of how to bring these systematic distortions under systematic control. It helps to stand back a bit from the problem and to cast a somewhat wider net. Accordingly, let the whole category of phenomena that are gathered around this issue be thematized under the family name of an obstruction to inquiry (OTI). This includes as a subordinate genus the panoply of systematic distortions, generated by disingenuous reflections, that can be hypothesized to have their source in protecting the favored assumptions and defending the implicit claims of a particular status quo, no matter whether the implicated propositions are held to be the prerogatives of a privileged POV or whether they are delivered up to indictment as the prejudices of a more widely sanctioned world view. The archetype of this behavior is appropriately addressed under the mythological or the psychological category of narcissism.

It is important to note that the family OTI and the genus narcissus differ in the levels of hypothesis that are involved in their concepts, both in their speculative formation and in their provisional attribution. The presence of an OTI is fairly easy to surmise from its distinguishing traits: the dissipative conduct and the rambling course that affect the inquiry in question. To the degree that the suspicion of its effect and the verification of its force can be assembled from superficial traces, this makes its maintenance supportable on circumstantial evidence alone. In a phrase, one says that the wider hypothesis lies “nearer to nature” than the narrower construction, or that it makes its appearance closer to the purely phenomenal sphere. In contrast, unraveling the precise nature of the obstruction requires a deeper investigation. There is an additional hypothesis involved in guessing the source of the resistance, no matter how prevalent a particular genus of distortion is found and no matter how likely an individual species of explanation is in fact.

Within this wider setting it may be possible to focus more clearly on the species of threats to accurate reflection that need to be clarified here. Already, besides the stigma of stubborn error that hangs over the whole refractory horde, there is a germ of paradox that hides within the very folds of this classification. Namely, it is that the first obstacle one finds to reflection, and hence to every form of reflective inquiry, is a kind of narcissism or self love. It begins naturally enough, ensconced in the not unnatural desire of every form of life to preserve itself in its present form. But the simple desire to remain as is can be diverted into a blinded esteem of the self, one that admires its present condition only as reflected in the array of disingenuous reflections and contrived presentations that make up a fixed, idealized, and very selective image. Finally and strangely enough, this unreflective form of narcissism even comes to prefer the simplistic and beautiful lies to the realistic forms that a veritable mirror would show.

The danger of narcissism, with respect to the prospects of a reflective inquiry, is not in the dynamic attractions and the realistic affections that a person or a society bears toward its truer self, and that in turn inform their respective bearings toward the selves they are meant to be, but in the static character of its attachment to a fixed, idealized, and partial image of that self.

Once again, the quality that distinguishes reflection from introspection, its fallibility, is a trait that sufficiently reflective agents can find reflected in their own conduct of reflection, and needless to say, their conduct in general. This quality of fallibility, thus cognized and thus converted, that is, once its application to oneself is acknowledged and its consequences for one's experience are recognized, becomes a type of self-recognizant character, an internalized trait that leads reflective agents to become more corrigible, more docile, and thus more educatable. This makes it possible for reflective agents to build up their images of reality from scratch materials, to proceed through steps that are always revisable and edifiable, and to leave the finishing of their forms to the work of future editions. In the final analysis, while this mannerism of aesthetic distance and tempered discretion prevents any affection or any impression from becoming too “immediate”, in the strictest sense of that word, it is just this mode of detachment that assures the sensible image of its eventual remediation.

The nature and use of reflection in inquiry, as it currently appears, can be described as follows. Reflection on conduct leads to a description of that conduct, posed in terms of a reflective image. Over an interval of time or an extended period of investigation, these descriptive images are accumulated into exhaustive theories and compiled into compact models of the conduct in question. To be useful in science, or empirical inquiry, these theories and models must be capable of being false with respect to their intentions, amenable to being tested in further experience, and subject to being amended on subsequent reflection.

In sum, the very feature of reflection that seems to be its chief defect, the fact that it can generate false images, casting reflections that are false to the actions they intend to represent and even leading to wholly distorted perspectives on the objectified scene of activity, is the very characteristic that saves its appearance in experience and the very trait that permits it to show its face at the court of inquiry, which all along admits that distortions acknowledged to be imperfect images can still be disclosed to subsequent experience and remedied in future reflections.

A Candid Point of View

This section discusses, in a general and informal way, the objectives inspiring and the requirements surrounding the elaboration of a RIF. This is approached, in part, by taking up the intuitive notions of a point of view (POV) and a point of development (POD), as they stake out, respectively, the intellectual repertoire and history of a typical agent of inquiry. Initially, these ideas serve in a familiar manner to characterize the intellectual skills and growth of agents, in particular, as they bear on the cultivation of the agents' reflective resources. Increasingly, these concepts are subjected to formalization, partly by analyzing their relations to each other and gradually by relating their inherent structures and referent involvements to the already formalized concepts of objective frameworks, genres, and motifs.

As I reflect on signs and texts, I am led to enumerate more and more phenomena associated with the process of interpretation and with the models of it that I find in sign relations. Some of the deepest and subtlest of these phenomena, at least, that I am able to observe and recount, take their theme from a certain “intermingling of categories” that is found at the basis of every real phenomenon. This issue comes to prominence and makes itself evident as topic of inquiry whenever one tries to organize the original chaos of phenomena through the imposition of a suitable scheme of categories.

What is the typical outcome of setting out such a scheme for nature? No sooner does one institute a provisional scheme of categories for organizing phenomena than one discovers every system with a stamp of reality to it steadfastly ignoring the lines of one's naive imagination. And yet it soon becomes clear that this seeming “perversity of nature” arises from an error of attribution on the part of the mind that casts the aspersion. Ultimately, it stems from the fact that every scheme of categories that the mind can forge and foist on nature, for instance, sign and object, self and other, remains, after all, the scene of a mere abstraction, implicating the pallid and the shadowy sides of the same dissention, but all the while circling about and turning on the complex but unitary reality that underlies the phenomenon in question.

In view of these complexities, that interfere with applying even the simplest of organizational paradigms to the material of signs and texts, it is necessary for me to pause a while and carefully contemplate how I can rehabilitate their use, at least, for the ends of this investigation. First, I examine the distinction between sign and object. Then, I consider the duality between self and other, or what amounts to the same thing, the relation between a first person and a second person POV. In each case, the task is to discover how a distinction that seems so easy to subvert can ultimately be developed into a useful instrument of analysis and articulation.

There's nought but care on ev'ry han',
	In every hour that passes, O;
What signifies the life o man,
	An 'twere na for the lasses, O.
— Robert Burns, Green Grow the Rashes, O

Any object, anything grasped as a whole, can be a sign. Indeed, the entire life of a person or a people can serve as sign unto itself or others and take on a significance all its own. In converse fashion, every sign token is an object in the world. In this role, a sign is forced to obey the ruling and relevant natural laws and empowered to take on a dynamics all its own.

In the contention between signs and objects, the answer initially given by the pragmatic theory of signs is that anything can potentially serve in any role of a sign relation. In particular, the distinction between sign and object is a pragmatic distinction, a mark of use, not an essential distinction, a mark of substance. This is the right answer as far as the beginning of the question goes, where it is the possible character of everything that is at issue. The pragmatic approach makes it possible to begin an investigation that would otherwise be obstructed by a futile search for non-existent essentials, as if it were necessary to divine them from prior considerations before any experience has been ventured and before a bit of empirical evidence has been collected.

Reason alone teaches us to know good and evil. Therefore conscience, which makes us love the one and hate the other, though it is independent of reason, cannot develop without it.

Rousseau, Emile, or On Education

But the form of answer that is sufficient to begin a study is not the form of answer that is necessary to end it. Even though it is useful for a general theory of signs to provide a patently indifferent form of answer at the preliminary phases of its investigation, this style of response is ultimately judged to be facile when it comes to questions about the good of a sign, the end of an inquiry, or the suitability of each thing to the role it is assigned. In the end, an all purpose brand of conceptual scheme, allowing for the equipotential coverage of every conceivable option, however useful or necessary to the task, is likely to be found insufficient for wrapping up these goods and delivering them into the service of the mind. Thus, by the round about way of this objection, one brings to mind the other meaning, the underlying nuance and the ultimate sense, of the word object, which suggests the end, the goal, or the good of something.

Questions about the good of something, and what must be done to get it, and what shows the way to do it, belong to the normative sciences of aesthetics, ethics, and logic, respectively. Aesthetic knowledge is a creature's most basic sense of what is good or bad for it, as signaled by the experiential features of pleasure or pain, respectively. Ethical knowledge deals with the courses of action and patterns of conduct that lead to these ends. Logical knowledge begins from the remoter signs of what actions are true and false to their ends, and derives the necessary consequences indicated by combinations of signs.

In pragmatic thought, the normative disciplines can be imagined as three concentric cylinders resting on their bases, increasing in height as they narrow, from aesthetics to ethics to logic, in that order. Considered with regard to the plane of their experiential bases, logic is subsumed by ethics, which is subsumed by aesthetics. And yet, in another sense, logic affords a perspective on ethics, while ethics affords a perspective on aesthetics.

That is about all I can say about normative considerations at this point. Further discussion is put off until this text has developed either the intuitive insight or the theoretical power to say something more definite.

Because a sign, so far as it can tell in the time it passes, addresses an unknown future interpretant, that is, an indefinite futurity of potential responses, there is always an aspect of dialogue about the sign relation, especially insofar as it is subject to extension. This is true no matter who, whether self or other, is ostensibly addressed by the sign or text at issue, and never mind what the chances are of a literal return in the communication. In this regard, it is recognizance enough for a sign to be issued or a text to be written in anticipation of its future result. And though it is never certain, it is always possible that the author of a text partially anticipates the use that others make of what is signed.

It is one of the rules of my system of general harmony, that the present is big with the future, and that he who sees all sees in that which is that which shall be.

G.W. Leibniz, Theodicy, paragraph 360

	When these prodigies
Do so conjointly meet, let not men say
“These are their reasons”, “they are natural”,
For I believe they are portentous things
Unto the climate that they point upon.
Julius Caesar: Casca—1.3.28–32

Indeed it is a strange-disposed time;

But men may construe things after their fashion,

Clean from the purpose of the things themselves.

Julius Caesar: Cicero—1.3.33–35

In order to recover the faculties supported by one's favorite categories and to maintain the proper use of their organizational schemes, it is incumbent on the part of the wary, conscientious, and duly circumspect schemer to recognize in every case how each part of the contention is implicated in the action of the other. In this connection, a triumvirate of closely related aspects of sign relations comes to the fore:

There is an aspect of futurity, marking the openness of signs to interpretation and the extensibility of sign relations in multitudes of novel but meaningful ways. This dimension of regard is staked out in anticipation of the possibility that perfectly fitting but previously unsuspected interpretants can be discovered within or added to any given sign relation, whether passed or present.
There is a factor that contemporary theorists call alterity, noting the quality of radical and reciprocal otherness that is involved in the dialogue of one self with another. Besides its invocation of the wholly other, this term subsumes all the ways that one being can be alien and unknown to itself, and it even suggests the host of alterations, deviations, distortions, errors, and transmutations that accompany all acts of record-keeping and interpretation.
There is a feature that C.S. Peirce called tuity, acknowledging the aspect of thouness or the prospect of a second person POV that is brought into play whenever one self addresses another. Along with the perspective of a genuine other, this recognizes all the referrals and deferrals that an interpretive agent can make to a past, present, or potential self.

All of these dimensions of concern focus on the circumstance that signs, especially written or recorded signs, moderate a complexly integrated sort of relationship between self and other, or between first person and second person POVs, in such a way that they render the paired categories of each scheme inextricably involved in one another.

There are well-known dangers of paradox, but not so well acknowledged risks of distortion, that arise in the interrogation of any reflection. Although its outward signs are obvious, the source of the difficulty is remarkably difficult to trace. Perhaps it can be approached as follows. Without trying to say what consciousness is, I can still speak sensibly of its contents, and talk of their structures in relation to each other. These contents, whether percepts or concepts or whatever, are all signs. And so I can study the effects of reflection in the medium of its texts and develop a model of reflection as a process that evolves these texts.

What generally happens when one tries to model reflective consciousness and to formalize the reflective discourses that signify its public life? In reaching for the available languages of logic and set theory, one is likely to use them as reductively as possible on the first attempt, and thus to state the relation of anything to awareness directly in terms of membership, in sum, by means of a globally overarching dyadic relation. What does this picture of reflection pretend about the relation of the world to the mind, or conversely, the relation of awareness to anything? Although it confuses the relation of content to consciousness with the relation of object to concept, this degree of play in the imagery is a forgivable, occasionally useful, and a probably inescapable analogy. In any case, it does not amount to the most serious distortion in the picture as a whole.

What is really wrong with the dyadic picture of reflection is the fact that it treats both of the relations it surveys, of minds to ideas and of ideas to things, on the model of a consummation and a containment, as if to place everything being related in an all embracing hierarchy and in opposition to all forms of reciprocal participation among its entities. This image renders a consciousness of contents and a concept of objects each in the likeness of a set and its elements, rather than presenting them as they most likely are, a relationship of systems or agents and of texts or signs to the ideals or objects that motivate them, constituting mutually embracing forms of participation in a unified textual activity. In all, the initial attempt at explaining reflection lays it out according to a conception that grasps it prey, and loses the creature in the process, rather than a conception that releases the potential of what it imprisons.

One of the reasons for bringing the pragmatic theory of signs to bear on this discussion is to deal with just these problems, constellated by the need for reflection and made acute by the defects of the dyadic picture. By means of triadic sign relations, and given a capacity to create and modify the interpretant signs that fill out its original set of semantic equivalence classes, an interpretive agent has the “elbow room” needed to stand aside from the ongoing process of interpretation, to reflect on its present determinants, and to consider its possible developments.

An inquiry that cannot clearly and completely comprehend itself as an object can at least inquire into the succession of signs that record its progress. The writer of a text can use that text to describe, at least partially, the process of writing and using it so. The reader of a text can understand that text to describe, at least partially, the process of reading and understanding it so. Further, a discussion can generate a record that describes, more than just the transient proceedings of that discussion, the principles and parameters that determine its creation. In each of these ways, a text can address the qualities that determine its intended character, comment on the context in which it takes a part, and act on behalf of its pretended objectives.

The procedural distinction just recognized, between the passing traces of a process and the permanent determinants of its generic character, informs a significant issue, on which is staked nothing less than the empirical feasibility of an inquiry into inquiry. From this point on, a certain figure of speech can be used to mark this distinction, when it is relevant to the course of discussion, and to signal a deliberate turn in the direction of consideration, when the corresponding exchange of its dialectical roles is intended. According to the nuances of this paradigm, one can distinguish a process intended in the substantive generative sense from a process intended in the genitive gerundive sense, and address oneself selectively, at turns, to the process that achieves versus the process of achieving any contemplated activity or result.

An inquiry at such a point of development that it cannot entirely grasp its ongoing process of inquiry as an object of thought, namely, as the process that inquires, can at least try to capture a representative sample of the signs that record its process of inquiring. Speaking metaphorically and with the proper apology, every thus generated and thus collected text of inquiry (TOI) can be addressed as a partial reflection of the generative process of inquiry. Moreover, it is not irredeemably illegitimate to say that a TOI can partly describe itself, since this merely personifies the circumstance that a process of inquiry can describe itself partly in the form of a TOI.

O jest unseen, inscrutable, invisible

As a nose on a man's face or a weathercock on a steeple.

My master sues to her, and she hath taught her suitor,

He being her pupil, to become her tutor.

O excellent device! Was there ever heard a better? —

That my master, being scribe, to himself should write the letter.

Two Gentlemen of Verona: Speed—2.1.127–132

When I write out my thinking in the form of a text, a critical thing happens: It faces me as the thought of another, and I start to think of what it says as though another person had said it. Almost unwittingly, a critical process comes into play. In regarding the text as expressing the thought of another, I begin to see it from different POVs than the one that led to its writing. As I find my own inquiry reflected in one or another TOI, it addresses me afresh as the question of another and I encounter it again as a novel line of investigation. This time around, though, the topic of concern and the style of expression become subject to directions of criticism that would probably not occur to me otherwise, since the angles of attack permitting them do not open up on their own, neither on first thinking nor ever, most likely, while merely speaking. This can be the beginning of critical reflection, but it can also stir up destructive forms of interference that inhibit and obstruct the very flow of thought itself.

If I can be granted the license to continue saying that a text says this or that about itself when what I really mean is that a person or process employs its text to say the corresponding thing about itself or its text, then I can begin to introduce a variety of descriptive terms and logical tools into this text that can be used to talk about what this or another TOI “thinks” or “believes” at various points in its development, that is, in order to detail what I or its proper author thinks or believes at the corresponding points of discussion.

Fourteen, a sonneteer thy praises sings;

What magic myst'ries in that number lie!

Your hen hath fourteen eggs beneath her wings

That fourteen chickens to the roost may fly.

Fourteen full pounds the jockey's stone must be;

His age fourteen — a horse's prime is past.

Fourteen long hours too oft the Bard must fast;

Fourteen bright bumpers — bliss he ne'er must see!

Before fourteen, a dozen yields the strife;

Before fourteen — e'en thirteen's strength is vain.

Fourteen good years — a woman gives us life;

Fourteen good men — we lose that life again.

What lucubrations can be more upon it?

Fourteen good measur'd verses make a sonnet.

Robert Burns, A Sonnet Upon Sonnets

One of the main problems that the present TOI has to address is how a TOI can address the problems of self-reference that an inquiry into inquiry involves. If a sonnet can say something true about sonnets, then a TOI, far less limited in the number and measure of its lines, ought to be able to say something true about TOIs in general, unless the removal of these limitations takes away the only things whereof and whereby it has to speak, the ends and means of its own form of speech.

Using the pragmatic theory of signs, the forms of self-reference that have to be addressed in this project can be divided into two kinds, or classified in accord with two dimensions of referential involvement. Roughly speaking, reference in the broader sense can suggest either a denotative reference to an object or a connotative reference to a sense. Therefore, a projected self-reference can be classified according to the ways that its components of reference propose to recur on themselves: how much pretends to be a self description along denotative lines and how much purports to be a self address in the connotative direction.

Under suitably liberalized conditions of interpretation, then, what is meant by “a self-referent text”, whether one that denotatively describes or connotatively addresses itself? Apparently, it can mean a text that addresses, describes, refers to, or speaks to either one of two issues: (1) the outwardly passing features of its own succession of signs, or (2) the inwardly relied on properties of its own regenerative sources.

It is one thing for a text to be generated according to the laws laid down in another. This takes place, for example, in devising or following a proof according to the axioms and rules of inference that are recorded in a proof system. It is another thing entirely for a text or a corpus of texts to derive or induce the very principles of their own generation and then return to disclose the process of derivation or induction itself according to which the whole text or corpus is divined or drafted.

What the discussion of reflection has so far been leading up to, if I stop to reflect on what might be the implicit project behind its scheme of development, is tantamount to a monadology, a project of a complete and total provision for a system of perfect but virtual self-reflections. But I suspect that such a project is unsupportable in reality outside the realm of infinite resources and pre-established harmonies, while my present aim is to see what can be done with finite and empirical means. A monadology, if it entertains itself with any form of investigation at all, addresses the task as a sheer masquerade, styling its inquiry after the fashion of a complete logical analysis (CLA).

On principle, there is nothing inherently the matter with the form of the CLA itself, but it does not embody all by itself the spirit of suspense that accompanies a genuine human inquiry. A real inquiry cannot know before it starts what the answer is and how the end will be achieved, and it cannot, if it wishes, merely trick out the foils of an already completed and pre-arranged survey, parading them as a passing series of plotted and transient complications in the guise of an honest quest. Some types of completeness are far more complete than others, however. Taken with respect to a properly limited and workably modest context, and treated as relatively complete rather than absolutely complete, the ideal substrate of the CLA forms a suitably plastic material for modeling many forms of concretely reasonable inquiry.

Invoked with a spirit of moderation, the idealized model envisioned in the CLA can nevertheless serve as virtual guide for practical inquiries, highlighting the space of conceivable models and projecting a standard against which to measure every approximate, likely, and partial result. An inquiry of this self controlled kind, that considers in addition the logical alternative to every hypothesis it finds itself making, if it is addressed appropriately to the conditions of its constraining resources, can achieve complete success only within a tightly circumscribed sphere of action. Thus, the ideal of CLA informs a workable genre of inquiry, but the experimental variations that it enables and permits an agent to contemplate are bound up with the experiences that can be expressed in a language of finite and discrete signs, and exactly to the extent that they are in fact expressible.

In principle and in effect, an inquiry pushing the envelope of CLA is restricted to a “universe of limited marks”. For all practical purposes, it must keep its remarks to a finite universe of discourse, and a small one at that. Beyond these bounds, every inquiry is forced to take its chances on a pure hypothesis, unmitigated by any consideration of the opposite case. Communities of inquiry, however, are likely to embody a distribution of individual inquiries that have placed their bets on opposing options. Diversity of interpretation leads to disjunctions of opinion that can render many heads much smarter than one, but it also engenders forms of disagreement, discord, and duplicity that, for all their practical inevitability, are not essentially necessary.

Engaging in practical inquiry in a community of partially informed and presumptively constrained reasoners, then, is a task that leads to the recognition of several critical needs, not only for ways of synthesizing fragmentary interpretations of the presumptive truth and for reconciling divergent accounts of the objective world, but also for strategies that make these methods of negotiating differences and resolving conflicts more commonly available to all the inquirers in a putative community. Finally, an agent attempting to be reasonable under these conditions needs to be permitted to exercise a number of editorial prerogatives. For example, there needs to be a way to retract projections, that is, to recognize the alienated aspects of oneself that appear to crop up in others and to reconsider the rejected options for thought and action that nevertheless are capable of leading to bona fide values.

In a striking analogy with visual perception, it is the reflections in the ambient flow of energy that make it possible for one complication in the medium, a living being, to see another variation in the density of the medium, animate or otherwise, as an object. Reflection permits one to render an experience as due to a separate entity, to regard its occasion as the appearance of an object, and to respond to its cause as a reality. The analogy is broken at the junctures where an agent attributes these reflections to the passive “reflectances” of the object itself rather than perceiving them as the active responsibility of every participant in the process as a whole.

In accord with this visual analogy, two factors frustrate the prospects of indefinitely extending and smoothly finishing any project of inquiry that works in a medium of CLA:

The transparent obstruction (TO), or obstacle of transparency, is due to an initial inability to discover and to render visible every assumption, category, or distinction that one automatically and implicitly acts according to.
The opaque obstruction (OO), whether it presents itself in the guise of an obvious or an obscure obstacle, arises on account of a final incapacity to consider both sides of every question posed. This can amount to either one of two shortcomings: (a) failing to identify a logical alternative to every presumption or thesis that one identifies with, or (b) failing to evaluate a logical alternative to every assumption or hypothesis that one does in fact identify.

In short, a finite information creature (FIC) is required to keep the contents of its forms within the range of a definite set of figures and to rest the forms of its contents within the scope of a certain cast of characters. To be sure, these are precisely the characters that can be modeled and the figures that can be cut within a circumscribed theater of operations that everyone calls a partial logical analysis (PLA).

A Projective Point of View

A necessary connection between signs and reflection gives the TOI its critical function as a transitional object in the development of inquiry. In the form of a TOI, I address my reflection as if it were the reflection of another. On the off chance that it renders me a bit more critical, as I eye both its sources of authority and its styles of presentation, I can regard the record of this reflection as a partially alienated object, an artifact of unknown origin, or a work of uncertain provenance. And so the very existence of a sign, that takes after another in a search for its meaning and ultimately takes its place in tracing the traces of that process of inquiry, is intimately bound up with the act of reflection.

There is, moreover, a connection between the act of reflection and the psychological mechanism called projection that is useful to notice here. As it happens in practice, the effect of reflection is frequently achieved, not directly, by means of a deliberate effort to observe and to evaluate one's own conduct, but more indirectly, through the initial observation and the subsequent criticism of another's behavior, finally followed up by the often delayed afterthought and usually reluctant insight that the properties ascribed to the other's behavior can also apply to one's own.

The relationship between the isolated components of behavior in this sort of projective situation amounts to a familiar kind of sign relation. In regard to the properties possessed in common, the other's pattern of behavior is an icon, at first unrecognized, of one's own form of conduct. The introspective act of recognizing and assimilating the significance of such a relationship is referred to as retracting or re-owning the projected attributes and descriptions. To sum things up in these terms, the retraction of a projection can bring about, in its composite fashion, the ultimate effect of a critical reflection, namely, the elicitation and application of a valid description to one's own conduct.

Before the usefulness of this insight can be appreciated, it is necessary to resolve an interdisciplinary conflict over the use of the term projection and to sort out the relationship between the psychological and the mathematical concepts of projection.

O time, thou must untangle this, not I.

It is too hard a knot for me t'untie.

Twelfth Night: Viola—2.3.39–40

There are a couple of contingencies surrounding the trials of learning from one's own experience, issuing from and bearing on the complexity of that very experience, that appear to be tangled up with each other. Echoing the mythology of the Gordian knot, the Herculean Hydra, the Laocoonian serpent, and the Persean Medusa, each of which accounts of perverse polymorphism seems to reflect a variant aspect but to capture a sheer fragment of the underlying archetype, these two factors can be addressed by means of the following allegory:

The Knot. It is frequently difficult to learn anything at all from the encounter with one's own experience, especially while one is still faced with the full complexity of that experience.
The Knife. One tends to establish a personal array of mental or conceptual frames, planes, or sections on which one can reliably and reductively chart, map", or project one's experience.

The relationship between these two factors is such that the Knot leads to the Knife as its adaptive or expedient remedy, but that the Knife affords only a transitory relief for the problems bound up in the Knot, and further, an excessive reliance on any fixed array of armaments and stratagems under the emblem of the Knife has the contrary tendency to worsen the troubles experienced under the category of the Knot.

Thus, it is fair to say that the difficulty of learning from the full complexity of one's own experience is a problem condition that partly leads to and partly arises from the very configurations of artificial sections and arbitrary coordinates that one contrives to project one's experience on and to judge one's experience by, respectively. Although one's idealizations, simplifications, and other pet schemes of reductive representation can serve to render one's experience initially manageable, they can ultimately and adversely interfere with seeing the obvious.

In this setting, it is possible to bring about an accommodation between the mathematical and the psychological concepts of projection and to reconcile their discordant uses of the term within a concerted paradigm. For example, in dealing with the joint configuration space of a multiple agent system, one considers this yoked extension space (YES) to fall within a common extension (CE) of all the single agent state spaces. Each agent involved in such a system projects, in a geometric sense, the total action of the system on its own section of the whole CE, its local outlook, mental plane, personal frame of reference (FOR), or point of view (POV).

What does the POV of an agent consist in? Generally speaking, agents are not dumb. They are not limited to a single view of their situation, nor are they restricted to a single scenario for its ongoing development. They can entertain many different possibilities as candidates for the so-called and partly self-describing “objective situation” and they can envision many different ways that these potential situations might be developing, both before and after their passage through the moment in question. Furthermore, under circumstances favorable to reflection, agents can invoke POVs that help them to contemplate many different possible developments in the constitution of these very same POVs.

Now, it is conceivable that all the POVs entertained by a single agent are predetermined as having the same collection of generic characters, and thus that this invariant constitution is what really limits the range of all possible POVs for the agent in question. If so, it leads to the idea that this invariant constitution defines a uniquely general POV, a highest order meta-POV, or a consummate POV of the agent involved. Still, the only points of access and the only paths of approach that an agent can have to its own consummate POV, if indeed such a goal does make sense, are through the agency and the medium of whatever POVs it happens to have at each passing moment in its developmental history. Consequently, a persistent enough search for a good POV opens up the investigation of each agent's prevailing point of development (POD).

In the best of all possible worlds, then, being under the influence of one POV does not render an agent incapacitated for considering others. Of course, there are practical limitations that affect both the capacity and the flexibility of a particular POV, and there can be found in force both logical constraints and resource constraints that leave a POV with a narrowly fixed and impoverished character, one that the agent opting for it can fail to represent reflectively enough within the scope of this POV itself. In particular, the finite information constructions (FICs) that are accessible from a computational standpoint are especially limited in the kinds of POVs they are able to attain.

This means that POVs and PODs have recursive constitutions and recursive involvements with one another, calling on and referring to other POVs and PODs, both for the exact definitions that are needed and also for the more illuminating elaborations that might be possible, both those belonging to the same agent, reflexively, and those possessed by other agents, vicariously. A large part of the task of building a RIF is taken up with formalizing POVs and PODs, in part by analyzing their intuitive notions in terms of their implicit recursive structures and their referential involvements with each other, and in part by exploring their potential relationships with the previously formalized concepts of objective concerns (OCs).

In settings where recursion is contemplated, it is possible to conceive of a distinction between well-founded recursions, that lead to determinate definitions of the entities in question, and buck-passing recursions, that lead one down the “garden path” to an interminable “run-around”. The catch, of course, is that it is not always possible to implement an effective procedure that can accomplish what it is possible to conceive. Thus, there are cases where the imagined distinction does not apply and times when the putative difference is not always detectable in practice.

In this connection, there are two or three fundamental questions that need to be addressed by this project:

What makes a POV or a POD well-founded?
Can buck-passing POVs and PODs be tolerated?
How should they be treated and regulated, if tolerated?

A tentative approach to these questions is tendered by the pragmatic theory of sign relations, where the definitive and the elaborative aspects of recursion correspond to the denotative and the connotative components of reference, respectively. Although it is always useful to organize the connotative realm in the species of a determinate ordering or a well-founded hierarchy, there is found in these parts generally a greater tolerance for the baroque proliferation of circuitous references and a broader acceptance of provincial, dialectic, and private coinages.

If all thought takes place in signs, as a tenet of pragmatism holds, then mental space is a space of signs and their interpretants, in other words, it is a connotative realm.

In this perspective, that is to say, in the POV of the present project and in the current opinion of its author, a POV is associated with an abstractly defined, but concretely embodied and frequently distributed, section of memory (SOM), where the signs constituting it are stored. In this rendition, a SOM is a curve, surface, volume, or more general subspace of the total memory space, in other words, a subset of memory that can be treated, under the appropriate change of coordinates, as being swept out by a set of variables, and ultimately addressed as being generated by a list of binary variables or bits. Working under the assumption that agents can engage in non-trivial developments, it must be granted that they have the ability to change their POVs in significant ways between the successive PODs in their progress, and thus to move or jump from one SOM to another, as dictated by will or as constrained by habit.

In this comparison, what is visualized as the geometric structure of a cone is commonly implemented through the data structure of a tree, that is, a set of memory addresses (along with their associated contents) that are accessible from a single location, namely, the root of the tree, or the literal point of the POV.

Typically, but not infallibly, an agent can reduce the complexity of what is projected on its personal POV by employing a reductive hypothesis or a simplifying assumption. Often, but not always, this idealization is arrived at by picking one agent to treat as nominal, in other words, whose actions and perceptions are regarded as natural, normal, or otherwise unproblematic. Usually, especially if one is a mature agent, this nominal agent is just oneself, but a novice agent, unsure of what to do in a novel situation, can chose another agent to fill the role of a nominal guide and to serve as a reference point.

It would be nice if one could ignore the sharper edge of knowledge that is brought to light at this point, and fret but lightly over the smooth and middling courses that gloss the conformal plateaus of established knowledge. However, it is the nature of the inquiry into inquiry that one cannot forever restrict one's attention to the generic, nominal, or unexceptional case, well away from the initial conditions of learning and the boundary conditions of reasoning. Still, for the purposes of a first discussion of POVs and PODs, I limit my concern to the nominal case, where the reductive strategy indicated is useful to some degree and where the nominal agent of choice is none other than oneself.

Under default conditions of operation, then, each one's POV embodies the reductive assumption that one's own particular actions and perceptions are nominal, that is, natural, normal, or otherwise “not a problem”. Relative to this ordinary setting, each one's POV is normally configured for tracking the more problematic courses of other agents and the drift of the residual system as a whole. Therefore, the natural setting of a POV can be pictured in terms of the perceptual gestalt it facilitates.

In unexceptional circumstances one always takes one's own agency and one's own experience for granted. This is tantamount to assuming that a synthetic balance is already in effect between the claims of conduct and the trials of bearing. Given this much free reign in arranging the play of forces, the artificial state of accord that results can present itself to be a neutral context of interpretation and the superimposed scene of rapport that prevails can pretend itself to be the unquestioned background of instrumental activity that is implicated in every notable objective contemplated or observation performed. Cast in the role of a stationary stage for the action, there is a whole body of interactions that reside in dynamic equilibrium with each other and that make this proving ground appear to be at rest, but the whole contrivance merely acts to place in relief and to render more obvious whatever else in the way of phenomenal experience is thereby permitted to figure against it as representing an object worthy of inquiry.

Loosely speaking, and operating under the usual anthropomorphism, one can say that an agent projects the joint state trajectory, the course that the whole system takes through a sufficiently well defined CE, onto a trajectory through its own proper space, the residual state space that is encompassed by its chosen POV. Strictly speaking, in another sense, all that is known of an agent is just what is projected on its space, and thus one can say that an agent is wholly constituted by this projection.

The difference between the two senses of projection can now be rationalized as follows. A psychological projection begins when a mathematical projection is employed to deal with a complex experience, that is, an overwhelmingly complicated trajectory of the total system. But the default assumption that one's own actions are not significantly implicated in what happens can occasionally turn out to be unjustified.

In a case of psychological or transverse projection, the significant aspects or motivating features of a problematic situation are attributed to the other actors, while one's own collusion in the relevant character of the total situation is ignored, denied, or otherwise relegated to the peripheral background of the configuration kept in focal awareness, the figure that is currently being attended as a content of consciousness. This form of strategic reorganization usually occurs reflexively among the automatic processes of perception, in spite one's full knowledge or token recognition of the times when it is just as likely that the salient quality of the situation is due to one's own conduct, and even when it is equally possible that the complexion of the moment cannot be resolved into separate components and rendered accountable to individuals at all.

A Formal Point of View

In this section the concept of a point of view (POV) is taken up in greater detail and subjected to the first few steps of a formalization process. This makes it possible to explore the wider implications of the idea, to pursue the lines of inquiry it suggests to greater lengths, and to apply the tentative formalism to an issue of pressing concern, namely, the question of what kind of distinction ought to be posed between the dynamic and the symbolic aspects of intelligent systems.

If there were nothing but a single POV to entertain, a diversion of attention to matters of perspective would hardly be worth the candle. Accordingly, the dimensions of change and diversity are intrinsic to the worth of the whole idea.

One of the reasons for trying to formalize the concept of a POV is so that this TOI, along with others on its model, can reflectively comment on its own POV, as it progresses from moment to moment, and critically examine it as it develops.

When it comes to the subject of systems theory, a particular POV is so widely propagated that it might as well be regarded as the established, received, or traditional POV. The POV in question says that there are dynamic systems and symbolic systems, and never the twain shall meet. I naturally intend to challenge this assumption, preferring to suggest that dynamic and symbolic attributes are better regarded as different aspects of a single underlying system, as “two sides of the same coin”. But first I have to express the assumption well enough to question it.

Beyond the dim inkling of an underlying influence, a sufficiently critical level of reflection on a POV requires a language that is articulate and analytic enough to transform each thesis posed in it into the form of a question. A deliberately reflective technology is needed to bring the prevailing, prejudicial, and hypocritical underpinnings of a POV to light, since biases due to assumptions obscurely held are seldom automatically revealed. This highlights the need for a critical apparatus that can be applied to the typical TOI, supplying its interpreter with the technical means to take up a critical POV with respect to it.

A logical calculus cannot initiate reflection on a text, but it can help to support and maintain it. The raw ability to perceive selected features of an ongoing text and the basic language of primitive terms, that allow one to mark the presence and note the passing of these features, have to be supplied from outside the calculus at the outset of its calculations. In the present text, the means to support critical reflection on its own POV and others are implemented in the form of a propositional calculus. Given the raw ability of a perceptive interpreter to form glosses on the text and to reflect on the contents of its current POV, a logical calculus can serve to augment the text and assist its critique by catalyzing the consideration of alternative POVs and facilitating reasoning about the wider implications of any particular POV.

The discussion so far has dwelt at length on a particular scene, returning periodically to the fragmentary but concrete situation of a dialogue between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{A}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{B},} poring over the formal setting and teasing out the casual surroundings of a circumscribed pair of sign relations. If the larger inquiry into inquiry is ever to lift itself off from these concrete and isolated grounds, then there is need for a way to extract the lessons of this exercise for reuse on other occasions. If items of knowledge with enduring value are to be found in this arena, then they ought to be capable of application to broader areas of interest and to richer domains of inquiry, and this demands ways to test their tentative findings in analogous and alternative situations of a more significant stripe. One way to do this is to identify properties and details of the selected examples that can be varied within the bounds of a common theme and treated as parameters whose momentary values convey the appearance of complete individuality to each particular case.

Typically, a movement from reduced examples to realistic exercises takes a definite but gradual progression of steps, moving forward through the paces of abstraction, generalization, transformation, and re application. The prospects of success in these stages of development are associated with the introduction of certain formal devices. Principal among these are the explicit recognition of sets of parameters and their expression in terms of lists of variables.

As I understand them, variables are a class of beneficially ambiguous or usefully equivocal signs. In effect, variables are just signs, but signs possessed of a more adaptive constitution or affected by a more flexible interpretation than signs of the usual, more constant variety. These forms of employment turn variables into a class of reusable signs, converting them into sustainable resources for meaning that can be used in a plurality of ways and deployed to articulate different choices at different times from among the available points of thematic variation.

The next major task of this discussion, while continuing to take its bearings from examples as concrete as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(\text{A})\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(\text{B}),\!} is to develop systematic methods for divining the bearing of such isolated examples on issues of real concern. This involves two stages:

One needs to detect the invariant features of the currently known examples, in other words, the dimensions along which their values are, knowingly or unknowingly, held to be constant.
One needs to try varying the features that are presently held to be constant by imagining new examples that are able to realize alternative features.

The larger issue at stake throughout these stages is how the agent of inquiry can find ways to express the lessons of individual exercises in ways that persist through and rise above their individual attachments to experience, thereby living through detailed experiences while remaining undiverted by their peculiar distractions.

There appears to be a practical necessity in drawing at least a tentative distinction between the role of an object and the role of an interpreter, even if a moment of reflection occasionally requires a single entity to fill both roles, and even though a mass of experience with systems that try to draw hard and fast distinctions between things, once and for all, leads one to see that a need exists for ways to withdraw every pretense of any distinction, redrawing it anew if possible, and drawing on new grounds if necessary. There is never anything initially or immediately obvious about a sign itself that says it destined to represent an object of a particular type, and this makes it necessary to infer the type that ought to be specified from the pattern of references in which the sign is actually observed to be engaged.

A distinction that one is initially tempted to treat as substantial but is later bound to discover as purely interpretive, like that between objects and signs, subjects and predicates, particles and waves, or dynamic and symbolic aspects of systems, can frequently bedevil sensible inquiry for quite some period of time. To deal with this problem, there needs to be a standardly available mechanism for introducing these staple but still provisional distinctions, accepting them on a par with axioms at first, but without precluding the opportunities to later revise the substantive imports of their interpretations.

On the way to integrating dynamic and symbolic approaches to systems there are several different sorts of things that can happen. It can happen that a certain distinction, a natural or artificial feature that separates the outlooks of the dynamic and symbolic perspectives, or the sheer appearance of a distinction, a suggestion of a line that leads an observer to see a difference between the two views in the first place, merely gets erased. Or it can happen that the ostensible distinction between the two standpoints marks in reality a naturally useful border, one that is well worth preserving, and yet a wealth of connections that constitutes the true relationship between the two realms can be marked and remarked with increasing visibility in the meantime. In any case, there are lines of pretended distinction and potential difference that must be crossed, and then recrossed, time after time, until their exact form and precise nature have become marked in their clarity or else transparent in their obliteration.

I would like to detach, for a moment, from the particular contrast of interest here, the one posed between dynamic and symbolic orientations, to examine the general question of relating contrasting aspects or views. In this connection, two distinct but correlated efforts at classification and organization arise in tandem with each other. One concern seeks to classify the attributes, categories, features, properties, or qualities that are used to describe the object observed, while the other project tries to organize the approaches, instruments, methods, perspectives, or views that are used to observe the object described.

To invoke the traditional terminology, natural classes of predicates are referred to as categories or predicaments, making it natural to call the classification and study of predicates by the name of categoric, while the classification and study of methods is classically referred to as heuristic or methodeutic (Peirce, CP 2.105–110 and 2.207).

Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: “Category” from Aristotle and Kant, “Functor” from Carnap …, and “natural transformation” from then current informal parlance.

Saunders Mac Lane, Cat.Work.Math. 29–30

Categoric. Although this subject is historically referred to as the “theory of categories”, in modern times it is necessary to distinguish it from the mathematical subject of category theory, whose claim to the title is confessedly derived by stealth. By way of suffering unto the older discipline the freshness of the younger subject, the original study and more general classification of predicates can be referred to as the doctrine of categories (DOC). This is a fair description, given that optional schemes of basic categories are commonly taken up, maintained, and transmitted in decidedly catechismic and rigidly dogmatic fashions.

Perhaps it is the mind's reluctance to revive the uncertainties and to relive the struggles that these schemes were made to resolve, but once the fundamental categories are settled it is nearly impossible to revise them, however poorly they come to fit the current circumstances of life. No matter how original the thinking that leads up to a site where a stable foundation can be poured, the foundation itself is typically laid down as if it were cut from unalterable stone.

I even hope that what I have done may prove a first step toward the resolution of one of the main problems of logic, that of producing a method for the discovery of methods in mathematics.

Charles S. Peirce, CP 3.364, “On the Algebra of Logic : A Contribution to the Philosophy of Notation”,
American Journal of Mathematics, 7(2), 180–202, (1885).

Methodeutic. This subject, that C.S. Peirce gave the alternate titles of “speculative rhetoric” or “formal rhetoric”, because it is a science that “would treat of the formal conditions of the force of symbols, or their power of appealing to a mind, that is, of their reference in general to interpretants” (CP 1.444 and 1.559), and that he assigned the task to find “a method of discovering methods” (CP 2.108 and 3.364), is one that clearly has a special relevance to the pursuit of an inquiry into inquiry.

In an effort to gradually begin formalizing these issues, I introduce the concept of a point of development (POD). This notion is intended to capture a particular moment in the history of a system or its agent, as it is reflected in the systems of propositions associated with each POD. Relative to a particular POD there can be distinguished, though neither exclusively nor exhaustively, two types of propositions that are said to be “associated” with it. Roughly speaking, these types of propositions reflect the thoughts that are applied to a POD and the thoughts that are attached to a POD, respectively.

A proposition that applies to a POD can be formulated in more detail as a proposition about or on a POD. This describes the corresponding POD as though observed from an outside perspective, stating features that locate it within a space of dynamic configurations or that place it in relation to some other medium of common description. This manner of associating propositions with PODs is tantamount to adopting a third person POV on the system or its agent, and it is commonly used to convey an impression of objectivity, no matter whether this standpoint is well taken or not.
A proposition that attaches to a POD can be formalized in more detail as a proposition at or in a POD. This represents what an agent thinks or believes, entertains or maintains, in sum, what an agent is aware of or willing to assert at a particular POD. By way of filling out the formula, this type of proposition expresses thoughts and is expressed in signs that are likewise regarded as attached to the POD in question. In general, propositions at a POD can be formed to express every conceivable modality. Collectively, they can state anything that an agent notes or thinks, observes or imagines at a given moment of its developmental history. They can reflect any aspect of an agent's awareness, belief, conjecture, doubt, expectation, intention, observation, or any other latitude of thought that is actively considered or faithfully preserved throughout the moment in question, and in this sense they are considered to be attached to, bound to, contained in, or localized at a particular POD.

In one sense, propositions about a POD are potentially the general case, since propositions at a POD can be incorporated within their formulation. That is, a proposition about a POD is allowed to make assertions about the propositions at that POD, plus assertions about their relation to propositions at other PODs. But propositions whose references are this involved, articulated as propositions about propositions at a POD, for instance, are classed as higher order propositions and need to be inferred through processes of hypothesis and experiment, conjecture and confirmation, instead of being observed outright. In another sense, propositions at a POD are intrinsically the prototype, since it is from their data that every other type must be constructed.

Propositions about PODs naturally collect into theories about PODs, and at the next level of aggregation these constitute the familiar sorts of dynamic theories that are used to describe the state spaces of systems and the trajectories of agents through them. Concentrating on these types of propositions leads to the kinds of theories about systems where a “neutral observer”, not involved in the system itself, is postulated or fancied to stand outside the dynamics of the “observable object system”: where this “objective reasoner” is supposedly able to theorize about the observable system without essentially becoming a part of its operations or necessarily being involved as a participant in its actual workings, and where the same “passive agent” never finds itself forced to interact in an irreversible or irrevocable manner with the autonomous course of the object system's action.

The thoughts attached to a POD, the things an agent thinks or believes, entertains or maintains at one POD, in relation to what the agent thinks or believes, is aware of or willing to assert at another POD, is the very form of subject matter that is bound to come to light and bound to fall into play whenever one studies the development of a reflective system, whether the focus of interest is the course of a particular inquiry or the emergence of a generic intelligence.

From a pragmatic point of view, a belief is a proposition that an agent is prepared to act on. In practice, this means that information about beliefs can be obtained from observations of action, as long as one remembers that this information is almost always partial information, contingent on the sample of actions that are actually observed and limited by the circumstance that not all preparations result in action.

It may be thought that there is an important distinction between belief and knowledge that ought to be recognized in the modes of maintaining propositions at or in a POD. Given the pragmatic definition of belief, however, there is no local mark that can tell belief and knowledge apart. That is, there is no practical difference that can be sustained, in the propositions attached to a single POD, between those that reflect items of contingent belief and those that reflect items of certain knowledge. Even if the propositions at or in a POD are artificially marked in ways that can later be reliably detected, the problem of constantly updating so fleeting a form of distinction makes the accumulating profusion of ephemeral distinctions as immaterial and unenlightening as every other genre of eracist obliterature.

A distinction between belief and knowledge appears to arise only in the interactions and comparisons that can be made between different PODs, either those enjoyed by a single agent in the history of a single system or those passed through by ostensibly different agents and systems. The sense of the distinction can be sustained only if the order of its relational context continues to be recognized, which means that the mark of the distinction cannot be strained to the point of being an absolute. In this context, different systems and their agents are said to be at comparable PODs precisely to the degree and exactly to the extent that the propositions at and about them, respectively, can be compared. In many respects, the comparison of propositions at different PODs is equally complex and problematic whether it is one agent or several that is being considered.

With all this in mind, I can give a formulation of what the practical difference between belief and knowledge consists in. Roughly speaking, an agent says that an agent knows something if and only if the one believes what the other believes. More precisely, an agent at one POD has reason to say that an agent at another POD (possibly a former self) knows something about something (or knew something about something) if and only if the one believes what the other believes about it, all things being relative to the PODs that the agents are at.

Propositions associated with a POD are often found in organized bodies, forming more or less logical systems of more or less logical statements. Whatever their type or modality with respect to a POD, “propositions of a feather gather together”. That is, they tend to collect into organized bodies of propositions that share compatible types of association and comparable modes of assertion. In logic, an arbitrary collection of propositions is called a theory, no matter how coherent, complete, or consistent it turns out to be when subjected in time to critical review. Taking up this liberalized notion of a theory in the present setting, a bunch of propositions at or in a POD forms a theory at or in a POD, while a bunch of propositions about or on a POD forms a theory about or on a POD.

A reasonably organized system is amenable to having its propositions sorted further, forming collections of propositions that are intended to be interpreted in the same light, and constellating theories that bear on single modes of contemplation or declaration among their propositions.

With respect to the propositions at a POD, the present inquiry into inquiry is mainly concerned with the modalities of expectation, intention, and observation. This is due to a couple of differential modalities, derived in pairs from among these three, that appear to drive every form of inquiry, at least, to some degree.

There is the moment of doubt or uncertainty that is encountered in a surprising phenomenon, providing an impulse for the component of inquiry that seeks an explanation to relieve the shock. This factor driving inquiry can be analyzed as deriving from the differences that occur between one's expectations and one's observations.
There is the moment of desire or difficulty that is countenanced in a problematic situation, providing an impulse for the component of inquiry that seeks a plan of action to resolve the trouble. This factor driving inquiry can be analyzed as deriving from the differences that occur between one's intentions and one's observations.

It should be obvious that these conceptions represent another attempt to formalize the relationship between dynamic and symbolic approaches to intelligent systems. Once again, the paradigms that are established for dealing with propositions at or about PODs are typically specialized to consider one or the other but seldom both. This leads to the familiar sorts of dichotomies being imposed on a subject matter where the types are more complementary and generative than exclusive and exhaustive. Thus, one finds methodologies in the field that can work well either from an “external” (dynamic, model-theoretic, empirical) perspective or from an “internal” (symbolic, proof-theoretic, rational) perspective, but that are seldom able to incorporate both technologies into an integrated methodology.

The concept of a POD in the history of a system, with its associated division of propositions into those that apply exterior to it and those that attach interior to it, is yet another way of approaching a recurring subject, “the being and the role of the interpreter”, that the general concept of an objective concern (OC) broached at an earlier point of development in this text, is also intended to capture. Advancing as if from a pair of complementary and convergent directions, the notion of a POD, in the way it supplies a footing to the propositions about or on it and serves to encapsulate the propositions at or in it, equips a growing SOI with all the pivotal, trophic, and vital functions that the notion of an objective motif (OM) realized in an interpretive moment (IM) is likewise meant to provide.

The relationship between a POD and an OM at an IM can be understood as follows. …

In order to continue formalizing the discussion of POVs and PODs within the text that uses them, I introduce the following notations:

${\begin{array}{lcccr}j:x\Delta y&,&x\Delta _{j}y&,&x\Delta y:j\\j:x\neq y&,&x\neq _{j}y&,&x\neq y:j\\j:(x~,~y)&,&(x~,~y)_{j}&,&(x~,~y):j\end{array}}$

All these expressions are intended to indicate a set of circumstances that could otherwise be described as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j ~\text{appears to see a distinction between}~ x ~\text{and}~ y.}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j ~\text{partitions a dimension of discourse between}~ x ~\text{and}~ y.}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j ~\text{sees}~ x ~\text{and}~ y ~\text{as mutually exclusive and exhaustive possibilities.}}

In this scheme, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} x {}^{\prime\prime}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} y {}^{\prime\prime}} indicate logical dimensions of variation or propositional features of description that govern an agent's possibilities of action and perception. Used as primitive logical terms they denote the distinctive features that determine an agent's spaces of performance and experience. In combination with logical operators they generate a descriptive framework that encompasses both: (1) the methodological approaches or perspectives toward objects that an agent can adopt, and (2) the categorical aspects of objects, the independently coherent systems of properties and qualities that characterize the hypothetically unified object system.

In practice, it does not matter whether one regards Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y\!} as logical features or as boolean variables, so long as the full set of positive and negative features Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{ x, (x), y, (y) \}\!} is initially available to classify the relevant space of object perceptions or interpretive actions. Analogous to its role in the staging relations { $\{\lessdot ~,~\gtrdot \},$ the label Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} j {}^{\prime\prime}} indicates the active interpreter, that is, the system and moment of interpretation or the state of the interpretive system that is held to be responsible for finding, making, testing, or following through the consequences of posing the contemplated distinctions.

Dual to the statements of momentary interpretive distinctions (MIDs) are the respective statements of momentary interpretive coincidences (MICs):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lcccr} j : x = y & , & x =_j y & , & x = y : j \\ j : x \Leftrightarrow y & , & x \Leftrightarrow_j y & , & x \Leftrightarrow y : j \\ j : ((x ~,~ y)) & , & ((x ~,~ y))_j & , & ((x ~,~ y)) : j \end{array}}

Each of these expressions is intended to indicate a set of circumstances that could otherwise be rendered by any one of the following, logically equivalent statements:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j ~\text{appears to see a coincidence between}~ x ~\text{and}~ y.}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j ~\text{draws no distinction between the dimensions}~ x ~\text{and}~ y.}

j~{\text{sees}}~x~{\text{and}}~y~{\text{as manifestly equivalent ranges of possibilities.}}

The introduction of explicit names for systems of interpretation, as well as for their interpretive moments, models of interpretation, objective concerns, points of development, and situations of use, is intended to flesh out the lifeless idiom or insipid brand of assignment statements that are currently found in CL settings, which are typically rendered so abstractly as to constitute a entire style of anonymous, passive, or unattributed excuses for fully executable commands.

In a related usage, one is permitted to reparse the anonymous or passive form of assignment statement,

{}^{\backprime \backprime }x:=y{}^{\prime \prime },

read as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} x ~\text{is set equal to}~ y {}^{\prime\prime},}

converting it into the corresponding attributive or active form of assignment statement,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} j : x = y {}^{\prime\prime},}

read as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} j ~\text{sets}~ x ~\text{equal to}~ y {}^{\prime\prime}.}

Returning to the present application, the categorical project leads one to seek something in the object itself, some factor that divides up its dynamic and symbolic aspects, some plane of cleavage that explains the natural divisions between different types of object system, while the methodeutic outlook leads one to wonder whether the specialized mode of being that is beheld in the object is not in fact due to something in the style and direction of approach, some artifact of method that is being cast on the object system from the eye of the beholder.

I would like to articulate a systematic hypothesis that prevails over the scene of this work, tacitly imposing the deliberately hopeful assumption that there is always some sort of hypostatic unity to be found beneath the manifold diversity of phenomena. It is not just my own presumption or personal preference to say this. I find it to be a likely and common assumption, constantly being used to address all sorts of interesting phenomena and almost every process of note, whether or not it is ever expressly enunciated.

This hypothesis is probably implicit in the very idea of a system, that is to say, in the notion of things standing together, and it is central to the very conception of a systematic universe or a universal system. Nevertheless, I will have to take responsibility for the particular way that this premiss is expressed and developed in this text. Because it amounts to the underlying hope that there is always a unified system, some one thing that subsists beneath every form of phenomenal process and that remains available to substantiate and explain whatever manner of diversity in appearances is encountered, something or other that is always ready to be explicated but seldom necessary to declare, I call this assumption the hypostatically unified system hypothesis (HUSH).

In accord with this tacit assumption, that rules the entire realm of systems theory, it can be presumed that there is an integral system, prior in its real status to the manifold of observable appearances, that is somehow able to manifest itself in the severally projected roles of a dynamic process and a symbolic purpose. But to harvest any practical consequences from the employment of this inchoative precept, the twin yoke of questions, categorical and methodeutic, must now be taken up:

What constitutes the differences between the dynamic and symbolic aspects of the hypothetically unified intelligent system?
What features divide the two perspectives that find these aspects respectively salient?

The integration of symbolic and dynamic approaches to systems thinking requires a significant level of reconstructive effort, one that is capable of extending its energies in both the analytic and synthetic directions. It may be nothing more than a metaphor to describe it this way, but there is something like a dynamic economy of energy exchanges that goes on in facilitating the required “metaboly of symbols” (Peirce).

In this vein, there seem to be laws analogous to conservation principles that govern the transactions between subordinate processes, determining the interactions that are most likely to occur between the breaking down of old conceptual bonds and the creation of new configurations of ideas at higher and lower levels of conceptual equilibrium. Brought to bear on the present task, the specific manifestations of “mental energy” that are called on to accomplish the current work of integration have a potential for raising questions about the relation of “logic” to “time”, and thus revive an issue that goes back to the very birth of thought.

The relation of logical and temporal realms, of rational ideas to real experiences, is an ancient and fundamental question, one whose initial answers were laid down in their present form at the very beginnings of reflective inquiry and whose sedimented contents now lie metamorphosed into the deepest bedrocks of our native and systematic philosophies. The distribution of current opinion on the matter regards the question as being (1) “previously settled” or (2) “incapable of solution”, with little thought given to a tertium quid, or a more fluid medium that could moderate between the extremes of these fixed alternatives.

Unfortunately, the customary and habitual classification of a problem as “insoluble”, even when justified, can work against the recognition of methods that are available to ameliorate its more objectionable impacts. When it comes to the relationship of logic to time, I believe that the resources are currently available that could advance our understanding of this issue in new directions. All it would take is the will to reconfigure those resources in the appropriate ways.

To expand the formula: The realm of logic is typified by rational concepts regarding invariant patterns, virtually, by ideas about forms, while the rule of time is filled out by realistic experiences with changing qualities, ultimately, by feelings of content and discontent. The application of the integrative effort to intelligent systems in general and to inquiry driven systems (IDSs) in particular only sharpens the question of logic and time to the point of self-application.

Considerations like these, as old and as constant as the hills, and as much over our heads as the eternally renewed and inconstant weather, are deserving of occasional notice, yet their relevance to the work of the moment is doomed by their very quality of necessity to fade into the background of present concerns, and their saliency as problematic phenomena quickly recedes from the scope of any perspective so bent on immediate application as that falling within my present focus.

Three Styles of Linguistic Usage

The theory of sign relations, in general, and the construction of a RIF, in particular, demands that this discussion strike a compromise among several styles of usage that are not normally brought together in the same forum or comprehended in the same frame. Under the rubric of a notion of style or a norm of significance (NOS) this text recognizes a collective need for three distinctive styles of linguistic usage, or three different attitudes toward the intentions of language.

These styles of usage, along with their correlated perspectives on usage and their appropriate contexts of usage, can be put into a graded series by noticing how the more finely grained perspectives on the matter of language use correspond to the more narrowly scoped areas of content that are swept out by their roughly concentric contexts of discussion. Accordingly, the styles, perspectives, and contexts of usage that I need to relate can be distinguished as follows, proceeding in order of their increasing formality:

Broadest of all is the informal language (IL) context, which incorporates the ordinary mathematical context within its compass. Relative to the aims of the present work, which are largely mathematical, these two contexts are roughly coextensive and can be treated as one. All of the more usual contexts are marked by the operation of a working assumption about the interpretation of formal symbols that I call the object convention. Loosely speaking, this takes it for granted that signs always refer to objects, not because of any credible guarantees that they do, but mostly due to a lack of interest in the cases where they do not. Failures of meaning, logical inconsistencies, and doubts about the foundations of the whole enterprise are treated as incidental problems to be discussed and corrected off line.
Next in order is the formal language (FL) context, where the syntax of expressions needs to be specified explicitly and where the semantics of expressions does not usually permit every combination of signs to have a meaning. All of the more formal contexts are marked by the operation of a working assumption about the interpretation of formal symbols that I call the sign convention. Roughly speaking, this views a sign primarily as a mere sign, putting it in question whether any sign has an object. In styles of usage at this or greater degrees of formality, the reception of signs is marked by a heightened suspicion, where the benefit of the doubt and the burden of proof in the matter of signs having meaning are critically reversed from their natural defaults. Signs are assumed to be innocent of meaning until shown otherwise.
Most constrained of all is the computational language (CL) context, which incorporates the interests of computational linguistics along with the aims of implementing and using programming languages. There are many styles of programming languages and many more styles of putting them to use. I concentrate here on a particular version of the Pascal language and describe the particular ways I have chosen to implement the concepts I need with the constructs it makes available.

Next I need to consider the complex of relationships that exists among these three styles of usage, along with the corresponding relationships that exist among their associated perspectives and contexts. In regard to the questions raised by these three norms of significance, the pragmatic theory of sign relations is intended to help reflective interpreters, and other students of language, maintain all the advantages of taking up abstract and isolated perspectives on language use, but to achieve this without losing a sense of the connection that each peculiar outlook has to the richly interwoven pattern of a larger unity.

In many places these variegated styles of usage express themselves not so much in isolated domains of influence or distinctive layers of context as in different perspectives on the same text. But different lights on a developing picture can cause different figures and patterns to emerge, and different ways of treating a developing text can lead it to grow in different directions. Thus, discrepant points of view on the emergence of a literature can stimulate different works to vie for its canon, and discriminating angles of approach to what seems like a level plain and a unified field of language can harvest a wealth of alternate appreciations. And so different styles of writing arise in correspondence with different styles of reading, and each rising style of readership engenders a new style of authorship in its wake.

At other times these degrees of formality play themselves out in a temporal process. Consider a typical scenario for solving problems through formalization:

One begins by approaching the problem informally, in other words, in IL-posed terms, drawing on the common resources of technical notions and mathematical methods that are available, familiar, intuitively understood, and that suggest themselves as possibly being relevant to the problem.
Next, the problem of interest and the array of methods selected for addressing it are both reformulated in FL terms, a process that requires many obscurities and omissions of the original problem statement to be weeded out and filled in, respectively.
Finally, the formalized version of the problem method constellation is reconstructed to the extent possible in a CL framework.

At any stage of this procedure one may discover, or begin to suspect, that the current representation of the problem or the present selection of methods is inadequate to the task or unlikely to lead to a solution. In this event one is forced to back track to an earlier stage of the problem's formulation and to look for ways of changing one's grasp of the situation.

Even though the styles of usage at the three degrees of formalization use overlapping vocabularies of technical terms, the interpretations that they put on some of these terms, together with the working attitudes that they promote toward the corresponding concepts, are tantamount in practice to the possession of distinct concepts for the very same terms.

Three issues of linguistic usage on which the three norms of significance get most out of joint are on the questions of (1) signs and their significance, (2) the utilization of set theory and set-theoretic constructions, and (3) the ontological or pragmatic status of variables. The rest of this section makes a cursory survey of the bearings that the three norms take toward these issues, in preparation for more detailed treatments in later sections.

In each perspective that an observer takes up, the natural attitude is to focus on a particular class of objects, to remain less aware of the signs being used to denote them, and to remain even less aware that these objects and signs can take up other roles in the same or other sign relations. In constantly shifting from one perspective to another, however, the transparent uses of signs and the ulterior circumstances that determine how objects and signs are cast start to become visible. Altogether, the interaction between casual and formal styles of usage is like an exchange carried on between radically different economies, where commodities and utilities that are freely traded in one kind of market are severely taxed in the other.

The IL perspective, along with its specialization to ordinary mathematical discourse, thinks itself to have a grasp of the unitary object itself, conveniently forgetting the multiplicity of abstract, arbitrary, and artificial constructions that are needed to make this impression possible. In particular, the ordinary mathematical attitude thinks itself to have a grasp of the one idea while its puts the many appearances out of mind, and it constantly exerts itself to neglect all the labor that goes into taking up this stance. It ignores the circumstance that numbers, however intuited, can only be indicated and rationalized to others as equivalence classes of constructions formed on the matter of numerals.

The FL perspective, along with its implementations in CL contexts, allows one to treat signs as objects, and thus to study syntactic domains as objective languages. This creates what seems like a higher order of discussion, but the designation of these objects as signs is purely token if their use as signs is forgotten in the process. Consequently, the FL perspective, together with the CL attitude that realizes it, has the job of recovering and reconstructing exactly what has been taken out of consideration in the IL context, namely, the details of actual usage that are taken for granted, abstracted away, and conveniently ignored.

Although the mathematical structures developed under the informal norm of significance can become incredibly sophisticated in their orders of complexity and degrees of formalization, from a pragmatic standpoint they are still construed under naive assumptions about language use. This is because discussions carried out under the IL perspective do not make it their business to reflect on the relations between objects and signs, but presume that these matters can be separated from their subject proper and relegated to preliminary stages of the ultimately refined treatment.

In order to make this discussion of styles and issues and more concrete, the next several sections examine the practical bearings of the three styles of usage as they work out with regard to each of the identified issues of usage. This will be done by choosing a theoretical subject to illustrate the ideals of each style of usage, and then by developing the bearing of this subject on each of the three issues mentioned.

In accord with this plan, the next three sections present the basic ideas of three subjects: group theory, formal language theory, and computation theory. The presentation of these subjects is intended to serve both illustrative and instrumental purposes, exemplifying the ideals of the IL, FL, and CL styles of usage, respectively, but also equipping subsequent discussion with a supply of ready tools that can be used in its further development. After the treatment of these three subjects, and following the introduction of higher order sign relations, the next three sections after that are finally able to take up the three issues mentioned above, concerning the theoretical standings of signs, sets, and variables, respectively, and to consider how each of these issues appears in the light of each style of usage.

Basic Notions of Group Theory

Many of the most salient themes that have a call to be played out in this work — the application of generic forms of operation to themselves and to each other, the relationship of invariant forms to their variant presentations, and the relationship of abstract forms to their concrete representations — all of these topics arise in a very instructive way within the mathematical subject of group theory. This is most likely due to the fact that group theory, as a mathematical tool, got its start and much of its later sharpening in the process of trying to clarify the physical and formal phenomena that involve these very same issues.

In group theory, fortunately, these themes arise in a slightly plainer fashion, and the otherwise mystifying questions they involve have been studied to the point that their original mysteries are barely observed. Thus, a good way to approach the construction of a RIF is to study the well understood versions of self-application and self-explanation that turn up in group theory. Given the simpler character and the familiar condition of these topics in that area, they supply a convenient basis for subsequent extensions and help to arrange a staging ground for the types of sign theoretic generalizations that are ultimately desired.

This section develops the aspects of group theory that are needed in this work, bringing together a fundamental selection of abstract ideas and concrete examples that are used repeatedly throughout the rest of the project. To start, I present an abstract formulation of the basic concepts of group theory, beginning from a very general setting in the theory of relations and proceeding in quick order to the definitions of groups and their representations. After that, I describe a couple of concrete examples that are designed mainly to illustrate the abstract features of groups, but that also appear in different guises at later stages of this discussion.

A sequence of domains (SOD) is a nonempty sequence of nonempty sets. A declarative indication of a sequence of sets, typically offered in staking out the grounds of a discussion, is taken for granted as a SOD. Thus, the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime}(X_i){}^{\prime\prime}} is assumed by default to refer to a SOD Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (X_i)_{i \in I},\!} where each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_i\!} is assumed to be a nonempty set.

Given a SOD Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (X_i),\!} its cartesian product, notated as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textstyle\prod_i (X_i)} or $\textstyle \prod _{i}X_{i},$ is defined as follows:

\prod _{i}(X_{i})=\prod _{i}X_{i}=\{(x_{i}):x_{i}\in X_{i}\}.

A relation is defined on a SOD as a subset of its cartesian product. In symbols, $L\!$ is a relation on $(X_{i}),\!$ if and only if $L\subseteq \textstyle \prod _{i}X_{i}.$

A Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\!} -ary relation or a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\!} -place relation is a relation on an ordered $k\!$ -tuple of nonempty sets. Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L\!} is a $k\!$ -place relation relation on the SOD Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (X_1, \ldots, X_k)\!} if and only if $L\subseteq X_{1}\times \ldots \times X_{k}.\!$ In various applications, the $k\!$ -tuple elements $(x_{1},\ldots ,x_{k})\!$ of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L\!} are called its elementary relations, individual transactions, ingredients, or effects.

Before continuing with the chain of definitions, a slight digression is needed at this point to loosen up the interpretation of relation symbols in what follows. Exercising a certain amount of flexibility with notation, and relying on a discerning interpretation of equivocal expressions, one can use the name ${}^{\backprime \backprime }L{}^{\prime \prime }$ or any other indication of a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\!} -place relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L\!} in a wide variety of different fashions, both logical and operational.

First, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L\!} can be associated with a logical predicate or a proposition that says something about the space of effects, being true of certain effects and false of all others. In this way, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} L {}^{\prime\prime}} can be interpreted as naming a function from $\textstyle \prod _{i}X_{i}$ to the domain of truth values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{B} = \{ 0, 1 \}.} With the appropriate understanding, it is permissible to let the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} L : X_1 \times \ldots \times X_k \to \mathbb{B} {}^{\prime\prime}} indicate this interpretation.

Second, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L\!} can be associated with a piece of information that allows one to complete various sorts of partial data sets in the space of effects. In particular, if one is given a partial effect or an incomplete Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\!} -tuple, say, one that is missing a value in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j^\text{th}\!} place, as indicated by the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} (x_1, \ldots, \hat{j}, \ldots, x_k) {}^{\prime\prime},} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} L {}^{\prime\prime}} can be interpreted as naming a function from the cartesian product of the domains at the filled places to the power set of the domain at the missing place. With this in mind, it is permissible to let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} L : X_1 \times \ldots \times \hat{j} \times \ldots \times X_k \to \mathrm{Pow}(X_j) {}^{\prime\prime}} indicate this use of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} L {}^{\prime\prime}.} If the sets in the range of this function are all singletons, then it is permissible to let ${}^{\backprime \backprime }L:X_{1}\times \ldots \times {\hat {j}}\times \ldots \times X_{k}\to X_{j}{}^{\prime \prime }$ specify the corresponding use of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} L {}^{\prime\prime}.}

In general, the indicated degrees of freedom in the interpretation of relation symbols can be exploited properly only if one understands the consequences of this interpretive liberality and is prepared to deal with the problems that arise from its “polymorphic” practices — from using the same sign in different contexts to refer to different types of objects. For example, one should consider what happens, and what sort of IF is demanded to deal with it, when the name Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} L {}^{\prime\prime}} is used equivocally in a statement like Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L = L^{-1}(1),\!} where a sensible reading requires it to denote the relational set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L \subseteq \textstyle\prod_i X_i} on the first appearance and the propositional function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L : \textstyle\prod_i X_i \to \mathbb{B}} on the second appearance.

A triadic relation is a relation on an ordered triple of nonempty sets. Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L\!} is a triadic relation on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (X, Y, Z)\!} if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L \subseteq X \times Y \times Z.\!} Exercising a proper degree of flexibility with notation, one can use the name of a triadic relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L \subseteq X \times Y \times Z\!} to refer to a logical predicate or a propositional function, of the type Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X \times Y \times Z \to \mathbb{B},\!} or any one of the derived binary operations, of the three types Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X \times Y \to \mathrm{Pow}(Z),\!} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X \times Z \to \mathrm{Pow}(Y),\!} and $Y\times Z\to \mathrm {Pow} (X).\!$

A binary operation or law of composition (LOC) on a nonempty set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X\!} is a triadic relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle * \subseteq X \times X \times X\!} that is also a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle * : X \times X \to X.\!} The notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} x * y {}^{\prime\prime}\!} is used to indicate the functional value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle *(x, y) \in X,~\!} which is also referred to as the product of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y\!} under $*.\!$

A binary operation or LOC $*\!$ on $X\!$ is associative if and only if $(x*y)*z=x*(y*z)\!$ for every $x,y,z\in X.\!$

A binary operation or LOC $*\!$ on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X\!} is commutative if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x*y = y*x\!} for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x, y \in X.\!}

A semigroup consists of a nonempty set with an associative LOC on it. On formal occasions, a semigroup is introduced by means a formula like Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X} = (X, *),\!} read to say that ${\underline {X}}\!$ is the ordered pair $(X,*).\!$ This form specifies $X\!$ as the nonempty set and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle *\!} as the associative LOC. By way of recalling the extra structure, this specification underscores the name of the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X\!} to form the name of the semigroup Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}.\!} In contexts where there is only one semigroup being discussed, or where the additional structure is otherwise understood, it is common practice to call the semigroup by the same name as the underlying set. In contexts where more than one semigroup is formed on the same set, indexed notations like ${\underline {X}}_{i}=(X,*_{i})\!$ may be used to distinguish them.

A unit element in a semigroup Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X} = (X, *)\!} is an element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e\!} in $X\!$ such that $x*e=x=e*x\!$ for all $x\in X.\!$ In other words, a unit element is a two-sided identity element. If a semigroup Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}\!} has a unit element, then it is unique, since if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e'\!} is also a unit element, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e' = e'*e = e.\!}

A monoid is a semigroup with a unit element. Formally, a monoid Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}\!} is an ordered triple $(X,*,e),\!$ where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X\!} is a set, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle *\!} is an associative LOC on the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e\!} is the unit element in the semigroup Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (X, *).\!}

An inverse of an element $x\!$ in a monoid Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X} = (X, *, e)\!} is an element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y \in X\!} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x*y = e = y*x.\!} An element that has an inverse in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}\!} is said to be invertible (relative to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle *\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e\!} ). If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x\!} has an inverse in ${\underline {X}},\!$ then it is unique to $x.\!$ To see this, suppose that $y'\!$ is also an inverse of $x.\!$ Then it follows that:

y'~=~y'*e~=~y'*(x*y)~=~(y'*x)*y~=~e*y~=~y.\!

A group is a monoid all of whose elements are invertible. That is, a group is a semigroup with a unit element in which every element has an inverse. Putting all the pieces together, then, a group ${\underline {X}}=(X,*,e)\!$ is a set $X\!$ with a binary operation $*:X\times X\to X\!$ and a designated element $e\!$ that is subject to the following three axioms:

G1.	(associative)	$x(yz)~=~(xy)z,\!$	for all $x,y,z\in X.\!$
G2.	(identity)	$ex~=~x~=~xe,\!$	for some $e\in X.\!$
G3.	(inverses)	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle xy ~=~ e ~=~ yx,\!}	for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y \in X,\!} for all $x\in X.\!$

It is customary to use a number of abbreviations and conventions in discussing semigroups, monoids, and groups. A system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X} = (X, *)\!} is given the adjective commutative if and only if $*\!$ is commutative. Commutative groups, however, are traditionally called abelian groups. By way of making comparisons with familiar systems and operations, the following usages are also common.

One says that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}\!} is written multiplicatively to mean that a raised dot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {(\cdot)}\!} or concatenation is used instead of a star for the LOC. In this case, the unit element is commonly written as an ordinary algebraic one, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1,\!} while the inverse of an element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x\!} is written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x^{-1}.\!} The multiplicative manner of presentation is the one that is usually taken by default in the most general types of situations. In the multiplicative idiom, the following definitions of powers, cyclic groups, and generators are also common.

In a semigroup, the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n^\text{th}\!} power of an element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x\!} is notated as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x^n\!} and defined for every positive integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n\!} in the following manner. Proceeding recursively, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x^1 = x\!} and let

x^{n}=x^{n-1}\cdot x\!

for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n > 1.\!}

In a monoid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x^n\!} is defined for every non-negative integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n\!} by letting

x^{0}=1\!

and proceeding the same way for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n > 0.\!}

In a group, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x^n\!} is defined for every integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n\!} by letting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x^n = (x^{-1})^{-n}\!} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n < 0\!} and proceeding the same way for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n \ge 0.\!}

A group Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}\!} is cyclic if and only if there is an element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g \in X\!} such that every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x \in X\!} can be written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x = g^n\!} for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{Z}.\!} In this case, an element such as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g\!} is called a generator of the group.

One says that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}\!} is written additively to mean that a plus sign Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (+)\!} is used instead of a star for the LOC. In this case, the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x + y\!} indicates a value in $X\!$ called the sum of $x\!$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y.\!} This involves the further conventions that the unit element is written as a zero, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0,\!} and may be called the zero element, while the inverse of an element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x\!} is written as $-x,\!$ and may be called the negative of $x.\!$ Usually, but not always, this manner of presentation is reserved for commutative systems and abelian groups. In the additive idiom, the following definitions of multiples, cyclic groups, and generators are also common.

In a semigroup written additively, the

n^{\text{th}}\!

multiple of an element

x\!

is notated as

nx\!

and defined for every positive integer

n\!

in the following manner. Proceeding recursively, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1x = x\!} and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle nx = (n-1)x + x\!} for all

n>1.\!

In a monoid written additively, the multiple Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle nx\!} is defined for every non-negative integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n\!} by letting

0x=0\!

and proceeding the same way for

n>0.\!

In a group written additively, the multiple Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle nx\!} is defined for every integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n\!} by letting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle nx = (-n)(-x)\!} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n < 0\!} and proceeding the same way for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n \ge 0.\!}

A group Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X} = (X, +, 0)\!} is cyclic if and only if there is an element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g \in X\!} such that every

x\in X\!

can be written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x = ng\!} for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{Z}.\!} In this case, an element such as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g\!} is called a generator of the group.

Mathematical systems, like the relations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L\!} and operational structures Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}\!} encountered above, are seldom comprehended in perfect isolation, but need to be viewed in relation to each other, as belonging to families of comparable systems. Systems are compared by finding or making correspondences between them, and this can be formalized as a task of setting up and probing various types of mappings between the sundry appearances of their objective structures. This requires techniques for exploring the spaces of mappings that exist between families of systems, for inquiring into and demonstrating the existence of specified types of functions between them, plus technical concepts for classifying and comparing their diverse representations. Therefore, in order to compare the structures of different objective systems and to recognize the same objective structure when it appears in different phenomenal or syntactic disguises, it helps to develop general forms of comparison that can organize the welter of possible associations between systems and single out those that represent a preservation of the designated forms.

The next series of definitions develops the mathematical concepts of homomorphism and isomorphism, special types of mappings between systems that serve to formalize the intuitive notions of structural analogy and abstract identity, respectively. In very rough terms, a homomorphism is a structure-preserving mapping between systems, but only in the sense that it preserves some part or some aspect of the structure mapped, whereas an isomorphism is a correspondence that preserves all of the relevant structure.

The induced action of a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f : X\to Y\!} on the cartesian power Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X^k\!} is the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f' : X^k \to Y^k\!} defined by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f'((x_1, \ldots, x_k)) ~=~ (f(x_1), \ldots, f(x_k)).\!}

Usually, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f'\!} is regarded as the natural, obvious, tacit, or trivial extension that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f : X \to Y\!} possesses in the space of functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X^k \to Y^k,\!} and is thus allowed to go by the same name as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f.\!} This convention, assumed by default, is expressed by the formula:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f((x_1, \ldots, x_k)) ~=~ (f(x_1), \ldots, f(x_k)).\!}

A relation homomorphism from a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\!} -place relation $P\subseteq X^{k}\!$ to a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\!} -place relation $Q\subseteq Y^{k}\!$ is a mapping between the underlying sets, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h : X \to Y,\!} whose induced action Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h : X^k \to Y^k\!} preserves the indicated relations, taking every element of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P\!} to an element of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q.\!} In other words:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x_1, \ldots, x_k) \in P ~\Rightarrow~ h((x_1, \ldots, x_k)) \in Q.\!}

Applying this definition to the case of two binary operations, say Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle *_1\!} on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_1\!} and $*_{2}\!$ on $X_{2},\!$ which are special kinds of triadic relations, say Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle *_1 \subseteq X_1^3\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle *_2 \subseteq X_2^3,\!} one obtains:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x, y, z) \in *_1 ~\Rightarrow~ h((x, y, z)) \in *_2.\!}

Under the induced action of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h : X_1 \to X_2,\!} or its tacit extension as a mapping Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h : X_1^3 \to X_2^3,\!} this implication yields the following:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x, y, z) \in *_1 ~\Rightarrow~ (h(x), h(y), h(z)) \in *_2.\!}

The left hand side of this implication is expressed more commonly as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x *_1 y = z.\!}

The right hand side of the implication is expressed more commonly as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h(x) *_2 h(y) = h(z).\!}

From these two equations one derives, by substituting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x *_1 y\!} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z\!} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h(z),\!} a succinct formulation of the condition for a mapping Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h : X_1 \to X_2\!} to be a relation homomorphism from a system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (X_1, *_1)\!} to a system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_2, *_2,\!} expressed in the form of a distributive law or linearity condition:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h(x *_1 y) ~=~ h(x) *_2 h(y).\!}

To sum up the development so far in a general way: A homomorphism is a mapping from a system to a system that preserves an aspect of systematic structure, usually one that is relevant to an understood purpose or context. When the pertinent aspect of structure for both the source and the target system is a binary operation or a LOC, then the condition that the LOCs be preserved in passing from the pre-image to the image of the mapping is frequently expressed by stating that the image of the product is the product of the images. That is, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h : X_1 \to X_2\!} is a homomorphism from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\underline{X}_1 = (X_1, *_1)}\!} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\underline{X}_2 = (X_2, *_2)},\!} then for every $x,y\in X_{1}\!$ the following condition holds:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h(x *_1 y) ~=~ h(x) *_2 h(y).\!}

Next, the concept of a homomorphism or structure-preserving map is specialized to the different kinds of structure of interest here.

A semigroup homomorphism from a semigroup Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\underline{X}_1 = (X_1, *_1)}\!} to a semigroup Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\underline{X}_2 = (X_2, *_2)}\!} is a mapping between the underlying sets that preserves the structure appropriate to semigroups, namely, the LOCs. This makes it a map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h : X_1 \to X_2\!} whose induced action on the LOCs is such that it takes every element of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle *_1\!} to an element of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle *_2.\!} That is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x, y, z) \in *_1 ~\Rightarrow~ h((x, y, z)) = (h(x), h(y), h(z)) \in *_2.\!}

A monoid homomorphism from a monoid Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}_1 = (X_1, *_1, e_1)\!} to a monoid Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}_2 = (X_2, *_2, e_2)\!} is a mapping between the underlying sets, $h:X_{1}\to X_{2},\!$ that preserves the structure appropriate to monoids, namely, the LOCs and the identity elements. This means that the map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h\!} is a semigroup homomorphism from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}_1\!} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}_2,\!} where these are considered as semigroups, but with the extra condition that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h\!} takes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e_1\!} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e_2.\!}

A group homomorphism from a group Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}_1 = (X_1, *_1, e_1)\!} to a group Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}_2 = (X_2, *_2, e_2)\!} is a mapping between the underlying sets, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h : X_1 \to X_2,\!} that preserves the structure appropriate to groups, namely, the LOCs, the identity elements, and the inverse elements. This means that the map $h\!$ is a monoid homomorphism from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_1\!} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_2,\!} where these are viewed as monoids, with the extra condition that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h(x^{-1}) = h(x)^{-1}\!} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x \in X_1.\!} As it happens, the inverse elements are automatically preserved if the LOCs and the identity elements are, so a monoid homomorphism suffices to constitute a group homomorphism for a monoid that is also a group. To see why this is so, consider the following chain of equalities:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h(x) *_2 h(x^{-1}) ~=~ h(x *_1 x^{-1}) ~=~ h(e_1) ~=~ e_2.\!}

An isomorphism is a homomorphism that is one to one and onto, or bijective. Systems that have an isomorphism between them are called isomorphic to each other and belong to the same isomorphism class. From an abstract point of view, isomorphic systems are tantamount to the same mathematical object, differing at most in their manner of presentation and the details of their representation. Usually these differences are regarded as purely notational, a mere change of names. Thus, they are seen as accidental or accessory features of the object, corresponding to different ways of grasping the objective structure that is the main interest of the study but not considered as essential parts of its ultimate constitution or even necessary to its final comprehension.

Finally, to introduce two pieces of language that are often useful: an endomorphism is a homomorphism from a system into itself, while an automorphism is an isomorphism from a system onto itself.

If nothing more succinct is available, a group can be specified by means of its operation table, usually styled either as a multiplication table or an addition table. Table 32.1 illustrates the general scheme of a group operation table. In this case the group operation, treated as a “multiplication”, is formally symbolized by a star Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (*),\!} as in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x * y = z.\!} In contexts where only algebraic operations are formalized it is common practice to omit the star, but when logical conjunctions (symbolized by a raised dot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {(\cdot)}\!} or by concatenation) appear in the same context, then the star is retained for the group operation.

Another way of approaching the study or presenting the structure of a group is by means of a group representation, in particular, one that represents the group in the special form of a transformation group. This is a set of transformations acting on a concrete space of “points” or a designated set of “objects”. In providing an abstractly given group with a representation as a transformation group, one is seeking to know the group by its effects, that is, in terms of the action it induces, through the representation, on a concrete domain of objects. In the type of representation known as a regular representation, one is seeking to know the group by its effects on itself.

Tables 32.2 and 32.3 illustrate the two conceivable ways of forming a regular representation of a group Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle G.\!}

The ante-representation of $x_{i}\!$ in $G\!$ is a function from $G\!$ to $G\!$ that is formed by considering the effects of $x_{i}\!$ on the elements of $G\!$ when $x_{i}\!$ acts in the role of the first operand of the group operation. Notating this function as $h_{1}(x_{i}):G\to G,\!$ the regular ante-representation of $G\!$ is a map $h_{1}:G\to (G\to G)\!$ that is schematized in Table 32.2. Here, each of the functions $h_{1}(x_{i}):G\to G\!$ is represented as a set of ordered pairs of the form $(x_{j}~,~x_{i}*x_{j}).\!$

The post-representation of $x_{i}\!$ in $G\!$ is a function from $G\!$ to $G\!$ that is formed by considering the effects of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_i\!} on the elements of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle G\!} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_i\!} acts in the role of the second operand of the group operation. Notating this function as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h_2(x_i) : G \to G,\!} the regular post-representation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle G\!} is a map $h_{2}:G\to (G\to G)\!$ that is schematized in Table 32.3. Here, each of the functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h_2(x_i) : G \to G\!} is represented as a set of ordered pairs of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x_j ~,~ x_j * x_i).\!}

${\text{Table 32.1}}~~{\text{Scheme of a Group Operation Table}}\!$
$*\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_0\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_j\!}	$\cdots \!$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_0\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_0 * x_0\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	$x_{0}*x_{j}\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_i\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_i * x_0\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_i * x_j\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	$\cdots \!$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Table 32.2} ~~ \text{Scheme of the Regular Ante-Representation}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Element}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Function as Set of Ordered Pairs of Elements}\!}
$x_{0}\!$	$\{\!$	$(x_{0}~,~x_{0}*x_{0}),\!$	$\cdots \!$	$(x_{j}~,~x_{0}*x_{j}),\!$	$\cdots \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
$\cdots \!$	$\{\!$	$\cdots \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	$\cdots \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	$\}\!$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_i\!}	$\{\!$	$(x_{0}~,~x_{i}*x_{0}),\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x_j ~,~ x_i * x_j),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	$\cdots \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Table 32.3} ~~ \text{Scheme of the Regular Post-Representation}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Element}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Function as Set of Ordered Pairs of Elements}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_0\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x_0 ~,~ x_0 * x_0),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	$(x_{j}~,~x_{j}*x_{0}),\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	$\cdots \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_i\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x_0 ~,~ x_0 * x_i),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	$(x_{j}~,~x_{j}*x_{i}),\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	$\}\!$
$\cdots \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	$\cdots \!$	$\cdots \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdots\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}

In following these maps, notice how closely one is treading in these representations to defining each element in terms of itself, but without quite going that far. There are a couple of catches that save this form of representation from falling into a “vicious circle”, that is, into a pattern of self-reference that would beg the question of a definition and vitiate its usefulness as an explanation of each group element's action. First, the regular representations do not represent that a group element is literally equal to a set of ordered pairs involving that very same group element, but only that it is mapped to something like this set. Second, careful usage would dictate that the something like that one finds in the image of a representation, being something that is specified only up to its isomorphism class, is a transformation that really acts, not on the group elements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_j\!} themselves, but only on their inert tokens, inactive images, partial symbols, passing names, or transitory signs of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} x_j {}^{\prime\prime}.\!}

These reservations are crucial to understanding the form of explanation that a regular representation provides, that is, what it explains and what it does not. If one is seeking an ontological explanation of what a group and its elements are, then one would have reason to object that it does no good to represent a group and its elements in terms of their actions on the group elements themselves, since one still does not know what the latter entities are. Notice that the form of this objection is reminiscent of a dilemma that is often thought to obstruct the beginning of an inquiry into inquiry. A similar pattern of knots occurs when one tries to explain the process of formalization in terms of its effects on the term formalization. In each case, the resolution of the difficulty turns on recognizing a distinction between the active and passive modes of existence that go with each nameable objective.

In order to have concrete materials available for future discussions of group theoretic issues, the remainder of this section takes up a pair of small examples, the groups of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4,\!} and uses them to illustrate the chain of definitions and the forms of representation given above.

There are just two groups of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4.\!} Both are abelian (commutative), but one is cyclic and the other is not. The cyclic group on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4\!} elements is commonly referred to as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_4.\!} (The German words Zahl for “number” and Zyklus for “cycle” together make the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_n\!} suggestive of the integers modulo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n,\!} which form a cyclic group of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n\!} under the addition operation.) The acyclic group on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4\!} elements is usually called the Klein 4 group and notated as $V_{4}.\!$ (The German word Vierbein is the substantive form of an adjective that means “four-legged”.)

For the sake of comparison, I give a discussion of both these groups.

The next series of Tables presents the group operations and regular representations for the groups $V_{4}\!$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_4.\!} If a group is abelian, as both of these groups are, then its Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h_1\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h_2\!} representations are indistinguishable, and a single form of regular representation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {h : G \to (G \to G)}\!} will do for both.

Table 33.1 shows the multiplication table of the group Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_4,\!} while Tables 33.2 and 33.3 present two versions of its regular representation. The first version, somewhat hastily, gives the functional representation of each group element as a set of ordered pairs of group elements. The second version, more circumspectly, gives the functional representative of each group element as a set of ordered pairs of element names, also referred to as objects, points, letters, or symbols.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Table 33.1} ~~ \text{Multiplication Operation of the Group} ~ V_4\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdot\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{f}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{g}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{h}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{f}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{g}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{h}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{f}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{f}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{h}\!}	$\mathrm {g} \!$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{g}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{g}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{h}\!}	$\mathrm {e} \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{f}\!}
$\mathrm {h} \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{h}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{g}\!}	$\mathrm {f} \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}\!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Table 33.2} ~~ \text{Regular Representation of the Group} ~ V_4\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Element}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Function as Set of Ordered Pairs of Elements}\!}
$\mathrm {e} \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{e}, \mathrm{e}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{f}, \mathrm{f}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{g}, \mathrm{g}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{h}, \mathrm{h})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{f}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{e}, \mathrm{f}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{f}, \mathrm{e}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{g}, \mathrm{h}),~\!}	$(\mathrm {h} ,\mathrm {g} )\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{g}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{e}, \mathrm{g}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{f}, \mathrm{h}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{g}, \mathrm{e}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{h}, \mathrm{f})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{h}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	$(\mathrm {e} ,\mathrm {h} ),\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{f}, \mathrm{g}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{g}, \mathrm{f}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{h}, \mathrm{e})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Table 33.3} ~~ \text{Regular Representation of the Group} ~ V_4\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Element}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Function as Set of Ordered Pairs of Symbols}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{f}\!}	$\{\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\!}	$({}^{\backprime \backprime }{\text{g}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{h}}{}^{\prime \prime }),\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
$\mathrm {g} \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\!}	$({}^{\backprime \backprime }{\text{f}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{h}}{}^{\prime \prime }),\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{h}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}

Tables 34.1 and 35.1 show two forms of operation table for the group Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_4,\!} presenting the group, for the sake of contrast, in multiplicative and additive forms, respectively. Tables 34.2 and 35.2 give the corresponding forms of the regular representation.

The multiplicative and additive versions of what is abstractly the same group, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_4,\!} can be used to illustrate the concept of a group isomorphism.

Let the multiplicative version of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_4\!} be formalized as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_4(\cdot) ~=~ \underline{X}_1 ~=~ (X_1, *_1, e_1) ~=~ ( \{1, a, b, c \}, \cdot, 1),\!}

where ${}^{\backprime \backprime }\cdot {}^{\prime \prime }\!$ denotes the operation in Table 34.1.

Let the additive version of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_4\!} be formalized as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_4(+) ~=~ \underline{X}_2 ~=~ (X_2, *_2, e_2) ~=~ ( \{0, 1, 2, 3 \}, +, 0),\!}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} + {}^{\prime\prime}\!} denotes the operation in Table 35.1.

Then the mapping Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h : X_1 \to X_2\!} whose ordered pairs are given by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h ~=~ \{ (1, 0), (a, 1), (b, 2), (c, 3) \}\!}

constitutes an isomorphism from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_4(\cdot)\!} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_4(+).\!}

This fact can be verified in several ways: (1) by checking that the map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h\!} is bijective and that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h(x \cdot y) = h(x) + h(y)\!} for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y\!} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_4(\cdot),\!} (2) by noting that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h\!} transforms the whole multiplication table for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_4(\cdot)\!} into the whole addition table for $Z_{4}(+)\!$ in a one-to-one and onto fashion, or (3) by finding that both systems share some collection of properties that are definitive of the abstract group, for example, being cyclic of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4.\!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Table 34.1} ~~ \text{Multiplicative Presentation of the Group} ~ Z_4(\cdot)~\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cdot\!}	$\mathrm {1}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{a}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{b}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{c}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{1}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{1}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{a}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{b}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{c}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{a}}	$\mathrm {a}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{b}}	$\mathrm {c}$	$\mathrm {1}$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{b}}	$\mathrm {b}$	$\mathrm {c}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{1}}	$\mathrm {a}$
$\mathrm {c}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{c}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{1}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{a}}	$\mathrm {b}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Table 34.2} ~~ \text{Regular Representation of the Group} ~ Z_4(\cdot)\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Element}\!}	${\text{Function as Set of Ordered Pairs of Elements}}\!$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{1}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	$(\mathrm {1} ,\mathrm {1} ),\!$	$(\mathrm {a} ,\mathrm {a} ),\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{b}, \mathrm{b}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{c}, \mathrm{c})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{a}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{1}, \mathrm{a}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{a}, \mathrm{b}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{b}, \mathrm{c}),\!}	$(\mathrm {c} ,\mathrm {1} )\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{b}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	$(\mathrm {1} ,\mathrm {b} ),\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{a}, \mathrm{c}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{b}, \mathrm{1}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{c}, \mathrm{a})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{c}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{1}, \mathrm{c}),\,\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{a}, \mathrm{1}),\!}	$(\mathrm {b} ,\mathrm {a} ),\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{c}, \mathrm{b})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Table 35.1} ~~ \text{Additive Presentation of the Group} ~ Z_4(+)\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle +\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{0}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{1}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{2}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{3}\!}
$\mathrm {0} \!$	$\mathrm {0} \!$	$\mathrm {1} \!$	$\mathrm {2} \!$	$\mathrm {3} \!$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{1}\!}	$\mathrm {1} \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{2}\!}	$\mathrm {3} \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{0}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{2}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{2}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{3}\!}	$\mathrm {0} \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{1}\!}
$\mathrm {3} \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{3}\!}	$\mathrm {0} \!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{1}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{2}\!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Table 35.2} ~~ \text{Regular Representation of the Group} ~ Z_4(+)\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Element}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Function as Set of Ordered Pairs of Elements}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{0}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{0}, \mathrm{0}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{1}, \mathrm{1}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{2}, \mathrm{2}),\!}	$(\mathrm {3} ,\mathrm {3} )~\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{1}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{0}, \mathrm{1}),~\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{1}, \mathrm{2}),\!}	$(\mathrm {2} ,\mathrm {3} ),\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{3}, \mathrm{0})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{2}\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{0}, \mathrm{2}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{1}, \mathrm{3}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{2}, \mathrm{0}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{3}, \mathrm{1})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{3}\!}	$\{\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{0}, \mathrm{3}),\!}	$(\mathrm {1} ,\mathrm {0} ),\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{2}, \mathrm{1}),\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\mathrm{3}, \mathrm{2})\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \}\!}

Standard references for the above material are:

Jacobson, N., Basic Algebra I, W.H. Freeman, San Francisco, CA, 1974.
Lang, S., Algebra, 2nd ed., Addison Wesley, Menlo Park, CA, 1984.
Rotman, J.J., An Introduction to the Theory of Groups, 3rd ed., Allyn & Bacon, Boston, MA, 1984.

Basic Notions of Formal Language Theory

This section collects the material on formal language theory that is needed for the rest of this work.

A formal language is a countable set of expressions, each of which is a finite sequence of elements taken from a finite set of symbols. The primitive symbols that are used to generate the expressions of a formal language are collectively called its alphabet or its lexicon, depending on whether the expressions of the language are regarded on analogy with words or sentences, respectively.

So long as one considers only words or only sentences, that is, only one level of finite sequences of symbols, it does not matter essentially what the sequences are called. Unless otherwise specified, a formal language is taken by default to be a one-level formal language, containing only a single level of sequences. If one wants to consider both words and sentences, that is, finite sequences of symbols and then finite sequences of these lower level sequences, all in the same context of discussion, then one has to move up to an essentially more powerful concept, that of a two-level formal language.

Until further notice, the next part of this discussion applies only to one-level formal languages. When this project reaches the stage of dealing with higher-level formal languages, a few of the following definitions and default assumptions will need to be adjusted slightly.

It is convenient to have a general term for referring to alphabets and lexicons, indifferently, without concern for their level of construction. Therefore, any finite set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}} is described as a syntactic resource for the syntactic domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X},} provided its elements can be used as syntactic primitives to construct the signs and expressions in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}.} If the primitive signs in a syntactic resource are interpreted to denote primitive objects or primitive operations, then a collection of such objects or operations is described as an objective or an operational resource, as the case may be.

It is always tempting to seek analogies between formal languages and algebraic structures, and it is often very useful to do so. But if one tries to forge an analogy between the relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}} ~\text{is a resource for}~ \underline{X},} in the formal language sense, and the relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}} ~\text{is a basis for}~ \underline{X},} in the algebraic sense, then it becomes necessary to observe important differences between the two perspectives, as they are currently applied.

In formal language theory one typically fixes the syntactic resource Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}} as the primary reality, that is, as the ruling parameter of discussion, and then considers each formal language Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}} that can be generated on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}} as a particular subset of the maximal language that is possible on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}.} This direction of approach can be contrasted with what is more usual in algebraic studies, where the generated object Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}} is taken as the primary reality, and a basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}} is defined secondarily as a minimal or independent spanning set, but generally serves as only one of many possible bases.

The linguistic relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}} ~\text{is a resource for}~ \underline{X}} is thus exploited in the opposite direction from the algebraic relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}} ~\text{is a basis for}~ \underline{X}.} There does not appear to be any reason in principle why either study cannot be cast the other way around, but it has to be noted that the current practices, and the preferences that support them, dictate otherwise.

By way of a general notation, I use doubly underlined capital letters to denote finite sets taken as the syntactic resources of formal languages, and I use doubly underlined lower case letters to denote their symbols. Schematically, this appears as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}} ~=~ \{ \underline{\underline{x}}_1, \ldots, \underline{\underline{x}}_k \}.}

In a formal language context, I use singly underlined capital letters to indicate the various formal languages being considered, that is, the countable sets of sequences over a given syntactic resource that are being singled out for attention, and I use singly underlined lower case letters to indicate various individual sequences in these languages. Schematically, this appears as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X} ~=~ \{ \underline{x}_1, \ldots, \underline{x}_\ell, \ldots \}.}

Usually, one compares different formal languages over a fixed resource, but since resources are finite it is no trouble to unite a finite number of them into a common resource. Without loss of generality, then, one typically has a fixed set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}} in mind throughout a given discussion and has to consider a variety of different formal languages that can be generated from the symbols of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}.} These sorts of considerations are aided by defining a number of formal operations on the resources Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}} and the languages Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{X}.}

The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k^\text{th}\!} power of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}},} written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}^k,} is defined as the set of all sequences of length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\!} over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}.}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}^k ~=~ \{ (u_1, \ldots, u_k) : u_i \in \underline{\underline{X}}, i = 1 ~\text{to}~ k \}.}

By convention for the case where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k = 0,\!} this gives Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}^0 = \{ () \},} that is, the singleton set consisting of the empty sequence. Depending on the setting, the empty sequence is referred to as the empty word or the empty sentence, and is commonly denoted by an epsilon Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {}^{\backprime\backprime} \varepsilon {}^{\prime\prime}} or a lambda ${}^{\backprime \backprime }\lambda {}^{\prime \prime }.$ In this text a variant epsilon symbol will be used for the empty sequence, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\varepsilon = ()}.\!} In addition, a singly underlined epsilon will be used for the language that consists of a single empty sequence, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline\varepsilon = \{ \varepsilon \} = \{ () \}.}

It is probably worth remarking at this point that all empty sequences are indistinguishable (in a one-level formal language, that is), and thus all sets that consist of a single empty sequence are identical. Consequently, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}^0 = \{ () \} = \underline{\varepsilon} = \underline{\underline{Y}}^0,} for all resources Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{Y}}.} However, the empty language Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varnothing = \{ \}} and the language that consists of a single empty sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline\varepsilon = \{ \varepsilon \} = \{ () \}} need to be distinguished from each other.

The surplus of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}},} written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}^+,} is defined as the set of all positive length sequences over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}.}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}^+ ~=~ \bigcup_{j = 1}^\infty \underline{\underline{X}}^j ~=~ \underline{\underline{X}}^1 \cup \ldots \cup \underline{\underline{X}}^k \cup \ldots}

The kleene star of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}},} written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}^*,} is defined as the set of all finite sequences over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{\underline{X}}.}

{\underline {\underline {X}}}^{*}~=~\bigcup _{j=0}^{\infty }{\underline {\underline {X}}}^{j}~=~{\underline {\underline {X}}}^{0}\cup {\underline {\underline {X}}}^{+}~=~{\underline {\underline {X}}}^{0}\cup \ldots \cup {\underline {\underline {X}}}^{k}\cup \ldots

A standard reference for the above material is:

Denning, P.J., Dennis, J.B., and Qualitz, J.E., Machines, Languages, and Computation, Prentice-Hall, Englewood Cliffs, NJ, 1978.

A Perspective on Computation

In this section, instead of presenting a standard foundation for computation theory, I focus on a single idea that captures the essence of the computational approach, given that the background assumptions of a formal approach are already in place, in others words, amounting to the specific difference that the CL style adds to the FL perspective.

The notion of computation that makes sense in this setting conceives it as a process that replaces signs with better signs of the same objects. For instance, a computation replaces arbitrary indications of numerical values and other formal entities with clearer and more concise signs of the same objects, ultimately resulting in the clearest and most concise signs of them, called their canonical interpretants or normal forms.

Viewed from a standpoint in the pragmatic theory of signs, computation is a process that trades a sign for a better sign of the same object. Thus, a computation is an interpretive process whose passage from sign to interpretant sign improves the indication of the object in some way. The dimensions along which signs can be compared are various, usually being described as measures of clarity, distinctness, or usability of the information conveyed, but all such measures are interpretive in character. That is, the sense in which a computation improves its signs is relative to the purpose actualized in a given moment of interpretation.

It is probably worth emphasizing this point. There need be nothing intrinsic to a sign itself that makes it better or worse than another. This is apparent from examples as simple as the sign relations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(\text{A})\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(\text{B}),\!} where nothing intrinsic to the grammatical categories of signs makes either the nouns or the pronouns essentially better than the others in every situation. In general, a preference defined on signs need reflect nothing more than the purpose or caprice of a particular interpreter at a given moment of interpretation. Of course, one is usually interested in cases where a measure of aptness, quality, or utility can be justified on more stable and substantial grounds.

Computation adds to the bare conception of a sign relation a notion of progress, which implies in turn: (1) the dynamic notion of a temporal process taking place between signs, and (2) the evaluative notion of a utility measure rating each sign's relative virtue as a sign.

A sign process or interpretive process is hypothesized to take place in the connotative plane of a particular sign relation, constituting a temporal process or a dynamic system that is responsible for changing signs into their interpretant signs. A sign utility is a comparative measure of sign quality, rating each sign's relative virtues as a sign of a given object. Progress in a sign process means that a change taking place between signs is one that acts in concert with increasing the sign's quality of indication.

• Overview • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Part 6 • Part 7 • Part 8 • Part 9 • Part 10 • Part 11 • Part 12 • Part 13 • Part 14 • Part 15 • Part 16 • Appendix A1 • Appendix A2 • References • Document History •

Inquiry Driven Systems • Part 11

Contents

Reflective Interpretive Frameworks (cont.)

The Phenomenology of Reflection

A Candid Point of View

A Projective Point of View

A Formal Point of View

Three Styles of Linguistic Usage

Basic Notions of Group Theory

Basic Notions of Formal Language Theory

A Perspective on Computation

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