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Inquiry Driven Systems • Part 12

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Author: Jon Awbrey

Reflective Interpretive Frameworks (cont.)

Higher Order Sign Relations

Higher Order Sign Relations : Introduction

When interpreters reflect on their own use of signs they require an appropriate technical language in which to pursue their reflections.  For this they need signs referring to sign relations, signs referring to elements and components of sign relations, and signs referring to properties and classes of sign relations.  The orders of signs developing as reflection evolves can be placed under the description of “higher order signs” and the extended sign relations involving them can be referred to as “higher order sign relations”.

Whether any forms of observation and reflection can be conducted outside the medium of language is not a question I can address here. It is apparent as a practical matter, however, that stable and sharable forms of knowledge depend on the availability of an adequate language. Accordingly, there is a relationship of practical necessity that binds the conditions for reflective interpretation to the possibility of extending sign relations through higher orders. At minimum, in addition to the signs of objects originally given, there must be signs of signs and signs of their interpretants, and each of these higher order signs requires a further occurrence of higher order interpretants to continue and complete its meaning within a higher order sign relation. In general, higher order signs can arise in a number of independent fashions, but one of the most common derivations is through the specialized devices of quotation. This establishes a contingent relation between reflection and quotation.

This entire topic, involving the relationship of reflective interpreters to the realm of higher order sign relations and the available operators for quotation, forms the subject of a recurring investigation that extends throughout the rest of this work. This section introduces only enough of the basic concepts, terminology, and technical machinery that is necessary to get the theory of higher order signs off the ground.

By way of a first definition, a higher order sign relation is a sign relation, some of whose signs are higher order signs. If an extra degree of precision is needed, higher order signs can be distinguished in a variety of different species or types, to be taken up next.

In devising a nomenclature for the required species of higher order signs, it is a good idea to generalize slightly, designing an analytic terminology that can be adapted to classify the higher order signs of arbitrary relations, not just the higher order signs of sign relations. The work of developing a more powerful vocabulary can be put to good account at a later stage of this project, when it is necessary to discuss the structural constituents of arbitrary relations and to reflect on the language that is used to discuss them. However, by way of making a gradual approach, it nonetheless helps to take up the classification of higher order signs in a couple of passes, first considering the categories of higher order signs as they apply to sign relations and then discussing how the same ideas are relevant to arbitrary relations.

Here are the species of higher order signs that can be used to discuss the structural constituents and intensional genera of sign relations:

  1. Signs that denote signs, that is, signs whose objects are signs in the same sign relation, are called higher ascent (HA) signs.
  2. Signs that denote dyadic components of elementary sign relations, that is, signs whose objects are elemental pairs or dyadic actions having any one of the forms or are called higher employ (HE) signs.
  3. Signs that denote elementary sign relations, that is, signs whose objects are elemental triples or triadic transactions having the form are called higher import (HI) signs.
  4. Signs that denote sign relations, that is, signs whose objects are themselves sign relations, are called higher upshot (HU) signs.
  5. Signs that denote intensional genera of sign relations, that is, signs whose objects are properties or classes of sign relations, are called higher yclept (HY) signs.

Analogous species of higher order signs can be used to discuss the structural constituents and intensional genera of arbitrary relations. In order to describe them, it is necessary to introduce a few extra notions from the theory of relations. This, in turn, occasions a recurring difficulty with the exposition that needs to be noted at this point.

The subject matters of relations, types, and functions enjoy a form of recursive involvement with one another which makes it difficult to know where to get on and where to get off the circle of explanation.  As I currently understand their relationship, it can be approached in the following order:

Relations have types.
Types are functions.
Functions are relations.

In this setting, a type is a function from the places of a relation, that is, from the index set of its components, to a collection of sets known as the domains of the relation.

When a relation is given an extensional representation as a collection of elements, these elements are called its elementary relations or its individual transactions. The type of an elementary relation is a function from an index set whose elements are called the places of the relation to a set of sets whose elements are called the domains of the relation. The arity or adicity of an elementary relation is the cardinality of this index set. In general, these cardinalities can be ranked as finite, denumerably infinite, or non-denumerable.

Elementary relations are also called the effects of a relation, more specifically, as its maximal or total effects, which are the kinds of effects that one usually intends in the absence of further qualification. More generally, a component relation or a partial transaction of a relation is a projection of one of its elementary relations on a subset of its places.

A homogeneous relation is a relation, all of whose elementary relations have the same type. In this case, the type and the arity are properties that are defined for the relation itself. The rest of this discussion is specialized to homogeneous relations.

When the arity of a relation is a finite number then the relation is called a -place relation. In this case, the elementary relations are just the -tuples belonging to the relation. In the finite case, for example, a non-trivial properly partial transaction is a -tuple extracted from a -tuple of the relation, where The first element of an elementary relation is called its object or relate, while the remaining elements are called its correlates.

  1. Signs that denote single correlates of an object in a relation are called higher ascent (HA) signs.
  2. Signs that denote moderate effects in a relation, that is, signs whose objects are partial transactions or -tuples involving more than one place but less than the full set of places in a relation, are called higher employ (HE) signs.
  3. Signs that denote elementary relations involving all the places of a relation are called higher import (HI) signs.
  4. Signs that denote relations are called higher upshot (HU) signs.
  5. Signs that denote properties or classes of relations are called higher yclept (HY) signs.

Whenever the sense is clear, it is usually convenient to stick with the more generic terms for higher order signs and higher order sign relations, letting context determine the appropriate meaning. For the rest of this section, it is mainly the categories of higher ascent signs and higher import signs that come into play.

Inquiry into inquiry is necessary because it is an unavoidable part of the inquiry into anything else, since critical reflection on the methods employed is implicit in the task. This means that inquiry into inquiry must be able to formulate and critique alternative descriptions of inquiry in general, including itself. Thus, there are notions of entelechy, of a self-referent objective, a completion in self-description, or an end to self-actualization, that are intrinsic to the conception of inquiry, whether or not its ends-in-view are ever achieved. If inquiry, as a manner of thinking, is carried on in sign relations and is ever to be supported by computational means, then these reflections raise the issue of self-describing sign relations and self-documenting data structures.

This is where higher order sign relations come in, making it possible to formalize sign relations that describe themselves and other sign relations, and thus enabling one to conceive of inquiries that inquire into themselves and other inquiries, at least in part. It is useful to approach these topics in a couple of stages, at first, by describing sign relations that describe other sign relations, and then, by describing sign relations that describe themselves. Although the implicit aim, or naive hope, is always to make these descriptions as complete as possible, it has to be recognized that partial success is all that is likely to be realized in practice. It seems to be something between rare and impossible that a non-trivial sign relation could completely describe itself with respect to every facet of its being and in all the ways that it does in fact exist.

Nevertheless, partially self-describing sign relations and partially self-documenting data structures do arise in practice, and so it is incumbent on this inquiry to look into the question of how they usually develop. That is, how does a sign get itself interpreted in a sign relation in such a way that it acts as a partial self-description of that selfsame sign relation? There appear to be two main ways that this can happen. Occasionally, it develops through the reflective operation or insightful turn of retracting projections, that is, by recognizing that a feature attributed to others is also (or primarily) an aspect of oneself. More commonly, partially self-describing sign relations are encountered already in place, as when a higher order sign relation has signs that describe lower orders, partial aspects, or previous stages of itself.

A further reduction in the number of different kinds of signs to worry about can be achieved by means of a special technique — some may call it an “artful dodge” — for referring indifferently to the elements of a set without referring to the set itself. Under the designation of a plural indefinite reference (PIR) is included all the various ways of dealing with denominations, multiple denotations, collective references, or objective multitudes that avail themselves of this trick.

By way of definition, a sign in a sign relation is said to be, to constitute, or to make a plural indefinite reference (PIR) to (every element in) a set of objects, if and only if denotes every element of This relationship can be expressed in a succinct formula by making use of one additional definition.

The denotation of in written is defined as follows:

Then makes a PIR to in if and only if Of course, this includes the limiting case where is a singleton, say In this case the reference is neither plural nor indefinite, properly speaking, but denotes uniquely.

The proper exploitation of PIRs in sign relations makes it possible to finesse the distinction between HI signs and HU signs, in other words, to provide a ready means of translating between the two kinds of signs that preserves all the relevant information, at least, for many purposes. This is accomplished by connecting the sides of the distinction in two directions. First, a HI sign that makes a PIR to many triples of the form can be taken as tantamount to a HU sign that denotes the corresponding sign relation. Second, a HU sign that denotes a singleton sign relation can be taken as tantamount to a HI sign that denotes its single triple. The relation of one sign being “tantamount to” another is not exactly a full-fledged semantic equivalence, but it is a reasonable approximation to it, and one that serves a number of practical purposes.

In particular, it is not absolutely necessary for a sign relation to contain a HU sign in order for it to contain a description of itself or another sign relation. As long the sign relation is “content” to maintain its reference to the object sign relation in the form of a constant name, then it suffices to use a HI sign that makes a PIR to all of its triples.

In the theory of sign relations, as in formal language theory, one tends to spend a lot of the time talking about signs as objects. Doing this requires one to have signs for denoting signs and ways of telling when a sign is being used as a sign or is just being mentioned as an object. Generally speaking, reflection on the usage of an established order of signs recruits another order of signs to denote them, and then another, and another, until a limit on one's powers of reflection is ultimately reached, and finally one is forced to conduct one's meaning in forms of interpretive practice that fail to be fully reflective in one critical respect or another. In the last resort one resigns oneself to letting the recourse of signs be guided by casually intuited inklings of their potential senses.

In this text a number of linguistic devices are used to assist the faculty of reflection, hopefully forestalling the relegation of its powers to its own natural resources for a long enough spell to observe its action. A discussion of these techniques and strategies follows.

In the declaration of higher order signs and the specification of their uses, one can employ the same terminology and technical distinctions that are found to be effective in describing sign relations. This turns the established terms for significant properties of world elements and the provisional terms for their relationships to each other to the ends of prescribing the relative orders of higher order signs and their objects. In short, the received theory of signs, however transient it may be at any given moment of inquiry, allows one to declare the absolute types and the relative roles that all of these entities are meant to take up.

For example, if I say that and that then it means I imagine myself to have an interpretation or a sign relation in mind where and are both signs belonging to a single order of signs and is a sign belonging to the next higher order of signs up from everything being relative to that particular moment of interpretation. Of course, as far as wholly arbitrary sign relations go, there is nothing to guarantee that the interpretation I think myself to have in mind at one moment can be integrated with the interpretation I think myself to have in mind at another moment, or that a just order can be founded in the end by any manner of interpretation that “just follows orders” in this way.

Ordinary quotation marks function as an operator on pieces of text to create names for the signs or expressions enclosed in them. In doing this the quotation marks delay, defer, or interrupt the normal use of their subtended contents, interfering with the referential use of a sign or the evaluation of an expression in order to create a new sign. The use of this constructed sign is to mention the immediate contents of the quotation marks in a way that can serve thereafter to indicate these contents directly or allude to them indirectly.

In the informal context, however, quotation marks are used equivocally for several other purposes. In particular, they are frequently used to call attention to the immediate use of a sign, to stress it or redress it for a definitive, emphatic, or skeptical service, but without necessarily intending to interrupt or seriously alter its ongoing use. Furthermore, ordinary quotation marks are commonly taken so literally that they can inadvertently pose an obstacle to functional abstraction. For instance, if I try to refer to the effect of quotation as a mapping that takes signs to higher order signs, thereby attempting to define its action by means of a lambda abstraction: then there are modes of IL interpretation that would read this literally as a constant map, one that sends every element of the functional domain into the single code for the letter

For these reasons I introduce the use of raised angle brackets also called “arches” or “supercilia”, to configure a form of quotation marker, but one that is subject to a more definite set of understandings about its interpretation. Namely, the arch marker denotes a function on signs that takes (the name of) a syntactic element located within it as (the name of) a functional argument and returns as its functional value the name of that syntactic element. The parenthetical operators in this statement reflect the optional readings that prevail in some cases, where the simple act of noticing a syntactic element as a functional argument is already tantamount to having a name for it. As a result, a quoting function that is designed to operate on the signs denoting and not on the objects denoted seems to do nothing at all, but merely uses up a moment of time to do it.

In IL contexts the arch quotes are construed together with their syntactic contents as forming a certain kind of term, one that achieves a naming function on syntactic elements by taking the enclosed text as a functional argument and giving a directly embedded indication of it. In this type of setting the name of a string of length is a string of length

In FL contexts the arch marker denotes a function that takes the literal syntactic element bounded by it as its argument and returns the name, code, annotation, gödel number, or unique numerical identifier of that syntactic element. In this setting there need be no straightforward relationship between the size or complexity of the syntactic element and the magnitude of its numerical code or the form of its symbolic code.

In CL implementations the arch operation is intended to do exactly what the principal uses of ordinary quotes are supposed to do, except that it obeys restrictions that are necessary to make it work as a notation for a computable function on the identified syntactic domain.

One further remark on the uses of quotation marks is pertinent here. When using HA signs with high orders of complexity and depth, it is often convenient to revert to the use of ordinary quotes at the outer boundary of a quotational expression, in this way marking a return to the ordinary context of interpretation. For example, one observes the colloquial equivalence:

In general, a good way to specify the meaning of a new notation is by means of a semantic equation, or a system of semantic equations, that expresses the function of the new signs in terms of familiar operations. If it is merely a matter of introducing new signs for old meanings, then this method is sufficient. In this vein, the intention and use of the “supercilious notation” for reflecting on signs could have its definition approximated in the following way.

Let as signs for the object and let as signs for the object an object that incidentally happens to be sign. An alternative way of putting this is to say that the members of the set are equivalent as signs for the object while the members of the set are equivalent as signs for the sign

Higher Order Sign Relations : Examples

In considering the higher order sign relations that stem from the examples and it appears that annexing the first level of HA signs is tantamount to adjoining or instituting an auxiliary interpretive framework, one that has the semantic equations shown in Table 36.

However, there is an obvious problem with this method of defining new notations. It merely provides alternate signs for the same old uses. But if the original signs are ambiguous, then equating new signs to them cannot remedy the problem. Thus, it is necessary to find ways of selectively reforming the uses of the old notation in the interpretation of the new notation.

The invocation of higher order signs raises an important point, having to do with the typical ways that signs can become the objects of further signs, and the relationship that this type of semantic ascent bears to the interpretive agent's capacity for so-called “reflection”. This is a topic that will recur again as the discussion develops, but a speculative foreshadowing of its character will have to serve for now.

Any object of an interpreter's experience and reasoning, no matter how vaguely and casually it initially appears, up to and including the merest appearance of a sign, is already, by virtue of these very circumstances, on its way to becoming the object of a formalized sign, so long as the signs are made available to denote it. The reason for this is rooted in each agent's capacity for reflection on its own experience and reasoning, and the critical question is only whether these transient reflections can come to constitute signs of a more permanent use.

The immediate purpose of the arch operation is to equip the text with a syntactic mechanism for constructing higher order signs, that is, signs denoting signs. But the step of reflection that the arch device marks corresponds to a definite change on the part of the interpreter, affecting the pragmatic stance or the intentional attitude that the interpreter takes up with respect to the affected signs. Accordingly, because of its connection to the interpreter's capacity for critical reflection, the arch operation, whether signified by arches or quotes, opens up a topic of wide importance to the larger question of inquiry. Unfortunately, there is much to do before this issue can be taken up in detail, and immediate concerns make it necessary to break off further discussion for now.

A general understanding of higher order signs would not depend on the special devices that are used to construct them, but would define them as any signs that behave in certain ways under interpretation, that is, as any signs that are interpreted in a particular manner, yet to be specified. A proper definition of higher order signs, including a generic description of the operations that construct them, cannot be achieved at the present stage of discussion. Doing this correctly depends on carrying out further developments in the theories of formal languages and sign relations. Until this discussion reaches that point, much of what it says about higher order signs will have to be regarded as a provisional compromise.

The development of reflection on interpretation leads to the generation of higher order signs that denote lower order signs as their objects. This process is illustrated by the following sequence of progressively higher order signs, all of which stem from a plain precursor and ultimately refer back to their initial ancestor, in this case,

The intent of this succession, as interpreted in FL environments, is that denotes or refers to which denotes or refers to Moreover, its computational realization, as implemented in CL environments, is that addresses or evaluates to which addresses or evaluates to

The designations higher order and lower order are attributed to signs in a casual, local, and transitory way. At this point they signify nothing beyond the occurrence in a sign relation of a pair of triples having the form shown in Table 37.

This is all it takes to make a lower order sign and a higher order sign in relation to each other at the moments in question. Whether a global ordering of a more generally justifiable sort can be constructed from an arbitrary series of such purely local impressions is another matter altogether.

Nevertheless, the preceding observations do show a way to give a definition of higher order signs that does not depend on the peculiarities of quotational devices. For example, consider the previously described sequence of increasingly higher order signs stemming from the object Table 38 shows how this succession can be transcribed into the form of a sign relation. But this is formally no different from the sign relation suggested in Table 39, one whose individual signs are not constructed in any special way. Both of these representations of sign relations, if continued in a consistent manner, would have the same abstract structure. If one of them is higher order then so is the other, at least, if the attributes of order are meant to have any formally invariant meaning.

The rest of this section discusses the relationship between higher order signs and a concept called the reflective extension of a sign relation. Reflective extensions will be subjected to a more detailed study in a later part of this work. For now, just to see how the process works, the sign relations and are taken as starting points to illustrate the more common forms of reflective development.

In the most typical scenario, higher order sign relations come into being as the reflective extensions of simpler, possibly unreflective sign relations. Conversely, the incorporation of higher order signs within a sign relation leads to a larger sign relation that constitutes one of its reflective extensions. In general, there are many different ways that a reflective extension can get started and many different structures that can result.

In the initial slice of semantics presented for the sign relations and the sign domain is identical to the interpretant domain and this set is disjoint from the object domain In order for this discussion to develop more interesting examples of sign relations these constraints will need to be generalized. As a start in this direction, one can preserve the identification of the syntactic domain as and contemplate ways of varying the pattern of intersection between and

One direction of generalization is motivated by the desire to give interpreters a measure of “reflective capacity”. This is a property of sign relations that can be associated with the overlap of and and gauged by the extent to which is contained in In intuitive terms, interpreters are said to have a reflective capacity to the extent that they can refer to their own signs independently of their denotations. An interpretive system with a sufficient amount of reflective capacity can support the maintenance and manipulation of textual objects like expressions and programs without necessarily having to evaluate the expressions or execute the programs.

In ordinary discourse HA signs are usually generated by means of linguistic devices for quoting pieces of text. In computational frameworks these quoting mechanisms are implemented as functions that map syntactic arguments into numerical or syntactic values. A quoting function, given a sign or expression as its single argument, needs to accomplish two things: first, to defer the reference of that sign, in other words, to inhibit, delay, or prevent the evaluation of its argument expression, and then, to exhibit or produce another sign whose object is precisely that argument expression.

The rest of this section considers the development of sign relations that have moderate capacities to reference their own signs as objects. In each case, these extensions are assumed to begin with sign relations like and that have disjoint sets of objects and signs and thus have no reflective capacity at the outset. The status of and as the reflective origins of the associated reflective developments is recalled by saying that and themselves are the zeroth order reflective extensions of and in symbols, and

The next set of Tables illustrates a few the most common ways that sign relations can begin to develop reflective extensions. For ease of reference, Tables 40 and 41 repeat the contents of Tables 1 and 2, respectively, merely replacing ordinary quotes with arch quotes.

Tables 42 and 43 show one way that the sign relations and can be extended in a reflective sense through the use of quotational devices, yielding the first order reflective extensions, and These extensions add one layer of HA signs and their objects to the sign relations and respectively. The new triples specify that, for each in the set the HA sign of the form connotes itself while denoting

Notice that the semantic equivalences of nouns and pronouns referring to each interpreter do not extend to semantic equivalences of their higher order signs, exactly as demanded by the literal character of quotations. Also notice that the reflective extensions of the sign relations and coincide in their reflective parts, since exactly the same triples were added to each set.

There are many ways to extend sign relations in an effort to develop their reflective capacities. The implicit goal of a reflective project is to reach a condition of reflective closure, a configuration satisfying the inclusion where every sign is an object. It is important to note that not every process of reflective extension can achieve a reflective closure in a finite sign relation. This can only happen if there are additional equivalence relations that keep the effective orders of signs within finite bounds. As long as there are higher order signs that remain distinct from all lower order signs, the sign relation driven by a reflective process is forced to keep expanding. In particular, the process that is freely suggested by the formation of and cannot reach closure if it continues as indicated, without further constraints.

Tables 44 and 45 present higher import extensions of and respectively. These are just higher order sign relations that add selections of higher import signs and their objects to the underlying set of triples in and One way to understand these extensions is as follows. The interpreters and each use nouns and pronouns just as before, except that the nouns are given additional denotations that refer to the interpretive conduct of the interpreter named. In this form of development, using a noun as a canonical form that refers indifferently to all the triples of a sign relation is a pragmatic way that a sign relation can refer to itself and to other sign relations.

Several important facts about the class of higher order sign relations in general are illustrated by these examples. First, the notations appearing in the object columns of and are not the terms that these newly extended interpreters are depicted as using to describe their objects, but the kinds of language that you and I, or other external observers, would typically make available to distinguish them. The sign relations and as extended by the transactions of and respectively, are still restricted to their original syntactic domain This means that there need be nothing especially articulate about a HI sign relation just because it qualifies as higher order. Indeed, the sign relations and are not very discriminating in their descriptions of the sign relations and referring to many different things under the very same signs that you and I and others would explicitly distinguish, especially in marking the distinction between an interpretive agent and any one of its individual transactions.

In practice, it does an interpreter little good to have the higher import signs for referring to triples of objects, signs, and interpretants if it does not also have the higher ascent signs for referring to each triple's syntactic portions. Consequently, the higher order sign relations that one is likely to observe in practice are typically a mixed bag, having both higher ascent and higher import sections. Moreover, the ambiguity involved in having signs that refer equivocally to simple world elements and also to complex structures formed from these ingredients would most likely be resolved by drawing additional information from context and fashioning more distinctive signs.

These reflections raise the issue of how articulate a higher order sign relation is in its depiction of its object signs and its object sign relations. For now, I can do little more than note the dimension of articulation as a feature of interest, contributing to the scale of aesthetic utility that makes some sign relations better than others for a given purpose, and serving as a drive that motivates their continuing development.

The technique illustrated here represents a general strategy, one that can be exploited to derive certain benefits of set theory without having to pay the overhead that is needed to maintain sets as abstract objects. Using an identified type of a sign as a canonical form that can refer indifferently to all the members of a set is a pragmatic way of making plural reference to the members of a set without invoking the set itself as an abstract object. Of course, it is not that one can get something for nothing by these means. One is merely banking on one's recurring investment in the setting of a certain sign relation, a particular set of elementary transactions that is taken for granted as already funded.

As a rule, it is desirable for the grammatical system that one uses to construct and interpret higher order signs, that is, signs for referring to signs as objects, to mesh in a comfortable fashion with the overall pragmatic system that one uses to assign syntactic codes to objects in general. For future reference, I call this requirement the problem of creating a conformally reflective extension (CRE) for a given sign relation. A good way to think about this task is to imagine oneself beginning with a sign relation and to consider its denotative component Typically one has a naming function, say that maps objects into signs:

Part of the task of making a sign relation more reflective is to extend it in ways that turn more of its signs into objects. This is the reason for creating higher order signs, which are just signs for making objects out of signs. One effect of progressive reflection is to extend the initial naming function through a succession of new naming functions and so on, assigning unique names to larger allotments of the original and subsequent signs. With respect to the difficulties of construction, the hard core or adamant part of creating extended naming functions resides in the initial portion that maps objects of the “external world” to signs in the “internal world”. The subsequent task of assigning conventional names to signs is supposed to be comparatively natural and easy, perhaps on account of the nominal nature of signs themselves.

The effect of reflection on the original sign relation can be analyzed as follows. Suppose that a step of reflection creates higher order signs for a subset of Then this step involves the construction of a newly extended sign relation:

In this construction is that portion of the original signs for which higher order signs are created in the initial step of reflection, thereby being converted into The sign domain is extended to a new sign domain by the addition of these higher order signs, namely, the set Using arch quotes, the mapping from to can be defined as follows:

Finally, the reflectively extended naming function is defined as

A few remarks are necessary to see how this way of defining a CRE can be regarded as legitimate.

In the present context an application of the arch notation, for example, is read on analogy with the use of any other functional notation, for example, where is the name of a function is the context of its application, is the name of an argument and where the functional abstraction is just another name for the function

It is clear that some form of functional abstraction is being invoked in the above definition of Otherwise, the expression would indicate a constant function, one that maps every in its domain to the same code or sign for the letter But if this is allowed, then it appears to pose a dilemma, either to invoke a more powerful concept of functional abstraction than the concept being defined, or else to attempt an improper definition of the naming function in terms of itself.

Although it appears that this form of functional abstraction is being used to define the CRE in terms of itself, trying to extend the definition of the naming function in terms of a definition that is already assumed to be available, in reality this only uses a finite function, a finite table look up, to define the naming function for an unlimited number of higher order signs.

In CL contexts, especially in the Lisp tradition, the quotation operator is recognized as an “evaluation inhibitor” and implemented as a function that maps each syntactic element into its unique numerical identifier or gödel number. Perhaps one should pause to marvel at the fact that a form of delay, deference, and interruption akin to an inhibition should be associated with the creation of signs that refer in meaningful ways.

On reflection, though, the connection between attribution and inhibition, or acknowledgment and deference, begins to appear less remarkable, and in time it can even be understood as natural and necessary. For one thing, psychoanalytic and psychodynamic theories of mental functioning have long recognized that symbol formation and symptom formation are closely akin, being the twin founders of civilization and many of its discontents. For another thing, the following etymology can be rather instructive: The English word memory derives from the Latin memor for mindful, which is akin to the Latin mora for delay, the Greek mermera for care, and the Sanskrit smarati for he remembers. To explore the verbal complex a bit further, it merits remembering that the ideas of merit and membership, besides being connected with the due proportions, earned shares, and just deserts that are parceled out on parchment, are also tied up with the particular kind of care that is needed to take account of things part for part. (The Latin merere for earn or deserve, along with membrana for skin or parchment and memor for mindful, are all akin to the Greek merizein for divide and meros for part.) Although the voices of psychology and etymology are seldom heard at this depth in the wilderness of formal abstraction, I think it is worth heeding them on this point.

In CL environments of the Pascal variety there are several different ways that higher order signs are created. In these settings higher order signs, or signs for referring to signs as objects, can be implemented as the codes that serve as numerical identifiers of characters or the pointers that serve as accessory indices of symbolic expressions.

But not all the signs that are needed for referring to other signs can be constructed by means of quotation. Other forms of higher order signs have to be generated de novo, that is, constructed independently of previous successions and introduced directly into their appropriate orders. Among other things, this obviates the usual strategy for telling the order of a sign by counting its quota of quotation marks. Failing the chances of exploiting such a measure in absolute terms, and in the absence of a natural order for the construction of signs, the relative orders of signs can be assessed only by examining the complex network of denotative and connotative relationships that connect them, or the gaps that arise when they fail to do so.

In a CL context this often occurs when a constant is declared equal or a variable is set equal to a quoted character, as in the following sequence of Pascal expressions:

const comma = ',' ;
var x; x := comma ;

In this passage, the sign “comma” is made to denote whatever it is that sign “','” denotes, and the variable is then set equal to this value.

Higher Order Sign Relations : Application

Given the language in which a notation like makes sense, or in prospect of being given such a language, it is instructive to ask: “What must be assumed about the context of interpretation in which this language is supposed to make sense?” According to the theory of signs that is being examined here, the relevant formal aspects of that context are embodied in a particular sign relation, call it With respect to the hypothetical sign relation commonly personified as the prospective reader or the ideal interpreter of the intended language, the denotation of the expression is given by:

If follows rules that are typical of many species of interpreters, then the value of this expression will depend on the values of the following three expressions:

What are the roles of the signs and what are they supposed to mean to ? Evidently, is a constant name that refers to a particular function, is a variable name that makes a PIR to a collection of signs, and is a variable name that makes a PIR to a collection of sign relations.

This is not the place to take up the possibility of an ideal, universal, or even a very comprehensive interpreter for the language indicated here, so I specialize the account to consider an interpreter that is competent to cover the initial level of reflections that arise from the dialogue of and

For the interpreter the sign variable need only range over the syntactic domain and the relation variable need only range over the set of sign relations These requirements can be accomplished as follows:

  1. The variable name is a HA sign that makes a PIR to the elements of
  2. The variable name is a HU sign that makes a PIR to the elements of
  3. The constant name is a HI sign that makes a PIR to the elements of
  4. The constant name is a HI sign that makes a PIR to the elements of

This results in a higher order sign relation for that is shown in Table 46.

Following the manner of construction in this extremely reduced example, it is possible to see how answers to the above questions, concerning the meaning of might be worked out. In the present instance:

Conventions and Consequences

Issue 1. The Status of Signs

This Section considers an issue that affects the status of signs and their mode of significance, as it appears under each of the three norms of significance. The concerns that arise with respect to this issue can be divided into two sets of questions. The first type of question has to do with the default assumptions that are made about the meanings of signs and the strategies that are used to deal with signs that fail to have meanings. The second type of question has to do with higher order signs, or signs that involve signs among their objects.

Only certain types of signs are able to make their appearance in a given medium or a particular style of text, while many others are not. But a sign is a sign by virtue of the fact that it is interpreted as a sign, and thus plays the role of a sign in a sign relation, and not of necessity because it has any special construction other than that of being construed as a sign.

The theory of formal languages, as pursued under the formal language perspective, is closely related to the theory of semigroups, as pursued under the IL perspective, in the sense that arbitrary formal languages can be studied as subsets of the semigroups that embody the primitive concatenation of linguistic symbols within their algebraic laws of composition. Thus, in staging any discussion of formal languages, the theory of semigroups is often taken for a neutral, indifferent, or undifferentiated background, but the wisdom of using this setting is contingent on understanding the distinct outlooks of the casual and formal norms of significance. What divides the two styles and their favorite subjects in practice is a certain difference in attitude toward the status and role of their subject materials. Namely, it turns on the question of whether their primitive and derived elements are valued as terminal objects in and of themselves or whether these syntactic objects and constructions are interpreted as mere signs and sundry expressions whose true value lies elsewhere.

In taking up the informal language attitude toward any mathematical system, semigroups in particular, one assumes that signs are available for denoting a class of formal objects, but the issue of how these notational matters come to be constellated is considered to be peripheral, lacking in a substantive weight of concern and enjoying a purely marginal interest.

In the discussion of formal languages the presumption of significance is shifted in the opposite direction. Signs are presumed to be innocent of meaning until it can be demonstrated otherwise. One begins with a set of primitive objects, formally called “signs”, but treated as meaningless tokens or as objects that are bare of all extraneous semantic trappings. From these simplest signs, a law of composition allows the construction of complex expressions in regular ways, but other than that anything goes, at least, at first.

A first cut taken in the space of expressions divides them into two classes: (a) the grammatical, well-formed, or meaningful, maybe, versus (b) the ungrammatical, ill-formed, or meaningless, for sure. This first bit of semantic information is usually regarded as marking a purely syntactic distinction. Typically one seeks a recursive function that computes this bit of meaningfulness as a property of its argument and thereby decides (or semi-decides) whether an arbitrary expression (string, strand, sequence) constitutes an expressive expression (word, sentence, message), or not. The means of computation is often presented in the form of various grammars or automata that can serve as acceptors or generators for the language.

Depending on one's school of thought, the syntactic bit of computation for interesting cases of natural languages is thought to be either (1) formally independent of all the more properly semantic features, or (2) heavily reliant on the construal of further bits of meaning to make its decision. Accordingly, the semantics proper for such a language ought to begin either (1) serially after or (2) concurrently while the syntactic bit is done. The first standpoint is usually described as a “declaration of syntactic independence”, while the second opinion is often called a “semantic bootstrapping hypothesis”.

Over and above both of these positions the pragmatic theory of signs poses a stronger thesis of irreducibility or non-independence that one might call a “pragmatic bootstrapping hypothesis”. Even though it is a more complex task initially to work with triadic relations themselves instead of their dyadic projections, this hypothesis suggests that the structural integrity of interesting natural languages, when taken over the long haul, may well depend on them. One part of this thesis is not a hypothesis but a fact. There do indeed exist triadic relations that cannot be reconstructed uniquely from their dyadic projections, and thus are called irreducibly triadic. The parts of the thesis that are hypothetical, and that need to be cleared up by empirical inquiry, suggest that many of the most important sign relations are irreducibly triadic, and that interesting cases of natural languages depend heavily on these kinds of sign relations for their salient properties, for example, their relevance and adaptability to the objective world, their structural integrity and internal coherence, and their learnability by human agents and other species of finitely informed creatures.

In practice, this question has little consequence for the present study, on account of the extremely simple and artificial kinds of languages that are needed to carry out its aims. If some reason develops to emulate the properties of interesting natural languages in this microcosm, then a decision about which strategy to use can be made at that time. For now it seems worthwhile to keep exploring all of the above options.

In a formal language context one begins with the imposition of general inhibition against the notion that a specific class of signs has any meaning at all, or at least, that its elements have the meanings one is accustomed to think they do. It is significant that one does not proscribe all signs from having meaning, or else there is no point in having a discussion, and no point from which to carry on a discussion of anything at all. Therefore, the arena of formal discussion is a limited one and, except for the occasional resonance that its action induces in the surrounding discursive universe, most of the signs outside its bounds continue to be used in the habitual ways.

What can be done with the signs in question? Apparently, signs viewed as objects in the formal arena, temporarily cut off from their usual associations, treated as terminal values in themselves, and put under review to suggest explanations for themselves, can still be discussed. Doing this involves the use of other signs for denoting the signs in question. These extra signs, whose sense and use are not in question at the moment in question, are called into play as higher order signs, and it is their very meaningfulness and effectiveness that one must rely on to carry out the investigation of the lower order signs that are in question.

Detailed discussion of a set of signs in question requires the ability to classify the tokens of these signs according to their types. Doing this calls on the use of other higher order signs to denote these tokens, the transient instances of signs, and their types, the propertied classes of tokens that correspond to what is typically valued as a sign. The invocation of higher order signs can be iterated in a succession of higher orders that extends as far as one pleases, but no matter how much of this order is progressively formalized one eventually must resort to signs of such a high order that they are taken for granted as resting, for the moment, in an informal context of interpretation.

What is the sense and use of such a proceeding? Evidently, the signs in question, as a class, must present the inquirer with phenomena that are somehow simpler than, and yet convey instructive information about, the phenomenon known as the whole objective world (WOW). If their orders of complexity and perplexity are just as great as the world at large, then their investigation affords no advantage over the general empirical problem of trying to account for the WOW. If they enjoy no informative connection with the greater wonders of why the world is the way it is, and therefore fail to present a significant representation of the original question, then their isolated inquiry can serve no larger purpose in the world.

In situations like the one just described, where functions and relations on one order of arguments are clarified, defined, or explained in terms of functions and relations on another order of arguments, it is natural to understand the effort at clarification, definition, or explanation as a recursive process. What raises the potential for confusion in the given arrangement of formal and casual contexts is the circumstance that what seems natural to call the lower order arguments are being discussed in terms of what seems natural to call the higher order arguments. What is going on here? As it happens, the ordering of signs from lower order to higher order that seems obvious from the standpoint of their typical construction and their order of appearance on the stage of discussion does not reflect the measure of complexity that is relevant to the effort at recursive exposition.

The measure of complexity that is relevant to the formal exposition is the measure of doubt, uncertainty, or perplexity that one entertains about the sense and use of a sign beset by questions, whether this occurs by force of a voluntary effort to bracket its habitual senses or by dint of a puzzling event that brings its automatic uses to a halt.

It is the language being discussed that is the formal one, to be treated initially as an object, while the language that is used to carry out the discussion tries to maintain its informal viability, expecting in effect to be taken on faith as not undermining or vitiating the effort at inquiry due to unexamined flaws of its own. Nevertheless, if inquiry in general is expected to be self correcting, then a continuing series of failures to conclude inquiries by means of a given arrangement, that is, an inability to resolve uncertainties through a particular division of labor between formal and informal contexts, must lead to the grounds of attack being shifted.

In working out compromises between the formal and informal styles of usage one faces all the problems usually associated with integrating different frameworks of interpretation, but compounded by the additional factors (1) that this conflict of attitudes, or its practical importance, is seldom openly acknowledged and (2) that the frameworks in and of the negotiation to be transacted are rarely capable of being formalized, or even of being made conscious, to the same degree at the same time. These circumstances make the consequences of the underlying conflict difficult to address, and thus they continue to obstruct the desired implementation of a common computational language environment that could serve as a resource for work on both sides of the frame.

Issue 2. The Status of Sets

That the word “set” is being used indiscriminately for completely different notions and that this is the source of the apparent paradoxes of this young branch of science, that, moreover, set theory itself can no more dispense with axiomatic assumptions than can any other exact science and that these assumptions, just as in other disciplines, are subject to a certain arbitrariness, even if they lie much deeper here — I do not want to represent any of this as something new.

Julius König (1905), “On the Foundations of Set Theory and the Continuum Problem”, in Jean van Heijenoort (ed., 1967)

Set theory is not as young as it used to be, and not half as naive as it was when this statement was originally made, but the statement itself is just as apt in its application to the present scene and just as fresh in its lack of novelty as it was then. In the current setting, though, I am not so concerned with potentially different theoretical notions of a set that are represented by conventionally different axiom systems as I am with the actual diversity of practical notions that are used to deal with sets under each of the three norms of significance identified.

Even though all three norms of significance use set-theoretic constructions, the implicit theories of sets that are involved in their different uses are so varied in their assumptions and intentions that it amounts to a major source of friction between the casual and formal styles to try to pretend that the same subject is being invoked in every case. In particular, it makes a huge difference whether these sets are treated objectively, as belonging to the OF, or treated syntactically, as belonging to the IF.

In practical terms it makes all the difference in the world whether a set is viewed as a set of objects or whether it is viewed as a set of signs. The same set can be contemplated in each type of placement, but it does not always fit as well into both types of role. A set of objects is properly a part of the objective framework, and this is intended in its typical parts to model those realities whose laws and vagaries can extend outside the means of an agent's control. A set of signs is properly part of the interpretive framework, and this is constructed in its typical parts so that its variations and selections are subject to control for the ends of interpretive indication. The relevant variable is one of control, and the measure of it tells how well matched are the proper placements and the typical assignments that a given set is given.

Things referred to the objective world are not things that one expects to have much control over, at least, not at first, even though a reason for developing a language is to gain more control over events in time. Things referred to the realm of signs are things that one thinks oneself to have under control, at least, at first, even though their complexity can evolve in time beyond one's powers of oversight.

In an ordinary mathematical context, when one writes out the expression for a finite set in the form one expects to see the names of objects appearing between the braces. Furthermore, even if these additional expectations are hardly ever formalized, these objects are typically expected to be the terminal objects of denotative value in the appropriate context of discussion and to inhabit a single order of objective existence. In other words, it is common to assume that all the objects named have the same type, with no relations of consequence, functional, semantic, or otherwise, obtaining among them. As soon as these assumptions are made explicit, of course, it is obvious that they do not have to be so.

In formal language contexts, when a set is taken as the alphabet or the lexicon of a formal language, then the objects named are themselves signs, but it is still only their names that are subject to appearing between the braces. Often one seeks to handle this case by saying that what really appears between the braces are signs of sort that can suffice to represent themselves, and thus that these signs literally constitute their own names, but this is not ultimately a sensible tactic to try. As always, only the tokens of signs can appear on the page, and these come and go as the pages are turned. Although these tokens, by representing the types that encase them, partly succeed in referring to themselves, what they denote on principle is something much more abstract, general, and invariant than their own concrete, particular, and transient selves. Nevertheless, the expectation that all of the elements in the set reside at the same level of syntactic existence is still in effect.

The construction of a reflective interpretive framework demands a closer examination of these assumptions and requires a single discussion that can refer to mixed types of elements with significant relations among them.

In a formal language context one needs to be more self-conscious about the use of signs, and, after an initially painful period during which critical reflection seems more to interfere with thought more than to facilitate understanding, it is hoped that the extra measure of reflection will pay off when it is time to mediate one's thinking in a computational language framework.

There are numerous devices that one can use to assist with the task of reflection. Rather than trying to divert the customary connections of informal language use and the conventional conduct of its interpretation, it is easier to introduce a collection of markedly novel signs, analogous to those already in use but whose interpretation is both free enough to be changed and controlled through a series of experimental variations and flexible enough to be altered when fitting and repaired when faulty.

If is a set of objects under discussion, then one needs to consider several sets of signs that might be associated, element by element, with the elements of

  1. The nominal resource (nominal alphabet or nominal lexicon) for is a set of signs that is notated and defined as follows:

    This concept is intended to capture the ordinary usage of this set of signs in one familiar context or another.

  2. The mediate resource (mediate alphabet or mediate lexicon) for is a set of signs that is notated and defined as follows:

    This concept provides a middle ground between the nominal resource above and the literal resource described next.

  3. The literal resource (literal alphabet or literal lexicon) for is a set of signs that is notated and defined as follows:

    This concept is intended to supply a set of signs that can be used in ways analogous to familiar usages, but which are more subject to free variation and thematic control.

Issue 3. The Status of Variables

Another issue on which the three styles of usage diverge most severely is with respect to a crucial problem about the status of variables. Often this is posed as a question about the ontological status of variables, what kinds of objects they are, but it is better treated as a question about the pragmatic status of variables, what kinds of signs they are used as. In this section, I try to accommodate common practices in the use of variables in the process of building a bridge to the pragmatic perspective. The goal is to reconstruct customary ways of regarding variables within a overarching framework of sign relations, while disentangling the many confusions about the status of variables that obstruct their clear and consistent formalization.

Variables are the most problematic entities that have to be dealt with in the process of formalization, and this makes it useful to explore several different ways of approaching their treatment, either of accounting for them or explaining them away. The various tactics available for dealing with variables can be organized according to how they respond to two questions: Are variables good or bad, and what kinds of things are variables anyway? In other words:

  1. Are variables good things to have in a purified system of interpretation or a target formal system, or should variables be eliminated by the work of formalization?
  2. What sorts of things should variables be construed as?

The answers given to these questions determine several consequences. If variables are good things, things that ought to be retained in a purified formal system, then it must be possible to account for their valid uses in a sensible fashion. If variables are bad things, things that ought to be eliminated from a purified formal system, then it must be possible to “explain away” their properties and utilities in terms of more basic concepts and operations.

One approach is to eliminate variables altogether from the primitive conceptual basis of one's formalism, replacing every form of substitution with a form of application. In the abstract, this makes applications of constant operators to one another the only type of combination that needs to be considered. This is the strategy of the so-called combinator calculus.

If it is desired to retain a notion of variables in the formalism, and to maintain variables as objects of reference, then there are a couple of partial explanations of variables that still afford them with various measures of objective existence.

In the elemental construal of variables, a variable is just an existing object that is an element of a set the catch being “which element?” In spite of this lack of information, one is still permitted to write as a syntactically well-formed expression and otherwise treat the variable name as a pronoun on a grammatical par with a noun. Given enough information about the contexts of usage and interpretation, this explanation of the variable as an unknown object would complete itself in a determinate indication of the element intended, just as if a constant object had always been named by

In the functional construal of variables, a variable is a function of unknown circumstances that results in a known range of definite values. This tactic pushes the ostensible location of the uncertainty back a bit, into the domain of a named function, but it cannot eliminate it entirely. Thus, a variable is a function that maps a domain of unknown circumstances, or a sample space into a range of outcome values. Typically, variables of this sort come in sets of the form collectively called coordinate projections and together constituting a basis for a whole class of functions sharing a similar type. This construal succeeds in giving each variable name an objective referent, namely, the coordinate projection but the explanation is partial to the extent that the domain of unknown circumstances remains to be explained. Completing this explanation of variables, to the extent that it can be accomplished, requires an account of how these unknown circumstances can be known exactly to the extent that they are in fact described, that is, in terms of their effects under the given projections.

As suggested by the whole direction of the present work, the ultimate explanation of variables is to be given by the pragmatic theory of signs, where variables are treated as a special class of signs called indices.

Because it was necessary to begin informally, I started out speaking of things called “variables” as if there really were such things, taking it for granted that a consistent concept of their existence could be formed that would substantiate the ordinary usages carried out in their name, and contemplating judgments of their worth as if it were a matter of judging existing objects rather than the very ideas of their existence, whereas it is precisely the whole question at issue whether any of these presumptions are justified. As concessions to common usage, encounters with these assumptions are probably unavoidable, but a formal approach requires one to backtrack a bit, to treat the descriptive term “variable” as nothing more substantial than a general name in common use, and to examine whether its uses can be maintained in a purely formal system. Further, each of the “variables” that is taken to fall under this term has to allow its various indications to be reconsidered in the guise of mere signs and to permit the question of their objective reference to be examined anew.

At this point, it is worth trying to apply the insights of nominalism to these questions, if only to see where they lead.

It is the general advice of nominalism not to confuse a general name with the name of a general, that is, a universal, or a property possessed in common by many individual things. To this, pragmatism adds the distinct recommendation not to confuse an individual name with the name of an individual, because a particular that seems perfectly determinate for some purposes may not be determinate enough for other purposes.

In the perspective that results from combining these two points of view, general properties and individual instances alike can take on from the start an equally provisional status as objects of discussion and thought, in the meantime treated as interpretive fictions, as mere potentials for meaning, awaiting the settlement of their reality at the end of inquiry. Meanwhile, the individual can be exactly as tentative as the general, and ultimately, the general can be precisely as real as the individual. Still, their provisional treatment as hypothetical objects of reasoning does not affect their yet to be determined status as realities. This is so because it is possible that a hypothesis hits the mark, and it remains so as long as a provisional fiction, something called a likely story on account of its origin, can still succeed in guessing the truth aright.

Unlike generals, individuals, and numerous other forms of logical and mathematical objects, whose treatment as fictions does not affect their status as realities, one way or the other, there does not seem to be any consistent way of treating variables as objects. Although each one of the elemental and the functional construals appears to work well enough when taken by itself in the appropriate context, trying to combine these two notions into a single concept of the variable can lead to the mistake of confusing a function with one of its values.

Whether one tries to account for variables or chooses to explain them away, it is still necessary to say what kinds of entities are really involved when one is using this form of speech and trying to reason with or about its terms, whether one is speaking about things described as “variables” or merely about their terms of description, whether there are really objects to be dealt with or merely signs to be dispensed with.

According to one way of understanding the term, there is no object called a “variable” unless that object is a sign, and so the name “variable name” is redundant. Variables, if they are anything at all, are analogous to numerals, not numbers, and thus they fall within the broad class of signs called identifiers, more specifically, indices. In the case of variables, the advice of nominalism, not to confuse a variable name with the name of a variable, seems to be well taken.

If the world of elements appropriate to this discussion is organized into objective and syntactic domains, then there are fundamentally just two different ways of regarding variables, as objects or as signs. One can say that a variable is a fictional object that is contrived to provide a variable name with a form of objective referent, or one can say that a variable is a sign itself, the same thing as a variable name. In the present setting, it is convenient to arrange these broad approaches to variables according to the respective norms of significance under which one finds them most often pursued.

  1. The informal language approach to the question takes the objective construal of variables as its most commonly chosen default. The informal language style that is used in ordinary mathematical discussion associates a variable with a determinate set, one that the variable is regarded as “ranging over”. As a result, this norm of significance is forced to invoke a version of set theory, usually naive, to account for its use of variables.
  2. The formal language styles are manifestly varied in their explanations of variables, since there are many ways to formalize their ordinary uses. Two of the main alternatives are: (a) formalizing the set theory that is invoked with the use of variables, and (b) formalizing the sign relations in which variables operate as indices. Since an index is a kind of sign that denotes its object by virtue of an actual connection with it, and since the nature and direction of these actual connections can vary immensely from moment to moment, a variable is an extremely flexible and adaptable kind of sign, hence its character as a “reusable sign”.
  3. The computational language styles are also legion in their approaches to variables, but they can be divided into those that eliminate variables as a primitive concept and those that retain a notion of variables in their conceptual basis.
    1. An instructive case is presented by what is the most complete working out of the computational programme, the combinator calculus. Here, the goal is to eliminate the notion of a variable altogether from the conceptual basis of a formal system. In other words, it is projected to reduce its status as a primitive concept, one that applies to symbols in the object language, and to reformulate it as a derived concept, one that is more appropriate to describing constructions in a metalanguage.
    2. In computational language contexts where variables are retained as a primitive notion, there is a form of distinction between variables and variable names, but here it takes on a different sense, being the distinction between a sign and its higher order sign. This is because a variable is conceived as a “store”, a component of state of the interpreting machine, that contains different values from time to time, while the variable name is a symbolic version of that store's address. The store when full, or the state when determinate, constitutes a form of numeral, not a number, and so it is still a sign, not the object itself. This makes the variable name in this setting a type of higher order sign.

It is not just the influence of different conventions about language use that forms the source of so much confusion. Different conventions that prevail in different contexts would generate conceptual turbulence only at their boundaries with each other, and not distribute the disturbance throughout the interiors of these contexts, as is currently the case. But there are higher order differential conventions, in other words, conventions about changing conventions, that apply without warning all throughout what is pretended to be a uniform context.

For example, suppose I make a casual reference to the following set of pronouns:

Chances are that the reader will automatically shift to what I have called the sign convention to interpret this reference. Even without the instruction to expect a set of pronouns, it makes very little sense in this setting to think I am referring to a set of people, and so a charitable assumption about my intentions to make sense will lead to the intended interpretation.

However, suppose I make a similar reference to the following set of variables:

In this case it is more likely that the reader will take the suggested set of variable names as though they were the names of some fictional objects called “variables”.

The rest of this section deals with the case of boolean variables, soon to be invoked in providing a functional interpretation of propositional calculus.

This discussion draws on concepts from two previous papers (Awbrey, 1989 and 1994), changing notations as needed to fit the current context. Except for a number of special sets like and I use plain capital letters for ordinary sets, singly underlined capitals for coordinate spaces and vector spaces, and doubly underlined capitals for the alphabets and lexicons that generate formal languages and logical universes of discourse.

If is a set of elements, it is possible to construct a formal alphabet of letters or a formal lexicon of words corresponding to the elements of and notated as follows:

The set is known in formal settings as the literal alphabet or the literal lexicon associated with but on more familiar grounds it can be called the double of Under conditions of careful interpretation, any finite set can be construed as its own double, but for now it is safest to preserve the apparent distinction in roles until the sense of this double usage has become second nature.

This construction is often useful in situations where has to deal with a set of signs with a fixed or a faulty interpretation. Here one needs a fresh set of signs that can be used in ways analogous to the original, but free enough to be controlled and flexible enough to be repaired. In other words, the interpretation of the new list is subject to experimental variation, freely controllable in such a way that it can follow or assimilate the original interpretation whenever it makes sense to do so, but critically reflected and flexible enough to have its interpretation amended whenever necessary.

Interpreted on a casual basis, the set can be treated as a list of boolean variables, or, according to another reading, as a list of boolean variable names, but both of these choices are subject to the eventual requirement of saying exactly what a “variable” is.

The overall problem about the “ontological status” of variables will also be the subject of an extended study at a later point in this project, but for now I am forced to side-step the whole issue, merely giving notice of a signal distinction that promises to yield a measure of effective advantage in finally disposing of the problem.

If a sign, as accepted and interpreted in a particular setting, has an existentially unique denotation, that is, if there exists a unique object that the sign denotes under the operative sign relation, then the sign is said to possess a EU-denotation, or to have a EU-object. When this is so, the sign is said to be eudenotational, otherwise it is said to be dysdenotational.

Using the distinction accorded to eudenotational signs, the issue about the ontological status of variables can be illustrated as turning on two different acceptations of the list

  1. The natural (or naive) acceptation is for a reader to interpret the list as referring to a set of objects, in effect, to pass without hesitation from impressions of the characters to thoughts of their respective EU-objects all taken for granted to exist uniquely. The whole set of interpretive assumptions that go into this acceptation will be referred to as the object convention.
  2. The reflective (or critical) acceptation is to see the list before all else as a list of signs, each of which may or may not have a EU-object. This is the attitude that must be taken in formal language theory and in any setting where computational constraints on interpretation are being contemplated. In these contexts it cannot be assumed without question that every sign, whose participation in a denotation relation would have to be indicated by a recursive function and implemented by an effective program, does in fact have an existential denotation, much less a unique object. The entire body of implicit assumptions that go to make up this acceptation, although they operate more like interpretive suspicions than automatic dispositions, will be referred to as the sign convention.

In the present context, I can answer questions about the ontology of a “variable” by saying that each variable is a kind of a sign, in the boolean case capable of denoting an element of as its object, with the actual value depending on the interpretation of the moment. Note that is a sign, and that is another sign that denotes it. This acceptation of the list corresponds to what was just called the sign convention.

In a context where all the signs that ought to have EU-objects are in fact safely assured to do so, then it is usually less bothersome to assume the object convention. Otherwise, discussion must resort to the less natural but more careful sign convention. This convention is only “artificial” in the sense that it recalls the artifactual nature and the instrumental purpose of signs, and does nothing more out of the way than to call an implement “an implement”.

I make one more remark to emphasize the importance of this issue, and then return to the main discussion. Even though there is no great difficulty in conceiving the sign to be interpreted as denoting different types of objects in different contexts, it is more of a problem to imagine that the same object can literally be both a value (in ) and a function (from to ).

In the customary fashion, the name of the variable is flexibly interpreted to serve two additional roles. In algebraic and geometric contexts is taken to name the coordinate function In logical contexts serves to name the basic property or simple proposition, also called that goes into the construction of a propositional universe of discourse, in effect, becoming one of the sentence letters of a truth table and being used to label one of the simple enclosures of a venn diagram.

Rationalizing the usage of boolean variables to represent propositional features and functions in this manner, I can now discuss these concepts in greater detail, introducing additional notation along the way.

  1. The sign appearing in the contextual frame or interpreted as belonging to that frame, denotes the coordinate function The entire collection of coordinate maps in contributes to the definition of the coordinate space or vector space notated as follows:

    Associated with the coordinate space are various families of boolean-valued functions

    1. The set of all functions has a cardinality of and is denoted as follows:

    2. The set of linear functions has a cardinality of and is known as the dual space in vector space contexts. In formal language contexts, in order to avoid conflicts with the use of the kleene star operator, it needs to be given an alternate notation:

    3. The set of positive functions has a cardinality of and is notated as follows: <p>

    4. The set of singular functions has a cardinality of and is notated as follows:

    5. The set of coordinate functions, also referred to as the set of basic or simple functions, has a cardinality of and is denoted in the following ways:

  2. The sign read or understood in a propositional context, can be interpreted as denoting one of the features, qualities, basic properties, or simple propositions that go to define the -dimensional universe of discourse also notated as follows:

Propositional Calculus

The order of reasoning called propositional logic, as it is pursued from various perspectives, concerns itself with three domains of objects, with all three domains having analogous structures in the relationships of their objects to each other. There is a domain of logical objects called properties or propositions, a domain of functional objects called binary-, boolean-, or truth-valued functions, and a domain of geometric objects called regions or subsets of the relevant universe of discourse. Each domain of objects needs a domain of signs to refer to its elements, but if one's interest lies mainly in referring to the common aspects of structure exhibited by these domains, then it serves to maintain a single notation, variously interpreted for all three domains.

The first order of business is to comment on the logical significance of the rhetorical distinctions that appear to prevail among these objects. My reason for introducing these distinctions is not to multiply the number of entities beyond necessity but merely to summarize the variety of entities that have been used historically, to figure out a series of conversions between them, and to integrate suitable analogues of them within a unified system.

For many purposes the distinction between a property and a proposition does not affect the structural aspects of the domains being considered. Both properties and propositions are tantamount to fictional objects, made up to supply general signs with singular denotations, and serving as indirect ways to explain the plural indefinite references (PIRs) of general signs to the multitudes of their ultimately denoted objects. A property is signified by a sign called a term that achieves by a form of indirection a PIR to all the elements in a class of things. A proposition is signified by a sign called a sentence that achieves by a form of indirection a PIR to all the elements in a class of situations. But things are any objects of discussion and thought, in other words, a perfectly general category, and situations" are just special cases of these things.

There is still something left to the logical distinction between properties and propositions, but it is largely immaterial to the order of reasoning that is found reflected in propositional logic. When it is useful to emphasize their commonalities, properties and propositions can both be referred to as Props. As a handle on the aspects of structure that are shared between these two domains and as a mechanism for ignoring irrelevant distinctions, it also helps to have a single term for a domain of properties (DOP) and a domain of propositions (DOP).

Because a Prop is introduced as an intermediate object of reference for a general sign, it factors a PIR of a general sign across two stages, the first appearing as a reference of a general sign to a singular Prop, and the second appearing as an application of a Prop to its proper objects. This affords a point of articulation that serves to unify and explain the manifold of references involved in a PIR, but it requires a distinction to be fashioned between the intermediate objects, whether real or invented, and the original, further, or ultimate objects of a general sign.

Next, it is necessary to consider the stylistic differences among the logical, functional, and geometric conceptions of propositional logic. Logically, a domain of properties or propositions is known by the axioms it is subject to. Concretely, one thinks of a particular property or proposition as applying to the things or situations it is true of. With the synthesis just indicated, this can be expressed in a unified form: In abstract logical terms, a DOP is known by the axioms to which it is subject. In concrete functional or geometric terms, a particular element of a DOP is known by the things of which it is true.

With the appropriate correspondences between these three domains in mind, the general term proposition can be interpreted in a flexible manner to cover logical, functional, and geometric types of objects. Thus, a locution like can be interpreted in three ways: (1) literally, to denote a logical proposition, (2) functionally, to denote a mapping from a space of propertied or proposed objects to the domain of truth values, and (3) geometrically, to denote the so-called fiber of truth as a region or a subset of For all of these reasons, it is desirable to set up a suitably flexible interpretive framework for propositional logic, where an object introduced as a logical proposition can be recast as a boolean function and understood to indicate the region of the space that is ruled by

Generally speaking, it does not seem possible to disentangle these three domains from each other or to determine which one is more fundamental. In practice, due to its concern with the computational implementations of every concept it uses, the present work is biased toward the functional interpretation of propositions. From this point of view, the abstract intention of a logical proposition is regarded as being realized only when a program is found that computes the function

The functional interpretation of propositional calculus goes hand in hand with an approach to logical reasoning that incorporates semantic or model-theoretic methods, as distinguished from the purely syntactic or proof-theoretic option. Indeed, the functional conception of a proposition is model-theoretic in a double sense, not only because its notations denote functions as their semantic objects, but also because the domains of these functions are spaces of logical interpretations for the propositions, with the points of the domain that lie in the inverse image of truth under the function being the models of the proposition.

One of the reasons for pursuing a pragmatic hybrid of semantic and syntactic approaches, rather than keeping to the purely syntactic ways of manipulating meaningless tokens according to abstract rules of proof, is that the model theoretic strategy preserves the form of connection that exists between an agent's concrete particular experiences and the abstract propositions and general properties that it uses to describe its experience. This makes it more likely that a hybrid approach will serve in the realistic pursuits of inquiry, since these efforts involve the integration of deductive, inductive, and abductive sources of knowledge.

In this approach to propositional logic, with a view toward computational realization, one begins with a space called a universe of discourse, whose points can be reasonably well described by means of a finite set of logical features. Since the points of the space are effectively known only in terms of their computable features, one can assume that there is a finite set of computable coordinate projections for for some that can serve to describe the points of This means that there is a computable coordinate representation for in other words, a computable map that describes the points of insofar as they are known. Thus, each proposition can be factored through the coordinate representation to yield a related proposition one that speaks directly about coordinate -tuples but indirectly about points of Composing maps on the right, the mapping is defined by the equation For all practical purposes served by the representation the proposition can be taken as a proxy for the proposition saying things about the points of by means of 's encoding to

Working under the functional perspective, the formal system known as propositional calculus is introduced as a general system of notations for referring to boolean functions. Typically, one takes a space and a coordinate representation as parameters of a particular system and speaks of the propositional calculus on a finite set of variables In objective terms, this constitutes the domain of propositions on the basis notated as Ideally, one does not want to become too fixed on a particular set of logical features or to let the momentary dimensions of the space be cast in stone. In practice, this means that the formalism and its computational implementation should allow for the automatic embedding of into whenever

The rest of this section presents the elements of a particular calculus for propositional logic. First, I establish the basic notations and summarize the axiomatic presentation of the calculus, and then I give special attention to its functional and geometric interpretations.

This section reviews the elements of a calculus for propositional logic that I initially presented in two earlier papers (Awbrey, 1989 and 1994). This calculus belongs to a family of formal systems that hark back to C.S. Peirce's existential graphs (ExG) and it draws on ideas from Spencer Brown's Laws of Form (LOF). A feature that distinguishes the use of these formalisms can be summed up by saying that they treat logical expressions primarily as elements of a language and only secondarily as elements of an algebra. In other words, the most important thing about a logical expression is the logical object it denotes. To the extent that the object can be represented in syntax, this attitude puts the focus on the logical equivalence class (LEC) to which the expression belongs, relegating to the background the whole variety of ways that the expression can be generated from algebraically conceived operations. One of the benefits of this notation is that it facilitates the development of a differential extension for propositional logic that can be used to reason about changing universes of discourse.

A propositional language is a syntactic system that mediates the reasonings of a propositional logic. The objects of the language and the logic, that is, the logical entities denoted by the language and invoked by the operations of the logic, can be conceived to rest at various levels of abstraction, residing in spaces of functions that are basically of the types and remaining subject only to suitable choices of the parameter

Persistently reflective engagement in logical reasoning about any domain of objects leads to the identification of generic patterns of inference that appear to be universally valid, never disappointing the trust that is placed in them. After a time, a formal system naturally arises that commemorates one's continuing commitment to these patterns of logical conduct, and acknowledges one's conviction that further inquiry into their utility can be safely put beyond the reach of everyday concerns. At this juncture each descriptive pattern becomes a normative template, regulating all future ventures in reasoning until such time as a clearly overwhelming mass of doubtful outcomes cause one to question it anew.

Propositions about a coherent domain of objects tend to gather together and express themselves collectively in organized bodies of statements known as theories. As theories grow in size and complexity, one is faced with massive collections of propositional constraints and complex chains of logical inferences, and it becomes useful to support reasoning with the implementation of a propositional calculator.

At this point, variations in common and technical usage of the term proposition require a few comments on terminology. The heart of the issue is how to maintain a proper distinction between the logical form and the rhetorical style of a proposition, that is, how best to mark the difference between its invariant contents and its variant expressions. There are many ways to draw the required form of distinction between the objective situation and the significant expression in this relation. Here, I outline a compromise strategy that incorporates the advantages of several options and makes them available to intelligent choice as best fits the occasion.

  1. According to a prevailing technical usage, a proposition is a categorical object of abstract thought, something that is tantamount to an objective situation, a statistical event, or a state of affairs of a specified type. In distinction to the abstract proposition, a statement that a situation of the proposed type is actually in force is expressed in the form of a syntactic formula called a sentence.
  2. Another option enjoys a set of incidental advantages that makes it worth mentioning here and also worth exploring in a future discussion. Under this alternative, one refers to the signifying expressions as propositions, deliberately conflating propositions and sentences, but then introduces the needed distinction at another point of articulation, referring to the signified objects as positions.
  3. Attempting to strike a compromise with common usage, I often allow the word proposition to exploit the full range of its senses, denoting either object or sign according to context, and resorting to the phrase propositional expression whenever it is necessary to emphasize the involvement of the sign.

The operative distinction in every case, propositional or otherwise, is the difference in roles between objects and signs, not the names they are called by. To reconcile a logical account with the pragmatic theory of signs, one entity is construed as the propositional object (PO) and the other entity is recognized as the propositional sign (PS) at each moment of interpretation in a propositional sign relation. Once these roles are assigned, all the technology of sign relations applies to the logic of propositions as a special case. In the context of propositional sign relations, a semantic equivalence class (SEC) is referred to as a logical equivalence class (LEC). Each propositional object can then be associated, or even identified for all informative and practical purposes, with the LEC of its propositional signs. Accordingly, the proposition is reconstituted from its sentences in the appropriate way, as an abstract object existing in a semantic relation to its signs.

Taking this topic, the representation of sign relations, and seeking a computational formulation of its theory, leads to certain considerations about the best approach to the subject. Computational formulations are those with no recourse but to finitary resources. In setting up a computational formulation of any theory, one has to specify the finite set of axioms that are constantly available to subsequent reasoning. This makes it advisable to approach the topic of representations at a level of generality that will give the resulting theory as much power as possible, the kind of power to which inductive hypotheses can have easy and constant recourse. In order to furnish these resources with an ample supply of theoretical power …

In doing this, it is expeditious, if not absolutely necessary, to broaden the focus on sign relations in two ways: (1) to expand its extension from a special class of triadic relations to the wider sphere of -place relations, and (2) to diffuse its intension from fully specified and concretely presented relations to incompletely specified and abstractly described relations.

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