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Functional Logic • Higher Order Propositions

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Author: Jon Awbrey

Functional Conception of Quantification Theory

Up till now quantification theory has been based on the assumption of individual variables ranging over universal collections of perfectly determinate elements.  The mere act of writing quantified formulas like and involves a subscription to such notions, as shown by the membership relations invoked in their indices.  As we reflect more critically on the conventional assumptions in the light of pragmatic and constructive principles, however, they begin to appear as problematic hypotheses whose warrants are not beyond question, as projects of exhaustive determination overreaching the powers of finite information and control to manage.  Thus it is worth considering how the scene of quantification theory might be shifted nearer to familiar ground, toward the predicates themselves which represent our continuing acquaintance with phenomena.

Higher Order Propositional Expressions

If functions of type are propositions about things in then functions of type are propositions about propositions about things in the first in a series of higher order propositions based on

To ground this inquiry in concrete material, let us begin with a consideration of higher order propositional expressions, in particular, those stemming from propositions on 1 and 2 variables.

Note on notation.  The discussion that follows uses minimal negation operations, expressed as parenthesized tuples of the form and logical conjunctions, expressed as concatenated tuples of the form as the sole expression-forming operations of a calculus for boolean-valued functions or propositions.  The expressions of this calculus parse into data structures whose underlying graphs are called cacti by graph theorists.  Hence the name cactus language for this dialect of propositional calculus.

Higher Order Propositions and Logical Operators (n = 1)

A higher order proposition is a proposition about propositions.  If the original order of propositions consists of maps of the form then the next higher order of propositions consists of maps of the form   It is often useful to think of a higher order proposition as a measure on propositions.

For example, consider the case where   Then there are exactly four propositions and exactly sixteen higher order propositions based on this set, all taking the form

Table 1 lists the 16 measures of the form

Higher Order Propositions (n=1) 2.0.png

The contents of Table 1 are organized as follows.  Columns 1 and 2 form a truth table for the four propositions of the form with the row leaders in Column 1 displaying the names of the functions for = 0 to 3, while the entries in Column 2 give the values of each function for the argument values that are listed in the corresponding column head.  Column 3 displays a more or less canonical expression for the proposition in question.  The last sixteen columns are headed by a series of conventional names for the measures as ranges from 0 to 15, where the entries in the body of the Table record the values assigned to each by each

Table 2 presents a sample of interpretive categories for higher order propositions of type but it's best to put off discussing this Table further until we get beyond the 1-dimensional case.  The lower dimensional cases tend to be extremely condensed or degenerate in their structures, and the pattern of what's going on here can be set in higher relief as soon as we get even two logical variables to play off each other.

Interpretive Categories for Higher Order Propositions (n=1) 2.0.png

Higher Order Propositions and Logical Operators (n = 2)

By way of reviewing notation and preparing to extend it to higher order universes of discourse, let's first consider the universe of discourse based on two logical features or boolean variables and

The universe of discourse consists of two parts, a set of points and a set of propositions.

The points of form a space notated as follows.

Each point in may be indicated by means of a singular proposition, that is, a proposition that describes it uniquely.  This form of representation leads to the following enumeration of points.

Each point in may also be described by means of its coordinates, that is, by the ordered pair of values in that the coordinate propositions and take on that point.  This form of representation leads to the following enumeration of points.

The propositions of form a space notated as follows.

As always, it is convenient to overlook the finer marks of distinction between isomorphic structures, so long as one is aware of their presence and knows when it is critical to call on them again.

The next higher order universe of discourse that is built on is which may be developed in the following way.  The propositions of become the points of and the mappings of the type become the propositions of   In addition, it is convenient to equip the discussion with a selected set of higher order operators on propositions, all of which have the form

To save a few words in the remainder of this discussion, let us use the terms measure and qualifier to refer to all types of higher order propositions and operators.  To describe the present setting in picturesque terms, the propositions of may be regarded as a gallery of sixteen venn diagrams, while the measures are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not.  In this way, each judge partitions the gallery of pictures into two aesthetic portions, the pictures that dislikes and the pictures that likes.

There are measures of the form   Table 3 shows the first 24 of their number in the style of higher order truth table used above.  The column headed shows the value of the measure on each of the propositions for = 0 to 15.  The arrangement of measures in the order indicated will be called their standard ordering.  In this scheme of things, the index of the measure is the decimal equivalent of the bit string in the corresponding column of the Table, reading the binary digits in order from bottom to top.

Higher Order Propositions (n=2) 2.0.png

Umpire Operators

The measures of type present a formidable array of propositions about propositions about 2-dimensional universes of discourse.  The early entries in their standard ordering define universes too amorphous to detain us for long on a first pass but as we turn toward the high end of the ordering we begin to recognize familiar structures worth examining from new angles.

Instrumental to our study we define a couple of higher order operators,

referred to as the relative and absolute umpire operators, respectively.  If either operator is defined in terms of more primitive notions then the remaining operator can be defined in terms of the one first established.

Let be a two-dimensional boolean space, generated by two boolean variables or logical features and

Given an ordered pair of propositions as arguments, the relative umpire operator reports the value if the first implies the second, otherwise it reports the value

Expressing it another way:

In writing this, however, it is important to observe that the appearing on the left side and the appearing on the right side of the logical equivalence have different meanings.  Filling in the details, we have the following.

Writing types as subscripts and using the fact that it is possible to express this a little more succinctly as follows.

Finally, it is often convenient to write the first argument as a subscript.  Thus we have the following equation.

The absolute umpire operator, also known as the umpire measure, is a higher order proposition defined by the equation   In this case the subscript on the left and the argument on the right both refer to the constant proposition   In most settings where is applied to arguments it is safe to omit the subscript since the number of arguments indicates which type of operator is meant.  Thus, we have the following identities and equivalents.

The umpire measure is defined at the level of boolean functions as mathematical objects but can also be understood in terms of the judgments it induces on the syntactic level.  In that interpretation recognizes theorems of the propositional calculus over giving a score of to tautologies and a score of to everything else, regarding all contingent statements as no better than falsehoods.

One remark in passing for those who might prefer an alternative definition.  If we had originally taken to mean the absolute measure then the relative measure could have been defined as

Measure for Measure

Let us define two families of measures,

by means of the following equations:

Table 4 shows the value of each on each of the 16 boolean functions   In terms of the implication ordering on the 16 functions, says that is above or identical to in the implication lattice, that is, in the implication ordering.


Qualifiers of Implication Ordering α 2.0.png


Table 5 shows the value of each on each of the 16 boolean functions   In terms of the implication ordering on the 16 functions, says that is below or identical to in the implication lattice, that is, in the implication ordering.


Qualifiers of Implication Ordering β 2.0.png


Applied to a given proposition the qualifiers and tell whether is above or below respectively, in the implication ordering.  By way of example, let us trace the effects of several such measures, namely, those which occupy the limiting positions in the Tables.

Expressed in terms of the propositional forms they value positively, is a totally indiscriminate measure, accepting all propositions whereas and are measures valuing the constant propositions and respectively, above all others.

Finally, in conformity with the use of fiber notation to indicate sets of models, it is natural to use notations like the following to denote sets of propositions satisfying the umpires in question.

Extending the Existential Interpretation to Quantificational Logic

One of the resources we have for this work is a formal calculus based on C.S. Peirce's logical graphs.  For now we'll adopt the existential interpretation of that calculus, fixing the meanings of logical constants and connectives at the core level of propositional logic.  To build on that core we'll need to extend the existential interpretation to encompass the analysis of quantified propositions, or quantifications.  That in turn will take developing two further capacities of our calculus.  On the formal side we'll need to consider higher order functional types, continuing our earlier venture above.  In terms of content we'll need to consider new species of elemental or singular propositions.

Let us return to the 2-dimensional universe   A bridge between propositions and quantifications is afforded by a set of measures or qualifiers defined by the following equations.

A higher order proposition tells us something about the proposition namely, which elements in the space of type are assigned a positive value by   Taken together, the operators give us a way to express many useful observations about the propositions in   Figure 6 summarizes the action of the operators on the propositions of type

Venn Diagram 4 Dimensions UV Cacti 8 Inch.jpg

Application of Higher Order Propositions to Quantification Theory

Our excursion into the expanding landscape of higher order propositions has come round to the point where we can begin to open up new perspectives on quantificational logic.

Though it may be all the same from a purely formal point of view, it does serve intuition to adopt a slightly different interpretation for the two-valued space we take as the target of our basic indicator functions.  In that spirit we declare a novel type of existence-valued functions where is a pair of values indicating whether anything exists in the cells of the underlying universe of discourse.  As usual, we won't be too picky about the coding of those functions, reverting to binary codes whenever the intended interpretation is clear enough.  With that interpretation in mind the next few Tables illustrate the correspondence between classical quantification theory and higher order indicator functions.

Table 7 exhibits a fourfold schema of quantified propositional forms traditionally known as a “Square of Opposition”, relating it to a quartet of higher order propositions which, depending on context, are also known as measures, qualifiers, or higher order indicator functions.



Table 8 develops the above ideas in further detail, expressing a larger set of quantified propositional forms by means of propositions about propositions.


1 1 1 1 0 0 0 0
1 1 1 0 1 0 0 0
1 1 0 1 0 1 0 0
1 1 0 0 1 1 0 0
1 0 1 1 0 0 1 0
1 0 1 0 1 0 1 0
1 0 0 1 0 1 1 0
1 0 0 0 1 1 1 0
0 1 1 1 0 0 0 1
0 1 1 0 1 0 0 1
0 1 0 1 0 1 0 1
0 1 0 0 1 1 0 1
0 0 1 1 0 0 1 1
0 0 1 0 1 0 1 1
0 0 0 1 0 1 1 1
0 0 0 0 1 1 1 1


Tables 9 and 10 present the same information as Table 8, sorting the rows in different orders to reveal other symmetries in the arrays.


1 1 1 1 0 0 0 0
1 1 1 0 1 0 0 0
1 1 0 1 0 1 0 0
1 0 1 1 0 0 1 0
0 1 1 1 0 0 0 1
1 1 0 0 1 1 0 0
0 0 1 1 0 0 1 1
1 0 0 1 0 1 1 0
0 1 1 0 1 0 0 1
1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1
1 0 0 0 1 1 1 0
0 1 0 0 1 1 0 1
0 0 1 0 1 0 1 1
0 0 0 1 0 1 1 1
0 0 0 0 1 1 1 1


1 1 1 1 0 0 0 0
1 1 1 0 1 0 0 0
1 1 0 1 0 1 0 0
1 0 1 1 0 0 1 0
0 1 1 1 0 0 0 1
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
0 1 1 0 1 0 0 1
1 0 0 1 0 1 1 0
1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0
1 0 0 0 1 1 1 0
0 1 0 0 1 1 0 1
0 0 1 0 1 0 1 1
0 0 0 1 0 1 1 1
0 0 0 0 1 1 1 1


Table 11 provides a thumbnail sketch of the relationships discussed in this section.


 
 
   
   
     
     
 
 


Generalized Umpire Operators

To get a better handle on the space of higher order propositions and continue developing our functional approach to quantification theory, we'll need a number of specialized tools.  To begin, we define a higher order operator called the umpire operator, which takes 1, 2, or 3 propositions as arguments and returns a single truth value as the result.  Operators with optional numbers of arguments are called multigrade operators, typically defined as unions over function types.  Expressing in that form gives the following formula.

In contexts of application, that is, where a multigrade operator is actually being applied to arguments, the number of arguments in the argument list tells which of the optional types is “operative”.  In the case of the first and last arguments appear as indices, the one in the middle serving as the main argument while the other two arguments serve to modify the sense of the operation in question.  Thus, we have the following forms.

The operation evaluates the proposition on each model of the proposition and combines the results according to the method indicated by the connective parameter   In principle, the index may specify any logical connective on as many as arguments but in practice we usually have a much simpler form of combination in mind, typically either products or sums.  By convention, each of the accessory indices is assigned a default value understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition for the lower index and the continued conjunction or continued product operation for the upper index   Taking the upper default value gives license to the following readings.

This means if and only if holds for all models of   In propositional terms, this is tantamount to the assertion that or that

Throwing in the lower default value permits the following abbreviations.

This means if and only if holds for the whole universe of discourse in question, that is, if and only is the constantly true proposition   The ambiguities of this usage are not a problem so long as we distinguish the context of definition from the context of application and restrict all shorthand notations to the latter.

References

  • Quine, W.V. (1969/1981), “On the Limits of Decision”, Akten des XIV. Internationalen Kongresses für Philosophie, vol. 3 (1969). Reprinted, pp. 156–163 in Quine (ed., 1981), Theories and Things, Harvard University Press, Cambridge, MA.

Related Topics

Resources

Document History

Note. The above material is taken from a project report on Charles Sanders Peirce's conceptions of inquiry and analogy.  Online formatting of the original document and continuation of the initial project are currently in progress under the title Functional Logic : Inquiry and Analogy.

1995 • Oakland University • Inquiry and Analogy

Author: Jon Awbrey November 1, 1995
Course: Engineering 690, Graduate Project Winter Term, January 1995
Supervisors: M.A. Zohdy and F. Mili Oakland University
| Version:  Draft 3.25
| Created:  01 Jan 1995
| Relayed:  01 Nov 1995
| Revised:  24 Dec 2001
| Revised:  12 Mar 2004

2004 • Inquiry List • Functional Logic

  1. http://web.archive.org/web/20090303000827/http://stderr.org/pipermail/inquiry/2004-March/001256.html
  2. http://web.archive.org/web/20090303001935/http://stderr.org/pipermail/inquiry/2004-March/001257.html
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  6. http://web.archive.org/web/20090303002026/http://stderr.org/pipermail/inquiry/2004-March/001261.html

2004 • Ontology List • Functional Logic

  1. http://web.archive.org/web/20090303202815/http://suo.ieee.org/ontology/msg05480.html
  2. http://web.archive.org/web/20090302145522/http://suo.ieee.org/ontology/msg05481.html
  3. http://web.archive.org/web/20090302145531/http://suo.ieee.org/ontology/msg05482.html
  4. http://web.archive.org/web/20090303203051/http://suo.ieee.org/ontology/msg05483.html
  5. http://web.archive.org/web/20090303203442/http://suo.ieee.org/ontology/msg05484.html
  6. http://web.archive.org/web/20090302145538/http://suo.ieee.org/ontology/msg05485.html

2004 • NKS Forum • Functional Logic

2004 • NKS Forum • Introduction to Inquiry Driven Systems

  1. http://web.archive.org/web/20090302150057/http://forum.wolframscience.com/showthread.php?postid=1957#post1957
  2. http://web.archive.org/web/20090302145607/http://forum.wolframscience.com/showthread.php?postid=1960#post1960
  3. http://web.archive.org/web/20090302150102/http://forum.wolframscience.com/showthread.php?postid=1961#post1961
  4. http://web.archive.org/web/20090302150134/http://forum.wolframscience.com/showthread.php?postid=1962#post1962
  5. http://web.archive.org/web/20090302145918/http://forum.wolframscience.com/showthread.php?postid=1964#post1964
  6. http://web.archive.org/web/20090302145303/http://forum.wolframscience.com/showthread.php?postid=1966#post1966
  7. http://web.archive.org/web/20090302150013/http://forum.wolframscience.com/showthread.php?postid=1968#post1968