Functional Logic : Quantification Theory

Toward a Functional Conception of Quantificational Logic

Up till now quantification theory has been based on the assumption of individual variables ranging over universal collections of perfectly determinate elements. Merely to write down quantified formulas like \(\forall_{x \in X} F(x)\) and \(\exists_{x \in X} F(x)\) involves a subscription to such notions, as shown by the membership relations invoked in their indices. Reflected on pragmatic and constructive principles, however, these ideas begin to appear as problematic hypotheses whose warrants are not beyond question, projects of exhaustive determination that overreach the powers of finite information and control to manage. Therefore, it is worth considering how we might shift the scene of quantification theory closer to familiar ground, toward the predicates themselves that represent our continuing acquaintance with phenomena.

Higher Order Propositional Expressions

By way of equipping this inquiry with a bit of concrete material, I begin with a consideration of higher order propositional expressions (HOPE's), in particular, those that stem from the propositions on 1 and 2 variables.

Higher Order Propositions and Logical Operators (n = 1)

A higher order proposition is, very roughly speaking, a proposition about propositions. If the original order of propositions is a class of indicator functions \(F : X \to \mathbb{B},\) then the next higher order of propositions consists of maps of the type \(m : (X \to \mathbb{B}) \to \mathbb{B}.\)

For example, consider the case where \(X = \mathbb{B}.\) Then there are exactly four propositions \(F : \mathbb{B} \to \mathbb{B},\) and exactly sixteen higher order propositions that are based on this set, all bearing the type \(m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.\)

Table 10 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion: Columns 1 and 2 form a truth table for the four \(F : \mathbb{B} \to \mathbb{B},\) turned on its side from the way that one is most likely accustomed to see truth tables, with the row leaders in Column 1 displaying the names of the functions \(F_i,\!\) for \(i\!\) = 1 to 4, 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 one of the more usual expressions for the proposition in question. The last sixteen columns are topped by a collection of conventional names for the higher order propositions, also known as the measures \(m_j,\!\) for \(j\!\) = 0 to 15, where the entries in the body of the Table record the values that each \(m_j\!\) assigns to each \(F_i.\!\)

Table 7. Higher Order Propositions (n = 1)

\ x	1 0	F	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m
F \			00	01	02	03	04	05	06	07	08	09	10	11	12	13	14	15
F0	0 0	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
F1	0 1	(x)	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
F2	1 0	x	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
F3	1 1	1	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

I am going to put off explaining Table 8, that presents a sample of what I call "Interpretive Categories for Higher Order Propositions", until after we get beyond the 1-dimensional case, since these lower dimensional cases tend to be a bit "condensed" or "degenerate" in their structures, and a lot of what is going on here will almost automatically become clearer as soon as we get even two logical variables into the mix.

Table 8. Interpretive Categories for Higher Order Propositions (n = 1)

Measure	Happening	Exactness	Existence	Linearity	Uniformity	Information
m0	nothing happens
m1		just false	nothing exists
m2		just not x
m3			nothing is x
m4		just x
m5			everything is x	F is linear
m6					F is not uniform	F is informed
m7		not just true
m8		just true
m9					F is uniform	F is not informed
m10			something is not x	F is not linear
m11		not just x
m12			something is x
m13		not just not x
m14		not just false	something exists
m15	anything happens

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 us first consider the universe of discourse X° = [X] = [x₁, x₂] = [x, y], based on two logical features or boolean variables x and y.

1.	The points of X° are collected in the space:
	X = <<x, y>> = {<x, y>} ≈ B2.
	In other words, written out in full:
	X = {<"(x)", "(y)">, <"(x)", "y">, <"x", "(y)">, <"x", "y">}
	X ≈ {<0, 0>, <0, 1>, <1, 0>, <1, 1>}
2.	The propositions of X° make up the space:
	X↑ = (X → B) = {f : X → B} ≈ (B2 → B).

As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again.

The next higher order universe of discourse that is built on X° is X°2 = [X°] = [[x, y]], which may be developed in the following way. The propositions of X° become the points of X°2, and the mappings of the type m : (X → B) → B become the propositions of X°2. 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 w : (B² → B)^k → B.

To save a few words in the remainder of this discussion, I will 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 [x, y] may be regarded as a gallery of sixteen venn diagrams, while the measures m : (X → B) → B 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 m_j partitions the gallery of pictures into two aesthetic portions, the pictures (m_j)^–1(1) that m_j likes and the pictures (m_j)^–1(0) that m_j dislikes.

There are 2¹⁶ = 65536 measures of the type m : (B² → B) → B. Table 9 introduces the first 24 of these measures in the fashion of the higher order truth table that I used before. The column headed "m_j" shows the values of the measure m_j on each of the propositions f_i : B² → B, for i = 0 to 23, with blank entries in the Table being optional for values of zero. The arrangement of measures that continues according to the plan indicated here is referred to as the "standard ordering" of these measures. In this scheme of things, the index j of the measure m_j is the decimal equivalent of the bit string that is associated with m_j's functional values, which can be obtained in turn by reading the j^th column of binary digits in the Table as the corresponding range of boolean values, taking them up in the order from bottom to top.

Table 9. Higher Order Propositions (n = 2)

x :	1100	f	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m	m
y :	1010		0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
f0	0000	( )	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
f1	0001	(x)(y)			1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
f2	0010	(x) y					1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
f3	0011	(x)									1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0
f4	0100	x (y)																	1	1	1	1	1	1	1	1
f5	0101	(y)
f6	0110	(x, y)
f7	0111	(x y)
f8	1000	x y
f9	1001	((x, y))
f10	1010	y
f11	1011	(x (y))
f12	1100	x
f13	1101	((x) y)
f14	1110	((x)(y))
f15	1111	(( ))

Umpire Operators

In order to get a handle on the space of higher order propositions and eventually to carry out a functional approach to quantification theory, it serves to construct some specialized tools. Specifically, I define a higher order operator Υ, called the "umpire operator", which takes up to three propositions as arguments and returns a single truth value as the result. Formally, this so-called "multi-grade" property of \(\Upsilon\!\) can be expressed as a union of function types, in the following manner:

\[\Upsilon : \cup^{m = 1, 2, 3}((\mathbb{B}^k \to \mathbb{B})^m \to \mathbb{B}).\]

In contexts of application the intended sense can be discerned by the number of arguments that actually appear in the argument list. Often, the first and last arguments appear as indices, the one in the middle being treated 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:

Υ_p^r q = Υ(p, q, r)

Υ_p^r : (B^k → B) → B

The intention of this operator is that we evaluate the proposition q on each model of the proposition p and combine the results according to the method indicated by the connective parameter r. In principle, the index r might specify any connective on as many as 2^k arguments, but usually we have in mind a much simpler form of combination, most often either collective products or collective sums. By convention, each of the accessory indices p, r is assigned a default value that is understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition 1 : B^k → B for the lower index p, and the continued conjunction or continued product operation Π for the upper index r. Taking the upper default value gives license to the following readings:

1.	Υp q = Υ(p, q) = Υ(p, q, Π).
2.	Υp = Υ(p, __, Π) : (Bk → B) → B.

This means that Υ_p q = 1 if and only if q holds for all models of p. In propositional terms, this is tantamount to the assertion that p ⇒ q, or that _(p (q))_ = 1.

Throwing in the lower default value permits the following abbreviations:

3.	Υq = Υ(q) = Υ1 q = Υ(1, q, Π).
4.	Υ = Υ(1, __, Π) : (Bk → B) → B.

This means that Υq = 1 if and only if q holds for the whole universe of discourse in question, that is, if and only q is the constantly true proposition 1 : B^k → B. 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.

Measure for Measure

An acquaintance with the functions of the umpire operator can be gained from Tables 10 and 11, where the 2-dimensional case is worked out in full.

The auxiliary notations:

α_i f = Υ(f_i, f),

β_i f = Υ(f, f_i),

define two series of measures:

α_i, β_i : (B² → B) → B,

incidentally providing compact names for the column headings of the next two Tables.

Table 10. Qualifiers of Implication Ordering: α_i f = Υ(f_i ⇒ f)

x :	1100	f	α	α	α	α	α	α	α	α	α	α	α	α	α	α	α	α
y :	1010		15	14	13	12	11	10	9	8	7	6	5	4	3	2	1	0
f0	0000	( )																1
f1	0001	(x)(y)															1	1
f2	0010	(x) y														1		1
f3	0011	(x)													1	1	1	1
f4	0100	x (y)												1				1
f5	0101	(y)											1	1			1	1
f6	0110	(x, y)										1		1		1		1
f7	0111	(x y)									1	1	1	1	1	1	1	1
f8	1000	x y								1								1
f9	1001	((x, y))							1	1							1	1
f10	1010	y						1		1						1		1
f11	1011	(x (y))					1	1	1	1					1	1	1	1
f12	1100	x				1				1				1				1
f13	1101	((x) y)			1	1			1	1			1	1			1	1
f14	1110	((x)(y))		1		1		1		1		1		1		1		1
f15	1111	(( ))	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1

Table 11. Qualifiers of Implication Ordering: β_i f = Υ(f ⇒ f_i)

x :	1100	f	β	β	β	β	β	β	β	β	β	β	β	β	β	β	β	β
y :	1010		0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
f0	0000	( )	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
f1	0001	(x)(y)		1		1		1		1		1		1		1		1
f2	0010	(x) y			1	1			1	1			1	1			1	1
f3	0011	(x)				1				1				1				1
f4	0100	x (y)					1	1	1	1					1	1	1	1
f5	0101	(y)						1		1						1		1
f6	0110	(x, y)							1	1							1	1
f7	0111	(x y)								1								1
f8	1000	x y									1	1	1	1	1	1	1	1
f9	1001	((x, y))										1		1		1		1
f10	1010	y											1	1			1	1
f11	1011	(x (y))												1				1
f12	1100	x													1	1	1	1
f13	1101	((x) y)														1		1
f14	1110	((x)(y))															1	1
f15	1111	(( ))																1

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

	α00 f = 1	iff	f00 ⇒ f,	iff	0 ⇒ f,	hence	α00 f = 1 for all f.
	α15 f = 1	iff	f15 ⇒ f,	iff	1 ⇒ f,	hence	α15 f = 1 ⇒ f = 1.
	β00 f = 1	iff	f ⇒ f00,	iff	f ⇒ 0,	hence	β00 f = 1 ⇒ f = 0.
	β15 f = 1	iff	f ⇒ f15,	iff	f ⇒ 1,	hence	β15 f = 1 for all f.

Thus, α₀ = β₁₅ is a totally indiscriminate measure, one that accepts all propositions f : B² → B, whereas α₁₅ and β₀ are measures that value the constant propositions 1 : B² → B and 0 : B² → B, respectively, above all others.

Finally, in conformity with the use of the fiber notation to indicate sets of models, it is natural to use notations like:

[| α_i |] = (α_i)^(–1)(1),

[| β_i |] = (β_i)^(–1)(1),

[| Υ_p |] = (Υ_p)^(–1)(1),

to denote sets of propositions that satisfy the umpires in question.

Extending the Existential Interpretation to Quantificational Logic

Previously I introduced a calculus for propositional logic, fixing its meaning according to what C.S. Peirce called the "existential interpretation". As far as it concerns propositional calculus this interpretation settles the meanings that are associated with merely the most basic symbols and logical connectives. Now we must extend and refine the existential interpretation to comprehend the analysis of "quantifications", that is, quantified propositions. In doing so we recognize two additional aspects of logic that need to be developed, over and above the material of propositional logic. At the formal extreme there is the aspect of higher order functional types, into which we have already ventured a little above. At the level of the fundamental content of the available propositions we have to introduce a different interpretation for what we may call "elemental" or "singular" propositions.

Let us return to the 2-dimensional case X° = [x, y]. In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers L_uv : (B² → B) → B that have the following characters:

	L00 f
	=	L"(x)(y)" f
	=	α1 f
	=	Υ"(x)(y)" f
	=	Υ"(x)(y) ⇒ f"
	=	"f likes (x)(y)"

	L01 f
	=	L"(x) y " f
	=	α2 f
	=	Υ"(x) y " f
	=	Υ"(x) y ⇒ f"
	=	"f likes (x) y "

	L10 f
	=	L" x (y)" f
	=	α4 f
	=	Υ"x (y)" f
	=	Υ"x (y) ⇒ f"
	=	"f likes x (y)"

	L11 f
	=	L" x y" f
	=	α8 f
	=	Υ"x y" f
	=	Υ"x y ⇒ f"
	=	"f likes x y"

Intuitively, the L_uv operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values. Taken together, these measures provide us with the means to express many useful observations about the propositions in X° = [x, y], and so they mediate a subtext [L₀₀, L₀₁, L₁₀, L₁₁] that takes place within the higher order universe of discourse X°2 = [X°] = [[x, y]]. Figure 12 summarizes the action of the L_uv on the f_i within X°2.

o-----------------------------------------------------------o
|                                                           |
|                             o                             |
|                            / \                            |
|                           /   \                           |
|                          /x   y\                          |
|                         / o---o \                         |
|                        o   \ /   o                        |
|                       / \   o   / \                       |
|                      /   \  |  /   \                      |
|                     /     \ @ /     \                     |
|                    / x   y \ / x   y \                    |
|                   o  o---o  o  o---o  o                   |
|                  / \  \    / \    /  / \                  |
|                 /   \  @  /   \  @  /   \                 |
|                /     \   /     \   /     \                |
|               /   y   \ /       \ /   y   \               |
|              o    @    o    @    o    o    o              |
|             / \       / \       / \   |   / \             |
|            /   \     /   \     /   \  @  /   \            |
|           /     \   /x   y\   /     \   /     \           |
|          /  x y  \ / o   o \ /  x y  \ / x   y \          |
|         o    @    o   \ /   o    o    o  o   o  o         |
|         |\       / \   o   / \   |   / \  \ /  /|         |
|         | \     /   \  |  /   \  @  /   \  @  / |         |
|         |  \   /     \ @ /     \   /     \   /  |         |
|         |   \ /   x   \ / x   y \ /   x   \ /   |         |
|         |    o    @    o  o---o  o    o    o    |         |
|         |    |\       / \  \ /  / \   |   /|    |         |
|         |    | \     /   \  @  /   \  @  / |    |         |
|         |    |  \   /     \   /     \   /  |    |         |
|         |L_11|   \ /   o y \ / x o   \ /   |L_00|         |
|         o---------o    |    o    |    o---------o         |
|              |     \ x @   / \   @ y /     |              |
|              |      \     /   \     /      |              |
|              |       \   /     \   /       |              |
|              |L_10    \ /   o   \ /    L_01|              |
|              o---------o    |    o---------o              |
|                         \   @   /                         |
|                          \     /                          |
|                           \   /                           |
|                            \ /                            |
|                             o                             |
|                                                           |
o-----------------------------------------------------------o
Figure 12.  Higher Order Universe of Discourse [L_uv] c [[x, y]]

Application of Higher Order Propositions to Quantification Theory

Our excursion into the vastening landscape of higher order propositions has finally come round to the stage where we can bring its returns to bear on opening up new perspectives for quantificational logic.

There is a question arising next that is still experimental in my mind. Whether it makes much difference from a purely formal point of view is not a question I can answer yet, but it does seem to aid the intuition to invent a slightly different interpretation for the two-valued space that we use as the target of our basic indicator functions. Therefore, let us declare a type of "existential-valued" functions f : B^k → E, where E = {–e, +e} = {"empty", "exist"} is a couple of values that we interpret as indicating whether of not anything exists in the cells of the underlying universe of discourse, venn diagram, or other domain. As usual, let us not be too strict about the coding of these functions, reverting to binary codes whenever the interpretation is clear enough.

With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:

Table 13. Syllogistic Premisses as Higher Order Indicator Functions

A	Universal Affirmative	All	x	is	y	Indicator of " x (y)" = 0
E	Universal Negative	All	x	is	(y)	Indicator of " x y " = 0
I	Particular Affirmative	Some	x	is	y	Indicator of " x y " = 1
O	Particular Negative	Some	x	is	(y)	Indicator of " x (y)" = 1

Tables 14 and 15 develop these ideas in more detail.

Table 14. Relation of Quantifiers to Higher Order Propositions

Mnemonic	Category	Classical Form	Alternate Form	Symmetric Form	Operator
E Exclusive	Universal Negative	All x is (y)		No x is y	(L11)
A Absolute	Universal Affirmative	All x is y		No x is (y)	(L10)
		All y is x	No y is (x)	No (x) is y	(L01)
		All (y) is x	No (y) is (x)	No (x) is (y)	(L00)
		Some (x) is (y)		Some (x) is (y)	L00
		Some (x) is y		Some (x) is y	L01
O Obtrusive	Particular Negative	Some x is (y)		Some x is (y)	L10
I Indefinite	Particular Affirmative	Some x is y		Some x is y	L11

Table 15. Simple Qualifiers of Propositions (n = 2)

x :	1100	f	(L11)	(L10)	(L01)	(L00)	L00	L01	L10	L11
y :	1010		no x is y	no x is (y)	no (x) is y	no (x) is (y)	some (x) is (y)	some (x) is y	some x is (y)	some x is y
f0	0000	( )	1	1	1	1	0	0	0	0
f1	0001	(x)(y)	1	1	1	0	1	0	0	0
f2	0010	(x) y	1	1	0	1	0	1	0	0
f3	0011	(x)	1	1	0	0	1	1	0	0
f4	0100	x (y)	1	0	1	1	0	0	1	0
f5	0101	(y)	1	0	1	0	1	0	1	0
f6	0110	(x, y)	1	0	0	1	0	1	1	0
f7	0111	(x y)	1	0	0	0	1	1	1	0
f8	1000	x y	0	1	1	1	0	0	0	1
f9	1001	((x, y))	0	1	1	0	1	0	0	1
f10	1010	y	0	1	0	1	0	1	0	1
f11	1011	(x (y))	0	1	0	0	1	1	0	1
f12	1100	x	0	0	1	1	0	0	1	1
f13	1101	((x) y)	0	0	1	0	1	0	1	1
f14	1110	((x)(y))	0	0	0	1	0	1	1	1
f15	1111	(( ))	0	0	0	0	1	1	1	1