41. The Scope of Propositions and Classes in Notation

Harold R. (Hal) Foster’s Prince Valiant

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In the case when the smallest proposition enclosed in dots or other brackets contains two or more descriptions, we shall assume, in the absence of any indication to the contrary, that one which typographically occurs earlier has a larger scope than one which typographically occurs later. Thus

     (⍳x)(φx) = (⍳x)(yx)

will mean  (∃c) : φx . ≡_x . x = c : [(⍳x)(yx)] . c = (⍳x)(yx),

while    (⍳x)(yx) = (⍳x)(φx)

will mean (∃d) : yx . ≡_x . x = d : [(⍳x)(φx)] . (⍳x)(φx) = d.

These two propositions are easily shown to be equivalent.

 (2) Classes. The symbols for classes, like those for descriptions, are, in our system, incomplete symbols: their uses are defined, but they themselves are not assumed to mean anything at all. That is to say, the uses of such symbols are so defined that, when the definiens is substituted for the definiendum, there no longer remains any symbol which could be supposed to represent a class. Thus classes, so far as we introduce them, are merely symbolic or linguistic conveniences, not genuine objects as their members are if they are individuals.

 It is an old dispute whether formal logic should concern itself mainly with intensions or with extensions. In general, logicians whose training was mainly philosophical have decided for intensions, while those whose training was mainly mathematical have decided for extensions. The facts seem to be that, while mathematical logic requires extensions, philosophical logic refuses to supply anything except intensions. Our theory of classes recognizes and reconciles these two apparently opposite facts, by showing that an extension (which is the same as a class) is an incomplete symbol, whose use always acquires its meaning through a reference to intension.

In the case of descriptions, it was possible to prove that they are incomplete symbols. In the case of classes, we do not know of any equally definite proof, though arguments of more or less cogency can be elicited from the ancient problem of the One and the Many*¹. It is not necessary for our purposes, however, to assert dogmatically that there are no such things as classes. It is only necessary for us to show that the incomplete symbols which we introduce as representatives of classes yield all the propositions for the sake of which classes might be thought essential. When this has been shown, the mere principle of economy of primitive ideas leads to the non-introduction of classes except as incomplete symbols.

To explain the theory of classes, it is necessary first to explain the distinction between extensional and intensional functions. This is seen in effect 'by the following definitions:

The truth-value of a proposition is truth if it is true, and falsehood if it is false. (This expression is due to Frege.)

Two propositions are said to be equivalent when they have the same truth-value, i.e. when they are both true or both false.

Two propositional functions are said to be formally equivalent when they are equivalent with every possible argument, i.e. when any argument which satisfies the one satisfies the other, and vice versa. Thus "x^ is a man" is formally equivalent to " x^  is a featherless biped"; "x^ is an even prime" is formally equivalent to "x^ is identical with 2."

A function of a function is called extensional when its truth-value with any argument is the same as with any formally equivalent argument. That is to say, ƒ(φz^) is an extensional function of φz^ if, provided yz^ is formally equivalent to φz^, ƒ(φz^) is equivalent to ƒ(yz^). Here the apparent variables φ and y are necessarily of the type from which arguments can significantly be supplied to ƒ. We find no need to use as apparent variables any functions of non-predicative types; accordingly in the sequel all extensional functions considered are in fact functions of predicative functions*².

*¹ Briefly, these arguments reduce to the following: If there is such an object as a class, it must be in some sense one object. Yet it is only of classes that many can be predicated. Hence, if we admit classes as objects, we must suppose that the same object can be both one and many, which seems impossible.

*² Cf post 34.

Story and Art: John Stanley and Irving Tripp  Little Lulu © Respective copyright/trademark holders.

Here the authors expand on ways of recognizing and reconciling the seemingly opposite natures of mathematics and philosophy. As with all intractable differences, reconciliation can come without changing a problem, or even by compromise for that matter. The trick to working things out is to change our perception out the issues involved. Looking around for aspects that agree together can be a road to seeing a formal equivalence of value. So many obstacles arise in simple misunderstandings from the imprecise use of language, even after multiple attempts at clarification.

Script: Hannigan  Artwork: Perlin & Marcos 

The Inc. Hulk and HellcatÔ & © Respective copyright/trademark holders.



Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.