Introduction to Symbolic Logic and Its Applications by Rudolf Carnap

Author:Rudolf Carnap
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 1958-04-14T16:00:00+00:00

The identity principle P8 of language B (see 22a,b) is in accord with what has just been said. With its help e.g. ‘Pa⊃Pb’ is derivable from ‘a ≡ b’ on the one hand, and (by substituting ‘∼P’) ‘∼Pa⊃∼Pb’ on the other; from this last by transposition (cf. T8-6i(l)) comes ‘Pb⊃Pa’, which together with ‘Pa⊃Pb’ leads us to ‘Pa≡Pb’. Thus we see that it is adequate to phrase P8 with the conditional sign.

The following theorem tells us that identity is (totally) reflexive, symmetric and transitive.

+T29-1. Suppose , and are expressions of the type system; then a sentential formula having one of the following forms is L-true:

a. = .

b. = ⊃ ≡ .

c. ( = · = ) ⊃ = .

As earlier (see D17-1b), so here we write ‘≠’ for “non-identical”; in the present context, of course, ‘≠’ can stand between two expressions of any one type. Non-identity is frequently used when the word “two” appears in a verbal text. E.g. “For any two points, there are ...” is rendered ‘(x)(y)[Pt(xPt(y)·(x≠y) ⊃ (∃z)(...)]’.

Instances of the use of the identity sign between predicate expressions may be found in T29-3, T30-1, and D30-2; and of similar usage respecting functor expressions in 33c.

Sometimes we find it convenient to use ‘I’ as a conventional predicate designating identity, and similarly ‘J’ for non-identity—a practice that has proved advantageous in connection with other two-place predicates. Moreover, we can use ‘J3(a, b, c)’ as a compact way of saying that a,b and c are three different individuals; ‘J4’ can have a corresponding role respecting four arguments, etc.

D29-1. Ixy ≡ x=y.

D29-2. a. Jxy ≡ x≠y.

b. J3xyz ≡ (x≠y · x≠z · y≠z).

c. J4xyzu ≡ (x≠y · x≠z · x≠u · y≠z · y≠u · z≠u).

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