# Poincarae's Philosophy: From Conventionalism to Phenomenology

By Elie Zahar | Go to book overview

APPENDIX II
Natural Axiomatisation
(A Revision of Wajsberg's Requirement)

Let S be a first-order system, and L the language of S. We suppose that L is countable and that for any truth-function p, L possesses a connective p corresponding to p. Where no confusion can arise, p will be identified with p. (Note that for any n∈ω, there is only a finite number, namely exp(2,2n). of truth-functions having n argument-places. Keep n fixed for the time being. After lexicographically ordering the 2n n-tuples which constitute the arguments of these functions, the latter can, in their turn, be similarly ordered. Denote by km,,nthe mth truth-function, i.e. connective, which has n argument-places. Since m and n are natural numbers, the set {km,,n: m < exp(2,2n)} is denumerable, which is consistent with our assumption that L is countable. For n = 0, we have m < exp(2,20) = exp(2,l) = 2. We take k0,0 and k1,0 to be the truth-values False and True respectively. Denote k0,0 by f and k1,0 by t. Call a well-formed formula (wff) a prepositional unit if it is either atomic or starts with a quantifier, i.e. with (∀ or with (∃. Thus every wff W is built from prepositional units by means of the km,n's; these prepositional units are called the prepositional components of W.

Let A = {Ai: i∈J} be an axiom system of S. Denote by Ω the set of all prepositional components of members of A. As far as assignments of truthvalues are concerned, syntactically different elements of Ω, e.g. (∀x)P(x) and (∀x)(P(x) ∧ P(X)), will be treated as different entities. A sentence Z whose prepositional components lie in Ω. is said to be tautological if it turns out true under all assignments of truth-values to members of Ω. In such a case, we also have, by completeness,

Z. X∼Y will henceforth mean that the sentences X and Y are tautologically equivalent; i.e. that X ⇔ Y is a tautology, which entails (X ⇔ Y).

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