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Cet article est une suite a Outline of a revised formulation of logic of sense and denotation (Nous, 1973-1974).

3. MODEL 2-0. In Model 2-0, we must distinguish in every type the members and the antimembers. In the types o, 0 1 0 ?2, ... the members and the antimembers are the same, and are the same as in Model 2-2. In the type l the members are 1 and 2; and there are no antimembers. In the type n+1 -the members are all the ordered pairs (aLn, bLn) where aLn may be 0 or any member of the type Ln and bL may be 0 or any antimember of the type 1n; and the antimembers are the same as the members. In any type cu3 the members are all functions from the members of the type ,3 to the members of the type ax; and the antimembers are all functions from the antimembers of the type f3 to the antimembers of the type a. As an example, the type Lo therefore has no antimembers, but the type oL has one antimember, the null function from antimembers of the type X to antimembers of the type o. This latter function is identified, under the original program of A Formulation [4], with the null class of individuals (i.e., of antimembers of the type i). Then we make the following recursive definition. Let (0, 0),yi be the ordered pair (0, 0) if yI is a simple type; and let (0, 0)al,, be the function , which is both a member and an antimember of the type (x I PI , such that I I as I = (0, 0)0j for every member (or antimember) a.1 of the type I3* Where is a member of the type af3 and .. is an antimember of the type cat, let (4 ) be the function k. I Iwhich is both a member and an antimember of the type a1 I3 , such that, if a. is any member of the type ,3 and by is any antimember of the type I, we have 4cLII (ad

In this paper I advocate a reexamination of mathematical logic in the tradition of Principia Mathematica and the early writings of Russell ([PoM], [OD], [ML]), to ask in what form it should now be studied in order to preserve certain important contributions of Russell and in order to compare it with the somewhat different Fregean tradition. This should not be as ramified type theory with axioms of reducibility, as this involves excessive complication,l and Russell himself conceded the objections against the axioms of reducibility (see [Intr2nd]); and not ramified type theory without axioms of reducibility, as this is insufficient for classical mathematics even if one adds the axioms of extensionality which Russell, influenced by Wittgenstein, suggests in [Intr2nd]. The modified version of Russell's mathematical logic which is now usual is simple theory of types ([Chwistek 1, 2], [Ramsey], [Carnap 1], [G6del]), with reliance on Tarski's resolution ([Tarski 2], [Church 6]) of the semantical antinomies.2 This is often taken as an extensional theory with explicit addition of axioms of extensionality ([Carnap 1], [G6del], [Tarski 1], [Quine 1], [Henkin], [Church 5]). If one is concerned primarily, or exclusively, with extensional logic, this is indeed the best approach, but it loses some of the characteristically Russellian contributions to logic. Especially the motivation for Russell's contextual definitions of class abstracts and of descriptions largely disappears, and it may seem better just to introduce primitive notations for both of these and to adopt a theory of descriptions which is more like Frege's than Russell's.3 Having formulated extensional logic in this way, if we then