Outline of a Revised Formulation of the Logic of Sense and Denotation (Part II)
Alonzo Church · Noûs · 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,