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31 INTERPRETATION AND CONSISTENCY OF Q 19 (8) Po is an instance of Axiom Schema 8, and so has the form [a = p] . 2 [ S 4 , ] = [S\$4,], where ct and p are in K: and all free occurrences of c in A . are free for all variables in ct and p. If W,[ct = f l ] = f, then W,P, = t by Lemma 5. If W,[a = p ] = t, then W,cc = W,fl. Let 'p be that assignment which agrees with x on all primitive variables and on all type variables except c, while 'pc = W,u. By Lemma6 W,[S:Ao] = W,Ao = W,[SCpAo], so W,[[S:A,] = [S',A,]I = t.

LEMMA 14: If 2 > y and all free occurrences of u in the wff A , are free for all type variables in y, then \$:Ao is a wff of type (Sip). The proof is similar to the proof of Lemma 11. CHAPTER IT BASIC LOGIC I N Q We shall henceforth let 2 stand for an arbitrary finite set of wffs of type 0. We do not exclude the possibility that 2 may be empty. We shall often refer to 2 as a set of hypotheses or prernisses. We shall write 2 I- A , as an abbreviation for the phrase “ A , is derivable from the set 2 of hypotheses”.

0 0 0 0 132 I- [To = To] = To; [To = 0 0 3'01 = Fo; 0 I- [F, = To] = F,; F [F, = FO] 9 To. 2 I- [To 2 FO]2 . [[To= Fo] A Fo] 9 [To 2 Fo] R: 100, 103, def. 8. 9 are the desired theorems. 133 I- -To = Fo; t -Fo = To by E-Rules (101): 103, 132, def. of 134 t [To v To]= To; F [To v Fo] = To; I- [Fo v To]= To; I- [Fo v Fo] = Fo. Proof: by Rule R, 100, 103, 133, 131, def. of v . m. 111 BASIC LOGIC IN Q 43 DEFINITIONS: The class ofpropositionalwfls is the smallest class of wffsof type 0 which contains T o ,Fo , and all wff-variablesof type 0, and which, 0 if it contains A .