By P. B Andrews

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**Extra resources for A Transfinite Type Theory with Type Variables**

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2. 3. 111 49 BASIC LOGIC IN Q 155 I- VaVbA, = VbVaAo, provided this is a wff. The proof, using 153, 147 and 144, is similar to the proof of 154. 156 tVaVx,A, = Vx,VaAo, provided this is a wff, and a does not occur in p. (Note that if this is a wff, either a does not occur in or xa does not occur free in Ao. 3 t- Vx,VaAo 3 VaVxpAo 152, 144: 147. 3. 157 t-VaAo = Vb[StAol,provided that (1) VaA, is a wff; (2) 6 does not occur free in A,,; (3) all free occurrences of a in A. are free for 6. 4 by Rule P.

1. 2 are the desired theorems. 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 .

Is a theorem of Q. Hence we shall write t- A . to indicate that A . is a theorem. We next turn to the task of proving certain theorems in Q, and proving certain metatheorems and derived rules of inference. It will be clear that the proofs of our derived rules of inference can easily be made effective in the sense of [5]. As a convenience to the reader, after each line in a proof (or condensed proof) we shall indicate by name or number the rule (or rules) of inference used to obtain it, and then, after a colon, the numbers of the theorems or (in the case of a proof from hypotheses) lines of the proof from which the given line was inferred.