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Unformatted text preview: > 0 connectives/quantifiers, and for every closed wf W shorter than A,
T W iff I W.
Case 1: A has form (∼B), for B closed and shorter than A.
1. "⇒". Supppose T A. (i.e., T ∼B)
Then: T B. (T is consistent.)
Hence: I B. (Inductive Hypothesis.)
So:
I ∼B. (Cor. 3.34, B is closed.) Thus I A.
2. "⇐". Suppose I A. (i.e., I ∼B)
Then: I B. (Cor. 3.34, B is closed.)
So:
T B. (Inductive Hypothesis.)
So:
T ∼B. (T is complete.) Thus T A.
Case 2: A has form (B → C), for B, C closed and shorter than A.
1. "⇒". Suppose I A.
Then: I B and I ∼C.
So:
T B and T C. (Inductive Hypothesis.)
So:
T B and T ∼C. (T is complete.)
Note: T (B → (∼C → ∼(B → C))). (Tautology of L, hence L. Thus theorem of T.)
So:
T ∼(B → C). So T ∼A.
Thus: T A. (T is consistent.)
2. "⇐". Suppose T A.
Then: T ∼A. (T is complete.) Or T ∼(B → C).
Note: T ∼(B → C) → B and T ∼(B → C) → ∼C. (Tautologies of L, hence theorems of T.)
So:
T B an...
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This document was uploaded on 03/25/2014 for the course PL 3014 at NYU Poly.
 Spring '06
 JonathanBain

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