Lecture 4 Notes

Inductive hypothesis so i b cor 334 b is closed

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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.

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