henkin_a

# henkin_a - Part(a of Hunter’s Proof of Henkin’s...

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Unformatted text preview: Part (a) of Hunter’s Proof of Henkin’s Completeness Theorem for PS Branden Fitelson 02/21/07 The first part of Henkin’s completeness proof involves proving the following seven theorem schemas . 1. ‘ PS A ⊃ A 2. ‘ PS A ⊃ (B ⊃ A) 3. ‘ PS (A ⊃ (B ⊃ C)) ⊃ ((A ⊃ B) ⊃ (A ⊃ C)) 4. ‘ PS ∼ A ⊃ (A ⊃ B) 5. ‘ PS A ⊃ ( ∼ B ⊃∼ (A ⊃ B)) 6. ‘ PS (A ⊃ B) ⊃ ((A ⊃∼ B) ⊃∼ A) 7. ‘ PS ( ∼ A ⊃ B) ⊃ (( ∼ A ⊃∼ B) ⊃ A) We have already seen a proof of (1). (2) and (3) are (PS1) and (PS2). So, we just need proofs of the four theorem schemas (4)–(7). Here is a sketch (!) of one such proof (4 goals in boldface ). Exercise: figure out the substitution instances of the formulas (listed on the right) required to generate each MP step. PS1 A ⊃ (B ⊃ A) PS2 A ⊃ (B ⊃ C)) ⊃ ((A ⊃ B) ⊃ (A ⊃ C)) PS3 ( ∼ A ⊃∼ B) ⊃ (B ⊃ A) 1. A ⊃ (B ⊃ (C ⊃ B)) [MP, PS1, PS1] 2. ((A ⊃ (B ⊃ C)) ⊃ (A ⊃ B)) ⊃ ((A ⊃ (B ⊃ C))...
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