8dedthmproof - Chapter 8 Hilbert Proof Systems and...

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Chapter 8: Hilbert Proof Systems and Deduction Theorem PROOF OF THE DEDUCTION THEOREM 1
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Hilbert System H 1 : H 1 = ( L {⇒} , F { A 1 ,A 2 } MP ) A1 ( A ( B A )) , A2 (( A ( B C )) (( A B ) ( A C ))) , MP ( MP ) A ; ( A B ) B , 2
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DEDUCTION THEOREM (Herbrand, 1930) For any formulas A,B , if A B, then ( A B ) . We are going to prove now that for our system H 1 is strong enough to prove the Deduc- tion Theorem for it. In fact we prove a more general version of Herbrand theorem. To formulate it we introduce the following notation. We write Γ ,A B for Γ ∪ { A } ‘ B 3
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In general we write Γ ,A 1 ,A 2 ,...,A n B for Γ ∪ { A 1 ,A 2 ,...,A n } ‘ B. Deduction Theorem for H 1 For any subset Γ of the set of formulas F of H 1 and for any formulas A,B ∈ F , Γ , A H 1 B if and only if Γ H 1 ( A B ) . In particular, A H 1 B if and only if H 1 ( A B ) . 4
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We use in the proof the symbol instead of H 1 . Part 1. We first prove: If Γ , A B then Γ ( A B ) . Assume that Γ , A B, i.e. that we have a formal proof B 1 ,B 2 ,...,B n of B from the set of formulas Γ ∪ { A } , we have to show that Γ ( A B ) . 5
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8dedthmproof - Chapter 8 Hilbert Proof Systems and...

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