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8slides(2ex) - CHAPTER 8 System H2 and Formal Proofs...

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CHAPTER 8 System H 2 and Formal Proofs Hilbert System H 2 The system H 1 is sound and strong enough to prove the Deduction Theorem, but it is not complete. We extend now its set of logical axioms to a complete set of axioms , i.e. we define a system H 2 that is complete with respect to classical semantics. The proof of completeness will be presented in the next chapter. 1
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Definition of the system H 2 . H 2 = ( L {⇒ , ¬} , A 1 , A 2 , A 3 , MP ) A1 ( A ( B A )) , A2 (( A ( B C )) (( A B ) ( A C ))) , A3 (( ¬ B ⇒ ¬ A ) (( ¬ B A ) B ))) MP (Rule of inference) ( MP ) A ; ( A B ) B , A, B, C are any formulas of the propositional language L {⇒ , ¬} . 2
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We write H 2 A to denote that a formula A has a formal proof in H 2 (from the set of logical axioms A 1 , A 2 , A 3), and Γ H 2 A to denote that a formula A has a formal proof in H 2 from a set of formulas Γ (and the set of logical axioms A 1 , A 2 , A 3). 3
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Observe that system H 2 is obtained by adding axiom A 3 to the system H 1 . Hence the Deduction Theorem holds for sys- tem H 2 . Deduction Theorem for H 2 For any subset Γ of the set of formulas F of H 2 and for any formulas A, B ∈ F , Γ , A H 2 B if and only if Γ H 2 ( A B ) . In particular, A H 2 B if and only if H 2 ( A B ) . 4
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Obviously, the axioms A 1 , A 2 , A 3 are tautolo- gies, and the Modus Ponens rule leads from tautologies to tautologies, hence our proof system H 2 is sound i.e. the following the- orem holds. Soundness Theorem for H 2 For every formula A ∈ F , if H 2 A, then | = A. 5
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The soundness theorem proves that the sys- tem ”produces” only tautologies. We show, in the next chapter, that our proof sys- tem H 2 ”produces” not only tautologies, but that all tautologies are provable in it. This is called a completeness theorem for classical logic. Completeness Theorem for H 2 For every A ∈ F , H 2 A, if and only if | = A. 6
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The proof of completeness theorem (for a given semantics) is always a main point in any logic creation. There are many ways (techniques) to prove it, depending on the proof system, and on the semantics we define for it. We present in the next chapter two proofs of the completeness theorem for our system H 2 . The proofs use very different techniques, hence the reason of presenting both of them. In fact the proofs are valid for any proof sys- tem for classical propositional logic in which one can prove all formulas proved in the next section. 7
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FORMAL PROOFS IN H 2 Examples and Exercises We present here some examples of formal proofs in H 2 . There are two reasons for present- ing them. First reason is that all formulas we prove here to be provable play a crucial role in the proof of Completeness Theorem for H 2 , or are needed to find formal proofs of those needed.
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