10slides(2) - Chapter 10 Introduction to Intuitionistic...

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Chapter 10: Introduction to Intuitionistic Logic PART 2: Hilbert Proof System for proposi- tional intuitionistic logic. Language is a propositional language L = L {∪ , , , ¬} with the set of formulas denoted by F . Axioms A1 (( A B ) (( B C ) ( A C ))), A2 ( A ( A B )), A3 ( B ( A B )), 1
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A4 (( A C ) (( B C ) (( A B ) C ))), A5 (( A B ) A ), A6 (( A B ) B ), A7 (( C A ) (( C B ) ( C ( A B ))), A8 (( A ( B C )) (( A B ) C )), A9 ((( A B ) C ) ( A ( B C )), A10 ( A ∩ ¬ A ) B ), A11 (( A ( A ∩ ¬ A )) ⇒ ¬ A ), where A, B, C are any formulas in L . 2
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Rules of inference: we adopt a Modus Po- nens rule ( MP ) A ; ( A B ) B as the only inference rule. A proof system I I = ( L , F , A1 - A11 , ( MP ) ) , for L , A1 - A11 defined above, is called Hilbert Style Formalization for Intuitionis- tic Propositional Logic. This set of axioms is due to Rasiowa (1959). It differs from Heyting original set of ax- ioms but they are equivalent. 3
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We introduce, as usual, the notion of a for- mal proof in I and denote by I A the fact that A has a formal proof in I , or that that A is intuitionistically provable . We write | = I A to denote that the formula A is intuition- istic tautology. Completeness Theorem for I For any formula A ∈ F , I A ı and only if | = I A.
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