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Unformatted text preview: 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 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 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 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....
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This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Bachmair,L
 Computer Science

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