exercise5

# exercise5 - ˆL logic i.e for any a,b ∈ F ⊥,T ¬ ⊥ =...

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CSE541 EXERCISE 5 SOLVE ALL PROBLEMS as PRACTICE and only AFTER look at the SOLUTIONS!! QUESTION 1 Given a proof system: S = ( L , ⇒} , E = F AX = { ( A A ) , ( A ( ¬ A B )) } , ( r ) ( A B ) ( B ( A B )) ) . Deﬁnition: System S is sound if and only if (i) Axioms are tautologies and (ii) rules of inference are sound, i.e lead from all true premisses to a true conclusion. 1. Prove that S is sound under classical semantics. 2. Prove that S is not sound under K semantics deﬁned as follows. The language is the same in case of classical logic. Connectives ¬ , , of K are deﬁned as in
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Unformatted text preview: ˆL logic, i.e. for any a,b ∈ { F, ⊥ ,T } , ¬ ⊥ = ⊥ , ¬ F = T, ¬ T = F, a ∪ b = max { a,b } , a ∩ b = min { a,b } . Implication in Kleene’s logic is deﬁned as follows. For any a,b ∈ { F, ⊥ ,T } , a ⇒ b = ¬ a ∪ b. QUESTION 2 Write a formal proof in S deﬁned in Question 1 with 2 applications of the rule ( r ). QUESTION 3 Prove, by constructing a formal proof that ‘ S (( ¬ A ⇒ B ) ⇒ ( A ⇒ ( ¬ A ⇒ B ))) . 1...
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