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exercise6sol

# exercise6sol - CSE451 EXERCISE 6 SOLUTIONS QUESTION 1 Given...

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CSE451 EXERCISE 6 SOLUTIONS QUESTION 1 Given a proof system: S = ( L {∪ , ⇒} , E = F AX = { A 1 ,A 2 } , R = { ( r ) } ) , where A 1 = ( A ( A B )) , A 2 = ( A ( B A )) and ( r ) ( A B ) ( A ( A B )) 1. Solution: Prove that S is sound under classical semantics. Solution: Axioms of S are basic classical tautologies. The proof of soundness of the rule of inference is the following. Assume ( A B ) = T . Hence the logical value of conclusion is ( A ( A B )) = ( A T ) = T for all A . 2. Determine whether S is sound or not sound under K semantics. K semantics diﬀer from ˆLukasiewicz’s semantics only in a case on implication only. This table is: K-Implication F T F T T T ⊥ ⊥ T T F T Solution 1: S is not sound under K semantics. Let’s take truth assignment such that A = ,B = . The logical value of axiom A1 is as follows. ( A ( A B )) = ( ⊥⇒ ( ⊥ ∪ ⊥ )) = and 6 | = K ( A ( A B )). Observe

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exercise6sol - CSE451 EXERCISE 6 SOLUTIONS QUESTION 1 Given...

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