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# mid1sol - CSE541 Midterm 1 SOLUTIONS Fall 2010 L semantics...

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CSE541 Midterm 1 SOLUTIONS Fall 2010 ˆL semantics for ˆL , , , ∪} is defined as follows ˆL Negation ¬ F T T F ˆL Conjunction F T F F F F F T F T ˆL Disjunction F T F F T T T T T T ˆL-Implication F T F T T T T T T F T QUESTION 1 (1) Use the fact that v : V AR -→ { F, , T } be such that v * (( a b ) ⇒ ¬ b ) = under ˆL semantics to evaluate v * ((( b ⇒ ¬ a ) ( a ⇒ ¬ b )) ( a b )). Use shorthand notation. (1) Solution : (( a b ) ⇒ ¬ b ) = in two cases. C1 ( a b ) = and ¬ b = F . C2 ( a b ) = T and ¬ b = . Case C1: ¬ b = F , i.e. b = T , and hence ( a T ) = iff a = . We get that v is such that v ( a ) = and v ( b ) = T . We evaluate: v * ((( b ⇒ ¬ a ) ( a ⇒ ¬ b )) ( a b )) = ((( T ⇒ ¬ ⊥ ) ( ⊥⇒ ¬ T )) ( ⊥⇒ T )) = (( ⊥⇒⊥ ) T ) = T . Case C2: ¬ b = , i.e. b = , and hence ( a ∩ ⊥ ) = T what is impossible, hence v from case C1 is the only one. (2) Prove that in classical semantics L , ⇒} ≡ L , , ∪} . 1

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We define the EQUIVALENCE of LANGUAGES as follows: Given two languages: L 1 = L CON 1 and L 2 = L CON 2 , for CON 1 6 = CON 2 . We say that they are logically equivalent , i.e. L 1 ≡ L 2 if and only if the following conditions C1, C2 hold. C1: For every formula A of L 1 , there is a formula B of L 2 , such that A B, C2: For every formula C of L 2 , there is a formula D of L 1 , such that C D. (2) Solution: (Classical case) C1 holds because any formula of L , ⇒} is a formula of L , , ∪} . C2 holds due to the following definability of connectives equivalence ( A B ) ( ¬ A B ) . (3) Prove that the equivalence defining in classical logic does not hold under ˆL semantics, but nevertheless L , ⇒} ≡ L , , ∪} . Solution ( A B ) 6≡ ˆL ( ¬ A B ) Take A = B = . We get ⊥ ∪ ⊥ = and ¬ ⊥⇒⊥ = ⊥⇒⊥ = T .
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mid1sol - CSE541 Midterm 1 SOLUTIONS Fall 2010 L semantics...

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