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

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Unformatted text preview: CSE541 Midterm 1 SOLUTIONS Fall 2010 L semantics for L { , , , } is defined as follows L Negation F T T F L Conjunction F T F F F F F T F T L Disjunction F T F F T T T T T T L-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 L 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 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 L semantics, but nevertheless L { , } L { , , } ....
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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.

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

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