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Unformatted text preview: LPL 3.4 EXTRA CREDIT RIK SENGUPTA NOTE: Throughout this proof we define “A is equivalent to B” as “the truth value of A is the truth value of B”. Let P be a true sentence and let Q be formed by putting some number of negation symbols in front of P. Case 1: Q is formed by putting an even number 2n of negation symbols in front of P. We want to prove that Q is true. Let R(n) be the statement “If Q is formed by putting 2n (where n is an integer) negation symbols in front of a true statement P, then Q is true”. Obviously, R(0) is true, as then Q ≡ P, and P is true, so obviously Q is also true (from Identity Elimination). Also, R(1) is true, because “P is true” implies “the negation of P is false” implies “the negation of the negation of P is true”, because the negation of the negation of P is equivalent to P itself. Let R(i) be true for some integer i. We want to show that R(i + 1) is also true. But, proving R(i + 1) is true is equivalent to proving that the negation of the negation of the Q that appeared in R(i) is true (because...
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- Logic, Rik Sengupta