Using Z by Woodcock, Davies.

By Woodcock, Davies.

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Q [∧−intro] The rest of the proof follows by symmetry: p [1] q [2] [∨−intro1] [∨−intro2] p∨q ¬(p ∨ q) p∨q ¬(p ∨ q) [¬−elim] [¬−elim] false false [1] [2] [¬−intro ] [¬−intro ] ¬p ¬q [∧−intro] ¬p ∧ ¬q Before a natural deduction system becomes really useful, it is necessary to prove results about negation. In the next theorem we use the variant of de Morgan’s law that we have just proved. 11 The Law of the Excluded Middle states that either a proposition is true or it is false. That is, p ∨ ¬p The major connective is a disjunction, so let’s try to introduce it.

Thus, when we match the consequent to our current goal, we still have to find an instantiation of p. 7 Tautologies and contradictions Propositions which evaluate to t in every combination of their propositional variables are known as tautologies: they are always true. If, on the other hand, they evaluate to f in every combination, then they are known as contradictions. Of course, the negation of a contradiction is a tautology, and vice versa. 12 The following propositions are tautologies: p ∨ ¬p p⇒p p ⇒ (q ⇒ p) while the following are contradictions: p ∧ ¬p p ¬p ¬(p ⇒ (q ⇒ p)) 2 / Propositional Logic 24 To prove that a proposition is a tautology, we have only to produce a truth table and check that the major connective takes the value t for each combination of propositional variables.

Q [∧−intro] The rest of the proof follows by symmetry: p [1] q [2] [∨−intro1] [∨−intro2] p∨q ¬(p ∨ q) p∨q ¬(p ∨ q) [¬−elim] [¬−elim] false false [1] [2] [¬−intro ] [¬−intro ] ¬p ¬q [∧−intro] ¬p ∧ ¬q Before a natural deduction system becomes really useful, it is necessary to prove results about negation. In the next theorem we use the variant of de Morgan’s law that we have just proved. 11 The Law of the Excluded Middle states that either a proposition is true or it is false. That is, p ∨ ¬p The major connective is a disjunction, so let’s try to introduce it.

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