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A b [a b] .. .. A logical equivalence may be used to justify rewriting even when the proposition involved is only part of the goal or assumption: .. (¬ p ∨ q) ⇒ r [(¬a ∨ b) (p ⇒ q) ⇒ r .. (a ⇒ b)] .. 7 / Tautologies and contradictions 25 Tautologies involving implications also correspond to inference rules: if a ⇒ b is a tautology, then a [a ⇒ b] b may be used as a derived rule. An implication alone is not enough to justify rewriting part of a goal. To see why not, consider the following proposition: (p ∧ q) ⇒ (r ∨ s) The proposition (a ∧ b) ⇒ a is a tautology, but the proof step (p ∧ q) ⇒ (r ∨ s) [(a ∧ b) ⇒ a] p ⇒ (r ∨ s) is invalid.

Again, this is the same as ∀ y : N • y > 5. This predicate is false because not every natural number x is greater than 5: consider 3. The universal quantifier may be thought of as a generalised conjunction: for example, ∀x : N • x > 5 0 > 5 ∧ 1 > 5 ∧ 2 > 5 ∧ 3 > 5 ∧ ... 3 / Predicate Logic 30 The predicate > 5 would have to be true of every natural number; of 0 and of 1 and of 2 and of 3, etcetera. It is not true of 0, for example, and thus the whole quantified expression is false. 1 may be formalised as follows: • Let Student stand for the set of all students, and let Submit (x) mean that x must hand in course work.

A logical equivalence may be used to justify rewriting even when the proposition involved is only part of the goal or assumption: .. (¬ p ∨ q) ⇒ r [(¬a ∨ b) (p ⇒ q) ⇒ r .. (a ⇒ b)] .. 7 / Tautologies and contradictions 25 Tautologies involving implications also correspond to inference rules: if a ⇒ b is a tautology, then a [a ⇒ b] b may be used as a derived rule. An implication alone is not enough to justify rewriting part of a goal. To see why not, consider the following proposition: (p ∧ q) ⇒ (r ∨ s) The proposition (a ∧ b) ⇒ a is a tautology, but the proof step (p ∧ q) ⇒ (r ∨ s) [(a ∧ b) ⇒ a] p ⇒ (r ∨ s) is invalid.