T1.5-7 - <?xml version="1.0"...

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A rule of inference consists in premises and a conclusion: Premises : One or more propositions given as True. Conclusion: Proposition whose truth follows logically from the premises. Follow logically: C follows logically from the premises P 1 . . P n when (P 1 /\ . . /\P n )-> C is a tautology. This is written: P 1 /\ . . /\P n C Note: A B means that A B is a tautology. It is the same type of situation as with logical equivalence, where A B means A B is a tautology. E.g. the following is a rule of inference: P 1 = p -> q P 2 = p C = q 1
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p q p->q p/\(p->q) [p/\(p->q)]->q T T T T T T F F F T F T T F T F F T F T 2
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This rule of inference is a form of simple reasoning, e.g., If it rains, the dog gets wet. p: it rains q: the dog gets wet p -> q: If it rains, the dog gets wet. Let: P 1 = p -> q P 2 = p C = q We also write rules of inference like this: p p->q ———— q p and q are propositional variables , so the rule of inference holds for complex propositions of all types as long as the form of the rule is met: E.g., 3
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((s/\r)\/t) -> (r<->q) ((s/\r)\/t) ———————— r<->q A rule of inference may have one premise: p -> q ——— -q -> -p Every logical equivalence is two rules of inference. For example (p -> q) <=> (-p \/ q) is both the rule of inference (p -> q) => (-p \/ q) and (-p \/ q) => (p -> q). 4
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A nonrule of inference or fallacy . p -> q (If it rains the dog gets wet) q (The dog got wet) _________ p (It rained) To prove invalidity (1) construct a truth table (2) find values for p,q that make the premises true and the conclusion false (1) p q p->q q/\(p->q) [q/\(p->q)]->p T T T T T T F F F T F T T T F F F T F T (2) For p is false and q is true the premises are all true but the conclusion is false 5
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[[In Class Exercise]] Verify that the following is a fallacy: p->q -p ——— -q 6
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Some basic rules of inference are on p 66 (p. 58 ed. 5). Each logical equivalence on p 24 provides two rules of inference. The commonly used rules of inference are named. “Modus Ponens” is p->q p ———— q Here is one called “Modus Tollens.” p -> q -q ______ -p One can actually derive Modus Tollens from Modus Ponens and Contrapositive: 1. p -> q Premise 2. -q Premise Conclusion: -p 3. -q -> -p 1, Contrapositive 4. -p 2,3, Modus Ponens 7
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used in automatic logic provers, and is called Resolution: p \/ q –p \/ r ____________ q \/ r Resolution is a cancellation law: the p and –p cancel each other, leaving behind q \/ r. Here is a derivation: 1. p \/ q Premise 2. –p \/ r Premise Conclusion: q \/ r 3. q \/ p 1, commutative law 4. –q -> p 3, implication equivalence 5. –(-p) -> r 2, implication . . 6. p -> r
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This note was uploaded on 02/09/2009 for the course CSC 226 taught by Professor Watkins during the Spring '08 term at N.C. State.

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T1.5-7 - &lt;?xml version=&quot;1.0&quot;...

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