Inference Rules: Tautologies •
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Transcript Inference Rules: Tautologies •
Inference Rules: Tautologies
•(p (p q)) q.
•(q (p q)) p.
•p (p q).
•(p q) p.
•((p) (q)) p q.
•((p q) q) p.
Modus Ponens
Modus Tollens
Addition
Simplification
Conjunction
Disjunctive Syllogism
•((p q) (q r)) (p r) Hypothetical Syllogism
• (aka, the transitive rule)
•((p q ) ( p r)) (q r)
Resolution
•((p q ) (p r) (q r)) r Cases
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Fallacies
• (q (p q)) p.
•
Fallacy of Affirming the Conclusion
• (p (p q)) q.
•
Fallacy of Denying the Hypothesis
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Inference Rules: Quantified Statements
x P(x)
P(c)
Universal Instantiation
P(c)___
x P(x)
Universal Generalization
(for an individual object c from UoD)
(for any arbitrary element c from UoD)
x P(x)
P(c)
Existential Instantiation
P(c)__
x P(x)
Existential Generalization
(for some specific object c from UoD)
(for some object c from UoD)
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Proof Methods
• Trivial Proof of p q given q (see truth table).
• Vacuous Proof of p q given p (see truth table).
• Direct Proof of p q given p, show q.
• Indirect Proof of p q (proof by contraposition, so
• show q p starting with q).
• Proof of p q by Contradiction: recall that p q Ξ
• p q. Assume the negation, p q, is true and
• show a contradiction.
• Proof of p1 p2 p3 … pn q by Cases: show that
• pi q is true for all i, where 1 ≤ i ≤ n.
• Proof of x P(x) by example: just find an instance.
• Proof that x P(x) is false by counterexample: just
• find an instance c where P(c) is false.
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