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05a-Rules-of-Inference

# 05a-Rules-of-Inference - Rules of Introduction Inference...

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Rules of Inference Discrete Mathematics

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Discrete Mathematics – Rules of Inference 5-2 Previous Lecture Logically equivalent statements Statements Φ and Ψ are equivalent iff Φ↔Ψ is a tautology Main logic equivalences double negation DeMorgan’s laws commutative, associative, and distributive laws idempotent, identity, and domination laws the law of contradiction and the law of excluded middle absorption laws
Discrete Mathematics – Rules of Inference 5-3 First Law of Substitution Suppose that the compound statement Φ is a tautology. If p is a primitive statement that appears in Φ and we replace each occurrence of p by the same statement q, then the resulting compound statement Ψ is also a tautology. Let Φ = (p q) (q p), and we substitute p by p (s r) Therefore ((p (s r)) q) (q (p (s r)) is a tautology

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Discrete Mathematics – Rules of Inference 5-4 Second Law of Substitution Let Φ be a compound statement, p an arbitrary (not necessarily primitive!) statement that appears in Φ , and let q be a statement such that p q. If we replace one or more occurrences of p by q, then for the resulting compound statement Ψ we have Φ Ψ . Let Φ = (p q) (q p), and we substitute the first occurrence of p by p (p q). Recall that p p (p q) by Absorption Law. Therefore (p q) (q p) ((p (p q)) q) (q p).
Discrete Mathematics – Rules of Inference 5-5 Logic Inference One of the main goals of logic is to distinguish valid and invalid arguments What can we say about the following arguments: ``If you have a current password, then you can log onto the network.

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05a-Rules-of-Inference - Rules of Introduction Inference...

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