17-Derivations

17-Derivations - Discussion#17 Chapter 2 Section 2.3 1/15...

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Unformatted text preview: Discussion #17 Chapter 2, Section 2.3 1/15 Discussion #17 Derivations Discussion #17 Chapter 2, Section 2.3 2/15 Topics • Derivations proofs in predicate calculus • Inference Rules with Quantifiers – Laws – Universal/Existential Instantiation – Universal/Existential Generalization • Unification Discussion #17 Chapter 2, Section 2.3 3/15 Laws as Inference Rules • Equivalence laws can be used as inference rules. • Show: if 5 xP(x) ⇒ Q then 2200 x(P(x) ⇒ Q) ∨ R. 1. 5 xP(x) ⇒ Q premise 2. ¬5 xP(x) ∨ Q implication law 3. 2200 x ¬ P(x) ∨ Q de Morgan’s law 4. 2200 x( ¬ P(x) ∨ Q) distributive law 5. 2200 x(P(x) ⇒ Q) implication law 6. 2200 x(P(x) ⇒ Q) ∨ R law of addition, A |= A ∨ B Note: the propositional laws of inference hold, as well (e.g. Step 6). Discussion #17 Chapter 2, Section 2.3 4/15 Universal Instantiation (UI) When A is true for every instantiation, it is certainly true for some particular instantiation. 2200 xA S x t A Example: from 2200 x Mortal(x), we can derive Mortal(Smith). Discussion #17 Chapter 2, Section 2.3 5/15 UI Example Given the following two premises: 1. All intelligent students succeed 2. John is an intelligent student Prove that John succeeds. Proof: 1. 2200 x(Intelligent(x) ⇒ Succeed(x)) premise 2. Intelligent(John) premise 3. Intelligent(John) ⇒ Succeed(John) 1, UI 4. Succeed(John) 2&3, modus ponens Discussion #17 Chapter 2, Section 2.3 6/15 When A is true for one or more instantiations, we can let a variable, say b, designate any one of the true instantiations....
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This note was uploaded on 03/02/2012 for the course C S 236 taught by Professor Michaelgoodrich during the Winter '12 term at BYU.

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17-Derivations - Discussion#17 Chapter 2 Section 2.3 1/15...

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