17-Derivations

17-Derivations - Discussion#17 Derivations Discussion#17...

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