11 Slides--Proof Methods

11 Slides--Proof Methods - . . .CS103 HO#9 Proof Methods...

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................................................. ...CS103 HO#9 Proof Methods 4/9/10 1 1. How do we prove conditionals? The truth table for p q tells us that the conditional holds iff q can't be false when p is true . So what we can do is this, when we are dealing only with propositions : Premises. .. Suppose p is true ... q p q We consider this to be a subproof , since for these steps we are using an additional assumption. 1. How do we prove conditionals? The truth table for p q tells us that the conditional holds iff q can't be false when p is true . So what we can do is this, when we are dealing only with propositions : Premises. .. p ... q p q Intro or: Conditional Proof The truth table for p q tells us that the conditional holds iff q can't be false when p is true . So what we can do is this, when we are dealing only with propositions : Premises. .. p ... q p q Intro (A B) C A C 1. How do we prove conditionals? Simple Example: The truth table for p q tells us that the conditional holds iff q can't be false when p is true . So what we can do is this, when we are dealing only with propositions : Premises. .. p ... q p q Intro (A B) C A C A C Intro 1. How do we prove conditionals? Simple Example: The truth table for p q tells us that the conditional holds iff q can't be false when p is true . So what we can do is this, when we are dealing only with propositions : Premises. .. p ... q p q Intro (A B) C A A B Addition, 2 C M.P.,1,3 A C Intro, 2-4 1. How do we prove conditionals? Simple Example: 1. 2. 3. 4. 5. x (P(x) Q(x)) z (Q(z) R(z)) x(P(x) R(x)) 2. How do we do conditional proofs with quantifiers? x (P(x) Q(x)) z (Q(z) R(z)) x(P(x) R(x))
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................................................. ...CS103 HO#9 Proof Methods 4/9/10 2 Conditional Proofs with Quantifiers The rule of Universal Generalization tells us we can do this: Premises. .. Suppose c is an arbitrary object ... P(c) x P(x) Suppose c is an arbitrary object P(c) Q(c) x (P(x) Q(x)) Conditional Proofs with Quantifiers Premises. .. Suppose c is an arbitrary object Suppose P(c) is true ... Q(c) P(c) Q(c) x (P(x) Q(x)) General conditional proof Conditional Proofs with Quantifiers Premises. .. x (P(x) Q(x)) z (Q(z) R(z)) x(P(x) R(x)) 1. 2. 3. 4. 5. 6. 7. x (P(x) Q(x)) z (Q(z) R(z)) Suppose a is an arbitrary object P(a) Q(a) Instantiation, 1 Q(a) R(a) Instantiation, 2 P(a) R(a) is transitive (Hyp. Syll.), 4, 5 x(P(x) R(x)) Generalization, 3 - 6 1.
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11 Slides--Proof Methods - . . .CS103 HO#9 Proof Methods...

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