LPL 13.1-13.2 lecture

LPL 13.1-13.2 lecture - x Large(x) and another 1....

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Formal proofs with quantifiers Each of the four rules of inference for the quantifier symbols corresponds to a formal rule with the same name. Universal quantifier rules: Universal Elimination ( " Elim): " x S(x) : S(c) Universal elimination allows us to remove the universal quantifier from any universal statement at any point in the proof and consistently replace the variables with the name of some object in the domain of discourse. " Elim in use: 1. " x (Cube(x) Dodec(x)) 2. " y (Cube(y) Small(y)) 3. " z (Dodec(z) Large(z)) 4. Cube(a) Dodec(a) " Elim: 1 5. Cube(b) Small(b) " Elim: 2 6. Dodec(a) Large(a) " Elim: 3 As with the informal rule, there are two ways to apply the formal rule of universal introduction . General Conditional Proof ( " Intro): c P(c) : Restriction: c must not occur outside the subproof : where it is introduced. Q(c) " x (P(x) Q(x))
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Universal Introduction ( " Intro): c : Restriction: c must not occur outside the subproof : where it is introduced. P(c) " x P(x) " Intro in use: 1. " x (Dodec(x) Large(x)) 2. " x Dodec(x) 3. 4. 5. 6. 7. "
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Unformatted text preview: x Large(x) and another 1. " x (Dodec(x) Large(x)) 2. " x (Large(x) FrontOf(x, f) 3. 4. 5. 6. 7. 8 . " x (Dodec(x) FrontOf(x, f) Exercises: 13.2 and 13.3 Existential quantifier rules: Existential introduction ( $ Intro): S(c) : $ x S(x) Existential introduction allows us to write a general existential claim for any statement that is true of some constant (named object) in our domain of discourse. $ Intro in use: 1. Dodec(a) 2. Cube(b) Dodec(b) 3. Small(c) Large(d) 4. Tet(e) Medium(f) 5. $ x Dodec(x) $ Intro: 1 6. $ y (Cube(y) Dodec(y)) $ Intro: 2 7. $ z (Small(z) Large(d)) $ Intro: 3 8. $ x $ y (Tet(x) Medium(y)) $ Intro: 4 Existential elimination ( $ Elim): $ x S(x) c S(c) : Restriction: c must not occur outside the subproof : where it is introduced. Q Q $ Elim in use: 1. " x (Cube(x) Large(x)) 2. " x (Large(x) LeftOf(x, b) 3. $ x Cube(x) 4. 5. 6. 7. 8. 9. 10. 11. $ x (Large(x) LeftOf(x, b)) Exercises: 13.13 (same as 12.4) and 12.7...
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LPL 13.1-13.2 lecture - x Large(x) and another 1....

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