Rules for Quantative Logic
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Linguistics 501 Fall ’05 Laws and Rules for Predicate Logic (1) Laws of Quantifier Distribution Law 1 ¬(∀x)ϕ(x) ⇐⇒ (∃x)¬ϕ(x) Law 2 (∀x)(ϕ(x) ∧ ψ(x)) ⇐⇒ ((∀x)ϕ(x) ∧ (∀x)ψ(x)) Law 3 (∃x)(ϕ(x) ∨ ψ(x)) ⇐⇒ ((∃x)ϕ(x) ∨ (∃x)ψ(x)) Law 4 ((∀x)ϕ(x) ∨ (∀x)ψ(x)) = ⇒ (∀x)(ϕ(x) ∨ ψ(x)) Law 5 (∃x)(ϕ(x) ∧ ψ(x)) = ⇒ ((∃x)ϕ(x) ∧ (∃x)ψ(x)) (2) Laws of Quantifier (In)Dependence Law 6 (∀x)(∀y)ϕ(x, y) ⇐⇒ (∀y)(∀x)ϕ(x, y) Law 7 (∃x)(∃y)ϕ(x, y) ⇐⇒ (∃y)(∃x)ϕ(x, y) Law 8 (∃x)(∀y)ϕ(x, y)= ⇒ (∀y)(∃x)ϕ(x, y) (3) Laws of Quantifier Movement Law 9 (ϕ → (∀x)ψ(x)) ⇐⇒ (∀x)(ϕ → ψ(x)) Provided that x is not free in ϕ. Law 10 (ϕ → (∃x)ψ(x)) ⇐⇒ (∃x)(ϕ → ψ(x)) Provided that x is not free in ϕ. Law 11 ((∀x)ϕ(x) → ψ) ⇐⇒ (∃x)(ϕ(x) → ψ) Provided that x is not free in ψ. Law 12 ((∃x)ϕ(x) → ψ) ⇐⇒ (∀x)(ϕ(x) → ψ) Provided that x is not free in ψ. (4) Universal Instantiation (U.I.) (∀x)ϕ(x) ∴ ϕ(c) (5) Universal Generalization (U.G.) ϕ(v) ∴ (∀x)ϕ(x) (6) Existential Generalization (E.G.) ϕ(c) ∴ (∃x)ϕ(x) (7) Existential Instantiation (E.I.) (∃x)ϕ(x) ∴ ϕ(w) where w is a new constant
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Rules for logic
Transcript of Rules for Quantative Logic
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Linguistics 501Fall 05
Laws and Rules for Predicate Logic
(1) Laws of Quantifier DistributionLaw 1 (x)(x) (x)(x)Law 2 (x)((x) (x)) ((x)(x) (x)(x))Law 3 (x)((x) (x)) ((x)(x) (x)(x))Law 4 ((x)(x) (x)(x)) = (x)((x) (x))Law 5 (x)((x) (x)) = ((x)(x) (x)(x))
(2) Laws of Quantifier (In)DependenceLaw 6 (x)(y)(x, y) (y)(x)(x, y)Law 7 (x)(y)(x, y) (y)(x)(x, y)Law 8 (x)(y)(x, y) = (y)(x)(x, y)
(3) Laws of Quantifier MovementLaw 9 ( (x)(x)) (x)( (x))
Provided that x is not free in .Law 10 ( (x)(x)) (x)( (x))
Provided that x is not free in .Law 11 ((x)(x) ) (x)((x) )
Provided that x is not free in .Law 12 ((x)(x) ) (x)((x) )
Provided that x is not free in .
(4) Universal Instantiation (U.I.)(x)(x)
(c)
(5) Universal Generalization (U.G.)(v)
(x)(x)
(6) Existential Generalization (E.G.)(c)
(x)(x)
(7) Existential Instantiation (E.I.)(x)(x)
(w)where w is a new constant
