Lecture_22_09 - Lecture 22 Lecture 22 Logical Implication...

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Unformatted text preview: Lecture 22 Lecture 22 Logical Implication Logical Implication Γ╞ ϕ iff there is no pair (M,g), where M is an interpretation and g is a variable assignment, such that (M,g) satisfies Γ but fails to satisfy ϕ. Note: If we restrict our attention to sentences, Note If then of course we can avoid the references to variable assignments. variable Counterexamples Counterexamples Show that the following do not hold: ∀x∃ yR(x,y)╞ ∀w∀x∃ y(R(w,y) ∧R(x,y)) yR(x,y)╞ y(R(w,y) ∃ xH(x) ∧∃ yG(y) ╞ ∃ x ∃ y¬(x ≈ y) xH(x) yG(y) (x ∀x(H(x) → G(x)) ╞ ¬∃ x(¬H(x) ∧G(x)) x(H(x) H(x) ¬∃ Implication Laws for FOL Implication Laws for FOL Substitution of Free Variables by New Constants Γ╞⊥ iff Sv,c Γ╞⊥ iff v,c where c is any individual constant new for Γ. where Implication Laws for FOL Implication Laws for FOL Universal and Existential Quantification (∀,╞) Γ,∀xϕ ╞ ⊥ iff Γ,∀xϕ , Sx,cϕ ╞ ⊥ where c is any individual constant. (∃ ,╞) Γ,∃ xϕ ╞ ⊥ iff Γ,Sx,cϕ ╞ ⊥ where c is any individual constant new for Γ and ∃ xϕ . Implication Laws for FOL Implication Laws for FOL Negated Quantification (¬∀,╞) Γ,¬∀xϕ ╞ ⊥ iff Γ, ¬Sx,cϕ ╞ ⊥ where c is any individual constant new for Γ and ∀xϕ . (¬∃ ,╞) Γ,¬∃ xϕ ╞ ⊥ iff Γ,¬∃ xϕ ,¬Sx,cϕ ╞ ⊥ where c is any individual constant. Examples Examples Provide top­down derivations for the following: ∀vR(v,v)╞ ∃ yR(x,y) ╞ ∃ x∀yR(x,y) ∨∀x∃ y¬R(x,y) yR(x,y) ∃ y∀xR(x,y)╞ ∀x∃ yR(x,y) xR(x,y)╞ ...
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