Truth, Equivalence and Consequence

Truth, Equivalence and Consequence - ⇔ SameSize(b a...

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Truths, Equivalencies, and Consequences Tautological (Taut Con) First-Order (FO Con) Logical (Ana Con) T R U T H Α ¬ Α ¬ Tet(b) Tet(b) A A Cube(c) Cube(c) (Tautology) a = a (Tet(a) a = b) Tet(b) xCube(x) Cube(e) Cube(e) yCube(y) z (Tet(z) ¬ Tet(z)) (FO Validity) ¬ Smaller(c, c) FrontOf(a, b) BackOf(b, a) (Tet(b) Cube(d)) b d xLarge(x) ¬ xSmall(x) (Logical Truth) E Q U I V A L E N C E ¬ (A B) ¬ A ¬ B ¬ (A B) ¬ A ¬ B P ¬¬ P P Q ¬ P Q P Q (P Q) (Q P) P Q (P Q) ( ¬ Q ¬ P) (Tautological Equivalence) a = b b = a ¬ xCube(x) x ¬ Cube(x) ¬ xCube(x) x ¬ Cube(x) xCube(x) ¬ x ¬ Cube(x) xCube(x) ¬ x ¬ Cube(x) (FO Equivalence) SameSize(a, b)
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Unformatted text preview: ⇔ SameSize(b, a) Larger(a, b) ⇔ Smaller(b, a) FrontOf(b, c) ⇔ BackOf(c, b) LeftOf(d, e) ⇔ RightOf(e, d) (Logical Equivalence) C O N S E Q U E N C E Dodec(a) → Small(a) Dodec(a) Small(a) ¬ (Dodec(a) ∨ Small(a)) ¬ Dodec(a) (Tautological Consequence) Cube(f) f = b Cube(b) ∀ x (Cube(x) → Medium(x)) ∀ x Cube(x) ∀ x Medium(x) ∀ x (Cube(x) → Medium(x)) ∃ x ¬ Medium(x) ∃ x ¬ Cube(x) (FO Consequence) SameShape(b, c) Tet(c) Tet(b) ∀ x (Cube(x) → Large(x)) ∀ x Small(x) ¬ ∃ x Cube(x) ∀ x (x = b → Cube(x)) ∃ x Tet(x) ∃ x (x ≠ b) (Logical Consequence)...
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