143 Understanding First-order Logic • ∃x(5< x ∨ x2 + x − =0) ≡ ∃y (5 < y ∨ y + y − = 0), as the formula on the right is the result of renaming of the variable x in the formula on the left • ¬∃xP (x) ≡ ∃x¬P (x) For example, “No student passed the exam” should not be equivalent to “There is a student who did not pass the exam.” • “The integer x is not less than 0” is not logically equivalent to “The integer x is greater than or equal to 0.” Mathematically these mean the same (due of the mathematical property of the ordering of integers called trichotomy, which arranges all integers in a line) but not because of logical reasons Likewise, 2+2=4 is a mathematical but not a logical truth To put it simply: Logic does not know any mathematics It is important to distinguish logical from non-logical truths, and logical from non-logical equivalences 3.4.6 Logical equivalences involving quantifiers Here is a summary of the most important logical properties relating the quantifiers To begin with, the negation swaps the quantifiers as follows: ¬∀xA ≡ ∃x¬A; ¬∃xA ≡ ∀x¬A For example: • “Not every student wrote the test” means the same as “There is a student who did not write the test.” • “There is no natural number less than 0” means “Every natural number is not less than 0.” By negating both sides of the equivalences above we find that each of the universal and existential quantifier is definable in terms of the other: ∀xA ≡ ¬∃x¬A; ∃xA ≡ ¬∀x¬A Universal quantifiers distribute over conjunctions: ∀xP ∧ ∀xQ ≡ ∀x(P ∧ Q) Existential quantifiers distribute over disjunctions: ∃xP ∨ ∃xQ ≡ ∃x(P ∨ Q) Assuming that x does not occur free in Q, the following distributive equivalences hold: (a) (b) (c) (d) ∀xP ∀xP ∃xP ∃xP ∧ Q ≡ Q ∧ ∀xP ∨ Q ≡ Q ∨ ∀xP ∨ Q ≡ Q ∨ ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∀x(P ≡ ∀x(P ≡ ∃x(P ≡ ∃x(P ∧ Q); ∨ Q); ∨ Q); ∧ Q) Two nested quantifiers of the same type commute: ∀x∀yA ≡ ∀y ∀xA; ∃x∃yA ≡ ∃y ∃xA