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first-order equivalences

Module by: John Greiner, Phokion Kolaitis, Moshe Vardi, Matthias Felleisen. E-mail the authors

Summary: Some equivalences for manipulation of first-order formulas.

The following equivalences are in addition to those of propositional logic. In these, φφ and ψψ each stand for any WFF, but θθ stands for any WFF with no free occurrences of xx .

Table 1: First-order Logic Equivalences
Equivalence ∀ Variant ∃ Variant
Complementation of Quantifiers ∀x:¬φ≡¬∃x:φ x φ x φ ∃x:¬φ≡¬∀x:φ x φ x φ
Interchanging Quantifiers ∀x:∀y:φ≡∀y:∀x:φ x y φ y x φ ∃x:∃y:φ≡∃y:∃x:φ x y φ y x φ
Distribution of Quantifiers ∀x:φ∧ψ≡∀x:φ∧∀x:ψ x φ ψ x φ x ψ ∃x:φ∨ψ≡∃x:φ∨∃x:ψ x φ ψ x φ x ψ
∀x:φ∨θ≡∀x:φ∨θ x φ θ x φ θ ∃x:φ∧θ≡∃x:φ∧θ x φ θ x φ θ
∀x:φ⇒θ≡∃x:φ⇒θ x φ θ x φ θ
∀x:θ⇒φ≡θ⇒∀x:φ x θ φ θ x φ
Distribution of Quantifiers —— with non-empty domain ∀x:φ∧θ≡∀x:φ∧θ x φ θ x φ θ ∃x:φ∨θ≡∃x:φ∨θ x φ θ x φ θ
∃x:φ⇒θ≡∀x:φ⇒θ x φ θ x φ θ
∃x:θ⇒φ≡θ⇒∃x:φ x θ φ θ x φ
Renaming ∀x:φ≡∀y:φ⁡ x↦y ↦ x φ y ⁡ φ ↦ x y ∃x:φ≡∃y:φ⁡ x↦y ↦ x φ y ⁡ φ ↦ x y
Simplification of Quantifiers —— with non-empty domain ∀x:θ≡θ x θ θ ∃x:θ≡θ x θ θ
Simplification of Quantifiers —— with empty domain ∀x:φ≡true x φ ∃x:φ≡false x φ

When citing Distribution of Quantifiers, say what you're distributing over what: e.g., “ distribute ∀ over ∨ (with θθ being xx-free) ”.

In renaming, the notation φ⁡ x↦y ↦ ⁡ φ ↦ x y means “ φφ with each free occurrence of xx replaced by yy ”. It is a meta-formula; when writing any particular formula you don't write any brackets, and instead just do the replacement.

This set of equivalences isn't actually quite complete. For instance, ∃x:∀y:R⁢xy⇒∀y:∃x:R⁢xy x y Rxy y x Rxy is equivalent to true, but we can't show it using only the rules above. It does become complete1 if we add some analogs of the first-order inference rules, replacing ⊢ with ⇒ (and carrying along their baggage of “arbitrary” and “free-to-substitute-in”).

Footnotes

  1. It's not obvious when this system is complete; that's Gödel's completeness theorem, his 1929 Ph.D. thesis. Don't confuse it with his more celebrated Incompleteness Theorem, on the other hand, which talks about the ability to prove formulas which are true in all interpretations which include arithmetic (as opposed to all interpretations everywhere.)

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