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first-order equivalances (tmp)

Module by: Ian Barland

Summary: the version whose pdf table looks acceptable.

warning:

The information in this module is outdated. Please see my course for a table of contents.
The following equivalences are in addition to those of propositional logic (.ps).

First-order Logic Equivalences
Complementation of Quantifiers ∀x.x. ¬φ ≡φ≡ ¬ ∃x. φx.φ ∃x.x. ¬φ ≡φ≡ ¬ ∀x. φx.φ
Interchanging Quantifiers ∀x.x. ∀y. φ ≡y.φ≡ ∀y.y. ∀x. φx.φ ∃x.x. ∃y. φ ≡y.φ≡ ∃y.y. ∃x. φx.φ
Distribution of Quantifiers ∀x. (φ ∧ ψ) ≡ (x.(φ∧ψ)≡(∀x. φ ∧x.φ∧ ∀x. ψ )x.ψ) ∃x. (φ ∨ ψ) ≡ (x.(φ∨ψ)≡(∃x. φ ∨x.φ∨ ∃x. ψ )x.ψ)
The following identities each assume that ψψ does not have any free occurrences of variable xx.
Simplification of Quantifiers ∀x. ψ ≡ ψx.ψ≡ψ ∃x. ψ ≡ ψx.ψ≡ψ
Distribution of Quantifiers ∀x. (φ ∧ ψ) ≡ (x.(φ∧ψ)≡(∀x. φ ∧ ψ)x.φ∧ψ) ∃x. (φ ∧ ψ) ≡ (x.(φ∧ψ)≡(∃x. φ ∧ ψ)x.φ∧ψ)
  ∀x. (φ ∨ ψ) ≡ (x.(φ∨ψ)≡(∀x. φ ∨ ψ)x.φ∨ψ) ∃x. (φ ∨ ψ) ≡ (x.(φ∨ψ)≡(∃x. φ ∨ ψ)x.φ∨ψ)
  ∀x. (φ → ψ) ≡ (x.(φ→ψ)≡(∃x. φ → ψ)x.φ→ψ) ∃x. (φ → ψ) ≡ (x.(φ→ψ)≡(∀x. φ → ψ)x.φ→ψ)
  ∀x. (ψ → φ) ≡ (ψ →x.(ψ→φ)≡(ψ→ ∀x. φ )x.φ) ∃x. (ψ → φ) ≡ (ψ →x.(ψ→φ)≡(ψ→ ∃x. φ )x.φ)

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