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testing table output

Module by: Sarah Trowbridge

Our first-order inference rules
Abbreviation Name If you know all of… …then you can infer
∀∀Intro ∀∀-introduction
φφ
yy arbitrary.
∀x. φ yx∀x.φ yx
∀∀Elim ∀∀-elimination
∀ x. φ∀x.φ
t t is any term that is free to replace in φφ.
Domain non-empty.
φ x tφx t
∃∃Intro ∃∃-introduction
φφ
t t is any term inφ φ that is free to be replaced.
Domain non-empty.
∃x. φtx∃x.φtx
∃∃Elim ∃∃-elimination
∃ x. φ∃x.φ
c c is a new constant in the proof.
c c does not occur in the proof's conclusion.
φ xcφxc
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
In the following, ψψ stands for any WFF with no free occurrences of xx.
Simplification of Quantifiers, when universe non-empty ∀x . ψ ≡ ψ∀x.ψ≡ψ ∃x. ψ ≡ ψ∃x.ψ≡ψ
Simplification of Quantifiers, when universe empty ∀x . φ ≡ true∀x.φ≡ ∃x. φ ≡ false∃x.φ≡
Distribution of Quantifiers (ψψ w/o xx)
∀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 . φ ∀x.ψ → φ≡ψ → ∀x . φ
∃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. φ ∃x.ψ → φ≡ψ → ∃x. φ

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