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# testing table output

Module by: Sarah Trowbridge. E-mail the author

Table 1: Our first-order inference rules
Abbreviation Name If you know all of… …then you can infer
Intro -introduction
 φφ yy arbitrary.
x. φ yxx.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. φtxx.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
Table 2: 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. φx.y.φy.x.φ x. y. φ y. x. φx.y.φy.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 ψx . 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 ψx.ψψ x. ψ ψx.ψψ
Simplification of Quantifiers, when universe empty x . φ truex.φ x. φ falsex.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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