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• #### First-Order Logic

• ##### Reasoning with inference rules
• Exercises for First-Order Logic

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# propositional inference rules

Summary: A set of inference rules for propositional logic

Table 1: Our propositional inference rules
Abbreviation Name If you know all of… …then you can infer
Intro and-introduction
 φφ ψψ
φψ φ ψ
Elim and-elimination (left) φψ φ ψ φφ
and-elimination (right) φψ φ ψ ψψ
Intro or-introduction (left) φφ φψ φ ψ
or-introduction (right) ψψ φψ φ ψ
Elim or-elimination
 φ ⊢ θφ ⊢ θ ψ ⊢ θψ ⊢ θ φ∨ψ φ ψ
θθ
Intro if-introduction φ, ψ, , θ ωφ, ψ, , θ ω φψθω φ ψ θ ω
Elim if-elimination (modus ponens)
 φ⇒ψ φ ψ φφ
ψψ
falseIntro false-introduction
 φφ ¬φ φ
false
falseElim false-elimination false φφ
RAA reductio ad absurdum (v. 1) ¬φ falseφ φφ
reductio ad absurdum (v. 2) φ falseφ ¬φφ
¬¬Intro negation-introduction φφ ¬¬φφ
¬¬Elim negation-elimination ¬¬φφ φφ
CaseElim case-elimination (left)
 φ∨ψφ ψ ¬φφ
ψψ
case-elimination (right)
 φ∨ψφ ψ ¬ψψ
φφ

As usual, φφ, ψψ, θθ, ωω are meta-variables standing for any WFF.

This is by no means the only possible inference system for propositional logic.

## Aside:

This set of inference rules is based upon Discrete Mathematics with a Computer by Hall and O'Donnell (Springer, 2000) and The Beseme Project.

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