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Inference Rules (tmp)

Module by: Ian Barland

Summary: A version of the inference-rules table that can generate a pdf form

warning:

The information in this module is outdated. Please see my course for a table of contents.
Our propositional inference rules
Abbreviation Name If you know… …then you can infer
∧Intro and-introduction φφ (φ ∧ ψ)(φ∧ψ)
ψψ
∧Elim and-elimination (left) (φ ∧ ψ)(φ∧ψ) φφ
and-elimination (right) (φ ∧ ψ)(φ∧ψ) ψψ
∨Intro or-introduction (left) φφ (φ ∨ ψ)(φ∨ψ)
or-introduction (right) ψψ (φ ∨ ψ)(φ∨ψ)
∨Elim or-elimination φφ ⊢ θθ θθ
ψψ ⊢ θθ
(φ ∨ ψ)(φ∨ψ)
→Intro if-introduction φ, ψ, φ, ψ, …, θ, θ ⊢ ωω ((φ ∧ ψ ∧((φ∧ψ∧ … ∧ θ) → ω)∧θ)→ω)
→Elim if-elimination (modus ponens) (φ → ψ)(φ→ψ) ψψ
φφ
falseIntro false-introduction φφ false
¬ φφ
falseElim false-elimination false φφ
RAA reductio ad absurdum ¬φφ ⊢ false φφ
¬∨Intro negated-or-introduction ¬φφ ¬ (φ ∨ ψ)(φ∨ψ)
¬ψψ
¬∨Elim negated-or-elimination (left) ¬ (φ ∨ ψ)(φ∨ψ) ¬φφ
negated-or-elimination (right) ¬ (φ ∨ ψ)(φ∨ψ) ¬ψψ
¬Intro negation-introduction φφ ¬¬φφ
¬Elim negation-elimination ¬¬φφ φφ
CaseElim case-elimination (left) (φ ∨ ψ)(φ∨ψ) ψψ
¬φφ
case-elimination (right) (φ ∨ ψ)(φ∨ψ) φφ
¬ψψ

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

This is by no means the only possible inference system for propositional logic. We will talk about others in lecture.

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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