List of rules of inference

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This is a list of rules of inference, logical laws that relate to mathematical formulae.

Contents

[edit] Introduction

Rules of inference are syntactical transformation rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.

Discharge rules permit inference from a subderivation based on a temporary assumption. Below, the notation

\varphi \vdash \psi\,\!

indicates such a subderivation from the temporary assumption \varphi\,\! to \psi\,\!.

[edit] Rules for classical sentential calculus

[edit] Rules for negations

Reductio ad absurdum (or Negation Introduction)
\varphi \vdash \psi\,\!
\underline{\varphi \vdash \lnot \psi}\,\!
\lnot \varphi\,\!
Reductio ad absurdum (related to the law of excluded middle)
\lnot \varphi \vdash \psi\,\!
\underline{\lnot \varphi \vdash \lnot \psi}\,\!
\varphi\,\!
Noncontradiction (or Negation Elimination)
\varphi\,\!
\underline{\lnot \varphi}\,\!
\psi\,\!
Double negation elimination
\underline{\lnot \lnot \varphi}\,\!
\varphi\,\!
Double negation introduction
\underline{\varphi \quad \quad}\,\!
\lnot \lnot \varphi\,\!

[edit] Rules for conditionals

Deduction theorem (or Conditional Introduction)
\underline{\varphi \vdash \psi}\,\!
\varphi \rightarrow \psi\,\!
Modus ponens (or Conditional Elimination)
\varphi \rightarrow \psi\,\!
\underline{\varphi \quad \quad \quad}\,\!
\psi\,\!
Modus tollens
\varphi \rightarrow \psi\,\!
\underline{\lnot \psi \quad \quad \quad}\,\!
\lnot \varphi\,\!

[edit] Rules for conjunctions

Adjunction (or Conjunction Introduction)
\varphi\,\!
\underline{\psi \quad \quad \ \ }\,\!
\varphi \land \psi\,\!
Simplification (or Conjunction Elimination)
\underline{\varphi \land \psi}\,\!
\varphi\,\!
\underline{\varphi \land \psi}\,\!
\psi\,\!

[edit] Rules for disjunctions

Addition (or Disjunction Introduction)
\underline{\varphi \quad \quad \ \ }\,\!
\varphi \lor \psi\,\!
\underline{\psi \quad \quad \ \ }\,\!
\varphi \lor \psi\,\!
Separation of Cases (or Disjunction Elimination)
\varphi \lor \psi\,\!
\varphi \rightarrow \chi\,\!
\underline{\psi \rightarrow \chi}\,\!
\chi\,\!
Disjunctive syllogism
\varphi \lor \psi\,\!
\underline{\lnot \varphi \quad \quad}\,\!
\psi\,\!
\varphi \lor \psi\,\!
\underline{\lnot \psi \quad \quad}\,\!
\varphi\,\!

[edit] Rules for biconditionals

Biconditional introduction
\varphi \rightarrow \psi\,\!
\underline{\psi \rightarrow \varphi}\,\!
\varphi \leftrightarrow \psi\,\!
Biconditional Elimination
\varphi \leftrightarrow \psi\,\!
\underline{\varphi \quad \quad}\,\!
\psi\,\!
\varphi \leftrightarrow \psi\,\!
\underline{\psi \quad \quad}\,\!
\varphi\,\!

[edit] Rules of classical predicate calculus

In the following rules, \varphi(\beta / \alpha)\,\! is exactly like \varphi\,\! except for having the term \beta\,\! everywhere \varphi\,\! has the free variable \alpha\,\!.

Universal Introduction (or Universal Generalization)
\underline{\varphi\quad\;\;}\,\!
\forall \alpha\, \varphi\,\!

Restriction: None.

Universal Elimination (or Universal Instantiation)
\forall \alpha\, \varphi\!
\overline{\varphi{(\beta / \alpha)}}\!

Restriction: No free occurrence of \alpha\,\! in \varphi\,\! falls within the scope of a quantifier quantifying a variable occurring in \beta\,\!.

Existential Introduction (or Existential Generalization)
\underline{\varphi(\beta / \alpha)}\,\!
\exists \alpha\, \varphi\,\!

Restriction: No free occurrence of \alpha\,\! in \varphi\,\! falls within the scope of a quantifier quantifying a variable occurring in \beta\,\!.

Existential Elimination (or Existential Instantiation)
\exists \alpha\, \varphi\,\!
\underline{\varphi \vdash \psi}\,\!
\psi\,\!

Restriction: There is no free occurrence of \alpha\,\! in \psi\,\!.

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