List of rules of inference

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List of rules of inference

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

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 $psi,!$.

=Rules for classical sentential calculus=

Rules for negations

;Reductio ad absurdum (or "Negation Introduction"):: $varphi vdash psi,!$: $underline\left\{varphi vdash lnot psi\right\},!$: $lnot varphi,!$

;Reductio ad absurdum (related to the law of excluded middle):: $lnot varphi vdash psi,!$: $underline\left\{lnot varphi vdash lnot psi\right\},!$: $varphi,!$

;Noncontradiction (or "Negation Elimination"):: $varphi,!$: $underline\left\{lnot varphi\right\},!$: $psi,!$

;Double negation elimination:: $underline\left\{lnot lnot varphi\right\},!$: $varphi,!$

;Double negation introduction:: $underline\left\{varphi quad quad\right\},!$: $lnot lnot varphi,!$

Rules for conditionals

;Deduction theorem (or "Conditional Introduction"):: $underline\left\{varphi vdash psi\right\},!$: $varphi ightarrow psi,!$

;Modus ponens (or "Conditional Elimination"):: $varphi ightarrow psi,!$: $underline\left\{varphi quad quad quad\right\},!$: $psi,!$

;Modus tollens:: $varphi ightarrow psi,!$: $underline\left\{lnot psi quad quad quad\right\},!$: $lnot varphi,!$

Rules for conjunctions

;Adjunction (or "Conjunction Introduction"):: $varphi,!$: $underline\left\{psi quad quad \right\},!$: $varphi land psi,!$

;Simplification (or "Conjunction Elimination"):: $underline\left\{varphi land psi\right\},!$ : $varphi,!$

: $underline\left\{varphi land psi\right\},!$ : $psi,!$

Rules for disjunctions

;Addition (or "Disjunction Introduction"):: $underline\left\{varphi quad quad \right\},!$: $varphi lor psi,!$

: $underline\left\{psi quad quad \right\},!$: $varphi lor psi,!$

;Separation of Cases (or "Disjunction Elimination"): $varphi lor psi,!$: $varphi ightarrow chi,!$: $underline\left\{psi ightarrow chi\right\},!$: $chi,!$

;Disjunctive syllogism:: $varphi lor psi,!$: $underline\left\{lnot varphi quad quad\right\},!$: $psi,!$

: $varphi lor psi,!$: $underline\left\{lnot psi quad quad\right\},!$: $varphi,!$

Rules for biconditionals

;Biconditional introduction:: $varphi ightarrow psi,!$: $underline\left\{psi ightarrow varphi\right\},!$: $varphi leftrightarrow psi,!$

;Biconditional Elimination:: $varphi leftrightarrow psi,!$: $underline\left\{varphi quad quad\right\},!$: $psi,!$

: $varphi leftrightarrow psi,!$: $underline\left\{psi quad quad\right\},!$: $varphi,!$

=Rules of classical predicate calculus=

In the following rules, is exactly like $varphi,!$ except for having the term everywhere $varphi,!$ has the free variable $alpha,!$.

;Universal Introduction (or "Universal Generalization"):: $underline\left\{varphiquad;;\right\},!$: $forall alpha, varphi,!$

Restriction: None.

;Universal Elimination (or "Universal Instantiation"):: $forall alpha, varphi!$:

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

;Existential Introduction (or "Existential Generalization"):: : $exists alpha, varphi,!$

Restriction: No free occurrence of $alpha,!$ in $varphi,!$ falls within the scope of a quantifier quantifying a variable occurring in . ;Existential Elimination (or "Existential Instantiation"):: $exists alpha, varphi,!$: $underline\left\{varphi vdash psi\right\},!$: $psi,!$

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

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