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| {{Portal|Logic}}
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| This is a list of [[Rule of inference|rules of inference]], logical laws that relate to mathematical formulae.
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| ==Introduction==
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| '''Rules of inference''' are syntactical '''transform''' 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.
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| ''Discharge rules'' permit inference from a subderivation based on a temporary assumption. Below, the notation
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| : <math>\varphi \vdash \psi\,\!</math> | |
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| indicates such a subderivation from the temporary assumption <math>\varphi\,\!</math> to <math>\psi\,\!</math>.
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| ==Rules for classical sentential calculus==
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| Sentential calculus is also known as [[propositional calculus]].
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| ===Rules for negations===
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| ;[[Reductio ad absurdum]] (or ''Negation Introduction''):
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| : <math>\varphi \vdash \psi\,\!</math>
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| : <math>\underline{\varphi \vdash \lnot \psi}\,\!</math>
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| : <math>\lnot \varphi\,\!</math>
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| ;Reductio ad absurdum (related to the law of [[excluded middle]]):
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| : <math>\lnot \varphi \vdash \psi\,\!</math>
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| : <math>\underline{\lnot \varphi \vdash \lnot \psi}\,\!</math>
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| : <math>\varphi\,\!</math>
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| ;[[Noncontradiction]] (or ''Negation Elimination''):
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| : <math>\varphi\,\!</math>
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| : <math>\underline{\lnot \varphi}\,\!</math>
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| : <math>\psi\,\!</math>
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| ;[[Double negative elimination|Double negation elimination]]:
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| : <math>\underline{\lnot \lnot \varphi}\,\!</math>
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| : <math> \varphi\,\!</math>
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| ;[[Double negative introduction|Double negation introduction]]:
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| : <math>\underline{\varphi \quad \quad}\,\!</math>
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| : <math> \lnot \lnot \varphi\,\!</math>
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| ===Rules for conditionals=== | |
| ;[[Deduction theorem]] (or ''[[Conditional proof|Conditional Introduction]]''):
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| : <math>\underline{\varphi \vdash \psi}\,\!</math>
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| : <math>\varphi \rightarrow \psi\,\!</math>
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| ;[[Modus ponens]] (or ''Conditional Elimination''):
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| : <math>\varphi \rightarrow \psi\,\!</math>
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| : <math>\underline{\varphi \quad \quad \quad}\,\!</math>
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| : <math>\psi\,\!</math>
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| ;[[Modus tollens]]:
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| : <math>\varphi \rightarrow \psi\,\!</math>
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| : <math>\underline{\lnot \psi \quad \quad \quad}\,\!</math>
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| : <math>\lnot \varphi\,\!</math>
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| ===Rules for conjunctions=== | |
| ;[[Conjunction introduction|Adjunction]] (or ''Conjunction Introduction''):
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| : <math>\varphi\,\!</math>
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| : <math>\underline{\psi \quad \quad \ \ }\,\!</math>
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| : <math>\varphi \land \psi\,\!</math>
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| ;[[Simplification (logic)|Simplification]] (or ''Conjunction Elimination''):
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| : <math>\underline{\varphi \land \psi}\,\!</math>
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| : <math>\varphi\,\!</math>
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| : <math>\underline{\varphi \land \psi}\,\!</math>
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| : <math>\psi\,\!</math>
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| ===Rules for disjunctions===
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| ;[[Addition (logic)|Addition]] (or ''Disjunction Introduction''):
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| : <math>\underline{\varphi \quad \quad \ \ }\,\!</math>
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| : <math>\varphi \lor \psi\,\!</math>
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| : <math>\underline{\psi \quad \quad \ \ }\,\!</math>
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| : <math>\varphi \lor \psi\,\!</math>
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| ;[[Case analysis]]
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| : <math>\varphi \lor \psi\,\!</math>
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| : <math>\varphi \rightarrow \chi\,\!</math>
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| : <math>\underline{\psi \rightarrow \chi}\,\!</math>
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| : <math>\chi\,\!</math>
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| ;[[Disjunctive syllogism]]:
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| : <math>\varphi \lor \psi\,\!</math>
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| : <math>\underline{\lnot \varphi \quad \quad}\,\!</math>
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| : <math>\psi\,\!</math>
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| : <math>\varphi \lor \psi\,\!</math>
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| : <math>\underline{\lnot \psi \quad \quad}\,\!</math>
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| : <math>\varphi\,\!</math>
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| ===Rules for biconditionals===
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| ;[[Biconditional introduction]]:
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| : <math>\varphi \rightarrow \psi\,\!</math>
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| : <math>\underline{\psi \rightarrow \varphi}\,\!</math>
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| ;Biconditional Elimination:
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| : <math>\underline{\varphi \quad \quad}\,\!</math>
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| : <math>\psi\,\!</math>
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| : <math>\underline{\psi \quad \quad}\,\!</math>
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| : <math>\varphi\,\!</math>
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| : <math>\underline{\lnot \varphi \quad \quad}\,\!</math>
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| : <math>\lnot \psi\,\!</math>
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| : <math>\underline{\lnot \psi \quad \quad}\,\!</math>
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| : <math>\lnot \varphi\,\!</math>
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| : <math>\underline{\psi \lor \varphi}\,\!</math>
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| : <math>\psi \land \varphi \,\!</math>
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| : <math>\varphi \leftrightarrow \psi\,\!</math>
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| : <math>\underline{\lnot \psi \lor \lnot \varphi}\,\!</math>
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| : <math>\lnot \psi \land \lnot \varphi \,\!</math>
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| ==Rules of classical [[First-order logic|predicate calculus]]==
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| In the following rules, <math>\varphi(\beta / \alpha)\,\!</math> is exactly like <math>\varphi\,\!</math> except for having the term <math>\beta\,\!</math> everywhere <math>\varphi\,\!</math> has the free variable <math>\alpha\,\!</math>.
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| ;[[Universal generalization|Universal Introduction]] (or ''Universal Generalization''):
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| : <math>\underline{\varphi{(\beta / \alpha)}}\,\!</math>
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| : <math>\forall \alpha\, \varphi\,\!</math>
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| Restriction 1: <math>\beta</math> does not occur in <math>\varphi</math>.
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| <br/>
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| Restriction 2: <math>\beta</math> is not mentioned in any hypothesis or undischarged assumptions.
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| ;[[Universal instantiation|Universal Elimination]] (or ''Universal Instantiation''):
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| : <math> \forall \alpha\, \varphi\!</math>
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| : <math>\overline{\varphi{(\beta / \alpha)}}\!</math>
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| Restriction: No free occurrence of <math>\alpha\,\!</math> in <math>\varphi\,\!</math> falls within the scope of a quantifier quantifying a variable occurring in <math>\beta\,\!</math>.
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| ;Existential Introduction (or ''Existential Generalization''):
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| : <math>\underline{\varphi(\beta / \alpha)}\,\!</math>
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| : <math>\exists \alpha\, \varphi\,\!</math>
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| Restriction: No free occurrence of <math>\alpha\,\!</math> in <math>\varphi\,\!</math> falls within the scope of a quantifier quantifying a variable occurring in <math>\beta\,\!</math>.
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|
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| ;Existential Elimination (or ''Existential Instantiation''):
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| : <math>\exists \alpha\, \varphi\,\!</math>
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| : <math>\underline{\varphi(\beta / \alpha) \vdash \psi}\,\!</math>
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| : <math>\psi\,\!</math>
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| Restriction 1: No free occurrence of <math>\alpha\,\!</math> in <math>\varphi\,\!</math> falls within the scope of a quantifier quantifying a variable occurring in <math>\beta\,\!</math>.
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| <br/> | |
| Restriction 2: There is no occurrence, free or bound, of <math>\beta\,\!</math> in <math>\psi\,\!</math>.
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| ==Table: Rules of Inference - a short summary==
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| The rules above can be summed up in the following table.<ref>Kenneth H. Rosen: ''Discrete Mathematics and its Applications'',Fifth Edition, p. 58.</ref> The "[[Tautology (logic)|Tautology]]" column shows how to interpret the notation of a given rule.
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| {| class="wikitable"
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| |-
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| ! Rule of inference
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| ! Tautology
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| ! Name
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| |-
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| |<math>\begin{align}
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| p \\
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| \therefore \overline{p \vee q} \\
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| \end{align}</math>
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| | <math>p \rightarrow (p \vee q)</math>
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| | Addition
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| |-
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| |<math>\begin{align}
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| p \wedge q \\
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| \therefore \overline{p \quad \quad \quad} \\
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| \end{align}</math>
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| | <math>(p \wedge q) \rightarrow p</math>
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| | Simplification
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| |-
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| |<math>\begin{align}
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| p\\
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| q\\
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| \therefore \overline{p \wedge q} \\
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| \end{align}</math>
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| | <math>((p) \wedge (q)) \rightarrow (p \wedge q)</math>
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| | Conjunction
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| |-
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| |<math>\begin{align}
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| p\\
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| p \rightarrow q\\
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| \therefore \overline{q \quad \quad \quad} \\
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| \end{align}</math>
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| | <math>((p \wedge (p \rightarrow q)) \rightarrow q</math>
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| | [[Modus ponens]]
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| |-
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| |<math>\begin{align}
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| \neg q\\
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| p \rightarrow q\\
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| \therefore \overline{\neg p \quad \quad \quad} \\
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| \end{align}</math>
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| | <math>((\neg q \wedge (p \rightarrow q)) \rightarrow \neg p</math>
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| | [[Modus tollens]]
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| |-
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| |<math>\begin{align}
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| p \rightarrow q\\
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| q \rightarrow r\\
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| \therefore \overline{p \rightarrow r} \\
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| \end{align}</math>
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| | <math>((p \rightarrow q) \wedge (q \rightarrow r)) \rightarrow (p \rightarrow r)</math>
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| | Hypothetical syllogism
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| |-
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| |<math>\begin{align}
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| p \vee q \\
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| \neg p \\
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| \therefore \overline{q \quad \quad \quad} \\
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| \end{align}</math>
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| | <math>((p \vee q) \wedge \neg p) \rightarrow q</math>
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| | Disjunctive syllogism
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| |-
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| |<math>\begin{align}
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| p \vee q \\
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| \neg p \vee r \\
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| \therefore \overline{q \vee r} \\
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| \end{align}</math>
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| | <math>((p \vee q) \wedge (\neg p \vee r)) \rightarrow (q \vee r)</math>
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| | Resolution
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| |}
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| All rules use the basic logic operators. A complete table of "logic operators" is shown by a [[truth table]], giving definitions of all the possible (16) truth functions of 2 [[Boolean algebra (logic)|boolean variables]] (''p'', ''q''):
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| {| class="wikitable" style="margin:1em auto 1em auto; text-align:center;"
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| ! ''p'' || ''q''
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| ! 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7
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| !| 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15
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| |-
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| ! T || T
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| | || F || F || F || F || F || F || F || F || || T || T || T || T || T || T || T || T
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| |-
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| ! T || F
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| | || F || F || F || F || T || T || T || T || || F || F || F || F || T || T || T || T
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| |-
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| ! F || T
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| | || F || F || T || T || F || F || T || T || || F || F || T || T || F || F || T || T
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| |-
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| ! F || F
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| | || F || T || F || T || F || T || F || T || || F || T || F || T || F || T || F || T
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| |}
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| where T = true and F = false, and, the columns are the logical operators: '''0''', false, [[Contradiction]]; '''1''', NOR, [[Logical NOR]]; '''2''', [[Converse nonimplication]]; '''3''', '''¬p''', [[Negation]]; '''4''', [[Material nonimplication]]; '''5''', '''¬q''', Negation; '''6''', XOR, [[Exclusive disjunction]]; '''7''', NAND, [[Logical NAND]]; '''8''', AND, [[Logical conjunction]]; '''9''', XNOR, [[If and only if]], [[Logical biconditional]]; '''10''', '''q''', [[Projection function]]; '''11''', if/then, [[Logical implication]]; '''12''', '''p''', Projection function; '''13''', then/if, [[Converse implication]]; '''14''', OR, [[Logical disjunction]]; '''15''', true, [[Tautology (logic)|Tautology]].
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| Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples:
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| * The column-14 operator (OR), shows ''Addition rule'': when ''p''=T (the hypothesis selects the first two lines of the table), we see (at column-14) that ''p''∨''q''=T.
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| *: We can see also that, with the same premise, another conclusions are valid: columns 12, 14 and 15 are T.
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| * The column-8 operator (AND), shows ''Simplification rule'': when ''p''∧''q''=T (first line of the table), we see that ''p''=T.
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| *: With this premise, we also conclude that ''q''=T, ''p''∨''q''=T, etc. as showed by columns 9-15.
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| * The column-11 operator (IF/THEN), shows ''Modus ponens rule'': when ''p''→''q''=T and ''p''=T only one line of the truth table (the first) satisfies these two conditions. On this line, ''q'' is also true. Therefore, whenever p → q is true and p is true, q must also be true.
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| Machines and well-trained people use this [[Lookup table|look at table approach]] to do basic inferences, and to check if other inferences (for the same premises) can be obtained.
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| ===Example 1===
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| Let us consider the following assumptions: "If it rains today, then we will not go on a canoe today. If we do not go on a canoe trip today, then we will go on a canoe trip tomorrow. Therefore (Mathematical symbol for "therefore" is <math>\therefore</math>), if it rains today, we will go on a canoe trip tomorrow.
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| To make use of the rules of inference in the above table we let <math>p</math> be the proposition "If it rains today", <math>q</math> be " We will not go on a canoe today" and let <math>r</math> be "We will go on a canoe trip tomorrow". Then this argument is of the form:
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| <math>\begin{align}
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| p \rightarrow q\\
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| q \rightarrow r\\
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| \therefore \overline{p \rightarrow r} \\
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| \end{align}</math>
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| ===Example 2===
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| Let us consider a more complex set of assumptions: "It is not sunny today and it is colder than yesterday". "We will go swimming only if it is sunny", "If we do not go swimming, then we will have a barbecue", and "If we will have a barbecue, then we will be home by sunset" lead to the conclusion "We will be home before sunset."
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| Proof by rules of inference: Let <math>p</math> be the proposition "It is sunny this today", <math>q</math> the proposition "It is colder than yesterday", <math>r</math> the proposition "We will go swimming", <math>s</math> the proposition "We will have a barbecue", and <math>t</math> the proposition "We will be home by sunset". Then the hypotheses become <math>\neg p \wedge q, r \rightarrow p, \neg r \rightarrow s</math> and <math>s \rightarrow t</math>. Using our intuition we conjecture that the conclusion might be <math>t</math>. Using the Rules of Inference table we can proof the conjecture easily:
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| {| class="wikitable"
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| |-
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| ! Step
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| ! Reason
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| | 1.<math>\neg p \wedge q</math>
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| | Hypothesis
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| |-
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| | 2. <math>\neg p</math>
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| | Simplification using Step 1
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| |-
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| | 3. <math>r \rightarrow p</math>
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| | Hypothesis
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| |-
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| | 4. <math>\neg r</math>
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| | Modus tollens using Step 2 and 3
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| |-
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| | 5. <math>\neg r \rightarrow s</math>
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| | Hypothesis
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| |-
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| | 6. <math>s</math>
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| | Modus ponens using Step 4 and 5
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| |-
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| | 7. <math>s \rightarrow t</math>
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| | Hypothesis
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| |-
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| | 8. <math>t</math>
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| | Modus ponens using Step 6 and 7
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| |}
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| ==References==
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| <references/>
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| {{Logic}}
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| {{DEFAULTSORT:List Of Rules Of Inference}}
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| [[Category:Rules of inference|*]]
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| [[Category:Mathematics-related lists|Rules of inference]]
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| [[Category:Philosophy-related lists|Rules of inference]]
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| [[de:Schlussregel]]
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| [[it:Elenco di regole di inferenza]]
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| [[he:חוקי היקש]]
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