Modus ponens
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| - | In [[logic]], a '''rule of inference''', '''inference rule''', or '''transformation rule''' is the act of drawing a conclusion based on the [[Logical form|form]] of [[premise]]s interpreted as a function which takes premises, analyses their [[Syntax (logic)|syntax]], and returns a conclusion (or [[multiple-conclusion logic|conclusions]]). For example, the rule of inference [[modus ponens]] takes two premises, one in the form of "If p then q" and another in the form of "p" and returns the conclusion "q". The rule is valid with respect to the semantics of [[classical logic]] (as well as the semantics of many other [[non-classical logic]]s), in the sense that if the premises are true (under an interpretation) then so is the conclusion. | + | In [[propositional calculus|propositional logic]], '''''modus ponens''''' ('''MP'''; also '''''modus ponendo ponens''''' ([[Latin]] for "mode that affirms by affirming") or '''implication elimination''') is a [[rule of inference]]. It can be summarized as "''P [[material conditional|implies]] Q'' and ''P'' are both asserted to be true, so therefore ''Q'' must be true." |
| + | ''Modus ponens'' is closely related to another valid form of argument, ''[[modus tollens]]''. Both have apparently similar but invalid forms such as [[affirming the consequent]], [[denying the antecedent]], and [[evidence of absence]]. [[Constructive dilemma]] is the [[Logical disjunction|disjunctive]] version of ''modus ponens''. [[Hypothetical syllogism]] is closely related to ''modus ponens'' and sometimes thought of as "double ''modus ponens''." | ||
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| + | The history of ''modus ponens'' goes back to [[Classical antiquity|antiquity]]. The first to explicitly describe the argument form ''modus ponens'' was [[Theophrastus]]. | ||
| ==See also== | ==See also== | ||
| - | * [[Inference objection]] | + | * [[Condensed detachment]] |
| - | * [[Immediate inference]] | + | * [[Stoic logic]] |
| - | * [[Law of thought]] | + | * "[[What the Tortoise Said to Achilles]]" |
| - | * [[Logical truth]] | + | |
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In propositional logic, modus ponens (MP; also modus ponendo ponens (Latin for "mode that affirms by affirming") or implication elimination) is a rule of inference. It can be summarized as "P implies Q and P are both asserted to be true, so therefore Q must be true."
Modus ponens is closely related to another valid form of argument, modus tollens. Both have apparently similar but invalid forms such as affirming the consequent, denying the antecedent, and evidence of absence. Constructive dilemma is the disjunctive version of modus ponens. Hypothetical syllogism is closely related to modus ponens and sometimes thought of as "double modus ponens."
The history of modus ponens goes back to antiquity. The first to explicitly describe the argument form modus ponens was Theophrastus.
See also
