Editing Modus tollens (section)

{{short description|Rule of logical inference}}
{{Use dmy dates|date=October 2020}}
{{Italic title}}
{{Infobox mathematical statement
| name = <em>Modus tollens</em>
| type = {{Plainlist|
* [[Deductive reasoning|Deductive]] [[argument form]]
* [[Rule of inference]]
}}
| field = {{Plainlist|
* [[Classical logic]]
* [[Propositional calculus]]
}}
| statement = <math>P</math> implies <math>Q</math>. <math>Q</math> is false. Therefore, <math>P</math> must also be false.
| symbolic statement = <math>P \rightarrow Q, \neg Q</math> <math>\therefore\neg P</math><ref name="KA">{{Cite web |url=https://www.khanacademy.org/partner-content/wi-phi/wiphi-critical-thinking/wiphi-fallacies/v/denying-the-antecedent |author=Matthew C. Harris |title=Denying the antecedent |publisher=[[Khan academy]]}}</ref>
}}

{{Transformation rules}}

In [[propositional calculus|propositional logic]], '''''modus tollens''''' ({{IPAc-en|ˈ|m|oʊ|d|ə|s|_|ˈ|t|ɒ|l|ɛ|n|z}}) ('''MT'''), also known as '''''modus tollendo [[wiktionary:tollens|tollens]]''''' ([[Latin language|Latin]] for "mode that by denying denies")<ref>{{cite book |last=Stone |first=Jon R. |year=1996 |title=Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language |location=London |publisher=Routledge |isbn=978-0-415-91775-9 |page=[https://archive.org/details/latinforillitera0000ston/page/60 60] |url=https://archive.org/details/latinforillitera0000ston |url-access=registration }}</ref> and '''denying the consequent''',<ref>{{cite book |last=Sanford |first=David Hawley |year=2003 |title=If P, Then Q: Conditionals and the Foundations of Reasoning |location=London |publisher=Routledge |edition=2nd |isbn=978-0-415-28368-7 |page=39 |quote=[Modus] tollens is always an abbreviation for modus tollendo tollens, the mood that by denying denies. |url=https://books.google.com/books?id=h_AUynB6PA8C&pg=PA39 }}</ref> is a [[Deductive reasoning|deductive]] [[Logical form|argument form]] and a [[rule of inference]]. ''Modus tollens'' is a mixed [[hypothetical syllogism]] that takes the form of "If ''P'', then ''Q''. Not ''Q''. Therefore, not ''P''." It is an application of the general truth that if a statement is true, then so is its [[contrapositive]]. The form shows that [[inference]] from ''P implies Q'' to ''the negation of Q implies the negation of P'' is a [[Validity (logic)|valid]] argument.

The history of the inference rule ''modus tollens'' goes back to [[Ancient history|antiquity]].<ref>[[Susanne Bobzien]] (2002). [https://dx.doi.org/10.1163/156852802321016541 "The Development of Modus Ponens in Antiquity"], ''Phronesis'' 47.</ref> The first to explicitly describe the argument form ''modus tollens'' was [[Theophrastus]].<ref>[http://plato.stanford.edu/entries/logic-ancient/#StoSyl "Ancient Logic: Forerunners of ''Modus Ponens'' and ''Modus Tollens''"]. ''[[Stanford Encyclopedia of Philosophy]]''.</ref>

''Modus tollens'' is closely related to ''[[modus ponens]]''. There are two similar, but [[fallacy|invalid, forms of argument]]: [[affirming the consequent]] and [[denying the antecedent]].  See also [[contraposition]] and [[proof by contrapositive]].