View source for Modus tollens

{{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 "method of removing by taking away")<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]].

== Explanation ==
The form of a ''modus tollens'' argument is a mixed [[hypothetical syllogism]], with two premises and a conclusion:

:If ''P'', then ''Q''.
:Not ''Q''.
:Therefore, not ''P''.

The first premise is a [[Material conditional|conditional]] ("if-then") claim, such as ''P'' implies ''Q''. The second premise is an assertion that ''Q'', the [[consequent]] of the conditional claim, is not the case. From these two premises it can be logically concluded that ''P'', the [[Antecedent (logic)|antecedent]] of the conditional claim, is also not the case.

For example:

:If the dog detects an intruder, the dog will bark.
:The dog did not bark.
:Therefore, no intruder was detected by the dog.

Supposing that the premises are both true (the dog will bark if it detects an intruder, and does indeed not bark), it follows that no intruder has been detected. This is a valid argument since it is not possible for the conclusion to be false if the premises are true. (It is conceivable that there may have been an intruder that the dog did not detect, but that does not invalidate the argument; the first premise is "if the dog ''detects'' an intruder". The thing of importance is that the dog detects or does not detect an intruder, not whether there is one.)

Example 1:
:If I am the burglar, then I can crack a safe.
:I cannot crack a safe.
:Therefore, I am not the burglar.
Example 2: 
:If Rex is a chicken, then he is a bird.
:Rex is not a bird.
:Therefore, Rex is not a chicken.

== Relation to ''modus ponens'' ==
Every use of ''modus tollens'' can be converted to a use of ''[[modus ponens]]'' and one use of [[transposition (logic)|transposition]] to the premise which is a material implication. For example:

:If ''P'', then ''Q''. (premise – material implication)
:If not ''Q'', then not ''P''. (derived by transposition)
:Not ''Q'' . (premise)
:Therefore, not ''P''. (derived by ''modus ponens'')

Likewise, every use of ''modus ponens'' can be converted to a use of ''modus tollens'' and transposition.

== Formal notation ==
The ''modus tollens'' rule can be stated formally as:

:<math>\frac{P \to Q, \neg Q}{\therefore \neg P}</math>

where <math>P \to Q</math> stands for the statement "P implies Q". <math>\neg  Q</math> stands for "it is not the case that Q" (or in brief "not Q"). Then, whenever "<math>P \to Q</math>" and "<math>\neg Q</math>" each appear by themselves as a line of a [[formal proof|proof]], then "<math>\neg P</math>" can validly be placed on a subsequent line.

The ''modus tollens'' rule may be written in [[sequent]] notation:

:<math>P\to Q, \neg Q \vdash \neg P</math>

where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>\neg P</math> is a [[logical consequence|syntactic consequence]] of <math>P \to Q</math> and <math>\neg Q</math> in some [[formal system|logical system]];

or as the statement of a functional [[Tautology (logic)|tautology]] or [[theorem]] of propositional logic:

:<math>((P \to Q) \land \neg Q) \to \neg P</math>

where <math>P</math> and <math>Q</math> are propositions expressed in some [[formal system]];

or including assumptions:

:<math>\frac{\Gamma \vdash P\to Q ~~~ \Gamma \vdash \neg Q}{\Gamma \vdash \neg P}</math>

though since the rule does not change the set of assumptions, this is not strictly necessary.

More complex rewritings involving ''modus tollens'' are often seen, for instance in [[set theory]]:

:<math>P\subseteq Q</math>
:<math>x\notin Q</math>
:<math>\therefore x\notin P</math>

("P is a subset of Q. x is not in Q. Therefore, x is not in P.")

Also in first-order [[predicate logic]]:

:<math>\forall x:~P(x) \to Q(x)</math>
:<math>\neg Q(y)</math>
:<math>\therefore ~\neg P(y)</math>

("For all x if x is P then x is Q. y is not Q. Therefore, y is not P.")

Strictly speaking these are not instances of ''modus tollens'', but they may be derived from ''modus tollens'' using a few extra steps.

== Justification via truth table ==
The validity of ''modus tollens'' can be clearly demonstrated through a [[truth table]].

{| class="wikitable" style="margin: 0 auto; text-align:center; width:45%"
|-
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p → q
|-
| T || T || T
|-
| T || F || F
|-
| F || T || T
|-
| F || F || T
|}

In instances of ''modus tollens'' we assume as premises that p → q is true and q is false. There is only one line of the truth table—the fourth line—which satisfies these two conditions. In this line, p is false. Therefore, in every instance in which p → q is true and q is false, p must also be false.

== Formal proof ==

=== Via disjunctive syllogism ===
{| class="wikitable"
! ''Step''
! ''Proposition''
! ''Derivation''
|-
| 1 || <math>P\rightarrow Q</math> || Given
|-
| 2 || <math>\neg Q</math> || Given
|-
| 3 || <math>\neg P\lor Q</math> || [[Material implication (rule of inference)|Material implication]] (1)
|-
| 4 || <math>\neg P</math> || [[Disjunctive syllogism]] (3,2)
|}

=== Via ''reductio ad absurdum'' ===
{| class="wikitable"
! ''Step''
! ''Proposition''
! ''Derivation''
|-
| 1 || <math>P\rightarrow Q</math> || Given
|-
| 2 || <math>\neg Q</math> || Given
|-
| 3 || <math>P</math> || Assumption
|-
| 4 || <math>Q</math> || [[Modus ponens]] (1,3)
|-
| 5 || <math>Q \land \neg Q</math> || [[Conjunction introduction]] (2,4)
|-
| 6 || <math>\neg P</math> || ''[[Reductio ad absurdum]]'' (3,5)
|-
| 7 || <math>\neg Q \rightarrow \neg P</math> || ''Conditional introduction'' (2,6)
|}

=== Via contraposition ===
{| class="wikitable"
! ''Step''
! ''Proposition''
! ''Derivation''
|-
| 1 || <math>P\rightarrow Q</math> || Given
|-
| 2 || <math>\neg Q</math> || Given
|-
| 3 || <math>\neg Q\rightarrow \neg P</math> || [[Contraposition]] (1)
|-
| 4 || <math>\neg P</math> || [[Modus ponens]] (2,3)
|}

==Correspondence to other mathematical frameworks==
===Probability calculus===
''Modus tollens'' represents an instance of the [[law of total probability]] combined with [[Bayes' theorem]] expressed as:

<math display="block">\Pr(P)=\Pr(P\mid Q)\Pr(Q)+\Pr(P\mid \lnot Q)\Pr(\lnot Q)\,,</math>

where the conditionals <math>\Pr(P\mid Q)</math> and <math>\Pr(P\mid \lnot Q)</math> are obtained with (the extended form of) [[Bayes' theorem]] expressed as:

<math display="block">\Pr(P\mid Q) = \frac{\Pr(Q \mid P)\,a(P)}{\Pr(Q\mid P)\,a(P)+\Pr(Q\mid \lnot P)\,a(\lnot P)}\;\;\;</math> and  <math display="block">\Pr(P\mid \lnot Q) = \frac{\Pr(\lnot Q \mid P)\,a(P)}{\Pr(\lnot Q\mid P)\,a(P)+\Pr(\lnot Q\mid \lnot P)\,a(\lnot P)}.</math>

In the equations above <math>\Pr(Q)</math> denotes the probability of <math>Q</math>, and <math>a(P)</math> denotes the [[base rate]] (aka. [[prior probability]]) of <math>P</math>. The [[conditional probability]] <math>\Pr(Q\mid P)</math> generalizes the logical statement <math>P \to Q</math>, i.e. in addition to assigning TRUE or FALSE we can also assign any probability to the statement. Assume that <math>\Pr(Q) = 1</math> is equivalent to <math>Q</math> being TRUE, and that <math>\Pr(Q) = 0</math> is equivalent to <math>Q</math> being FALSE.  It is then easy to see that <math>\Pr(P) = 0</math> when <math>\Pr(Q\mid P) = 1</math> and <math>\Pr(Q) = 0</math>. This is because <math>\Pr(\lnot Q\mid P) = 1 - \Pr(Q\mid P) = 0</math> so that <math>\Pr(P\mid \lnot Q) = 0</math> in the last equation. Therefore, the product terms in the first equation always have a zero factor so that <math>\Pr(P) = 0</math> which is equivalent to <math>P</math> being FALSE. Hence, the [[law of total probability]] combined with [[Bayes' theorem]] represents a generalization of ''modus tollens''.<ref>Audun Jøsang 2016:p.2</ref>

===Subjective logic===
''Modus tollens'' represents an instance of the abduction operator in [[subjective logic]] expressed as:

<math display="block">\omega^{A}_{P\tilde{\|}Q}= (\omega^{A}_{Q|P},\omega^{A}_{Q|\lnot P})\widetilde{\circledcirc} (a_{P},\,\omega^{A}_{Q})\,,</math>

where <math>\omega^{A}_{Q}</math> denotes the subjective opinion about <math>Q</math>, and <math>(\omega^{A}_{Q|P},\omega^{A}_{Q|\lnot P})</math> denotes a pair of binomial conditional opinions, as expressed by source <math>A</math>. The parameter <math>a_{P}</math> denotes the [[base rate]] (aka. the [[prior probability]]) of <math>P</math>. The abduced marginal opinion on <math>P</math> is denoted <math>\omega^{A}_{P\tilde{\|}Q}</math>. The conditional opinion <math>\omega^{A}_{Q|P}</math> generalizes the logical statement <math>P \to Q</math>, i.e. in addition to assigning TRUE or FALSE the source <math>A</math> can assign any subjective opinion to the statement. The case where <math>\omega^{A}_{Q}</math> is an absolute TRUE opinion is equivalent to source <math>A</math> saying that <math>Q</math> is TRUE, and the case where <math>\omega^{A}_{Q}</math> is an absolute FALSE opinion is equivalent to source <math>A</math> saying that <math>Q</math> is FALSE.  The abduction operator <math>\widetilde{\circledcirc}</math> of [[subjective logic]] produces an absolute FALSE abduced opinion <math>\omega^{A}_{P\widetilde{\|}Q}</math> when the conditional opinion <math>\omega^{A}_{Q|P}</math> is absolute TRUE and the consequent opinion <math>\omega^{A}_{Q}</math> is absolute FALSE. Hence, subjective logic abduction represents a generalization of both ''modus tollens'' and of the [[Law of total probability]] combined with [[Bayes' theorem]].<ref>Audun Jøsang 2016:p.92</ref>

== See also ==
* {{annotated link|Evidence of absence}}
* {{annotated link|Latin phrases}}
* {{annotated link|Modus operandi|''Modus operandi''}}
* {{annotated link|Modus ponens|''Modus ponens''}}
* {{annotated link|Modus vivendi|''Modus vivendi''}}
* {{annotated link|Non sequitur (logic)|''Non sequitur''}}
* {{annotated link|Proof by contradiction}}
* {{annotated link|Proof by contrapositive}}
* {{annotated link|Stoic logic}}

== Notes ==
{{Reflist}}

== Sources ==
* Audun Jøsang, 2016, ''[https://books.google.com/books?id=nqRlDQAAQBAJ&q=%22Modus+tollens%22 Subjective Logic; A formalism for Reasoning Under Uncertainty]'' Springer, Cham, {{ISBN|978-3-319-42337-1}}

== External links ==
* ''[http://mathworld.wolfram.com/ModusTollens.html Modus Tollens]'' at Wolfram MathWorld

{{DEFAULTSORT:Modus Tollens}}
[[Category:Classical logic]]
[[Category:Rules of inference]]
[[Category:Latin logical phrases]]
[[Category:Theorems in propositional logic]]