View source for Material implication (rule of inference)

{{short description|Rule of replacement in propositional logic}}
{{use dmy dates|date=March 2025}}
{{other uses|Material implication (disambiguation)}}{{distinguish|Material inference}}
{{Infobox mathematical statement
| name = Material implication
| type = [[Rule of replacement]]
| field = [[Propositional calculus]]
| statement = ''P implies Q'' is [[Logical equivalence|logically equivalent]] to ''not-<math>P</math> or <math>Q</math>''. Either form can replace the other in [[formal proof|logical proofs]].
| symbolic statement = <math>P \to Q \Leftrightarrow \neg P \lor Q</math>
}}
{{Transformation rules}}

In '''classical''' [[propositional logic]], '''material implication'''<ref name="Hurley2011">{{cite book |author=Patrick J. Hurley |title=A Concise Introduction to Logic |url=https://books.google.com/books?id=Ikp2dGWT5O4C&q=%22Material+implication%22 |date=1 January 2011 |publisher=Cengage Learning |isbn=978-0-8400-3417-5}}</ref><ref>{{cite book |last1=Copi |first1=Irving M. |authorlink1=Irving Copi |last2=Cohen |first2=Carl |authorlink2=Carl Cohen (professor)|title=Introduction to Logic |url=https://archive.org/details/isbn_9780536179364 |url-access=registration |publisher=Prentice Hall |year=2005 |page=[https://archive.org/details/isbn_9780536179364/page/371 371]}}</ref> is a [[Validity (logic)|valid]] [[rule of replacement]] that allows a [[material conditional|conditional statement]] to be replaced by a [[logical disjunction|disjunction]] in which the [[antecedent (logic)|antecedent]] is [[negation|negated]]. The rule states that ''P implies Q'' is [[Logical equivalence|logically equivalent]] to ''not-<math>P</math> or <math>Q</math>'' and that either form can replace the other in [[formal proof|logical proofs]]. In other words, if <math>P</math> is true, then <math>Q</math> must also be true, while if <math>Q</math> is {{em|not}} true, then <math>P</math> cannot be true either; additionally, when <math>P</math> is not true, <math>Q</math> may be either true or false.

: <math>P \to Q \Leftrightarrow \neg P \lor Q,</math>

where "<math>\Leftrightarrow</math>" is a [[metalogic]]al [[symbol (formal)|symbol]] representing "can be replaced in a proof with", ''P'' and ''Q'' are any given logical [[Statement (logic)|statements]], and <math>\neg P \lor Q</math> can be read as "(not ''P'') or ''Q''". To illustrate this, consider the following statements:
* <math>P</math>: Sam ate an [[Orange (fruit)|orange]] for lunch.
* <math>Q</math>: Sam ate a [[fruit]] for lunch.

Then, to say "Sam ate an orange for lunch" {{em|implies}} "Sam ate a fruit for lunch" (<math>P \to Q</math>). Logically, if Sam did not eat a fruit for lunch, then Sam also cannot have eaten an orange for lunch (by [[contraposition]]). However, merely saying that Sam did not eat an orange for lunch provides no information on whether or not Sam ate a fruit (of any kind) for lunch.

==Partial proof==
Suppose we are given that <math>P \to Q</math>. Then we have <math>\neg P \lor P</math> by the [[law of excluded middle]] (i.e. either <math>P</math> must be true, or <math>P</math> must not be true). 

Subsequently, since <math>P \to Q</math>, <math>P</math> can be replaced by <math>Q</math> in the statement, and thus it follows that <math>\neg P \lor Q</math> (i.e. either <math>Q</math> must be true, or <math>P</math> must not be true).

Suppose, conversely, we are given <math>\neg P \lor Q</math>. Then if <math>P</math> is true, that rules out the first disjunct, so we have <math>Q</math>. In short, <math>P \to Q</math>.<ref>{{cite web |url=https://math.stackexchange.com/q/243949 |website=Mathematics Stack Exchange |title=Equivalence of a→b and ¬ a ∨ b}}</ref> However, if <math>P</math> is false, then this entailment fails, because the first disjunct <math>\neg P</math> is true, which puts no constraint on the second disjunct <math>Q</math>. Hence, nothing can be said about <math>P \to Q</math>. In sum, the equivalence in the case of false <math>P</math> is only conventional, and hence the formal proof of equivalence is only partial. 

This can also be expressed with a [[truth table]]:
{| class="wikitable" style="text-align:center"
! P !! Q !! ¬P !! P → Q !! ¬P ∨ Q
|-
| T || T || F  || T     || T
|-
| T || F || F  || F     || F
|-
| F || T || T  || T     || T
|-
| F || F || T  || T     || T
|}

== Example ==
An example: we are given the conditional fact that if it is a bear, then it can swim. Then, all 4 possibilities in the truth table are compared to that fact.
# If it is a bear, then it can swim — T
# If it is a bear, then it can not swim — F
# If it is not a bear, then it can swim — T because it doesn’t contradict our initial fact.
# If it is not a bear, then it can not swim — T (as above)

Thus, the conditional fact can be converted to <math>\neg P \vee Q</math>, which is "it is not a bear" or "it can swim",
where <math>P</math> is the statement "it is a bear" and <math>Q</math> is the statement "it can swim".

==The equivalence does not hold in intuitionistic logic==

[[Intuitionistic logic]] does not treat <math>P \to Q</math> as equivalent to <math>\neg P \vee Q</math> because
: <math> P \to Q \cancel{\Rightarrow} \neg P \or Q</math>

Given <math>P \to Q</math>, one can constructively transform a proof of <math>P</math> into a proof of <math>Q</math>. In particular, <math>P \to P</math> holds in intuitionistic logic. If <math> P \to Q \Rightarrow \neg P \or Q</math> would hold, then <math>\neg P \or P</math> could be derived. However, the latter is the [[law of excluded middle]], which is not accepted by intuitionistic logic (one cannot assume <math>\neg P \or P</math> without knowing which case applies).

==References==
{{Reflist}}

{{Classical logic}}
[[Category:Rules of inference]]
[[Category:Theorems in propositional logic]]