Material implication (rule of inference)

In classical <a href="/facts/Propositional_logic/MNt5v31V">propositional logic</a>, material implication is a <a href="/facts/Validity_(logic)/oMGkywGC">valid</a> <a href="/facts/Rule_of_replacement/uTyqA71f">rule of replacement</a> that allows a <a href="/facts/Material_conditional/hYf4b7AE">conditional statement</a> to be replaced by a <a href="/facts/Logical_disjunction/jbI2EGlM">disjunction</a> in which the <a href="/facts/Antecedent_(logic)/8VVVqa06">antecedent</a> is <a href="/facts/Negation/q5AQBvBj">negated</a>. The rule states that P implies Q is <a href="/facts/Logical_equivalence/ysCWmsxA">logically equivalent</a> to not-
 
 
 
 P
 
 
 {\displaystyle P}
 
 or 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 and that either form can replace the other in <a href="/facts/Formal_proof/6nEhV4cw">logical proofs</a>. In other words, if 
 
 
 
 P
 
 
 {\displaystyle P}
 
 is true, then 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 must also be true, while if 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 is not true, then 
 
 
 
 P
 
 
 {\displaystyle P}
 
 cannot be true either; additionally, when 
 
 
 
 P
 
 
 {\displaystyle P}
 
 is not true, 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 may be either true or false.

P
        →
        Q
        ⇔
        ¬
        P
        ∨
        Q
        ,
      
    
    {\displaystyle P\to Q\Leftrightarrow \neg P\lor Q,}

where "
 
 
 
 ⇔
 
 
 {\displaystyle \Leftrightarrow }
 
" is a <a href="/facts/Metalogic/zfooQjCo">metalogical</a> <a href="/facts/Symbol_(formal)/NS2oUqEk">symbol</a> representing "can be replaced in a proof with", P and Q are any given logical <a href="/facts/Statement_(logic)/CjpWAyxk">statements</a>, and 
 
 
 
 ¬
 P
 ∨
 Q
 
 
 {\displaystyle \neg P\lor Q}
 
 can be read as "(not P) or Q". To illustrate this, consider the following statements:

<ul><li>
 
 
 
 P
 
 
 {\displaystyle P}
 
: Sam ate an <a href="/facts/Orange_(fruit)/qQdDNeL4">orange</a> for lunch.</li>
<li>
 
 
 
 Q
 
 
 {\displaystyle Q}
 
: Sam ate a <a href="/facts/Fruit/PBpuzk60">fruit</a> for lunch.</li></ul>
Then, to say "Sam ate an orange for lunch" implies "Sam ate a fruit for lunch" (
 
 
 
 P
 →
 Q
 
 
 {\displaystyle P\to Q}
 
). Logically, if Sam did not eat a fruit for lunch, then Sam also cannot have eaten an orange for lunch (by <a href="/facts/Contraposition/ENPumbyv">contraposition</a>). 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.

Material implication (rule of inference) open-in-new

Material implication (rule of inference)