Material implication (rule of inference)

<h2 id="partial-proof">Partial proof</h2>
<p>Suppose we are given that 
  
    
      
        P
        →
        Q
      
    
    {\displaystyle P\to Q}
  
. Then we have 
  
    
      
        ¬
        P
        ∨
        P
      
    
    {\displaystyle \neg P\lor P}
  
 by the <a href="/facts/Law_of_excluded_middle/tDhbugCN">law of excluded middle</a> (i.e. either 
  
    
      
        P
      
    
    {\displaystyle P}
  
 must be true, or 
  
    
      
        P
      
    
    {\displaystyle P}
  
 must not be true). 
</p><p>Subsequently, since 
  
    
      
        P
        →
        Q
      
    
    {\displaystyle P\to Q}
  
, 
  
    
      
        P
      
    
    {\displaystyle P}
  
 can be replaced by 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
 in the statement, and thus it follows that 
  
    
      
        ¬
        P
        ∨
        Q
      
    
    {\displaystyle \neg P\lor Q}
  
 (i.e. either 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
 must be true, or 
  
    
      
        P
      
    
    {\displaystyle P}
  
 must not be true).
</p><p>Suppose, conversely, we are given 
  
    
      
        ¬
        P
        ∨
        Q
      
    
    {\displaystyle \neg P\lor Q}
  
. Then if 
  
    
      
        P
      
    
    {\displaystyle P}
  
 is true, that rules out the first disjunct, so we have 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
. In short, 
  
    
      
        P
        →
        Q
      
    
    {\displaystyle P\to Q}
  
.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> However, if 
  
    
      
        P
      
    
    {\displaystyle P}
  
 is false, then this entailment fails, because the first disjunct 
  
    
      
        ¬
        P
      
    
    {\displaystyle \neg P}
  
 is true, which puts no constraint on the second disjunct 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
. Hence, nothing can be said about 
  
    
      
        P
        →
        Q
      
    
    {\displaystyle P\to Q}
  
. In sum, the equivalence in the case of false 
  
    
      
        P
      
    
    {\displaystyle P}
  
 is only conventional, and hence the formal proof of equivalence is only partial. 
</p><p>This can also be expressed with a <a href="/facts/Truth_table/4KsmArDW">truth table</a>:
</p>
<table><tbody><tr><th>P</th><th>Q</th><th>¬P</th><th>P → Q</th><th>¬P ∨ Q</th></tr><tr><td>T</td><td>T</td><td>F</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>F</td><td>F</td><td>F</td></tr><tr><td>F</td><td>T</td><td>T</td><td>T</td><td>T</td></tr><tr><td>F</td><td>F</td><td>T</td><td>T</td><td>T</td></tr></tbody></table>
<h2 id="example">Example</h2>
<p>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.
</p>
<ol><li>If it is a bear, then it can swim — T</li>
<li>If it is a bear, then it can not swim — F</li>
<li>If it is not a bear, then it can swim — T because it doesn’t contradict our initial fact.</li>
<li>If it is not a bear, then it can not swim — T (as above)</li></ol>
<p>Thus, the conditional fact can be converted to 
  
    
      
        ¬
        P
        ∨
        Q
      
    
    {\displaystyle \neg P\vee Q}
  
, which is "it is not a bear" or "it can swim",
where 
  
    
      
        P
      
    
    {\displaystyle P}
  
 is the statement "it is a bear" and 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
 is the statement "it can swim".
</p>
<h2 id="the-equivalence-does-not-hold-in-intuitionistic-logic">The equivalence does not hold in intuitionistic logic</h2>
<p><a href="/facts/Intuitionistic_logic/zI7ulCnM">Intuitionistic logic</a> does not treat 
  
    
      
        P
        →
        Q
      
    
    {\displaystyle P\to Q}
  
 as equivalent to 
  
    
      
        ¬
        P
        ∨
        Q
      
    
    {\displaystyle \neg P\vee Q}
  
 because
</p>

P
        →
        Q
        
          
            ⇒
          
        
        ¬
        P
        ∨
        Q
      
    
    {\displaystyle P\to Q{\cancel {\Rightarrow }}\neg P\lor Q}

<p>Given 
  
    
      
        P
        →
        Q
      
    
    {\displaystyle P\to Q}
  
, one can constructively transform a proof of 
  
    
      
        P
      
    
    {\displaystyle P}
  
 into a proof of 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
. In particular, 
  
    
      
        P
        →
        P
      
    
    {\displaystyle P\to P}
  
 holds in intuitionistic logic. If 
  
    
      
        P
        →
        Q
        ⇒
        ¬
        P
        ∨
        Q
      
    
    {\displaystyle P\to Q\Rightarrow \neg P\lor Q}
  
 would hold, then 
  
    
      
        ¬
        P
        ∨
        P
      
    
    {\displaystyle \neg P\lor P}
  
 could be derived. However, the latter is the <a href="/facts/Law_of_excluded_middle/tDhbugCN">law of excluded middle</a>, which is not accepted by intuitionistic logic (one cannot assume 
  
    
      
        ¬
        P
        ∨
        P
      
    
    {\displaystyle \neg P\lor P}
  
 without knowing which case applies).
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Patrick J. Hurley (1 January 2011). A Concise Introduction to Logic. Cengage Learning. ISBN 978-0-8400-3417-5. <a href="978-0-8400-3417-5" target="_blank">978-0-8400-3417-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 371. <a href="/wiki/Irving_Copi" target="_blank">/wiki/Irving_Copi</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"Equivalence of a→b and ¬ a ∨ b". Mathematics Stack Exchange. <a href="https://math.stackexchange.com/q/243949" target="_blank">https://math.stackexchange.com/q/243949</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

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

Material implication (rule of inference)