Absorption (logic)

<h2 id="formal-notation">Formal notation</h2>
<p>The <i>absorption</i> rule may be expressed as a <a href="/facts/Sequent/7YoB2hYb">sequent</a>:
</p>

P
        →
        Q
        ⊢
        P
        →
        (
        P
        ∧
        Q
        )
      
    
    {\displaystyle P\to Q\vdash P\to (P\land Q)}

<p>where 
  
    
      
        ⊢
      
    
    {\displaystyle \vdash }
  
 is a <a href="/facts/Metalogic/zfooQjCo">metalogical</a> symbol meaning that 
  
    
      
        P
        →
        (
        P
        ∧
        Q
        )
      
    
    {\displaystyle P\to (P\land Q)}
  
 is a <a href="/facts/Logical_consequence/AyDXp4dn">syntactic consequence</a> of 
  
    
      
        (
        P
        →
        Q
        )
      
    
    {\displaystyle (P\rightarrow Q)}
  
 in some <a href="/facts/Formal_system/KaeDAjK7">logical system</a>;
</p><p>and expressed as a truth-functional <a href="/facts/Tautology_(logic)/p82wWMaN">tautology</a> or <a href="/facts/Theorem/f2Pe1FMD">theorem</a> of <a href="/facts/Propositional_calculus/MNt5v31V">propositional logic</a>. The principle was stated as a theorem of propositional logic by <a href="/facts/Bertrand_Russell/SRpwNX2U">Russell</a> and <a href="/facts/Alfred_North_Whitehead/7KuylmP3">Whitehead</a> in  <i><a href="/facts/Principia_Mathematica/7ufeceV4">Principia Mathematica</a></i> as:
</p>

(
        P
        →
        Q
        )
        ↔
        (
        P
        →
        (
        P
        ∧
        Q
        )
        )
      
    
    {\displaystyle (P\to Q)\leftrightarrow (P\to (P\land Q))}

<p>where 
  
    
      
        P
      
    
    {\displaystyle P}
  
, and 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
 are propositions expressed in some <a href="/facts/Formal_system/KaeDAjK7">formal system</a>.
</p>
<h2 id="examples">Examples</h2>
<p>If it will rain, then I will wear my coat.
Therefore, if it will rain then it will rain and I will wear my coat.
</p>
<h2 id="proof-by-truth-table">Proof by truth table</h2>
<table><tbody><tr><th><i>                    P              {\displaystyle P}  </i></th><th><i>                    Q              {\displaystyle Q}  </i></th><th><i>                    P        →        Q              {\displaystyle P\rightarrow Q}  </i></th><th><i>                    P        →        (        P        ∧        Q        )              {\displaystyle P\rightarrow (P\land Q)}  </i></th></tr><tr><td>T</td><td>T</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>F</td><td>F</td></tr><tr><td>F</td><td>T</td><td>T</td><td>T</td></tr><tr><td>F</td><td>F</td><td>T</td><td>T</td></tr></tbody></table>
<h2 id="formal-proof">Formal proof</h2>
<table><tbody><tr><th><i>Proposition</i></th><th><i>Derivation</i></th></tr><tr><td>                    P        →        Q              {\displaystyle P\rightarrow Q}  </td><td>Given</td></tr><tr><td>                    ¬        P        ∨        Q              {\displaystyle \neg P\lor Q}  </td><td><a href="/facts/Material_implication_(rule_of_inference)/1WHxa8Se">Material implication</a></td></tr><tr><td>                    ¬        P        ∨        P              {\displaystyle \neg P\lor P}  </td><td><a href="/facts/Law_of_Excluded_Middle/tDhbugCN">Law of Excluded Middle</a></td></tr><tr><td>                    (        ¬        P        ∨        P        )        ∧        (        ¬        P        ∨        Q        )              {\displaystyle (\neg P\lor P)\land (\neg P\lor Q)}  </td><td><a href="/facts/Conjunction_introduction/XGt4Ie0R">Conjunction</a></td></tr><tr><td>                    ¬        P        ∨        (        P        ∧        Q        )              {\displaystyle \neg P\lor (P\land Q)}  </td><td><a href="/facts/Distribution_(logic)/0LNBrVJl">Reverse Distribution</a></td></tr><tr><td>                    P        →        (        P        ∧        Q        )              {\displaystyle P\rightarrow (P\land Q)}  </td><td>Material implication</td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Absorption_law/ffTPsQj7">Absorption law</a></li></ul>

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

<ol>
<li id="fn:1"><p>Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 362. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Rules of Inference". <a href="http://www.philosophypages.com/lg/e11a.htm" target="_blank">http://www.philosophypages.com/lg/e11a.htm</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Whitehead and Russell, Principia Mathematica, p. 14. <a href="/wiki/Principia_Mathematica" target="_blank">/wiki/Principia_Mathematica</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Absorption (logic) open-in-new

Absorption (logic)