Absorption (logic)

<p>Absorption is a <a href="/facts/Validity_(logic)/oMGkywGC">valid</a> <a href="/facts/Argument_form/foVBtMX4">argument form</a> and <a href="/facts/Rule_of_inference/QV5NklQ9">rule of inference</a> of <a href="/facts/Propositional_logic/MNt5v31V">propositional logic</a>. The rule states that if 
  
    
      
        P
      
    
    {\displaystyle P}
  
 implies 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
, then 
  
    
      
        P
      
    
    {\displaystyle P}
  
 implies 
  
    
      
        P
      
    
    {\displaystyle P}
  
 and 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
. The rule makes it possible to introduce <a href="/facts/Logical_conjunction/62uyyn1d">conjunctions</a> to <a href="/facts/Formal_proof/6nEhV4cw">proofs</a>. It is called the law of absorption because the term 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
 is "absorbed" by the term 
  
    
      
        P
      
    
    {\displaystyle P}
  
 in the <a href="/facts/Consequent/UIzcuaKF">consequent</a>. The rule can be stated:
</p>

P
              →
              Q
            
            
              ∴
              P
              →
              (
              P
              ∧
              Q
              )
            
          
        
      
    
    {\displaystyle {\frac {P\to Q}{\therefore P\to (P\land Q)}}}

<p>where the rule is that wherever an instance of "
  
    
      
        P
        →
        Q
      
    
    {\displaystyle P\to Q}
  
" appears on a line of a proof, "
  
    
      
        P
        →
        (
        P
        ∧
        Q
        )
      
    
    {\displaystyle P\to (P\land Q)}
  
" can be placed on a subsequent line.
</p>

Absorption (logic) open-in-new

Absorption (logic)