Disjunction introduction

Disjunction introduction or addition (also called or introduction) is a <a href="/facts/Rule_of_inference/QV5NklQ9">rule of inference</a> of <a href="/facts/Propositional_calculus/MNt5v31V">propositional logic</a> and almost every other <a href="/facts/Deduction_system/KaeDAjK7">deduction system</a>. The rule makes it possible to introduce <a href="/facts/Logical_disjunction/jbI2EGlM">disjunctions</a> to <a href="/facts/Formal_proof/6nEhV4cw">logical proofs</a>. It is the <a href="/facts/Inference/RBbe95bP">inference</a> that if P is true, then P or Q must be true.
An example in <a href="/facts/English_language/tpkxGt7b">English</a>:

Socrates is a man.
Therefore, Socrates is a man or pigs are flying in formation over the English Channel.
The rule can be expressed as:

P
            
              ∴
              P
              ∨
              Q
            
          
        
      
    
    {\displaystyle {\frac {P}{\therefore P\lor Q}}}

where the rule is that whenever instances of "
 
 
 
 P
 
 
 {\displaystyle P}
 
" appear on lines of a proof, "
 
 
 
 P
 ∨
 Q
 
 
 {\displaystyle P\lor Q}
 
" can be placed on a subsequent line. 
More generally it's also a simple <a href="/facts/Validity_(logic)/oMGkywGC">valid</a> <a href="/facts/Logical_form/foVBtMX4">argument form</a>, this means that if the premise is true, then the conclusion is also true as any rule of inference should be, and an <a href="/facts/Immediate_inference/yq5L6lQz">immediate inference</a>, as it has a single proposition in its premises. 
Disjunction introduction is not a rule in some <a href="/facts/Paraconsistent_logic/6HLVEvrr">paraconsistent logics</a> because in combination with other rules of logic, it leads to <a href="/facts/Principle_of_explosion/qMyRl2Ya">explosion</a> (i.e. everything becomes provable) and paraconsistent logic tries to avoid explosion and to be able to reason with contradictions. One of the solutions is to introduce disjunction with over rules. See <a href="/facts/Paraconsistent_logic/6HLVEvrr">Paraconsistent logic § Tradeoffs</a>.

Disjunction introduction open-in-new

Disjunction introduction