Commutativity of conjunction

<h2 id="formal-notation">Formal notation</h2>
Commutativity of conjunction can be expressed in <a href="/facts/Sequent/7YoB2hYb">sequent</a> notation as:

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

and

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

where 
 
 
 
 ⊢
 
 
 {\displaystyle \vdash }
 
 is a <a href="/facts/Metalogic/zfooQjCo">metalogical</a> symbol meaning that 
 
 
 
 (
 Q
 ∧
 P
 )
 
 
 {\displaystyle (Q\land P)}
 
 is a <a href="/facts/Logical_consequence/AyDXp4dn">syntactic consequence</a> of 
 
 
 
 (
 P
 ∧
 Q
 )
 
 
 {\displaystyle (P\land Q)}
 
, in the one case, and 
 
 
 
 (
 P
 ∧
 Q
 )
 
 
 {\displaystyle (P\land Q)}
 
 is a syntactic consequence of 
 
 
 
 (
 Q
 ∧
 P
 )
 
 
 {\displaystyle (Q\land P)}
 
 in the other, in some <a href="/facts/Formal_system/KaeDAjK7">logical system</a>;
or in <a href="/facts/Rule_of_inference/QV5NklQ9">rule form</a>:

P
              ∧
              Q
            
            
              ∴
              Q
              ∧
              P
            
          
        
      
    
    {\displaystyle {\frac {P\land Q}{\therefore Q\land P}}}

and

Q
              ∧
              P
            
            
              ∴
              P
              ∧
              Q
            
          
        
      
    
    {\displaystyle {\frac {Q\land P}{\therefore P\land Q}}}

where the rule is that wherever an instance of "
 
 
 
 (
 P
 ∧
 Q
 )
 
 
 {\displaystyle (P\land Q)}
 
" appears on a line of a proof, it can be replaced with "
 
 
 
 (
 Q
 ∧
 P
 )
 
 
 {\displaystyle (Q\land P)}
 
" and wherever an instance of "
 
 
 
 (
 Q
 ∧
 P
 )
 
 
 {\displaystyle (Q\land P)}
 
" appears on a line of a proof, it can be replaced with "
 
 
 
 (
 P
 ∧
 Q
 )
 
 
 {\displaystyle (P\land Q)}
 
";
or as the statement of a truth-functional tautology or <a href="/facts/Theorem/f2Pe1FMD">theorem</a> of propositional logic:

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

and

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

where 
 
 
 
 P
 
 
 {\displaystyle P}
 
 and 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 are <a href="/facts/Proposition/0877MFMD">propositions</a> expressed in some formal system.

<h2 id="generalized-principle">Generalized principle</h2>
For any propositions H1, H2, ... Hn, and permutation σ(n) of the numbers 1 through n, it is the case that:

H1 
 
 
 
 ∧
 
 
 {\displaystyle \land }
 
 H2 
 
 
 
 ∧
 
 
 {\displaystyle \land }
 
 ... 
 
 
 
 ∧
 
 
 {\displaystyle \land }
 
 Hn
is equivalent to

Hσ(1) 
 
 
 
 ∧
 
 
 {\displaystyle \land }
 
 Hσ(2) 
 
 
 
 ∧
 
 
 {\displaystyle \land }
 
 Hσ(n).
For example, if H1 is

It is raining
H2 is

<a href="/facts/Socrates/djIKjImv">Socrates</a> is mortal
and H3 is

2+2=4
then
It is raining and Socrates is mortal and 2+2=4
is equivalent to
Socrates is mortal and 2+2=4 and it is raining
and the other orderings of the predicates.

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

<ol>
<li id="fn:1">Elliott Mendelson (1997). Introduction to Mathematical Logic. CRC Press. ISBN 0-412-80830-7. <a href="0-412-80830-7" target="_blank">0-412-80830-7</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Commutativity of conjunction open-in-new

Commutativity of conjunction