Editing Distributive property (section)

== Propositional logic ==
{{Transformation rules}}

=== Rule of replacement ===

In standard truth-functional propositional logic, {{em|distribution}}<ref>[[Elliott Mendelson]] (1964) ''Introduction to Mathematical Logic'', page 21, D. Van Nostrand Company</ref><ref>[[Alfred Tarski]] (1941) ''Introduction to Logic'', page 52, [[Oxford University Press]]</ref> in logical proofs uses two valid [[Rule of replacement|rules of replacement]] to expand individual occurrences of certain [[logical connective]]s, within some [[Logical formula|formula]], into separate applications of those connectives across subformulas of the given formula. The rules are
<math display="block">(P \land (Q \lor R)) \Leftrightarrow ((P \land Q) \lor (P \land R)) \qquad \text{ and } \qquad (P \lor (Q \land R)) \Leftrightarrow ((P \lor Q) \land (P \lor R))</math>
where "<math>\Leftrightarrow</math>", also written <math>\,\equiv,\,</math> is a [[metalogic]]al [[Symbol (formal)|symbol]] representing "can be replaced in a proof with" or "is [[Logical equivalence|logically equivalent]] to".

=== Truth functional connectives ===

{{em|Distributivity}} is a property of some logical connectives of truth-functional [[propositional logic]]. The following logical equivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functional [[Tautology (logic)|tautologies]].
<math display="block">\begin{alignat}{13}
&(P &&\;\land &&(Q \lor R)) &&\;\Leftrightarrow\;&& ((P \land Q) &&\;\lor (P \land R)) && \quad\text{ Distribution of } && \text{ conjunction } && \text{ over } && \text{ disjunction } \\
&(P &&\;\lor &&(Q \land R)) &&\;\Leftrightarrow\;&& ((P \lor Q) &&\;\land (P \lor R)) && \quad\text{ Distribution of } && \text{ disjunction } && \text{ over } && \text{ conjunction } \\
&(P &&\;\land &&(Q \land R)) &&\;\Leftrightarrow\;&& ((P \land Q) &&\;\land (P \land R)) && \quad\text{ Distribution of } && \text{ conjunction } && \text{ over } && \text{ conjunction } \\
&(P &&\;\lor &&(Q \lor R)) &&\;\Leftrightarrow\;&& ((P \lor Q) &&\;\lor (P \lor R)) && \quad\text{ Distribution of } && \text{ disjunction } && \text{ over } && \text{ disjunction } \\
&(P &&\to &&(Q \to R)) &&\;\Leftrightarrow\;&& ((P \to Q) &&\to (P \to R)) && \quad\text{ Distribution of } && \text{ implication } && \text{  } && \text{  } \\
&(P &&\to &&(Q \leftrightarrow R)) &&\;\Leftrightarrow\;&& ((P \to Q) &&\leftrightarrow (P \to R)) && \quad\text{ Distribution of } && \text{ implication } && \text{ over } && \text{ equivalence } \\
&(P &&\to &&(Q \land R)) &&\;\Leftrightarrow\;&& ((P \to Q) &&\;\land (P \to R)) && \quad\text{ Distribution of } && \text{ implication } && \text{ over } && \text{ conjunction } \\
&(P &&\;\lor &&(Q \leftrightarrow R)) &&\;\Leftrightarrow\;&& ((P \lor Q) &&\leftrightarrow (P \lor R)) && \quad\text{ Distribution of } && \text{ disjunction } && \text{ over } && \text{ equivalence } \\
\end{alignat}</math>

;Double distribution: 
<math display="block">\begin{alignat}{13}
&((P \land Q) &&\;\lor (R \land S)) &&\;\Leftrightarrow\;&& (((P \lor R) \land (P \lor S)) &&\;\land ((Q \lor R) \land (Q \lor S))) && \\
&((P \lor Q) &&\;\land (R \lor S)) &&\;\Leftrightarrow\;&& (((P \land R) \lor (P \land S)) &&\;\lor ((Q \land R) \lor (Q \land S))) && \\
\end{alignat}</math>