Properties of Sentential Connectives

Reflexivity

Conditional and biconditional (but not conjunction and disjunction) are reflexive.

${\displaystyle \vDash \ \mathrm {P} \rightarrow \mathrm {P} \,\!}$

${\displaystyle \vDash \ \mathrm {P} \leftrightarrow \mathrm {P} \,\!}$

Commutativity

Conjunction, disjunction, and biconditional (but not conditional) are commutative.

${\displaystyle \mathrm {P} \land \mathrm {Q} \,\!}$       ${\displaystyle \mathrm {Q} \land \mathrm {P} \,\!}$

${\displaystyle \mathrm {P} \lor \mathrm {Q} \,\!}$       ${\displaystyle \mathrm {Q} \lor \mathrm {P} \,\!}$

${\displaystyle \mathrm {P} \leftrightarrow \mathrm {Q} \,\!}$       ${\displaystyle \mathrm {Q} \leftrightarrow \mathrm {P} \,\!}$

Associativity

Conjunction, disjunction, and biconditional (but not conditional) are associative.

${\displaystyle (\mathrm {P} \land \mathrm {Q} )\land \mathrm {R} \,\!}$       ${\displaystyle \mathrm {P} \land (\mathrm {Q} \land \mathrm {R} )\,\!}$

${\displaystyle (\mathrm {P} \lor \mathrm {Q} )\lor \mathrm {R} \,\!}$       ${\displaystyle \mathrm {P} \lor (\mathrm {Q} \lor \mathrm {R} )\,\!}$

${\displaystyle (\mathrm {P} \leftrightarrow \mathrm {Q} )\leftrightarrow \mathrm {R} \,\!}$       ${\displaystyle \mathrm {P} \leftrightarrow (\mathrm {Q} \leftrightarrow \mathrm {R} )\,\!}$

Distribution

We list ten distribution laws. Of these, probably the most important are that conjunction and disjunction distribute over each other and that conditional distributes over itself.

${\displaystyle \mathrm {P} \land (\mathrm {Q} \land \mathrm {R} )\,\!}$       ${\displaystyle (\mathrm {P} \land \mathrm {Q} )\land (\mathrm {P} \land \mathrm {R} )\,\!}$

${\displaystyle \mathrm {P} \land (\mathrm {Q} \lor \mathrm {R} )\,\!}$       ${\displaystyle (\mathrm {P} \land \mathrm {Q} )\lor (\mathrm {P} \land \mathrm {R} )\,\!}$

${\displaystyle \mathrm {P} \lor (\mathrm {Q} \land \mathrm {R} )\,\!}$       ${\displaystyle (\mathrm {P} \lor \mathrm {Q} )\land (\mathrm {P} \lor \mathrm {R} )\,\!}$

${\displaystyle \mathrm {P} \lor (\mathrm {Q} \lor \mathrm {R} )\,\!}$       ${\displaystyle (\mathrm {P} \lor \mathrm {Q} )\lor (\mathrm {P} \lor \mathrm {R} )\,\!}$

${\displaystyle \mathrm {P} \lor (\mathrm {Q} \rightarrow \mathrm {R} )\,\!}$       ${\displaystyle (\mathrm {P} \lor \mathrm {Q} )\rightarrow (\mathrm {P} \lor \mathrm {R} )\,\!}$

${\displaystyle \mathrm {P} \lor (\mathrm {Q} \leftrightarrow \mathrm {R} )\,\!}$       ${\displaystyle (\mathrm {P} \lor \mathrm {Q} )\leftrightarrow (\mathrm {P} \lor \mathrm {R} )\,\!}$

${\displaystyle \mathrm {P} \rightarrow (\mathrm {Q} \land \mathrm {R} )\,\!}$       ${\displaystyle (\mathrm {P} \rightarrow \mathrm {Q} )\land (\mathrm {P} \rightarrow \mathrm {R} )\,\!}$

${\displaystyle \mathrm {P} \rightarrow (\mathrm {Q} \lor \mathrm {R} )\,\!}$       ${\displaystyle (\mathrm {P} \rightarrow \mathrm {Q} )\lor (\mathrm {P} \rightarrow \mathrm {R} )\,\!}$

${\displaystyle \mathrm {P} \rightarrow (\mathrm {Q} \rightarrow \mathrm {R} )\,\!}$       ${\displaystyle (\mathrm {P} \rightarrow \mathrm {Q} )\rightarrow (\mathrm {P} \rightarrow \mathrm {R} )\,\!}$

${\displaystyle \mathrm {P} \rightarrow (\mathrm {Q} \leftrightarrow \mathrm {R} )\,\!}$       ${\displaystyle (\mathrm {P} \rightarrow \mathrm {Q} )\leftrightarrow (\mathrm {P} \rightarrow \mathrm {R} )\,\!}$

Transitivity

Conjunction, conditional, and biconditional (but not disjunction) are transitive.

${\displaystyle \vDash \ (\mathrm {P} \land \mathrm {Q} )\land (\mathrm {Q} \land \mathrm {R} )\rightarrow \mathrm {P} \land \mathrm {R} \,\!}$

${\displaystyle \vDash \ (\mathrm {P} \rightarrow \mathrm {Q} )\land (\mathrm {Q} \rightarrow \mathrm {R} )\rightarrow (\mathrm {P} \rightarrow \mathrm {R} )\,\!}$

${\displaystyle \vDash \ (\mathrm {P} \leftrightarrow \mathrm {Q} )\land (\mathrm {Q} \leftrightarrow \mathrm {R} )\rightarrow (\mathrm {P} \leftrightarrow \mathrm {R} )\,\!}$

Conditionals

These tautologies and equivalences are mostly about conditionals.

${\displaystyle \vDash \ \mathrm {P} \land \mathrm {Q} \rightarrow \mathrm {P} \qquad \qquad \vDash \ \mathrm {P} \land \mathrm {Q} \rightarrow \mathrm {Q} \,\!}$

${\displaystyle \vDash \ \mathrm {P} \rightarrow \mathrm {P} \lor \mathrm {Q} \qquad \qquad \vDash \ \mathrm {Q} \rightarrow \mathrm {P} \lor \mathrm {Q} \,\!}$

${\displaystyle \vDash \ (\mathrm {P} \rightarrow \mathrm {Q} )\lor (\mathrm {Q} \rightarrow \mathrm {R} )\,\!}$

${\displaystyle \vDash \ \lnot \mathrm {P} \rightarrow (\mathrm {P} \rightarrow \mathrm {Q} )\,\!}$      Conditional addition

${\displaystyle \vDash \ \mathrm {Q} \rightarrow (\mathrm {P} \rightarrow \mathrm {Q} )\,\!}$      Conditional addition

${\displaystyle \mathrm {P} \rightarrow \mathrm {Q} \,\!}$       ${\displaystyle \lnot \mathrm {Q} \rightarrow \lnot \mathrm {P} \,\!}$      Contraposition

${\displaystyle \mathrm {P} \land \mathrm {Q} \rightarrow \mathrm {R} \,\!}$       ${\displaystyle \mathrm {P} \rightarrow (\mathrm {Q} \rightarrow \mathrm {R} )\,\!}$      Exportation

Biconditionals

These tautologies and equivalences are mostly about biconditionals.

${\displaystyle \vDash \ \mathrm {P} \land \mathrm {Q} \rightarrow (\mathrm {P} \leftrightarrow \mathrm {Q} )\,\!}$      Biconditional addition

${\displaystyle \vDash \ \lnot \mathrm {P} \land \lnot \mathrm {Q} \rightarrow (\mathrm {P} \leftrightarrow \mathrm {Q} )\,\!}$      Biconditional addition

${\displaystyle \vDash (\mathrm {P} \leftrightarrow \mathrm {Q} )\lor (\mathrm {P} \leftrightarrow \lnot \mathrm {Q} )\,\!}$

${\displaystyle \lnot (\mathrm {P} \leftrightarrow \mathrm {Q} )\,\!}$       ${\displaystyle \lnot \mathrm {P} \leftrightarrow \mathrm {Q} \,\!}$       ${\displaystyle \mathrm {P} \leftrightarrow \lnot \mathrm {Q} \,\!}$

Miscellaneous

We repeat DeMorgan's Laws from the Interdefinability of connectives section of Expressibility and add two additional forms. We also list some additional tautologies and equivalences.

${\displaystyle \vDash \ \mathrm {P} \leftrightarrow \mathrm {P} \land \mathrm {P} \,\!}$      Idempotence for conjunction

${\displaystyle \vDash \ \mathrm {P} \leftrightarrow \mathrm {P} \lor \mathrm {P} \,\!}$      Idempotence for disjunction

${\displaystyle \vDash \ \mathrm {P} \rightarrow \mathrm {P} \lor \mathrm {Q} \,\!}$      Disjunctive addition

${\displaystyle \vDash \ \mathrm {Q} \rightarrow \mathrm {P} \lor \mathrm {Q} \,\!}$      Disjunctive addition

${\displaystyle \vDash \ \mathrm {P} \land \lnot \mathrm {P} \rightarrow \mathrm {Q} \,\!}$

${\displaystyle \mathrm {P} \land \mathrm {Q} \,\!}$       ${\displaystyle \lnot (\lnot \mathrm {P} \lor \lnot \mathrm {Q} )\,\!}$      Demorgan's Laws

${\displaystyle \mathrm {P} \lor \mathrm {Q} \,\!}$       ${\displaystyle \lnot (\lnot \mathrm {P} \land \lnot \mathrm {Q} )\,\!}$      Demorgan's Laws

${\displaystyle \lnot (\mathrm {P} \land \mathrm {Q} )\,\!}$       ${\displaystyle \lnot \mathrm {P} \lor \lnot \mathrm {Q} \,\!}$      Demorgan's Laws

${\displaystyle \lnot (\mathrm {P} \lor \mathrm {Q} )\,\!}$       ${\displaystyle \lnot \mathrm {P} \land \lnot \mathrm {Q} \,\!}$      Demorgan's Laws

${\displaystyle \mathrm {P} \,\!}$       ${\displaystyle \lnot \lnot \mathrm {P} \,\!}$      Double Negation

Deduction principle

Let ${\displaystyle \varphi \,\!}$ and ${\displaystyle \psi \,\!}$ both be formulae, and let ${\displaystyle \Gamma \,\!}$ be a set of formulae.

${\displaystyle \mathrm {If} \ \ \Gamma \cup \{\varphi \}\vDash \psi \mathrm {,\ then} \ \ \Gamma \vDash (\varphi \rightarrow \psi )\,\!}$

Reduction principle

Let ${\displaystyle \varphi \,\!}$ and ${\displaystyle \psi \,\!}$ both be formulae, and let ${\displaystyle \Gamma \,\!}$ be a set of formulae.

${\displaystyle \mathrm {If} \ \ \Gamma \cup \{\varphi \}\vDash \psi \ \ \mathrm {and} \ \ \Gamma \cup \{\varphi \}\vDash \lnot \psi \mathrm {,\ then} \ \ \Gamma \vDash \lnot \varphi ,\!}$

${\displaystyle \mathrm {If} \ \ \Gamma \cup \{\lnot \varphi \}\vDash \psi \ \ \mathrm {and} \ \ \Gamma \cup \{\lnot \varphi \}\vDash \lnot \psi \mathrm {,\ then} \ \ \Gamma \vDash \varphi \,\!}$