nLab
disjunction

Definitions

In logic, logical disjunction is the join in the poset of truth values.

Assuming that (as in classical logic) the only truth values are true ($T$) and false ($F$), then the disjunction $p\vee q$ of the truth values $p$ and $q$ may be defined by a truth table:

$p$	$q$		$p\vee q$
$T$	$T$		$T$
$T$	$F$		$T$
$F$	$T$		$T$
$F$	$F$		$F$

That is, $p\wedge q$ is true if and only if at least one of $p$ and $q$ is true. Disjunction also exists in nearly every non-classical logic.

More generally, if $p$ and $q$ are any two relations on the same domain, then we define their disjunction pointwise, thinking of a relation as a function to truth values. If instead we think of a relation as a subset of its domain, then disjunction becomes union.

Remarks

Disjunction as defined above is sometimes called inclusive disjunction to distinguish it from exclusive disjunction, where exactly one of $p$ and $q$ must be true.

Disjunction is de Morgan dual to conjunction.

Like any join, disjunction is an associative operation, so we can take the disjunction of any finite positive whole number of truth values; the disjunction is true if and only if at least one of the various truth values is true. Disjunction also has an identity element, which is the false truth value. Some logics allow a notion of infinitary disjunction. Indexed disjunction is existential quantification.

Rules of inference

The rules of inference? for disjunction in sequent calculus are dual to those for conjunction:

$$\begin{array}{c}\frac{\Gamma ,p,\Delta ⊢\Sigma ;\Gamma ,q,\Delta ⊢\Sigma }{\Gamma ,p\vee q,\Delta ⊢\Sigma }\text{left additive}\\ \frac{\Gamma ⊢\Delta ,p,\Sigma }{\Gamma ⊢\Delta ,p\vee q,\Sigma }\text{right additive 0}\\ \frac{\Gamma ⊢\Delta ,q,\Sigma }{\Gamma ⊢\Delta ,p\vee q,\Sigma }\text{right additive 1}\\ \end{array}$$\begin {gathered} \frac { \Gamma , p , \Delta \vdash \Sigma ; \; \Gamma , q , \Delta \vdash \Sigma } { \Gamma , p \vee q , \Delta \vdash \Sigma } \; \text {left additive} \\ \frac { \Gamma \vdash \Delta , p , \Sigma } { \Gamma \vdash \Delta , p \vee q , \Sigma } \; \text {right additive 0} \\ \frac { \Gamma \vdash \Delta , q , \Sigma } { \Gamma \vdash \Delta , p \vee q , \Sigma } \; \text {right additive 1} \\ \end {gathered}

Equivalently, we can use the following rules with weakened contexts:

$$\begin{array}{c}\frac{\Gamma ,p⊢\Delta ;q,\Sigma ⊢\Pi }{\Gamma ,p\vee q,\Sigma ⊢\Delta ,\Pi }\text{left multiplicative}\\ \frac{\Gamma ⊢\Delta ,p,q,\Sigma }{\Gamma ⊢\Delta ,p\vee q,\Sigma }\text{right multiplicative}\\ \end{array}$$\begin {gathered} \frac { \Gamma , p \vdash \Delta ; \; q , \Sigma \vdash \Pi } { \Gamma , p \vee q , \Sigma \vdash \Delta , \Pi } \; \text {left multiplicative} \\ \frac { \Gamma \vdash \Delta , p , q , \Sigma } { \Gamma \vdash \Delta , p \vee q , \Sigma } \text {right multiplicative} \\ \end {gathered}

The rules above are written so as to remain valid in logics without the exchange rule. In linear logic, the first batch of sequent rules apply to additive disjunction (interpret $p\vee q$ in these rules as $p\oplus q$), while the second batch of rules apply to multiplicative disjunction (interpret $p\vee q$ in those rules as $p⅋q$).

The natural deduction rules for disjunction are a little more complicated than those for conjunction:

$$\begin{array}{c}\frac{\Gamma ,p⊢r;\Gamma ,q⊢r}{\Gamma ,p\vee q⊢r}\text{elimination}\\ \frac{\Gamma ⊢p}{\Gamma ⊢p\vee q}\text{introduction 0}\\ \frac{\Gamma ⊢q}{\Gamma ⊢p\vee q}\text{introduction 1}\\ \end{array}$$\begin {gathered} \frac { \Gamma , p \vdash r ; \; \Gamma , q \vdash r } { \Gamma , p \vee q \vdash r } \; \text {elimination} \\ \frac { \Gamma \vdash p } { \Gamma \vdash p \vee q } \; \text {introduction 0} \\ \frac { \Gamma \vdash q } { \Gamma \vdash p \vee q } \; \text {introduction 1} \\ \end {gathered}