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, $p \vee q$ 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.

In natural deduction the inference rules for disjunction are given as

$$\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma \vdash P \vee Q \; \mathrm{prop}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma, P \; \mathrm{true} \vdash P \vee Q \; \mathrm{true}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma, Q \; \mathrm{true} \vdash P \vee Q \; \mathrm{true}}$$

$$\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop} \quad \Gamma, P \vee Q \; \mathrm{true} \vdash R \; \mathrm{prop} \quad \Gamma, P \; \mathrm{true} \vdash R \; \mathrm{true} \quad \Gamma, Q \; \mathrm{true} \vdash R \; \mathrm{true}}{\Gamma, P \vee Q \; \mathrm{true} \vdash R \; \mathrm{true}}$$

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.

In the context of substructural logics such as linear logic, we often have both additive disjunction $\oplus$ and multiplicative disjunction $\parr$; see the Rules of Inference below for the distinction. In linear logic, additive disjunction is the join under the entailment relation, just like disjunction in classical logic (and intuitionistic logic), while multiplicative disjunction is something different.

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.

In dependent type theory

In dependent type theory, the disjunction of two mere propositions, $P$ and $Q$, is the bracket type of their sum type, $\| P + Q \|$. Disjunction types in general could also be regarded as a particular sort of higher inductive type. In Coq syntax:

Inductive disjunction (P Q:Type) : Type :=
| inl : P -> disjunction P Q
| inr : Q -> disjunction P Q
| contr0 : forall (p q : disjunction P Q) p == q

If the dependent type theory has a type of propositions $\mathrm{Prop}$, such as the one derived from a type universe $U$ - $\sum_{A:U} \mathrm{isProp}(A)$, then the disjunction of two types $A$ and $B$ is defined as the dependent function type

$$A \vee B \equiv \prod_{P:\mathrm{Prop}} ((A \to P) \times (B \to P)) \to P$$

By weak function extensionality, the disjunction of two types is a proposition.

Theorem

The two definitions above are equivalent.

Proof

The propositional truncation of a type $A$ is equivalent to the following dependent function type

$$\| A \| \simeq \prod_{P:\mathrm{Prop}} (A \to P) \to P$$

Substituting the sum type $A + B$ for $A$, we have

$$\| A + B \| \simeq \prod_{P:\mathrm{Prop}} ((A + B) \to P) \to P$$

Given any type $C$, there is an equivalence

$$((A + B) \to C) \simeq (A \to C) \times (B \to C)$$

and if $A \simeq B$, then $(A \to C) \simeq (B \to C)$. In addition, for all type families $x:A \vdash B(x)$, and $x:A \vdash C(x)$, if there is a family of equivalences $e:\prod_{x:A} B(x) \simeq C(x)$, then there is an equivalence $\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A} C(x)\right)$. All this taken together means that there are equivalences

$$\| A + B \| \simeq \left(\prod_{P:\mathrm{Prop}} ((A + B) \to P) \to P\right) \simeq \left(\prod_{P:\mathrm{Prop}} ((A \to P) \times (B \to P)) \to P\right)$$

If one has the type of booleans and the existential quantifier, then the disjunction of two types $A$ and $B$ is given by the following type:

$$A \vee B \coloneqq \exists b:\mathrm{bool}.((b = \mathrm{true}) \to A) \times ((b = \mathrm{false}) \to B)$$

The disjunction $P \vee Q$ of two mere propositions $P$ and $Q$ is also the join type of the two types $P * Q$. This is because every mere proposition is a subtype of the unit type, and the disjunction of $P$ and $Q$ is the union of $P$ and $Q$ as two subtypes of the unit types, and the union of $P$ and $Q$ as subtypes of the unit type is defined to be the join type of $P$ and $Q$, the pushout type of the two product projection functions from the product type $P \times Q$ to $P$ and $Q$ respectively.

Classical vs constructive

There are a variety of connectives that are distinct in intuitionistic logic but are all equivalent to disjunction in classical logic. Here is a Hasse diagram of some of them, with the strongest statement at the bottom and the weakest at the top (so that each statement entails those above it):

$$\array { & & \neg(\neg{P} \wedge \neg{Q}) \\ & ⇗ & & ⇖ \\ \neg{P} \rightarrow Q & & & & P \leftarrow \neg{Q} \\ & ⇖ & & ⇗ \\ & & (\neg{P} \rightarrow Q) \wedge (P \leftarrow \neg{Q}) \\ & & \Uparrow \\ & & P \vee Q }$$

(A single arrow is implication in the object language; a double arrow is entailment in the metalanguage.) Note that $\neg{P} \wedge \neg{Q}$ is the negation of every item in this diagram.

In the double-negation interpretation? of classical logic in intuitionistic logic, $\neg(\neg{P} \wedge \neg{Q})$ is the interpretation in intuitionistic logic of disjunction in classical logic. For this reason, $\neg(\neg{P} \wedge \neg{Q})$ is sometimes called classical disjunction. But this doesn't mean that it should always be used when turning classical mathematics into constructive mathematics. Indeed, a stronger statement is almost always preferable, if one is valid; $\neg(\neg{P} \wedge \neg{Q})$ is merely the fallback position when nothing better can be found. (And as can be seen in the example in the paragraph after next, sometimes even this is not valid.)

In the antithesis interpretation of affine logic in intuitionistic logic, $(\neg{P} \rightarrow Q) \wedge (P \leftarrow \neg{Q})$ is the interpretation of the multiplicative disjunction $P \parr Q$ for affirmative propositions. More generally, a statement $P$ in affine logic is interpreted as a pair $(P^+,P^-)$ of mutually contradictory statements in intuitionistic logic; $P^-$ is simply the negation of $P^+$ for affirmative propositions, but in general, $P^-$ only entails $\neg{P^+}$. Then $P \parr Q$ is interpreted as $\big((P^- \rightarrow Q^+) \wedge (P^+ \leftarrow Q^-), P^- \wedge Q^-\big)$; that is, $(P \parr Q)^+$ is $(P^- \rightarrow Q^+) \wedge (P^+ \leftarrow Q^-)$, and $(P \parr Q)^-$ is $P^- \wedge Q^-$. (In contrast, the additive disjunction $P \oplus Q$ is interpreted as $(P^+ \vee Q^+, P^- \wedge Q^-)$. Note that $P \oplus Q$ entails $P \parr Q$ in affine logic, even though they are independent in linear logic.)

For a non-affirmative example, in the arithmetic of (located) real numbers, it is not constructively valid to derive $(a = 0) \vee (b = 0)$ from $a b = 0$, and it's not even valid to derive $\neg\big(\neg(a = 0) \wedge \neg(b = 0)\big)$ without Markov's principle (or at least some weak version of it), but it is valid to derive $(a \# 0) \rightarrow (b = 0)$ (and conversely), where $\#$ is the usual apartness relation between real numbers. (Here, $P^+$ is $a = 0$ and $P^-$ is $a \# 0$, and similarly for $Q$ and $b$.) Of course, it's also valid to derive $\neg\big((a \# 0) \wedge (b \# 0)\big)$ (which is actually equivalent).

Rules of inference

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

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

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

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

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 $p \parr q$).

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

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

Related concepts

$\phantom{-}$symbol$\phantom{-}$	$\phantom{-}$in logic$\phantom{-}$
$\phantom{A}$$\in$	$\phantom{A}$element relation
$\phantom{A}$$\,:$	$\phantom{A}$typing relation
$\phantom{A}$$=$	$\phantom{A}$equality
$\phantom{A}$$\vdash$$\phantom{A}$	$\phantom{A}$entailment / sequent$\phantom{A}$
$\phantom{A}$$\top$$\phantom{A}$	$\phantom{A}$true / top$\phantom{A}$
$\phantom{A}$$\bot$$\phantom{A}$	$\phantom{A}$false / bottom$\phantom{A}$
$\phantom{A}$$\Rightarrow$	$\phantom{A}$implication
$\phantom{A}$$\Leftrightarrow$	$\phantom{A}$logical equivalence
$\phantom{A}$$\not$	$\phantom{A}$negation
$\phantom{A}$$\neq$	$\phantom{A}$negation of equality / apartness$\phantom{A}$
$\phantom{A}$$\notin$	$\phantom{A}$negation of element relation $\phantom{A}$
$\phantom{A}$$\not \not$	$\phantom{A}$negation of negation$\phantom{A}$
$\phantom{A}$$\exists$	$\phantom{A}$existential quantification$\phantom{A}$
$\phantom{A}$$\forall$	$\phantom{A}$universal quantification$\phantom{A}$
$\phantom{A}$$\wedge$	$\phantom{A}$logical conjunction
$\phantom{A}$$\vee$	$\phantom{A}$logical disjunction
symbol	in type theory (propositions as types)
$\phantom{A}$$\to$	$\phantom{A}$function type (implication)
$\phantom{A}$$\times$	$\phantom{A}$product type (conjunction)
$\phantom{A}$$+$	$\phantom{A}$sum type (disjunction)
$\phantom{A}$$0$	$\phantom{A}$empty type (false)
$\phantom{A}$$1$	$\phantom{A}$unit type (true)
$\phantom{A}$$=$	$\phantom{A}$identity type (equality)
$\phantom{A}$$\simeq$	$\phantom{A}$equivalence of types (logical equivalence)
$\phantom{A}$$\sum$	$\phantom{A}$dependent sum type (existential quantifier)
$\phantom{A}$$\prod$	$\phantom{A}$dependent product type (universal quantifier)
symbol	in linear logic
$\phantom{A}$$\multimap$$\phantom{A}$	$\phantom{A}$linear implication$\phantom{A}$
$\phantom{A}$$\otimes$$\phantom{A}$	$\phantom{A}$multiplicative conjunction$\phantom{A}$
$\phantom{A}$$\oplus$$\phantom{A}$	$\phantom{A}$additive disjunction$\phantom{A}$
$\phantom{A}$$\&$$\phantom{A}$	$\phantom{A}$additive conjunction$\phantom{A}$
$\phantom{A}$$\invamp$$\phantom{A}$	$\phantom{A}$multiplicative disjunction$\phantom{A}$
$\phantom{A}$$\;!$$\phantom{A}$	$\phantom{A}$exponential conjunction$\phantom{A}$
$\phantom{A}$$\;?$$\phantom{A}$	$\phantom{A}$exponential disjunction$\phantom{A}$

References

• Landon D. C. Elkind, Richard Zach, The Genealogy of $\vee$ (arXiv:2012.06072)

The definition of the disjunction of two types in dependent type theory as the propositional truncation of the sum type is found in:

• Univalent Foundations Project, §3.7 in: Homotopy Type Theory – Univalent Foundations of Mathematics (2013) [web, pdf]

The definition of the disjunction of two mere propositions in dependent type theory as the join type of propositions is found in:

• Mike Shulman, Cohomology, Homotopy type theory blog (web)

And the disjunction of two types defined from the type of propositions and dependent product types can be found in:

• Madeleine Birchfield, Constructing coproduct types and boolean types from universes, MathOverflow (web)