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 () and false (), then the disjunction of the truth values and may be defined by a truth table:
That is, is true if and only if at least one of and is true. Disjunction also exists in nearly every non-classical logic.
More generally, if and 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
Disjunction as defined above is sometimes called inclusive disjunction to distinguish it from exclusive disjunction, where exactly one of and must be true.
In the context of substructural logics such as linear logic, we often have both additive disjunction and multiplicative disjunction ; 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, the disjunction of two mere propositions, and , is the bracket type of their sum type, . 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 , such as the one derived from a type universe - , then the disjunction of two types and is defined as the dependent function type
By weak function extensionality, the disjunction of two types is a proposition.
The two definitions above are equivalent.
The propositional truncation of a type is equivalent to the following dependent function type
Substituting the sum type for , we have
Given any type , there is an equivalence
and if , then . In addition, for all type families , and , if there is a family of equivalences , then there is an equivalence . All this taken together means that there are equivalences
If one has the boolean domain and the existential quantifier, then the disjunction of two types and is given by the following type:
The disjunction of two mere propositions and is also the join type of the two types . This is because every mere proposition is a subtype of the unit type, and the disjunction of and is the union of and as two subtypes of the unit types, and the union of and as subtypes of the unit type is defined to be the join type of and , the pushout type of the two product projection functions from the product type to and respectively.
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):
(A single arrow is implication in the object language; a double arrow is entailment in the metalanguage.) Note that is the negation of every item in this diagram.
In the double-negation interpretation? of classical logic in intuitionistic logic, is the interpretation in intuitionistic logic of disjunction in classical logic. For this reason, 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; 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, is the interpretation of the multiplicative disjunction for affirmative propositions. More generally, a statement in affine logic is interpreted as a pair of mutually contradictory statements in intuitionistic logic; is simply the negation of for affirmative propositions, but in general, only entails . Then is interpreted as ; that is, is , and is . (In contrast, the additive disjunction is interpreted as . Note that entails 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 from , and it's not even valid to derive without Markov's principle (or at least some weak version of it), but it is valid to derive (and conversely), where is the usual apartness relation between real numbers. (Here, is and is , and similarly for and .) Of course, it's also valid to derive (which is actually equivalent).
The rules of inference for disjunction in sequent calculus are dual to those for conjunction:
Equivalently, we can use the following rules with weakened contexts:
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 in these rules as ), while the second batch of rules apply to multiplicative disjunction (interpret in those rules as ).
The natural deduction rules for disjunction are a little more complicated than those for conjunction:
The definition of the disjunction of two types in dependent type theory as the propositional truncation of the sum type is found in:
The definition of the disjunction of two mere propositions in dependent type theory as the join type of propositions is found in:
And the disjunction of two types defined from the type of propositions and dependent product types can be found in:
Last revised on September 13, 2024 at 21:08:08. See the history of this page for a list of all contributions to it.