Heyting algebras are bounded lattices that are also equipped with
an additional binary operation imp (for impliciation, also
written as →).
Implication obeys the following laws:
a → a = 1
a ∧ (a → b) = a ∧ b
b ∧ (a → b) = b
a → (b ∧ c) = (a → b) ∧ (a → c)
In heyting algebras, and is equivalent to meet and or is
equivalent to join; both methods are available.
Heyting algebra also define complement operation (sometimes
written as ¬a). The complement of a is equivalent to (a → 0),
and the following laws hold:
a ∧ ¬a = 0
However, in Heyting algebras this operation is only a
pseudo-complement, since Heyting algebras do not necessarily
provide the law of the excluded middle. This means that there is no
guarantee that (a ∨ ¬a) = 1.
Heyting algebras model intuitionistic logic. For a model of
classical logic, see the boolean algebra type class implemented as
Bool.
Heyting algebras are bounded lattices that are also equipped with an additional binary operation
imp
(for impliciation, also written as →).Implication obeys the following laws:
In heyting algebras,
and
is equivalent tomeet
andor
is equivalent tojoin
; both methods are available.Heyting algebra also define
complement
operation (sometimes written as ¬a). The complement ofa
is equivalent to(a → 0)
, and the following laws hold:However, in Heyting algebras this operation is only a pseudo-complement, since Heyting algebras do not necessarily provide the law of the excluded middle. This means that there is no guarantee that (a ∨ ¬a) = 1.
Heyting algebras model intuitionistic logic. For a model of classical logic, see the boolean algebra type class implemented as
Bool
.