Boolean algebras are Heyting algebras with the additional constraint that the law of the excluded middle is true (equivalently, double-negation is true).
Boolean algebras are Heyting algebras with the additional constraint that the law of the excluded middle is true (equivalently, double-negation is true).
This means that in addition to the laws Heyting algebras obey, boolean algebras also obey the following:
- (a ∨ ¬a) = 1
- ¬¬a = a
Boolean algebras generalize classical logic: one is equivalent to
"true" and zero is equivalent to "false". Boolean algebras provide
additional logical operators such as xor, nand, nor, and
nxor which are commonly used.
Every boolean algebras has a dual algebra, which involves reversing true/false as well as and/or.
- Companion
- object
Value members
Concrete methods
Every Boolean algebra is a BoolRing, with multiplication defined as
and and addition defined as xor. Bool does not extend BoolRing
because, e.g. we might want a Bool[Int] and CommutativeRing[Int] to
refer to different structures, by default.
Every Boolean algebra is a BoolRing, with multiplication defined as
and and addition defined as xor. Bool does not extend BoolRing
because, e.g. we might want a Bool[Int] and CommutativeRing[Int] to
refer to different structures, by default.
Note that the ring returned by this method is not an extension of
the Rig returned from BoundedDistributiveLattice.asCommutativeRig.
- Definition Classes
Inherited methods
Return a CommutativeRig using join and meet. Note this must obey the commutative rig laws since meet(a, one) = a, and meet and join are associative, commutative and distributive.
Return a CommutativeRig using join and meet. Note this must obey the commutative rig laws since meet(a, one) = a, and meet and join are associative, commutative and distributive.
- Inherited from
- BoundedDistributiveLattice