trait Bool[A] extends Heyting[A] with GenBool[A]
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.
- Self Type
- Bool[A]
- Source
- Bool.scala
- Alphabetic
- By Inheritance
- Bool
- GenBool
- Heyting
- BoundedDistributiveLattice
- DistributiveLattice
- BoundedLattice
- BoundedJoinSemilattice
- BoundedMeetSemilattice
- Lattice
- MeetSemilattice
- JoinSemilattice
- Serializable
- Any
- Hide All
- Show All
- Public
- Protected
Abstract Value Members
- abstract def and(a: A, b: A): A
- Definition Classes
- GenBool
- abstract def complement(a: A): A
- Definition Classes
- Heyting
- abstract def getClass(): Class[_ <: AnyRef]
- Definition Classes
- Any
- abstract def one: A
- Definition Classes
- BoundedMeetSemilattice
- abstract def or(a: A, b: A): A
- Definition Classes
- GenBool
- abstract def zero: A
- Definition Classes
- BoundedJoinSemilattice
Concrete Value Members
- final def !=(arg0: Any): Boolean
- Definition Classes
- Any
- final def ##: Int
- Definition Classes
- Any
- final def ==(arg0: Any): Boolean
- Definition Classes
- Any
- def asBoolRing: BoolRing[A]
Every Boolean algebra is a BoolRing, with multiplication defined as
andand addition defined asxor.Every Boolean algebra is a BoolRing, with multiplication defined as
andand addition defined asxor. 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
Rigreturned fromBoundedDistributiveLattice.asCommutativeRig. - final def asInstanceOf[T0]: T0
- Definition Classes
- Any
- def dual: Bool[A]
This is the lattice with meet and join swapped
This is the lattice with meet and join swapped
- Definition Classes
- Bool → BoundedDistributiveLattice → BoundedLattice → Lattice
- def equals(arg0: Any): Boolean
- Definition Classes
- Any
- def hashCode(): Int
- Definition Classes
- Any
- def imp(a: A, b: A): A
- final def isInstanceOf[T0]: Boolean
- Definition Classes
- Any
- def isOne(a: A)(implicit ev: Eq[A]): Boolean
- Definition Classes
- BoundedMeetSemilattice
- def isZero(a: A)(implicit ev: Eq[A]): Boolean
- Definition Classes
- BoundedJoinSemilattice
- def join(a: A, b: A): A
- Definition Classes
- GenBool → JoinSemilattice
- def joinPartialOrder(implicit ev: Eq[A]): PartialOrder[A]
- Definition Classes
- JoinSemilattice
- def joinSemilattice: BoundedSemilattice[A]
- Definition Classes
- BoundedJoinSemilattice → JoinSemilattice
- def meet(a: A, b: A): A
- Definition Classes
- GenBool → MeetSemilattice
- def meetPartialOrder(implicit ev: Eq[A]): PartialOrder[A]
- Definition Classes
- MeetSemilattice
- def meetSemilattice: BoundedSemilattice[A]
- Definition Classes
- BoundedMeetSemilattice → MeetSemilattice
- def nand(a: A, b: A): A
- Definition Classes
- Heyting
- def nor(a: A, b: A): A
- Definition Classes
- Heyting
- def nxor(a: A, b: A): A
- Definition Classes
- Heyting
- def toString(): String
- Definition Classes
- Any
- def without(a: A, b: A): A
The operation of relative complement, symbolically often denoted
a\b(the symbol for set-theoretic difference, which is the meaning of relative complement in the lattice of sets). - def xor(a: A, b: A): A
Logical exclusive or, set-theoretic symmetric difference.