trait DeMorgan[A] extends Logic[A]

De Morgan algebras are bounded lattices that are also equipped with a De Morgan involution.

De Morgan involution obeys the following laws:

  • ¬¬a = a
  • ¬(x∧y) = ¬x∨¬y

However, in De Morgan algebras this involution does not necessarily provide the law of the excluded middle. This means that there is no guarantee that (a ∨ ¬a) = 1. De Morgan algebra do not not necessarily provide the law of non contradiction either. This means that there is no guarantee that (a ∧ ¬a) = 0.

De Morgan algebras are useful to model fuzzy logic. For a model of classical logic, see the boolean algebra type class implemented as Bool.

Self Type
DeMorgan[A]
Source
DeMorgan.scala
Linear Supertypes
Logic[A], BoundedDistributiveLattice[A], DistributiveLattice[A], BoundedLattice[A], BoundedJoinSemilattice[A], BoundedMeetSemilattice[A], Lattice[A], MeetSemilattice[A], JoinSemilattice[A], Serializable, Serializable, Any
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Inherited
  1. DeMorgan
  2. Logic
  3. BoundedDistributiveLattice
  4. DistributiveLattice
  5. BoundedLattice
  6. BoundedJoinSemilattice
  7. BoundedMeetSemilattice
  8. Lattice
  9. MeetSemilattice
  10. JoinSemilattice
  11. Serializable
  12. Serializable
  13. Any
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Visibility
  1. Public
  2. All

Abstract Value Members

  1. abstract def and(a: A, b: A): A
    Definition Classes
    Logic
  2. abstract def getClass(): Class[_]
    Definition Classes
    Any
  3. abstract def not(a: A): A
    Definition Classes
    Logic
  4. abstract def one: A
    Definition Classes
    BoundedMeetSemilattice
  5. abstract def or(a: A, b: A): A
    Definition Classes
    Logic
  6. abstract def zero: A
    Definition Classes
    BoundedJoinSemilattice

Concrete Value Members

  1. final def !=(arg0: Any): Boolean
    Definition Classes
    Any
  2. final def ##(): Int
    Definition Classes
    Any
  3. final def ==(arg0: Any): Boolean
    Definition Classes
    Any
  4. def asCommutativeRig: CommutativeRig[A]

    Return a CommutativeRig using join and meet.

    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.

    Definition Classes
    BoundedDistributiveLattice
  5. final def asInstanceOf[T0]: T0
    Definition Classes
    Any
  6. def dual: BoundedDistributiveLattice[A]

    This is the lattice with meet and join swapped

    This is the lattice with meet and join swapped

    Definition Classes
    BoundedDistributiveLattice → BoundedLattice → Lattice
  7. def equals(arg0: Any): Boolean
    Definition Classes
    Any
  8. def hashCode(): Int
    Definition Classes
    Any
  9. def imp(a: A, b: A): A
  10. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  11. def isOne(a: A)(implicit ev: Eq[A]): Boolean
    Definition Classes
    BoundedMeetSemilattice
  12. def isZero(a: A)(implicit ev: Eq[A]): Boolean
    Definition Classes
    BoundedJoinSemilattice
  13. def join(a: A, b: A): A
    Definition Classes
    DeMorgan → JoinSemilattice
  14. def joinPartialOrder(implicit ev: Eq[A]): PartialOrder[A]
    Definition Classes
    JoinSemilattice
  15. def joinSemilattice: BoundedSemilattice[A]
    Definition Classes
    BoundedJoinSemilattice → JoinSemilattice
  16. def meet(a: A, b: A): A
    Definition Classes
    DeMorgan → MeetSemilattice
  17. def meetPartialOrder(implicit ev: Eq[A]): PartialOrder[A]
    Definition Classes
    MeetSemilattice
  18. def meetSemilattice: BoundedSemilattice[A]
    Definition Classes
    BoundedMeetSemilattice → MeetSemilattice
  19. def nand(a: A, b: A): A
    Definition Classes
    Logic
  20. def nor(a: A, b: A): A
    Definition Classes
    Logic
  21. def nxor(a: A, b: A): A
    Definition Classes
    Logic
  22. def toString(): String
    Definition Classes
    Any
  23. def xor(a: A, b: A): A
    Definition Classes
    Logic

Inherited from Logic[A]

Inherited from BoundedDistributiveLattice[A]

Inherited from DistributiveLattice[A]

Inherited from BoundedLattice[A]

Inherited from BoundedJoinSemilattice[A]

Inherited from BoundedMeetSemilattice[A]

Inherited from Lattice[A]

Inherited from MeetSemilattice[A]

Inherited from JoinSemilattice[A]

Inherited from Serializable

Inherited from Serializable

Inherited from Any

Ungrouped