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sealed abstract class Polarity extends AnyRef

Polarity of a formula.

There are two polarities, positive/in-succedent and negative/in-antecedent. Polarities serve multiple purposes:

  • They distinguish strong and weak quantifiers. A universal (∀) quantifier in positive/in-succedent polarity is strong (requires an eigenvariable/Skolem inference), and is weak in negative/in-antecedent polarity (can be instantiated with any term).
  • They guide the construction of expansion trees and their deep formulas. A merge in positive polarity has a disjunction as deep formula, in negative polarity it has a conjunction.
  • They specify the side/cedent of a sequent.

Our convention is based on proofs in LK:

  • formulas in the succedent are positive
  • formulas in the antecedent are negative

This is used consistently, except for a major exception: in resolution proofs, the polarity is reversed. A formula in the antecedent of a clause in a resolution proof has the negative/in-antecedent polarity (as it is in the antecedent of a sequent). However upon conversion to LK/ET, the polarity switches to positive/in-succedent polarity.

Source
Polarity.scala
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Abstract Value Members

  1. abstract def inSuc: Boolean
  2. abstract def unary_!: Polarity

Concrete Value Members

  1. final def !=(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  2. final def ##(): Int
    Definition Classes
    AnyRef → Any
  3. def +(other: String): String
    Implicit
    This member is added by an implicit conversion from Polarity to any2stringadd[Polarity] performed by method any2stringadd in scala.Predef.
    Definition Classes
    any2stringadd
  4. def ->[B](y: B): (Polarity, B)
    Implicit
    This member is added by an implicit conversion from Polarity to ArrowAssoc[Polarity] performed by method ArrowAssoc in scala.Predef.
    Definition Classes
    ArrowAssoc
    Annotations
    @inline()
  5. final def ==(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  6. final def asInstanceOf[T0]: T0
    Definition Classes
    Any
  7. def clone(): AnyRef
    Attributes
    protected[java.lang]
    Definition Classes
    AnyRef
    Annotations
    @native() @throws( ... )
  8. def ensuring(cond: (Polarity) ⇒ Boolean, msg: ⇒ Any): Polarity
    Implicit
    This member is added by an implicit conversion from Polarity to Ensuring[Polarity] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  9. def ensuring(cond: (Polarity) ⇒ Boolean): Polarity
    Implicit
    This member is added by an implicit conversion from Polarity to Ensuring[Polarity] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  10. def ensuring(cond: Boolean, msg: ⇒ Any): Polarity
    Implicit
    This member is added by an implicit conversion from Polarity to Ensuring[Polarity] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  11. def ensuring(cond: Boolean): Polarity
    Implicit
    This member is added by an implicit conversion from Polarity to Ensuring[Polarity] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  12. final def eq(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  13. def equals(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  14. def finalize(): Unit
    Attributes
    protected[java.lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( classOf[java.lang.Throwable] )
  15. def formatted(fmtstr: String): String
    Implicit
    This member is added by an implicit conversion from Polarity to StringFormat[Polarity] performed by method StringFormat in scala.Predef.
    Definition Classes
    StringFormat
    Annotations
    @inline()
  16. final def getClass(): Class[_]
    Definition Classes
    AnyRef → Any
    Annotations
    @native()
  17. def hashCode(): Int
    Definition Classes
    AnyRef → Any
    Annotations
    @native()
  18. def inAnt: Boolean
  19. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  20. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  21. def negative: Boolean
  22. final def notify(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native()
  23. final def notifyAll(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native()
  24. def positive: Boolean
  25. final def synchronized[T0](arg0: ⇒ T0): T0
    Definition Classes
    AnyRef
  26. def toString(): String
    Definition Classes
    AnyRef → Any
  27. final def wait(): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )
  28. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )
  29. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @throws( ... )
  30. def [B](y: B): (Polarity, B)
    Implicit
    This member is added by an implicit conversion from Polarity to ArrowAssoc[Polarity] performed by method ArrowAssoc in scala.Predef.
    Definition Classes
    ArrowAssoc

Inherited from AnyRef

Inherited from Any

Inherited by implicit conversion any2stringadd from Polarity to any2stringadd[Polarity]

Inherited by implicit conversion StringFormat from Polarity to StringFormat[Polarity]

Inherited by implicit conversion Ensuring from Polarity to Ensuring[Polarity]

Inherited by implicit conversion ArrowAssoc from Polarity to ArrowAssoc[Polarity]

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