Polarity

This file defines the polarity of an expression, positive or negative. The two polarities form the group of order two, the units of ℤ, and a homomorphism out of it is fixed by an element whose square is one, so that the polarities act through any involution. The polarity of a sentence acts on propositions by complement, and the orbit of a proposition is the pair of its polarity alternatives, the focus value of a focused polarity and the answers of a polar question. The polarity of an antonym acts on degrees by negation. A polarity directs a function, and directions compose as polarities multiply.

Implementation notes

A polarity coerces into SignType, whose third value records the absence of a direction, as for a non-monotone context. The action on an additive group goes through ℤˣ, so that on a function space it is pointwise by definition.

References

• Wilder 2013, English ‘emphatic do’ formalized
• Gutzmann 2012, Verum – Fokus – Verum-Fokus? Fokus-basierte und lexikalische Ansätze