Defining double negation elimination

Abstract

In his paper "Generalised Ortho Negation" [2] J. Michael Dunn mentions a claim of mine to the effect that there is no condition on 'perp frames' equivalent to the holding of double negation elimination ∼∼A A. That claim is wrong. In this paper I correct my error and analyse the behaviour of conditions on frames for negations which verify a number of different theses. 1 Compatibility FramesDunn's work on general models for negation has been a significant advance in our understanding of negation in non-classical logics [1,2]. These models generalise Kripke models for intuitionistic logic and Routley-Meyer models for relevant implication. I will recount the essential details of these models for negation here before we look at the behaviour of inference patterns such as double negation elimination. Definition 1.1 A frame is a triple P, C, consisting of a set P of points, and two binary relations C and on P such that• is a partial order on P . That is, is reflexive, transitive and antisymmetric on P .• C is antitone in both places. That is, for any x and y in P , if xCy, x x and y y then x Cy .The relation between points may be interpreted as one of information inclusion. A point x is informationally included in y if everything warranted by x (or encoded by x or made true by x or included in x or however else information is thought to be related to points) is also warranted by y. This is a transitive, reflexive relation on points. If the information content of a point is all you care about (as it is here) then antisymmetry is also an appropriate condition for inclusion.C is the relation of compatibility between points. 2 A point x is compatible with y if and only if all claims excluded by x are not warranted by y. It is clear that this merits the antitonicity condition -if x is compatible with y, then x (excluding no more than x) must be compatible with y (warranting no more than y).It is important to realise what "excluding" amounts to here. A point x excludes a piece of information when it rules it out. A point might neither warrant nor exclude 1 Thanks to anonymous referees of the journal for comments which helped improve the presentation of this paper. 2 I use 'C' for a relation of compatibility instead of '⊥' for the relation of incompatibility used by Dunn, simply because I use '⊥' as the false proposition which entails all others. The difference is merely notational. 853L.

Negation on the Australian Plan

“…In standard possible worlds semantics, the proposition expressed by formula A is taken as |A| ∈ P(W ), the set of worlds where A is true. But when points in frames can stand in non-trivial information-inclusion relations, one should take the set of propositions in a frame F -call that set P rop(F), a subset of the power set of worlds P(W ) -as including only sets closed upwards with respect to : [33,35].…”

Substructural Negations

“…We obtain de Morgan negation. The constraint for DNE implies the star postulate (found by Restall [34], the name taken from [24]), in effect that every state has a maximal state among those compatible with x, namely:…”

Another plan for negation

“…This term has been used extensively in the literature, alas in different senses. For example, Restall [13] uses it as a relation between states (points) in an incompatibility frame, in the spirit of the Australian plan. Brandom [2] uses incompatibility as relation between sets of sentences and subsets of the power-set of sentences; and there are more.…”