Defining Double Negation Elimination

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 1 Compatibility Frames Dunn’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,v〉 consisting of a set P of points, and two binary relations C and v on P such that • v is a partial order on P . That is, v 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′ v x and y′ v y then x′Cy′. The relation v 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 1Thanks to anonymous referees of the journal for comments which helped improve the presentation of this paper. 2I 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. 853 L. J. of the IGPL, Vol. 8 No. 6, pp. 853–86