Abstract
We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities between such points. We defuse a number of objections to this Plan, raised by supporters of the American Plan for negation, in which negation is handled via a many-valued semantics. We show that the Australian Plan has substantial advantages over the American Plan.
Highlights
The last third of the Twentieth century saw a flowering of work in non-classical logics: the study of relevant logics, paraconsistent logics, orthologic, constructive logics, fuzzy logics, substructural logics and their cousins gave rise to a plethora of different kinds of models for such logics
We focus in this paper on the more general approach to negation; because we take modal approaches to negation to be important and interesting, and because D&O’s arguments apply not just to the Routley star semantics in particular, and to compatibility semantics in general
5We will come back below to the version of (S¬) using compatibility, and its contraposed, using incompatibility, after we have presented a formal semantics to serve as the target for some of D&O’s objections. 6‘No reduction of the identity relation has ever succeeded. [...] Nor yet is it called for, once we realize how much can be achieved in philosophy by means of elucidations that put concepts to use without attempting to reduce it but, in using the concept, exhibit its connections with other concepts that are established, genuinely coeval or collateral, and independently intelligible.’ [49, p. 5]
Summary
The last third of the Twentieth century saw a flowering of work in non-classical logics: the study of relevant logics, paraconsistent logics, orthologic, constructive logics, fuzzy logics, substructural logics and their cousins gave rise to a plethora of different kinds of models for such logics. Meyer and Martin’s paper was directed towards understanding the costs and benefits of different semantic schemes for relevant logics. Simplicity for semantic values (the Boolean yes/no answer, at each point) comes at the cost of complexity for the evaluation clause for negation. Once the truth conditions for a concept are given on the set of points, this automatically determines the interaction between that concept and negation All of this has been well understood since Meyer and Martin’s original mapping of the terrain. Attention has shifted from Meyer and Martin’s treatment of a de Morgan negation, modelled by the distinctive semantic device of the Routley star [42]. Semantics for negation, but the entire approach of treating negation as a point shift operator—survives unscathed, that its advantages over the American Plan remain intact, and substantial.
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