A Modality Called ‘Negation’:

Abstract

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for (the admissible) negations. The approach unifies in a philosophically motivated picture the following results: nothing can be called a negation properly if it does not satisfy (Minimal) Contraposition and Double Negation Introduction; the pair consisting of two split or Galois negations encodes a distinction without a difference; some paraconsistent negations also fail to count as real negations, but others may; intuitionistic negation qualifies as real negation, and classical Boolean negation does as well, to the extent that constructivist and paraconsistent doubts on it do not turn on the basic concept of compatibility but rather on the interpretation of worlds. When it is asserted that a negative signifies a contrary, we shall not agree, but admit no more than this: that the prefix 'not' indicates something different from the words that follow, or rather from the things designated by the words pronounced after the negative. Plato, Sophist 257b-c

Impossible Worlds

“…On this definition, ¬ is a 'negative modality': a quantifier over worlds, restricted by an accessibility relation interpreted as compatibility. Because we utter negations to express incompatibilities and exclusions (Berto 2015), a semantics for negation grounded in compatibility makes intuitive sense. Restall (1999) then shows how to get the Routley Star negation (S¬) from §6.1 out of (SC¬), by imposing conditions on compatibility.…”

Negation on the Australian Plan

“…However, the Australian Plan is no attempt to define negation away by reducing it to some other notion which makes no mention of negation. One of us, FB, has expressed, in print and in the very paper [8] targeted by D&O, skepticism (on which GR agrees) on any attempted definitional reduction of fundamental notions like reference, identity, necessity, or negation. Any elucidation of such notions is likely to make essential use of those very notions somewhere, in such a way that the explanation as a whole cannot count as a reductive definition.…”

“…To expect that the model theoretic semantics describes some process of cognition is to ask it to perform tasks beyond its remit. 8…”

“…A first one is connected to logical pluralism. FB's paper, [8], targeted by D&O, entertains a certain pluralism for negation, connected to the logical pluralism explored by GR in various works. The connection is the following.…”

“…Negation is understood as a modal operator, whose semantics is given in terms of relations of compatibility and incompatibility between points [15]. It has been connected to broader themes in philosophical logic, such as logical pluralism [4,5,8] and the interpretation and applications of substructural logics [33,34]. Furthermore, claims to the fundamental role of compatibility and incompatibility relations in giving an account of negation, also independently of direct appeal to the model theory of the Australian Plan, have been made in Aristotelian metaphysics [6,7,44] and in normative pragmatic accounts of semantics [10,11,22,29,36].…”