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Negation in the light of modal logic. (English) Zbl 0974.03019
Gabbay, Dov M. (ed.) et al., What is negation? Dordrecht: Kluwer Academic Publishers. Appl. Log. Ser. 13, 77-86 (1999).
If one thinks of modal logic generally as the theory of monadic propositional operators, then negation naturally falls within its domain, and one should expect the semantical methods developed for modal logics to provide insight into negation. This paper investigates especially a logic \({\mathbf N}\) that adds a minimal negation to positive intuitionistic logic, and then extensions of \({\mathbf N}\). \({\mathbf N}\) adds rule-contraposition and the axiom \((\neg A\wedge\neg B)\to \neg(A\vee B)\) to positive intuitionistic logic. Semantically it is characterized in terms of models on frames \(F= \langle W,R_I, R_N\rangle\) where \(W\) is a non-empty set of points, \(R_I\) is a weak ordering on \(W\) to interpret intuitionistic implication, and \(R_N\) is a modal accessibility relation to interpret negation, by the rule \(x\models\neg A\) iff \(\forall y(xR_N y\to\text{not }y\models A)\). For \({\mathbf N}\), frames are required to meet \(R_I R_N\subseteq R_N R^{-1}_I\). \({\mathbf N}\) is sound and complete with respect to all such frames. By imposing various other conditions on \(R_N\), or \(R_N\) and \(R_I\) in combination, the range of principles associated with negation are validated, and the systems formed by adding them to positive intuitionistic logic are complete with respect to the frames meeting those requirements. It is plausible that a similar treatment of the many forms of negation would apply in relevant logic or other substructural logics.
For the entire collection see [Zbl 0957.00012].

03B45 Modal logic (including the logic of norms)
03B20 Subsystems of classical logic (including intuitionistic logic)
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)