N. Kamide introduced a pair of classical and constructive logics, each with a peculiar type of negation: its double negation behaves as classical and intuitionistic negation, respectively. A consequence of this is that the systems prove contradictions but are non-trivial. The present paper aims at giving insights into this phenomenon by investigating subsystems of Kamide’s logics, with a focus on a system in which the double negation behaves as the negation of minimal logic. We establish the negation inconsistency of the (...) system and embeddability of contradictions from other systems. In addition, we attempt at an informational interpretation of the negation using the dimathematical framework of H. Wansing. (shrink)

The totality of extensional 1-ary connectives distinguishable in a logical framework allowing sequents with multiple or empty (alongside singleton) succedents form a lattice under a natural partial ordering relating one connective to another if all the inferential properties of the former are possessed by the latter. Here we give a complete description of that lattice; its Hasse diagram appears as Figure 1 in §2. Simple syntactic descriptions of the lattice elements are provided in §3; §§4 and 5 give some additional (...) remarks on matrix methods and on alternative terminology. Background: The size of this lattice was underestimated in [3]; some missing cases were noted in [4] in the course of correcting an example from [3] purporting to show the non-distributivity of the lattice. All the 'missing cases' (as well as those originally noted) are covered here. (The present discussion is self-contained.). (shrink)

This discussion explores the possibility of distinguishing a tighter notion of contrariety evident in the Square of Opposition, especially in its modal incarnations, than as that binary relation holding statements that cannot both be true, with or without the added rider ‘though can both be false’. More than one theorist has voiced the intuition that the paradigmatic contraries of the traditional Square are related in some such tighter way—involving the specific role played by negation in contrasting them—that distinguishes them from (...) other pairs of incompatible statements constructed from the same conceptual materials. Prominent among examples, these other nonstandard pairs are the ‘new contraries’ presented by Robert Blanché’s hexagon(s) of opposition. With special, though not exclusive, attention to these cases, we investigate whether contrariety in the distinguished sense can be captured by adding to the incompatibility condition the further demand that the pair of statements concerned can be represented as the results of applying some sentence operator to the content in its scope, for one of the pair, and, for the other, the application of that same operator to the negation of that content. For one of the two cases, a Blanché case, of nonstandard contrariety singled out for attention, the question of whether such a representation is available is settled at the end of Section 4, and then in a more satisfying way in Section 5, though for the other case, noticed by Peter Simons, the question remains open, after some tentative discussion in one subsection, 6.2, of an Appendix (Section 6). (shrink)

Only propositional logics are at issue here. Such a logic is contra-classical in a superficial sense if it is not a sublogic of classical logic, and in a deeper sense, if there is no way of translating its connectives, the result of which translation gives a sublogic of classical logic. After some motivating examples, we investigate the incidence of contra-classicality (in the deeper sense) in various logical frameworks. In Sections 3 and 4 we will encounter, originally as an example of (...) what (in Section 2) we call a contra-classical modal logic, an unusual logic boasting a connective (" demi-negation" ) whose double application is equivalent to a single application of the negation connective. Pondering the example points the way to a general characterization of contra-classicality (Theorems 3.3 and 4.6). In an Appendix (Section 5), we look at one alternative to classical logic as the target for such translational assimilation, intuitionistic logic, calling logics which resist the assimilation, in this case, contra- intuitionistic. We will show that one such logic is classical logic itself, thereby strengthening a result of Wojcicki's to the effect that the consequence relation of classical logic cannot be faithfully embedded by any connective-by-connective translation into that of intuitionistic logic. (What the "faithfully" means here is that not only is the translation of anything provable in the 'source' logic.. (shrink)

Sections 1 and 2 respectively raise and settle the question of whether, if an affirmative modality collapses (reduces to the null modality, that is) in a normal modal logic, then all modalities of the same length collapse in that logic, while Section 3 considers some special cases of an analogous phenomenon for congruential modal logics, closing with a general question about collapsing modalities in this broader range of logics.

We consider an example of four valued semantics partially inspired by quantum computations and negation-like operations occurred therein. In particular we consider a representation of so called square root of negation within this four valued semantics as an operation which acts like a cycling negation. We define two variants of logical matrices performing different orders over the set of truth values. Purely formal logical result of our study consists in axiomatizing the logics of defined matrices as the systems of binary (...) consequence relation and proving correctness and completeness theorems for these deductive systems. (shrink)