Heinrich Wansing

In this paper, we introduce two new semantic presentations of Abelian logic, the non-trivial negation inconsistent logic of Abelian lattice-ordered groups, which was independently developed by Ettore Casari, and by Robert Meyer and John Slaney. Abelian logic is presented through a methodology that combines elements of what is sometimes referred to as the “Bochum Plan” and the “American Plan.” While the Bochum Plan is an approach to defining contra-classical logics, the American Plan-developed by Nuel Belnap and Michael Dunn-in particular offers a conception of negation that invites an application of the Bochum Plan. The first semantics is a ternary frame Kripke semantics, and the second is based on ideas from Edwin Mares’ work. Thereby emerges a condition for the falsity of Abelian implication to be supported, which we analyse further in the separate context of the first-degree entailment logic. The perspectives are united in the end to provide a defence against the scepticism concerning the status of the Abelian negation as a negation.

Modal logic is one of the most widely applied logical formalisms. Systems of modal logic are being used in many disciplines, ranging from artificial intelligence, computer science, mathematics, formal grammar and semantics to philosophy. This volume presents substantial recent advances in the relational and the algorithmic treatment of modal logics. It contains papers from the fifth conference on "Advances in Modal logic," held in Manchester (UK) in September 2004. Written by leading experts in the field, the present book is indispensable for any advanced student and researcher in pure and applied modal logic.

The present note contains a critical discussion of the methodology of paraconsistent logic in general and “the central optimisation problem of paraconsistent logics” in particular. It is argued that there exist several reasons not to consider classical logic as the reference logic for developing systems of paraconsistent logic, and it is suggested to weaken a certain maximality condition that may be seen as essential for “optimisation”, which is a methodology in the tradition of Newton da Costa. It is argued that the guiding motivation for the development of paraconsistent logics should be neither epistemological nor ontological, but informational. Moreover, it is pointed out that there are other notions of maximality and other methodologies. A methodology due to Graham Priest and Richard Routley and another methodology that focuses on a minimal shrinkage of expressiveness relative to a given reference logic are considered in some detail.

In this paper we study logical bilateralism understood as a theory of two primitive derivability relations, namely provability and refutability, in a language devoid of a primitive strong negation and without a falsum constant, $\bot $, and a verum constant, $\top $. There is thus no negation that toggles between provability and refutability, and there are no primitive constants that are used to define an “implies falsity” negation and a “co-implies truth” co-negation. This reduction of expressive power notwithstanding, there remains some interaction between provability and refutability due to the presence of (i) a conditional and the refutability condition of conditionals and (ii) a co-implication and the provability condition of co-implications. Moreover, assuming a hyperconnexive understanding of refuting conditionals and a dual understanding of proving co-implications, neither non-trivial negation inconsistency nor hyperconnexivity is lost for unary negation connectives definable by means of certain surrogates of falsum and verum. Whilst a critical attitude towards $\bot $ and $\top $ can be justified by problematic aspects of the Brouwer-Heyting-Kolmogorov interpretation of the logical operations for these constants, the aim to reduce the availability of a toggling negation and observations on undefinability may also give further reasons to abandon $\bot $ and $\top $.

Connexive logic has room for two pairs of universal and particular quantifiers: one pair, $\forall $ and $\exists $, are standard quantifiers; the other pair, $\mathbb{A}$ and $\mathbb{E}$, are unorthodox, but we argue, are well-motivated in the context of connexive logic. Both non-standard quantifiers have been introduced previously, but in the context of connexive logic they have a natural semantic and proof-theoretic place, and plausible natural language readings. The results are logics that are negation inconsistent but non-trivial.

A theory of substructural negations as impossibility and as unnecessity based on bi-intuitionistic logic, also known as Heyting-Brouwer logic, has been developed by Takuro Onishi. He notes two problems for that theory and offers the identification of the two negations as a solution to both problems. The first problem is the lack of a structural rule corresponding with double negation elimination for negation as impossibility, DNE, and the second problem is a lack of correspondence between certain sequents and a characterizing frame property. While the identification of negation as impossibility and negation as unnecessity does solve Onishi’s problems, in general it nevertheless seems desirable to keep the two notions separate. The present paper addresses the first problem by introducing a reformulation of Onishi’s display sequent calculus in a language of sequents that incorporates Boolean negation. Instead of identifying negation as impossibility and negation as unnecessity, a notion of weak frame correspondence is defined. It is observed that DNE weakly corresponds with a certain frame property, and two structural sequent rules are presented that together also weakly correspond with that frame property and allow one to derive DNE. Moreover, the reformulated display calculus has an independent motivation by considerations on proof-theoretic semantics.

We present a logic which deals with connexive exclusion. Exclusion (also called “co-implication”) is considered to be a propositional connective dual to the connective of implication. Similarly to implication, exclusion turns out to be non-connexive in both classical and intuitionistic logics, in the sense that it does not satisfy certain principles that express such connexivity. We formulate these principles for connexive exclusion, which are in some sense dual to the well-known Aristotle’s and Boethius’ theses for connexive implication. A logical system in a language containing exclusion and negation can be called a logic of connexive exclusion if and only if it obeys these principles, and, in addition, the connective of exclusion in it is asymmetric, thus being different from a simple mutual incompatibility of propositions. We will develop a certain approach to such a logic of connexive exclusion based on a semantic justification of the connective in question. Our paradigm logic of connexive implication will be the connexive logic \({\textsf{C}}\), and exactly like this logic the logic of connexive exclusion turns out to be contradictory though not trivial.