Heinrich Wansing

We study a propositional bimodal logic consisting of two S4 modalities £ and [a], together with the interaction axiom scheme a £ϕ → £ aϕ. In the intended semantics, the plain £ is given the McKinsey-Tarski interpretation as the interior operator of a topology, while the labelled [a] is given the standard Kripke semantics using a reflexive and transitive binary relation a. The interaction axiom expresses the property that the Ra relation is lower semi-continuous with respect to the topology. The class of topological Kripke frames characterised by the logic includes all frames over Euclidean space where Ra is the positive flow relation of a differential equation. We establish the completeness of the axiomatisation with respect to the intended class of topological Kripke frames, and investigate tableau calculi for the logic, although tableau completeness and decidability are still open questions.

The trilattice SIXTEEN3 introduced in Shramko & Wansing (2005) is a natural generalization of the famous bilattice FOUR2. Some Hilbert-style proof systems for trilattice logics related to SIXTEEN3 have recently been studied (Odintsov, 2009; Shramko & Wansing, 2005). In this paper, three sequent calculi GB, FB, and QB are presented for Odintsovs coordinate valuations associated with valuations in SIXTEEN3. The equivalence between GB, FB, and QB, the cut-elimination theorems for these calculi, and the decidability of B are proved. In addition, it is shown how the sequent systems for B can be extended to cut-free sequent calculi for Odintsov’s LB, which is an extension of B by adding classical implication and negation connectives.

In this paper we discuss Gabbay's idea of basing nonmonotonic deduction on semantic consequence in intuitionistic logic extended by a consistency operator and Turner's suggestion of replacing the intuitionistic base system by Kleene's three-valued logic. It is shown that a certain counterintuitive feature of these approaches can be avoided by using Nelson's constructive logic N instead of intuitionistic logic or Kleene's system. Moreover, in N a more general notion of consistency can be defined and nonmonotonic deduction can thus be based on a logical system satisfying the Deduction Theorem

The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Behthem''s modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap''s display logic by introduction rules for the existential and the universal quantifier. These rules for x and x are analogous to the display introduction rules for the modal operators and and do not themselves allow the Barcan formula or its converse to be derived. En route from the minimal modal predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules.

Anti-realistic conceptions of truth and falsity are usually epistemic or inferentialist. Truth is regarded as knowability, or provability, or warranted assertability, and the falsity of a statement or formula is identified with the truth of its negation. In this paper, a non-inferentialist but nevertheless anti-realistic conception of logical truth and falsity is developed. According to this conception, a formula (or a declarative sentence) A is logically true if and only if no matter what is told about what is told about the truth or falsity of atomic sentences, A always receives the top-element of a certain partial order on non-ontic semantic values as its value. The ordering in question is a told-true order. Analogously, a formula A is logically false just in case no matter what is told about what is told about the truth or falsity of atomic sentences, A always receives the top-element of a certain told-false order as its value. Here, truth and falsity are pari passu , and it is the treatment of truth and falsity as independent of each other that leads to an informational interpretation of these notions in terms of a certain kind of higher-level information

Modal Logic, originally conceived as the logic of necessity and possibility, has developed into a powerful mathematical and computational discipline. It is the main source of formal languages aimed at analyzing complex notions such as common knowledge and formal provability. Modal and modal-like languages also provide us with families of restricted description languages for relational and topological structures; they are being used in many disciplines, ranging from artificial intelligence, computer science and mathematics via natural language syntax and semantics to philosophy. This volume presents a broad and up-to-date view of the field, with contributions covering both the foundations of modal logic itself and each of the aforementioned application areas. Complemented with an editorial introduction covering the roots of modal logic, this book is indispensable for any advanced student and researcher in non-classical logic and its applications.

The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution offered is in the spirit of the constructivist attitude usually maintained by defenders of the anti-realist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete