A modal framework for modelling abductive reasoning

CONCEPTUAL FOUNDATIONS OF QUANTUM MECHANICS: THE ROLE OF EVIDENCE THEORY, QUANTUM SETS, AND MODAL LOGIC

1 Modal logic: a semantic perspective

Maximal Kripke-type semantics for modal and superintuitionistic predicate logics

Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of G\"odel logic. The calculi are used to establish completeness and complexity results for these fragments.

The famous van Benthem theorem states that modal logic corresponds exactly to the fragment of first-order logic that is invariant under bisimulation. In this article we prove an exact analogue of this theorem in the framework of modal dependence logic (MDL) and team semantics. We show that Modal Team Logic (MTL) extending MDL by classical negation captures exactly the FO-definable bisimulation invariant properties of Kripke structures and teams. We also compare the expressive power of MTL to most of the variants and extensions of MDL recently studied in the area.

The use of the Hypo-Deductive model in management research has been questioned because it is very difficult to properly pre-specify hypotheses and tests prior to examining data. Abduction, and its close cousin, Inference to the Best Explanation (IBE), have been proposed as an alternative approach (King et al., 2019). But how does one “abduct”? There has been limited discussion in management research about the process of engaging with abduction, the challenges and opportunities it presents, how currently used research methodologies fit in an abductive framework, and the aspects that management scholars can take advantage of from the discussions in the philosophy of science about the abductive process. The purpose of this panel symposium is to have leading scholars to share their perspectives on the future of abductive studies in management.

Bi-intuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Bi-intuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent sequent calculus for BiInt has recently been shown by Uustalu to fail cut-elimination. We present a new cut-free sequent calculus for BiInt , and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between implication and its dual, similarly to future and past modalities in tense logic. Our calculus handles this interaction using extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of failed derivation trees. Our simple termination argument allows our calculus to be used for automated deduction, although this is not its main purpose.

The delicate point in the formalistic position is to explain how the non-intuitionistic classical mathematics is significant, after having initially agreed with the intuitionists that its theorems lack a real meaning in terms of which they are true (S. C. Kleene, 1952).

A binary modal logic for the intersection types of lambda-calculus

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (1979). Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency.

This paper, we propose a modal logic satisfying minimal requirements for reasoning about diagrams via collection of sets and relations between them, following Harel’s proposal. We first give an axiomatics of such a theory and then provide its Kripke semantics. Then we extend previous works of ours in order to obtain a decision procedure based on tableaux for this logic. Beside soundness and completeness of our tableaux, we manage to define a strategy of rule application ensuring termination by extending the usual loop test of modal logic S4 to whole sub-structures of the model being computed.

We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics.

We investigate the (multiagent) infinitary version \(\mathbf {K}_{\omega _1}\) of the propositional modal logic \(\mathbf {K}\), in which conjunctions and disjunctions over countably infinite sets of formulas are allowed. It is known that the natural infinitary extension \(\mathbf {LK}_{\omega _1}^{\Box }\) (here presented as a Tait-style calculus, \(\mathbf {TK}^{\sharp }_{\omega _1}\)) of the standard sequent calculus \(\mathbf {LK}_p^{\Box }\) for \(\mathbf {K}\) is incomplete with respect to Kripke semantics. It is also known that in order to axiomatize \(\mathbf {K}_{\omega _1}\) one has to add to \(\mathbf {LK}_{\omega _1}^{\Box }\) new initial sequents corresponding to the infinitary propositional counterpart \( BF _{\omega _1}\) of the Barcan Formula. We introduce a generalization of standard Kripke semantics, and prove that \(\mathbf {TK}^{\sharp }_{\omega _1}\) is sound and complete with respect to it. By the same proof strategy, we show that the stronger system \(\mathbf {TK}_{\omega _1}\), allowing countably infinite sequents, axiomatizes \(\mathbf {K}_{\omega _1}\), although it provably does not admit cut-elimination.

In this paper I shall present three solutions to the Knower Paradox which, despite important points in common, differ in several respects. The first solution, proposed by C. A. Anderson [1] is a hierarchical solution, developed in the framework of first-order arithmetic. However I will try to show that this solution is based on an incorrect argument. The second solution, inspired by a book of R.M. Smullyan [14], is developed in the framework of modal logic and it is based on the idea of interpreting one of the basic systems of the modal logic of provability in an epistemic way. I shall give arguments in support of this solution. The third solution, proposed by P. Egree [8] attempts to connect the first and the second solutions. I will show that this attempt fails for philosophical and formal reasons.

We explore the features of rewriting logic and, in particular, of the rewriting logic language Maude as a logical and semantic framework for representing and executing inference systems. In order to illustrate the general ideas we consider two substantial case studies. In the first one, we represent both the semantics of Milner's CCS and a modal logic for describing local capabilities of CCS processes. Although a rewriting logic representation of the CCS semantics is already known, it cannot be directly executed in the default interpreter of Maude. Moreover, it cannot be used to answer questions such as which are the successors of a process after performing an action, which is used to define the semantics of Hennessy-Milner modal logic. Basically, the problems are the existence of new variables in the righthand side of the rewrite rules and the nondeterministic application of the semantic rules, inherent to CCS. We show how these problems can be solved in a general, not CCS dependent way by controlling the rewriting process by means of reflection. This executable specification plus the reflective control of rewriting can be used to analyze CCS processes. The same techniques are also used to implement a symbolic semantics for LOTOS in our second case study. The good properties of Maude as a metalanguage allow us to implement a whole formal tool where LOTOS specifications without restrictions in their data types (given as ACT ONE specifications) can be executed. In summary, we present Maude as an executable semantic framework by providing easy-tool-building techniques for a language given its operational semantics.