Hypersequent calculi for non-normal modal and deontic logics: Countermodels and optimal complexity

We present some hypersequent calculi for all systems of the classical cube and their extensions with axioms $T$, $P$, $D$, and, for every $n\geq 1$, rule $RD^+_n$. The calculi are internal as they only employ the language of the logic, plus additional structural connectives. We show that the calculi are complete with respect to the corresponding axiomatisation by a syntactic proof of cut elimination. Then we define a terminating root-first proof search strategy based on the hypersequent calculi and show that it is optimal for coNP-complete logics. Moreover, we obtain that from every saturated leaf of a failed proof it is possible to define a countermodel of the root hypersequent in the bi-neighbourhood semantics, and for regular logics also in the relational semantics. We finish the paper by giving a translation between hypersequent rule applications and derivations in a labelled system for the classical cube

On the Logic of Belief and Propositional Quantification

We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is some-thing that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss

G\"odel-Dummett linear temporal logic

We investigate a version of linear temporal logic whose propositional fragment is G\"odel-Dummett logic (which is well known both as a superintuitionistic logic and a t-norm fuzzy logic). We define the logic using two natural semantics: first a real-valued semantics, where statements have a degree of truth in the real unit interval and second a `bi-relational' semantics. We then show that these two semantics indeed define one and the same logic: the statements that are valid for the real-valued semantics are the same as those that are valid for the bi-relational semantics. This G\"odel temporal logic does not have any form of the finite model property for these two semantics: there are non-valid statements that can only be falsified on an infinite model. However, by using the technical notion of a quasimodel, we show that every falsifiable statement is falsifiable on a finite quasimodel, yielding an algorithm for deciding if a statement is valid or not. Later, we strengthen this decidability result by giving an algorithm that uses only a polynomial amount of memory, proving that G\"odel temporal logic is PSPACE-complete. We also provide a deductive calculus for G\"odel temporal logic, and show this calculus to be sound and complete for the above-mentioned semantics, so that all (and only) the valid statements can be proved with this calculus.Comment: arXiv admin note: substantial text overlap with arXiv:2205.00574, arXiv:2205.0518

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

Deductive Systems in Traditional and Modern Logic

The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic

Interim research assessment 2003-2005 - Computer Science

This report primarily serves as a source of information for the 2007 Interim Research Assessment Committee for Computer Science at the three technical universities in the Netherlands. The report also provides information for others interested in our research activities

A validation process for a legal formalization method

peer reviewedThis volume contains the papers presented at LN2FR 2022: The International Workshop on Methodologies for Translating Legal Norms into Formal Representations, held on December 14, 2022 in a hybrid form (in person workshop was held in Saarland University, Saarbrucken) in association with 35th International Conference on Legal Knowledge and Information Systems (JURIX 2022). Using symbolic logic or similar methods of knowledge representation to formalise legal norms is one of the most traditional goals of legal informatics as a scientific discipline. More than mere theoretical value, this approach is also connected to promising real-world applications involving, e.g., the observance of legal norms by highly automated machines or even the (partial) automatisation of legal reasoning, leading to new automated legal services. Albeit the long research tradition on the use of logic to formalise legal norms-be it by using classic logic systems (e.g., first-order logic), be it by attempting to construct a specific system of logic of norms (e.g., deontic logic)-, many challenges involved in the development of an adequate methodology for the formalisation of concrete legal regulations remain unsolved. This includes not only the choice of a sufficiently expressive formal language or model, but also the concrete way through which a legal text formulated in natural language is to be translated into the formal representation. The workshop LN2FR seeked to explore the various challenges connected with the task of using formal languages and models to represent legal norms in a machine-readable manner. We had 13 submissions, which were reviewed by 2 or 3 reviewers. Among these, we selected 11 papers (seven long papers, three short papers, one published paper) for presentation and discussion