Logic of Proofs (LP) has been introduced in [2] as a collection of all valid formulas in the propositional language with labeled logical connectives [[t]]( ) where t is a proof term with the intended reading of [[t]]F as \t is a proof of F". LP is supplied with a natural axiom system, completeness and decidability theorems. LP may express some constructions of logic which have been formulated o...

The boxdot conjecture asserts that every normal modal logic that faithfully interprets T by the well-known boxdot translation is in fact included in T. We confirm that the conjecture is true. More generally, we present a simple semantic condition on modal logics L0 which ensures that the largest logic where L0 embeds faithfully by the boxdot translation is L0 itself. In particular, this natural...

The decision problem for provability logic remains PSPACE -complete even if the number of propositional atoms is restricted to one. In some cases the set of all tautologies of a modal logic is in coNP. An example of a logic like that is the well-known S5. However, most of the traditional modal systems, including S4 and T, have PSPACE -complete decision problem. So one can say that adding modali...

Modal dependence logic (MDL) was introduced recently by Väänänen. It enhances the basic modal language by an operator =(·). For propositional variables p1, . . . , pn the atomic formula =(p1, . . . , pn−1, pn) intuitively states that the value of pn is determined solely by those of p1, . . . , pn−1. We show that model checking for MDL formulae over Kripke structures is NPcomplete and further co...

This paper presents matching logic, a first-order logic (FOL) variant for specifying and reasoning about structure by means of patterns and pattern matching. Its sentences, the patterns, are constructed using variables, symbols, connectives and quantifiers, but no difference is made between function and predicate symbols. In models, a pattern evaluates into a power-set domain (the set of values...

We prove completeness of the propositional modal logic S4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connectives and necessity and possibility operators, and ♦. Propositional modal formulae are assigned to Lebesgue-me...

We extend propositional Gödel logic by a unary modal operator, which we interpret as 5 Gödel homomorphisms, i.e. functions [0, 1] → [0, 1] that distribute over the interpretations of the 6 binary connectives of Gödel logic. We show that validity in the propositional fragment has a simple 7 superintuitionistic Hilbert-type proof system, which is not structurally complete, and that validity 8 doe...

People reason about possibilities routinely, and reasoners can infer “modal” conclusions, i.e., conclusions that concern what is possible or necessary, from premises that make no mention of modality. For instance, given that Cullen was born in New York or Kentucky, it is intuitive to infer that it’s possible that Cullen was born in New York, and a recent set of studies on modal reasoning bear o...

The paper studies basic graded properties of unary and binary fuzzy connectives, i.e., unary and binary operations on the set of truth degrees of a background fuzzy logic extending the logic MTL of left-continuous t-norms. The properties studied in this paper are graded generalizations of monotony, Lipschitz continuity, null and unit elements, idempotence, commutativity, and associativity. The ...

We prove completeness of the propositional modal logic S4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connectives and necessity and possibility operators, and ♦. Propositional modal formulae are assigned to Lebesgue-me...