The paper considers the standard concept description language ALC augmented with various kinds of modal operators which can be applied to concepts and axioms. The main aim is to develop methods of proving decidability of the satis ability problem for this language and apply them to description logics with most important temporal and epistemic operators, thereby obtaining satis ability checking ...

The modal logic LL was introduced by Halpern and Rabin [HR] as a means of doing qualitative reasoning about likelihood. Here the relationship between LL and probability theory is examined. It is shown that there is a way of translating probability assertions into LL in a sound manner, so that LL in some sense can capture the probabilistic interpretation of likelihood. However, the translation i...

We discuss first order theories in which individual concepts are admitted as mathematical objects along with the things that reify them. This allows very straightforward formalizations of knowledge, belief, wanting, and necessity in ordinary first order logic without modal operators. Applications are given in philosophy and in artificial intelligence. We do not treat general concepts, and we do...

The theory of rough sets is an extension of set theory with two additional unary set-theoretic operators deened based on a binary relation on the universe. These two operators are related to the modal operators in modal logics. By exploring the relationship between rough sets and modal logics, this paper proposes and examines a number of extended rough set models. By the properties satissed by ...

In this paper we consider the class of truth-functional modal many-valued logics with the complete lattice of truth-values. The conjunction and disjunction logic operators correspond to the meet and join operators of the lattices, while the negation is independently introduced as a hierarchy of antitonic operators which invert bottom and top elements. The non-constructive logic implication will...

We introduce a class of algebras, called twist-structures, whose members are built as special squares of an arbitrary residuated lattice. We show how our construction relates to and encompasses results obtained by several authors on the algebraic semantics of non-classical logics. We define a logic that corresponds to our twist-structures and show how to expand it with modal operators, obtainin...

Several extensions of the basic modal language are characterized in terms of interpolation. Our main results are of the following form: Language L′ is the least expressive extension of L with interpolation. For instance, let M(D) be the extension of the basic modal language with a difference operator [7]. First-order logic is the least expressive extension ofM(D) with interpolation. These chara...

In this paper, I give the English progressive a semantic analysis as an intensional operator within the framework of modal semantics proposed by Kratzer. This treatment allows a combination of the central idea of Dowty’s influential analysis, that the progressive’s meaning has a major modal component, with the insights of other scholars (Parsons, Vlach, and Landman). My claim is that using a mo...

Logic is generally defined as the science of reasoning. Mathematical logic is mainly concerned with forms of reasoning that lead from true premises to true conclusions. Thus we say that the argument from σ0;σ1; · · · ;σn−1 to δ is logically correct if whenever σi is true for all i < n, then so is δ. In place of ‘argument’ one also speaks of a ‘rule’ or an ‘inference’ and says that the rule is v...

We investigate a novel intuitionistic modal logic, called Propositional Lax Logic, with promising applications to the formal veri cation of computer hardware. The logic has emerged from an attempt to express correctness `up to' behavioural constraints | a central notion in hardware veri cation | as a logical modality. The resulting logic is unorthodox in several respects. As a modal logic it is...