We present an algorithmically efficient criterion of modal definability for first-order existential conjunctive formulas with several free variables. Then we apply it to establish modal definability of some family of first-order $\forall\exists$-formulas. Finally, we use our definability results to show that, in any expressive description logic, the problem of answering modally definable conjunctive queries is polynomially reducible to the problem of knowledge base consistency.
One of natural combinations of Kripke complete modal logics is the product, an operation that has been extensively investigated over the last 15 years. In this paper we consider its analogue for arbitrary modal logics: to this end, we use product-like constructions on general frames and modal algebras. This operation was first introduced by Y. Hasimoto in 2000; however, his paper remained unnoticed until recently. In the present paper we quote some important Hasimoto’s results, and reconstruct the product operation in an algebraic setting: the Boolean part of the resulting modal algebra is exactly the tensor product of original algebras (regarded as Boolean rings). Also, we propose a filtration technique for Kripke models based on tensor products and obtain some decidability results.
This paper presents two major aspects of Frege’s and Peirce’s views on assertion and denial: first, their arguments for the notational choices concerning the representation of assertion and denial in Begriffsschrift (BS) and Existential Graphs (EGs), respectively; and second, those properties of BS and EGs which reflect their inventors’ views on assertion and denial. We show that while Frege’s notation has an ad hoc sign of assertion and an ad hoc sign of negation, Peirce has a sign of assertion which is also a sign of logical conjunction, and a sign of scope which is also a sign of negation.
The problem of truth-values of indirect meanings is discussed within the semantical theory of indirect meaning proposed by the present authors in a dialogue with Hintikka and Sandu’s theory. The authors preserve the key notion of the latter, the meaning line, but put it into different semantics (non-Fregean situational) and logic (paraconsistent). Like the contradictions, the indirect meanings tend to an explosion (there are always such possible worlds where they are true); to make them meaningful, there is a need of singling out the only relevant transworld connexion among the infinite number of the possible ones. The meaning line serves to this purpose. An analysis of the simplest semantical constructions with indirect meaning (tropes, humour, hints, riddles, etc.) is proposed.