Online research in philosophy

In the Three Dialogues, Hylas challenges Philonous to give a plausible account of the mosaic account of creation in subjective idealistic terms. Strangely, when faced with two alternative strategies, Berkeley chooses the less viable option and explicates the mosaic account of creation in terms of perceptibility. I shall show that Berkeley’s account of creation trivializes the affair, if it does not fail outright.

The simplest combination of unimodal logics L1 and L2 into a bimodal logic is their fusion, L1 ⊗ L2, axiomatized by the theorems of L1 for 1 and of L2 for 2. Shehtman introduced combinations that are not only bimodal, but two-dimensional: he defined 2-d Cartesian products of 1-d Kripke frames, using these Cartesian products to define the frame product L1 × L2 of L1 and L2. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman’s idea and introduced the topological (...) product L1 ×t L2, using Cartesian products of topological spaces rather than of Kripke frames. Frame products have been extensively studied, but much less is known about topological products. The goal of the current paper is to give necessary and conditions for the topological product to match the frame product, for Kripke complete extensions of S4: L1×tL2 = L1× L2 iff L1 S5 or L2 S5 or L1, L2 = S5. (shrink)

We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion S4 ⊕ S4. We axiomatize the modal logic of products of spaces with horizontal, vertical, and standard product topologies.We prove that both of these logics are complete for the product of rational numbers ℚ × ℚ with the appropriate topologies.

We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion ${\bf S4}\oplus {\bf S4}$ . We axiomatize the modal logic of products of spaces with horizontal, vertical, and standard product topologies. We prove that both of these logics are complete for the product of rational numbers ${\Bbb Q}\times {\Bbb Q}$ (...) with the appropriate topologies. (shrink)

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 of M(D) with interpolation. These characterizations are subsequently used to derive new results about hybrid logic, relation algebra and the guarded fragment.