No abstract

Relational lattices are obtained by interpreting lattice connectives as natural join and inner union between database relations. Our study of their equational theory reveals that the variety generated by relational lattices has not been discussed in the existing literature. Furthermore, we show that addition of just the header constant to the lattice signature leads to undecidability of the quasiequational theory. Nevertheless, we also demonstrate that relational lattices are not as intangible as one may fear: for example, they do form a pseudoelementary class. We also apply the tools of Formal Concept Analysis and investigate the structure of relational lattices via their standard contexts. Furthermore, we show that the addition of typing rules and singleton constants allows a direct comparison with monotonic relational expressions of Sagiv and Yannakakis.

C. I. Lewis invented modern modal logic as a theory of "strict implication" . Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than P and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction has been discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of intuitionistic in terms of preservativity in extensions of HA, i.e., Heyting's Arithmetic. 65The basic option is to read § § 2-4 to get the basics of motivational background, the Kripke semantics and an impression of possible reasoning systems. ? The reader who wants more solid treatment of Kripke semantics can extend the basic option with § 6. | The computer science package consists of the basic option and § 7.= The reader who wants to go somewhat more deeply into the history of the subject can extend the basic option with Appendix D. « The reader who wants to understand the basics of arithmetical interpretations can extend the basic option with § 5. » An extended package for arithmetical interpretations combines « with § 8.-The full arithmetical package extends » with Appendices A, B and C. The rise and fall of the house of Lewis "The error of philosophers"We are reflecting on L.E.J. Brouwer's heritage half a century after his passing. Given his negative views on the rôle of logic and formalisms in mathematics, it seems somewhat paradoxical that these days the name of intuitionism survives mostly in the context of intuitionistic logic. 3 One is reminded in this context of what Nietzsche called the error of philosophers:The philosopher believes that the value of his philosophy lies in the whole, in the structure. Posterity finds it in the stone with which he built and with which, from that time forth, men will build oftener and better-in other words, in the fact that the structure may be destroyed and yet have value as material. 4

I overview the work of the Tbilisi school on intuitionistic modal logics of well-founded/scattered structures and its connections with contemporary theoretical computer science. Fixed-point theorems and their consequences are of particular interest.Comment: The paper is a modified and extended version ("Author's Cut") of my contribution for the "Leo Esakia on Duality in Modal and Intuitionistic Logics.

Coalgebraic modal logic serves as a unifying framework to study a wide range of modal logics beyond the relational realm, including probabilistic and graded logics as well as conditional logics and logics based on neighbourhoods and games. Coalgebraic predicate logic (CPL), a generalization of a neighbourhoodbased first-order logic introduced by Chang, has been identified as a natural first-order extension of coalgebraic modal logic, which in particular coincides with the standard first-order correspondence language when instantiated to Kripke-style relational modal operators. Here, we generalize to the CPL setting the classical van Benthem/Rosen theorem stating that both over arbitrary and over finite models, modal logic is precisely the bisimulation-invariant fragment of first-order logic. As instances of this generic result, we obtain corresponding characterizations for, e.g., conditional logic, neighbourhood logic (i.e., classical modal logic), and monotone modal logic.