We introduce a complete many-valued semantics for two normal lattice-based modal logics. This semantics is based on reflexive many-valued graphs. We discuss an interpretation and possible applications of this logical framework in the context of the formal analysis of the interaction between (competing) scientific theories.

In recent years, unified correspondence has been developed as a generalized Sahlqvist theory which applies uniformly to all signatures of normal and regular (distributive) lattice expansions. A fundamental tool for attaining this level of generality and uniformity is a principled way, based on order theory, to define the Sahlqvist and inductive formulas and inequalities in every such signature. This definition covers in particular all (bi-)intuitionistic modal logics. The theory of these logics has been intensively studied over the past seventy years in connection with classical polyadic modal logics, using versions of Gödel-McKinsey- Tarski translations, suitably defined in each signature, as main tools. In view of this state-of-the-art, it is natural to ask (1) whether a general perspective on Gödel-McKinsey- Tarski translations can be attained, also based on order-theoretic principles like those underlying the general definition of Sahlqvist and inductive formulas and inequalities, which accounts for the known Gödel-McKinsey-Tarski translations and applies uniformly to all signatures of normal (distributive) lattice expansions; (2) whether this general perspective can be used to transfer correspondence and canonicity theorems for Sahlqvist and inductive formulas and inequalities in all signatures described above under Gödel-McKinsey-Tarski translations. In the present paper, we set out to answer these questions. We answer (1) in the armative; as to (2), we prove the transfer of the correspondence theorem for inductive inequalities of arbitrary signatures of normal distributive lattice expansions. We also prove the transfer of canonicity for inductive inequalities, but only restricted to arbitrary normal modal expansions of bi-intuitionistic logic. We also analyze the difficulties involved in obtaining the transfer of canonicity outside this setting, and indicate a route to extend the transfer of canonicity to all signatures of normal distributive lattice expansions.

By ‘informational entropy’, we understand an inherent boundary to knowability, due e.g. to perceptual, theoretical, evidential or linguistic limits. In this paper, we discuss a logical framework in which this boundary is incorporated into the semantic and deductive machinery, and outline how this framework can be used to model various situations in which informational entropy arises.

We extend the theory of unified correspondence to a broad class of logics with algebraic semantics given by varieties of normal lattice expansions (LEs), also known as ‘lattices with operators’. Specifically, we introduce a syntactic definition of the class of Sahlqvist formulas and inequalities which applies uniformly to each LE-signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives. We also introduce the algorithm ALBA, parametric in each LE-setting, which effectively computes first-order correspondents of LE-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which significantly extend the Sahlqvist class. Further, we show that every inequality on which ALBA succeeds is canonical. Projecting these results on specific signatures yields state-of-the-art correspondence and canonicity theory for many well known modal expansions of classical and intuitionistic logic and for substructural logics, from classical poly-modal logics to (bi-)intuitionistic modal logics to the Lambek calculus and its extensions, the Lambek-Grishin calculus, orthologic, the logic of (not necessarily distributive) De Morgan lattices, and the multiplicative-additive fragment of linear logic.

In the present paper, we prove canonicity results for lattice-based fixed point logics in a constructive meta-theory. Specifically, we prove two types of canonicity results, depending on how the fixed-point binders are interpreted. These results smoothly unify the constructive canonicity results for inductive inequalities, proved in a general lattice setting, with the canonicity results for fixed point logics on a bi-intuitionistic base, proven in a non-constructive setting.

Categorization systems are widely studied in psychology, sociology, and organization theory as information-structuring devices which are critical to decision-making processes. In the present paper, we introduce a sound and complete epistemic logic of categories and agents' categorical perception. The Kripke-style semantics of this logic is given in terms of data structures based on two domains: one domain representing objects (e.g. market products) and one domain representing the features of the objects which are relevant to the agents' decision-making. We use this framework to discuss and propose logic-based formalizations of some core concepts from psychological, sociological, and organizational research in categorization theory.

RS-frames were introduced by Gehrke as relational semantics for substructural logics. They are two-sorted structures, based on RS-polarities with additional relations used to interpret modalities. We propose an intuitive, epistemic interpretation of RS-frames for modal logic, in terms of categorization systems and agents’ subjective interpretations of these systems. Categorization systems are a key to any decision-making process and are widely studied in the social and management sciences. A set of objects together with a set of properties and an incidence relation connecting objects with their properties forms a polarity which can be ‘pruned’ into an RS-polarity. Potential categories emerge as the Galois-stable sets of this polarity, just like the concepts of Formal Concept Analysis. An agent’s beliefs about objects and their properties (which might be partial) is modelled by a relation which gives rise to a normal modal operator expressing the agent’s beliefs about category membership. Fixed-points of the iterations of the belief modalities of all agents are used to model categories constructed through social interaction.