This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

In the present paper, we study the correspondence and canonicity theory of modal subordination algebras and their dual Stone space with two relations, generalizing correspondence results for subordination algebras in [13–15, 25]. Due to the fact that the language of modal subordination algebras involves a binary subordination relation, we will find it convenient to use the so-called quasi-inequalities and $\varPi _{2}$-statements. We use an algorithm to transform (restricted) inductive quasi-inequalities and (restricted) inductive $\varPi _{2}$-statements to equivalent first-order correspondents on the (...) dual Stone spaces with two relations with respect to arbitrary (resp. admissible) valuations. (shrink)

A Heyting algebra is supplemented if each element a has a dual pseudo-complement \, and a Heyting algebra is centrally supplement if it is supplemented and each supplement is central. We show that each Heyting algebra has a centrally supplemented extension in the same variety of Heyting algebras as the original. We use this tool to investigate a new type of completion of Heyting algebras arising in the context of algebraic proof theory, the so-called hyper-MacNeille completion. We show that the (...) hyper-MacNeille completion of a Heyting algebra is the MacNeille completion of its centrally supplemented extension. This provides an algebraic description of the hyper-MacNeille completion of a Heyting algebra, allows development of further properties of the hyper-MacNeille completion, and provides new examples of varieties of Heyting algebras that are closed under hyper-MacNeille completions. In particular, connections between the centrally supplemented extension and Boolean products allow us to show that any finitely generated variety of Heyting algebras is closed under hyper-MacNeille completions. (shrink)

This paper presents an overview of the methods of hypersequents and display sequents in the proof theory of non-classical logics. In contrast with existing surveys dedicated to hypersequent calculi or to display calculi, our aim is to provide a unified perspective on these two formalisms highlighting their differences and similarities and discussing applications and recent results connecting and comparing them.

In this paper, we establish the first-order definability of sequents with consistent variable occurrence on bi-approximation semantics by means of the Sahlqvist–van Benthem algorithm. Then together with the canonicity results in Suzuki (2011), this allows us to establish a Sahlqvist theorem for substructural logic. Our result is not limited to substructural logic but is also easily applicable to other lattice-based logics.

In the present paper, we develop the algorithmic correspondence theory for hybrid logic with binder |$\mathcal {H}(@, \downarrow )$|⁠. We define the class of Sahlqvist inequalities for |$\mathcal {H}(@, \downarrow )$|⁠, and each inequality of which is shown to have a first-order frame correspondent effectively computable by an algorithm |$\textsf {ALBA}^{\downarrow }$|⁠.

The work is concerned with the so called display property of display logic. The motivation behind it is discussed and challenged. It is shown using one display calculus for intuitionistic logic as an example that the display property can be abandoned without losing subformula, cut elimination and completeness properties in such a way that results in additional expressive power of the system. This is done by disentangling structural connectives so that they are no longer context-sensitive. A recipe for characterizing structural (...) extensions of display calculi is provided. (shrink)

In the present paper, we start studying epistemic updates using the standard toolkit of duality theory. We focus on public announcements, which are the simplest epistemic actions, and hence on Public Announcement Logic without the common knowledge operator. As is well known, the epistemic action of publicly announcing a given proposition is semantically represented as a transformation of the model encoding the current epistemic setup of the given agents; the given current model being replaced with its submodel relativized to the (...) announced proposition. We dually characterize the associated submodel-injection map as a certain pseudo-quotient map between the complex algebras respectively associated with the given model and with its relativized submodel. As is well known, these complex algebras are complete atomic BAOs . The dual characterization we provide naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras . Thanks to this construction, the benefits and the wider scope of applications given by a point-free, intuitionistic theory of epistemic updates are made available. As an application of this dual characterization, we axiomatize the intuitionistic analogue of PAL, which we refer to as IPAL, prove soundness and completeness of IPAL w.r.t. both algebraic and relational models, and show that the well known Muddy Children Puzzle can be formalized in IPAL. (shrink)

In the present paper, we continue the research in Zhao (2021, Logic J. IGPL) to develop the Sahlqvist completeness theory for hybrid logic with satisfaction operators and downarrow binders |$\mathcal {L}( @, {\downarrow })$|⁠. We define the class of restricted Sahlqvist formulas for |$\mathcal {L}( @, {\downarrow })$| following the ideas in Conradie and Robinson (2017, J. Logic Comput., 27, 867–900), but we follow a different proof strategy which is purely proof-theoretic, namely showing that for every restricted Sahlqvist formula |$\varphi (...) $| and its hybrid pure correspondence |$\pi $|⁠, |$\textbf {K}_{\mathcal {H}( @, {\downarrow })}+\varphi $| proves |$\pi $|⁠; therefore, |$\textbf {K}_{\mathcal {H}( @, {\downarrow })}+\varphi $| is complete with respect to the class of frames defined by |$\pi $|⁠, using a modified version |$\textsf {ALBA}^{{\downarrow }}_{\textsf {Modified}}$| of the algorithm |$\textsf {ALBA}^{{\downarrow }}$| defined in Zhao (2021, Logic J. IGPL). (shrink)

We introduce the logic LRC, designed to describe and reason about agents’ abilities and capabilities in using resources. The proposed framework bridges two—up to now—mutually independent strands of literature: the one on logics of abilities and capabilities, developed within the theory of agency, and the one on logics of resources, motivated by program semantics. The logic LRC is suitable to describe and reason about key aspects of social behaviour in organizations. We prove a number of properties enjoyed by LRC and (...) its associated analytic calculus. These results lay at the intersection of the algebraic theory of unified correspondence and the theory of multitype calculi in structural proof theory. Case studies are discussed which showcase several ways in which this framework can be extended and enriched while retaining its basic properties, so as to model an array of issues, both practically and theoretically relevant, spanning from planning problems to the logical foundations of the theory of organizations. (shrink)