We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces isK4and the modal logic of weakly scattered Stone spaces isK4G. As a corollary, we obtain thatK4is also the modal logic of compact Hausdorff spaces andK4Gis the modal logic of weakly scattered compact Hausdorff spaces.

This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations (...) to weak Heyting algebras and implicative semilattices. The book also dives into the Blok-Esakia theorem and provides an outline of the intuitionistic modal logic KM which is closely related to the Gödel-Löb provability logic GL. One chapter scrutinizes Esakia’s work interpreting modal diamond as the derivative of a topological space within the setting of point-free topology. The final chapter in the volume is dedicated to the derivational semantics of modal logic and other related issues. (shrink)

The finite model property (FMP) in weakly transitive tense logics is explored. Let \(\mathbb {S}=[\textsf{wK}_t\textsf{4}, \textsf{K}_t\textsf{4}]\) be the interval of tense logics between \(\textsf{wK}_t\textsf{4}\) and \(\textsf{K}_t\textsf{4}\). We introduce the modal formula \(\textrm{t}_0^n\) for each \(n\ge 1\). Within the class of all weakly transitive frames, \(\textrm{t}_0^n\) defines the class of all frames in which every cluster has at most _n_ irreflexive points. For each \(n\ge 1\), we define the interval \(\mathbb {S}_n=[\textsf{wK}_t\textsf{4T}_0^{n+1}, \textsf{wK}_t\textsf{4T}_0^{n}]\) which is a subset of \(\mathbb {S}\). There are (...) \(2^{\aleph _0}\) logics in \(\mathbb {S}_n\) lacking the FMP, and there are \(2^{\aleph _0}\) logics in \(\mathbb {S}_n\) having the FMP. Then we explore the FMP in finitely alternative tense logics \(L_{n,m}=L\oplus \{\textrm{Alt}_n^F, \textrm{Alt}_m^P\}\) with \(n,m\ge 0\) and \(L\in \mathbb {S}\). For all \(k\ge 0\) and \(n,m\ge 1\), we define intervals \(\mathbb {F}^k_{n,m}\), \(\mathbb {P}^k_{n,m}\) and \(\mathbb {S}^k_{n,m}\) of tense logics. The number of logics lacking the FMP in them is either 0 or \(2^{\aleph _0}\), and the number of logics having the FMP in them is either finite or \(2^{\aleph _0}\). (shrink)

In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalising the well-known Gödel–McKinsey–Tarski translation of intuitionistic logic into S4. Došen's translation worked well for (variants of) BCI and stronger systems (BCW, BCK), but not for systems below BCI. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a corresponding sorted, residuated (multi) (...) modal logic. At the conceptual and philosophical level, the translation provides a classical interpretation of the meaning of the logical operators of various non-distributive propositional calculi. Technically, it allows for an effortless transfer of results, such as compactness and the Löwenheim-Skolem property and it opens up new directions for deducing properties of substructural logic systems by establishing appropriate transfer theorems. (shrink)

ABSTRACTCorrespondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components,, and, then frames are at the same time the basis for models of non-distributive lattice logic and of two-sorted, residuated modal logic. This suggests that (...) a reduction of the first to the latter may be possible, encoding Positive Lattice Logic as a fragment of Two-Sorted, Residuated Modal Logic. The reduction is analogous to the well-known Gödel-McKinsey-Tarski translation of Intuitionistic Logic into the S4 system of normal modal logic. In this article, we carry out this reduction in detail and we derive some properties of PLL from corresponding properties of First-Order Logic. The reduction we present is extendible to the case of lattices with operators, making use of recent results by this author on the relational representation of normal lattice expansions. (shrink)

Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are not normal. The set of extensions (...) of S4.3 is radically altered by the addition of a constant: we use it to construct continuum many such normal extensions of S4.3, and continuum many non-normal ones, none of which have the finite model property. But for logics with weakly transitive frames there are only eight maximally normal ones, of which five extend K4 and three extend S4. (shrink)

Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are not normal. The set of extensions (...) of S4.3 is radically altered by the addition of a constant: we use it to construct continuum many such normal extensions of S4.3, and continuum many non-normal ones, none of which have the finite model property. But for logics with weakly transitive frames there are only eight maximally normal ones, of which five extend K4 and three extend S4. (shrink)

We investigate mathematical structures that provide natural semantics for families of (quantified) non-classical logics featuring special unary connectives, known as recovery operators, that allow us to ‘recover’ the properties of classical logic in a controlled manner. These structures are known as topological Boolean algebras, which are Boolean algebras extended with additional operations subject to specific conditions of a topological nature. In this study, we focus on the paradigmatic case of negation. We demonstrate how these algebras are well-suited to provide a (...) semantics for some families of paraconsistent Logics of Formal Inconsistency and paracomplete Logics of Formal Undeterminedness. These logics feature recovery operators used to earmark propositions that behave ‘classically’ when interacting with non-classical negations. Unlike traditional semantical investigations, which are carried out in natural language (extended with mathematical shorthand), our formal meta-language is a system of higher-order logic (HOL) for which automated reasoning tools exist. In our approach, topological Boolean algebras are encoded as algebras of sets via their Stone-type representation. We use our higher-order meta-logic to define and interrelate several transformations on unary set operations, which naturally give rise to a topological cube of opposition. Additionally, our approach enables a uniform characterization of propositional, first-order and higher-order quantification, including restrictions to constant and varying domains. With this work, we aim to make a case for the utilization of automated theorem proving technology for conducting computer-supported research in non-classical logics. All the results presented in this paper have been formally verified, and in many cases obtained, using the Isabelle/HOL proof assistant. (shrink)

Both early analytic philosophy and the branch of mathematics now known as topology were gestated and born in the early part of the 20th century. It is not well recognized that there was early interaction between the communities practicing and developing these fields. We trace the history of how topological ideas entered into analytic philosophy through two migrations, an earlier one conceiving of topology geometrically and a later one conceiving of topology algebraically. This allows us to reassess the influence and (...) significance of topological methods for philosophy, including the possible fruitfulness of a third conception of topology as a structure determining similarity. (shrink)

In this paper we define an augmentation mHC of the Heyting propositional calculus HC by a modal operator ?. This modalized Heyting calculus mHC is a weakening of the Proof-Intuitionistic Logic KM of Kuznetsov and Muravitsky. In Section 2 we present a short selection of attractive (algebraic, relational, topological and categorical) features of mHC. In Section 3 we establish some close connections between mHC and certain normal extension K4.Grz of the modal system K4. We define a translation of mHC into (...) K4.Grz and prove that this translation is exact, i. e. theorem-preserving and deducibility-invariant. We have established (however, in this note we do not present a proof of this) that the lattice of all extensions of mHC is isomorphic to the lattice of normal extensions of K4.Grz (a generalization of the Kuznetsov and Muravitsky theorem). (shrink)

We present some results on algebraic and modal analysis of polynomial distortions of the standard provability predicate in Peano Arithmetic PA, and investigate three provability-like modal systems related to the Gödel–Löb modal system GL. We also present a short review of relational and topological semantics for these systems, and describe the dual category of algebraic models of our main modal system.

The Gödel translation provides an embedding of the intuitionistic logic$\mathsf {IPC}$into the modal logic$\mathsf {Grz}$, which then embeds into the modal logic$\mathsf {GL}$via the splitting translation. Combined with Solovay’s theorem that$\mathsf {GL}$is the modal logic of the provability predicate of Peano Arithmetic$\mathsf {PA}$, both$\mathsf {IPC}$and$\mathsf {Grz}$admit provability interpretations. When attempting to ‘lift’ these results to the monadic extensions$\mathsf {MIPC}$,$\mathsf {MGrz}$, and$\mathsf {MGL}$of these logics, the same techniques no longer work. Following a conjecture made by Esakia, we add an appropriate version (...) of Casari’s formula to these monadic extensions (denoted by a ‘+’), obtaining that the Gödel translation embeds$\mathsf {M^{+}IPC}$into$\mathsf {M^{+}Grz}$and the splitting translation embeds$\mathsf {M^{+}Grz}$into$\mathsf {MGL}$. As proven by Japaridze, Solovay’s result extends to the monadic system$\mathsf {MGL}$, which leads us to a provability interpretation of both$\mathsf {M^{+}IPC}$and$\mathsf {M^{+}Grz}$. (shrink)

Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces.

Stalnaker, 169–199 2006) introduced a combined epistemic-doxastic logic that can formally express a strong concept of belief, a concept of belief as ‘subjective certainty’. In this paper, we provide a topological semantics for belief, in particular, for Stalnaker’s notion of belief defined as ‘epistemic possibility of knowledge’, in terms of the closure of the interior operator on extremally disconnected spaces. This semantics extends the standard topological interpretation of knowledge with a new topological semantics for belief. We prove that the belief (...) logic KD45 is sound and complete with respect to the class of extremally disconnected spaces and we compare our approach to a different topological setting in which belief is interpreted in terms of the derived set operator. We also study belief revision as well as belief dynamics by providing a topological semantics for conditional belief and belief update modalities, respectively. Our setting based on extremally disconnected spaces, however, encounters problems when extended with dynamic updates. We then propose a solution consisting in interpreting belief in a similar way based on hereditarily extremally disconnected spaces, and axiomatize the belief logic of hereditarily extremally disconnected spaces. Finally, we provide a complete axiomatization of the logic of conditional belief and knowledge, as well as a complete axiomatization of the corresponding dynamic logic. (shrink)

This book illustrates the program of Logical-Informational Dynamics. Rational agents exploit the information available in the world in delicate ways, adopt a wide range of epistemic attitudes, and in that process, constantly change the world itself. Logical-Informational Dynamics is about logical systems putting such activities at center stage, focusing on the events by which we acquire information and change attitudes. Its contributions show many current logics of information and change at work, often in multi-agent settings where social behavior is essential, (...) and often stressing Johan van Benthem's pioneering work in establishing this program. However, this is not a Festschrift, but a rich tapestry for a field with a wealth of strands of its own. The reader will see the state of the art in such topics as information update, belief change, preference, learning over time, and strategic interaction in games. Moreover, no tight boundary has been enforced, and some chapters add more general mathematical or philosophical foundations or links to current trends in computer science. The theme of this book lies at the interface of many disciplines. Logic is the main methodology, but the various chapters cross easily between mathematics, computer science, philosophy, linguistics, cognitive and social sciences, while also ranging from pure theory to empirical work. Accordingly, the authors of this book represent a wide variety of original thinkers from different research communities. And their interconnected themes challenge at the same time how we think of logic, philosophy and computation. Thus, very much in line with van Benthem's work over many decades, the volume shows how all these disciplines form a natural unity in the perspective of dynamic logicians (broadly conceived) exploring their new themes today. And at the same time, in doing so, it offers a broader conception of logic with a certain grandeur, moving its horizons beyond the traditional study of consequence relations. (shrink)

-/- Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. The logic has been inspired by developments in meta-mathematics such as Gödel’s incompleteness theorems of 1931 and Löb’s theorem of 1953. As a modal logic, provability logic has been studied since the early seventies, and has had important applications in the foundations of mathematics. -/- From a philosophical point of view, provability logic is interesting because (...) the concept of provability in a fixed theory of arithmetic has a unique and non-problematic meaning, other than concepts like necessity and knowledge studied in modal and epistemic logic. Furthermore, provability logic provides tools to study the notion of self-reference. (shrink)

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)