Van Benthem Research Articles

A correspondence theorem for interpretability logic with respect to Verbrugge semantics

Abstract Interpretability logic is a modal logic that can be used to describe relative interpretability between extensions of a given first-order arithmetical theory. Verbrugge semantics is a generalization of the basic semantics for interpretability logic. Bisimulation is the basic equivalence between models for modal logic. The Van Benthem Correspondence Theorem establishes modal logic as the bisimulation invariant fragment of first-order logic. In this paper we show that a special type of bisimulations, the so-called w-bisimulations, enable an analogue of the Van Benthem Theorem for interpretability logic w.r.t. Verbrugge semantics.

Concepts of Knowability

Many philosophical discussions hinge on the concept of knowability. For example, there is a blooming literature on the so-called paradox of knowability. How to understand this notion, however? In this paper, we examine several approaches to the notion: the naive approach to take knowability as the possibility to know, the counterfactual approach endorsed by Edgington (1985) and Schlöder (2019) , approaches based on the notion of a capacity or ability to know (Fara 2010, Humphreys 2011), and finally, approaches that make use of the resources of dynamic epistemic logic (van Benthem 2004, Holliday 2017).

Realizing impossibilities

It is common in epistemic modal logic to model the epistemic states of agents via box operators in the normal logic S5. However, this approach treats agents as logically omniscient by requiring their knowledge to be closed under classical logical consequence. A promising way of avoiding logical omniscience consists in extending epistemic models with impossible states, that is states, where complex formulas are not evaluated recursively. However, this approach faces the dual problem of logical ignorance by modeling agents as not even minimally logically competent. In this paper I will outline an epistemic logic that combines impossible states with dynamic realization modalities akin to the dynamic announcement operators from public announcement logic (PAL). I will show that this epistemic logic avoids both the problem of logical omniscience and the problem of logical ignorance. Furthermore, I prove that so-called successful updates in my logic can be characterized in the same way as in PAL and that a similar logic due to Johan van Benthem can be simulated in my logic. Finally, I will compare my epistemic logic with a similar one, which has recently been advanced by JC Bjerring and Mattias Skipper.

Modal Information Logics: Axiomatizations and Decidability

The present paper studies formal properties of so-called modal information logics (MILs)—modal logics first proposed in (van Benthem 1996) as a way of using possible-worlds semantics to model a theory of information. They do so by extending the language of propositional logic with a binary modality defined in terms of being the supremum of two states. First proposed in 1996, MILs have been around for some time, yet not much is known: (van Benthem 2017, 2019) pose two central open problems, namely (1) axiomatizing the two basic MILs of suprema on preorders and posets, respectively, and (2) proving (un)decidability. The main results of the first part of this paper are solving these two problems: (1) by providing an axiomatization [with a completeness proof entailing the two logics to be the same], and (2) by proving decidability. In the proof of the latter, an emphasis is put on the method applied as a heuristic for proving decidability ‘via completeness’ for semantically introduced logics; the logics lack the FMP w.r.t. their classes of definition, but not w.r.t. a generalized class. These results are build upon to axiomatize and prove decidable the MILs attained by endowing the language with an ‘informational implication’—in doing so a link is also made to the work of (Buszkowski 2021) on the Lambek Calculus.

An (almost) fuzzy logic of action and preferences, its quasi-model interpretations, and the problem of its decidability

The logical depiction of the concept of action forms a unique method of grasping its semantic content. It finds its reflection in different proposals of epistemic and deontic-logic systems describing actions as the operational behaviour of an agent as a game played over a given game tree. This perspective establishes some unique connection between actions and the agent's preferences. One of the logical-game-theoretic approaches to their modelling has been elaborated in van Benthem's school in the common area of game-theory, epistemic and fixed-point logic. In reference to some ideas of van Benthem's school – the article aims to propose a new (almost) fuzzy logic system AFLAP for actions and preferences describing a game-theoretic behaviour of an agent over a given game tree. Nevertheless, the backward induction strategy is idealised and exchanged for the Church-Rosser property, and a piece of fuzziness is introduced to the system by considering two unique ‘epsilon’ relations, Move and BI, with their approximation sequences. An abstract model in the form of the so-called quasi-model is proposed for such a system. It forms a multi-level (potentially infinite) construction built due to the pattern recently elaborated and enhanced by Wolter-Kurucz's school. This construction is exploited in the article as a convenient basis for proving the decidability of AFLAP by showing its ‘abstract’ finite model property. This task is performed by constructing an additional workable quasi-model of the size dependent on the size of the set of subformulae of a given formula of AFLAP. Finally, some R-programming-oriented applications of the earlier theoretic considerations are presented.

TESTING DEFINITIONAL EQUIVALENCE OF THEORIES VIA AUTOMORPHISM GROUPS

Abstract Two first-order logic theories are definitionally equivalent if and only if there is a bijection between their model classes that preserves isomorphisms and ultraproducts (Theorem 2). This is a variant of a prior theorem of van Benthem and Pearce. In Example 2, uncountably many pairs of definitionally inequivalent theories are given such that their model categories are concretely isomorphic via bijections that preserve ultraproducts in the model categories up to isomorphism. Based on these results, we settle several conjectures of Barrett, Glymour and Halvorson.

Prevention of surgical site infections: a personal odyssey

Prevention of surgical site infections: a personal odyssey

Light at the end of the tunnel of Corti

Light at the end of the tunnel of Corti

Energy transitions and technology change: “Leapfrogging” reconsidered

Understanding the relationship of economic growth, energy consumption, and pollution is critical for forecasting energy demand and environmental impacts. Energy technology innovations create opportunities for low- and middle-income countries to “leapfrog” over older technologies that dominated the development paths of today’s industrialized countries. Yet technology change may also increase consumption at earlier development stages, and institutions and policy choices can hinder energy-efficient development. Extending a recent paper by van Benthem (2015), I build a panel dataset over 136 countries and nearly 60 years to show that over this wider and longer panel, the long-run energy intensity of economic growth is substantially lower for today’s developing countries than in developing countries in the past.

Three faces of recursion axioms: the case of constructive dynamic logic of relation changers

Abstract This paper proposes an intuitionistic generalization of van Benthem and Liu’s dynamic logic of relation changers, where relation changers are dynamic operators that rewrite each agent’s accessibility relation. We employ Nishimura’s Kripke semantics for a constructive propositional dynamic logic to define the semantics of relation changers. Strong completeness results of Hilbert-style axiomatizations for constructive dynamic logic of relation changers are established by two types of rewriting strategies via recursion axioms: inside-out and outside-in strategies. To establish the semantic completeness by outside-in strategy, we propose the notion of composition of relation changers and an appropriately defined notion of length of a formula. Moreover, we reveal that recursion axioms have additional two faces: semantic and proof-theoretic ones. As for the semantic face, we introduce an alternative semantics for dynamic logic of relation changers to specify the semantic meaning of recursion axioms used in the inside-out strategy. This leads us to a semantic completeness proof of the axiomatization for the original semantics, which does not require a rewriting strategy based on recursion axioms. As for the proof-theoretic face, we transform each of recursion axioms required in the outside-in strategy into two (left and right) inference rules to provide a sequent calculus for dynamic logic of relation changers. Since the resulting sequent calculus is cut-free but semi-analytic, it still enjoys the Craig interpolation theorem by Maehara’s method.

Sahlqvist correspondence theory for second-order propositional modal logic

AbstractModal logic with propositional quantifiers (i.e. second-order propositional modal logic ($\textsf {SOPML}$)) has been considered since the early time of modal logic. Its expressive power and complexity are high, and its van Benthem–Rosen theorem and Goldblatt–Thomason theorem have been proved by ten Cate (2006, J. Philos. Logic, 35, 209–223). However, the Sahlqvist theory of $\textsf {SOPML}$ has not been considered in the literature. In the present paper, we fill in this gap. We develop the Sahlqvist correspondence theory for $\textsf {SOPML}$, which covers and properly extends existing Sahlqvist formulas in basic modal logic. We define the class of Sahlqvist formulas for $\textsf {SOMPL}$ step by step in a hierarchical way, each formula of which is shown to have a first-order correspondent over Kripke frames effectively computable by an algorithm $\textsf {ALBA}^{\textsf {SOMPL}}$. In addition, we show that certain $\varPi _2$-rules correspond to $\varPi _2$-Sahlqvist formulas in $\textsf {SOMPL}$, which further correspond to first-order conditions, and that even for very simple $\textsf {SOMPL}$ Sahlqvist formulas, they could already be non-canonical.

Data-informed knowledge and strategies

The article proposes a new approach to reasoning about knowledge and strategies in multiagent systems. It emphasizes data, not agents, as the source of strategic knowledge. The approach brings together Armstrong's functional dependency from database theory, a data-informed knowledge modality based on a recent work by Baltag and van Benthem, and a newly proposed data-informed strategy modality. The main technical result is a sound and complete logical system that describes the interplay between these three logical operators.

A van Benthem Theorem for Atomic and Molecular Logics

After recalling the definitions of atomic and molecular logics, we show how notions of bisimulation can be automatically defined from the truth conditions of the connectives of any of these logics. Then, we prove a generalization of van Benthem modal characterization theorem for molecular logics. Our molecular connectives should be uniform and contain all conjunctions and disjunctions. We use modal logic, the Lambek calculus and modal intuitionistic logic as case study and compare in particular our work with Olkhovikov's work.

Excluded entailments and the de se/de re partition

ABSTRACT We show that some PRO-sentences (sentences containing an unpronounced subject in an infinitive clause) appear to receive de re interpretations when they occur in suitable discourse contexts or linguistic environments. This finding is surprising given the received view that such sentences are unambiguously de se [Morgan. 1970. “On the Criterion of Identity for Noun Phrase Deletion.” Papers from the Sixth Regional Meeting of the Chicago Linguistics Society, Chicago, IL, 380–389; Chierchia. 1990. “Anaphora and Attitudes de se.” In Semantics and Contextual Expression, edited by R. Bartsch, J. van Benthem, and P. van Emde Boas, 1–32. Dordrecht: Foris]. We argue that these apparent de re readings are not genuine readings assigned by the grammar, but rather arise via a pragmatic process that results in what we call ‘excluded entailments’. The interpretation of PRO-sentences thus shows a certain amount of pragmatic latitude that is reminiscent of other cases of non-literal language, such as examples of ‘pragmatic slack’ first discussed in Lasersohn [1999. “Pragmatic Halos.” Language 75: 522–551]. However, we show that there are crucial differences between the puzzling data at hand and more familiar cases of pragmatic slack, and argue that they constitute a different class of non-literal meaning that has received less attention in the literature. This class of meanings is akin to that introduced by the modifier ‘in effect’. Our view has consequences for semantic methodology, understanding the limits of non-literal language use and interpretation, and the division of labour between semantics and pragmatics in the analysis of attitude reports.

Hennessy-Milner and Van Benthem for Instantial Neighbourhood Logic

Hennessy-Milner and Van Benthem for Instantial Neighbourhood Logic

Finite Model Property and Bisimulation for LFD

Recently, Baltag and van Benthem introduced a decidable logic of functional dependence (LFD) that extends the logic of Cylindrical Relativized Set Algebras (CRS) with atomic local dependence statements. Its semantics can be given in terms of generalised assignment models or their modal counterparts, hence the logic is both a first-order and a modal logic. We show that LFD has the finite model property (FMP) using Herwig's theorem on extending partial isomorphisms, and prove a bisimulation invariance theorem characterizing LFD as a fragment of first-order logic.

A FIRST-ORDER FRAMEWORK FOR INQUISITIVE MODAL LOGIC

AbstractWe present a natural standard translation of inquisitive modal logic $\mathrm{InqML}$ into first-order logic over the natural two-sorted relational representations of the intended models, which captures the built-in higher-order features of $\mathrm{InqML}$ . This translation is based on a graded notion of flatness that ties the inherent second-order, team-semantic features of $\mathrm{InqML}$ over information states to subsets or tuples of bounded size. A natural notion of pseudo-models, which relaxes the non-elementary constraints on the intended models, gives rise to an elementary, purely model-theoretic proof of the compactness property for $\mathrm{InqML}$ . Moreover, we prove a Hennessy-Milner theorem for $\mathrm{InqML}$ , which crucially uses $\omega $ -saturated pseudo-models and the new standard translation. As corollaries we also obtain van Benthem style characterisation theorems.

Logics of dependence and independence: The local variants

Abstract Modern logics of dependence and independence are based on team semantics, which means that formulae are evaluated not on a single assignment of values to variables, but on a set of such assignments, called a team. This leads to high expressive power, on the level of existential second-order logic. As an alternative, Baltag and van Benthem have proposed a local variant of dependence logic, called logic of functional dependence ($ {\textsf {LFD}}$). While its semantics is also based on a team, the formulae are evaluated locally on just one of its assignments, and the team just serves as the supply of the possible assignments that are taken into account in the evaluation process. This logic thus relies on the modal perspective of generalized assignments semantics and can be seen as a fragment of first-order logic. For the variant of $ {\textsf {LFD}}$ without equality, the satisfiability problem is decidable. We extend the idea of localizing logics of dependence and independence in a systematic way, taking into account local variants of standard atomic dependency properties: besides dependence and independence, also inclusion, exclusion and anonymity. We study model-theoretic and algorithmic questions of the localized logics and also resolve some of the questions that had been left open by Baltag and van Benthem. In particular, we study decidability issues of the local logics and prove that satisfiability of $ {\textsf {LFD}}$ with equality is undecidable. Further, we establish characterization theorems via appropriate notions of bisimulation and study the complexity of model checking problems for these logics.

Bimodal Logic with Contingency and Accident: Bisimulation and Axiomatizations

In this paper, a suitable notion of bisimulation is proposed for the bimodal logic with contingency and accident. We obtain several van Benthem Characterization Theorems, and axiomatize the bimodal logic over the class of Eulidean frames and over some more restricted classes, showing their strong completeness via a novel strategy, thereby answering two open questions raised in the literature. With the new bisimulation notion, we also correct an error in the expressivity results in the literature.

Frame definability, canonicity and cut elimination in common sense modal predicate logics

Abstract The paper presents semantically complete Hilbert-style systems for some variants of common sense modal predicate logic proposed by van Benthem and further developed by Seligman. The paper also investigates frame definability in the logics and shows what axiom schema is canonical in the logics. In addition to these semantic investigations on the logics, the paper provides the sequent calculi for some of the logics which enjoy cut elimination theorem.