Framework Of Modal Logic Research Articles

A Medieval Controversy about Entailments between Categorical and ‘Continuing’ Propositions

The early thirteenth century tract Ars Meliduna deals with the issue whether categorical propositions entail, or are entailed by, ‘continuing’ propositions, i.e. by implications. From the perspective of modern logic, with implication interpreted as a material, truth-functional connective, the first question has to be answered in the affirmative because, e.g. β entails (α ⊃ β). But conversely (α ⊃ β) ‘normally’ doesn’t entail the truth (or the falsity) of any of the components α, β; hence the second question should be answered in the negative. These results cannot, however, be directly transferred to medieval logic because implication was there usually interpreted as a strict implication. Yet, within the framework of modal logic, one obtains parallel results. In particular, whenever β is necessary, then (α → β) must be true. Furthermore, if (α → β) is itself necessary, then it is entailed by α. Thus, in particular, ‘Socrates is a man’ entails the analytically true implication ‘If Socrates is a man, Socrates is an animal’, and the same premiss also entails the tautological consequent ‘If Socrates is white, Socrates is white’.

Modified Numerals and Split Disjunction: The First-Order Case

We present a number of puzzles arising for the interpretation of modified numerals. Following Büring and others we assume that the main difference between comparative and superlative modifiers is that only the latter convey disjunctive meanings. We further argue that the inference patterns triggered by disjunction and superlative modifiers are hard to capture in existing semantic and pragmatic analyses of these phenomena (neo-Gricean or grammatical alike), and we propose a novel account of these inferences in the framework of bilateral state-based modal logic defining a first order extension of Aloni (Semant Pragmat 15:5-EA, 2022. https://doi.org/10.3765/sp.15.5)’s BSML.

Expertise and information: an epistemic logic perspective

In this paper we present a modal logic framework to reason about the expertise of information sources. A source is considered an expert on a proposition varphi if they are able to correctly refute varphi in any possible world where varphi is false. Closely connected with expertise is a notion of soundness of information: varphi is said to be “sound” if it is true up to lack of expertise of the source. That is, any statement logically weaker than varphi on which the source has expertise must in fact be true. This is relevant for modelling situations in which sources make claims beyond their domain of expertise. Particular attention is paid to the connection between expertise and knowledge: we show that expertise and soundness admit precise interpretations in terms of S4 and S5 epistemic logic, under certain conditions. We go on to extend the framework to multiple sources, defining two notions of collective expertise. These also have epistemic interpretations via distributed and common knowledge from multi-agent epistemic logic. On the technical side, we give several sound and complete axiomatisations of various classes of expertise models.

Quantitative and Qualitative Approaches to Generalization and Replication-A Representationalist View.

In this paper, we provide a re-interpretation of qualitative and quantitative modeling from a representationalist perspective. In this view, both approaches attempt to construct abstract representations of empirical relational structures. Whereas quantitative research uses variable-based models that abstract from individual cases, qualitative research favors case-based models that abstract from individual characteristics. Variable-based models are usually stated in the form of quantified sentences (scientific laws). This syntactic structure implies that sentences about individual cases are derived using deductive reasoning. In contrast, case-based models are usually stated using context-dependent existential sentences (qualitative statements). This syntactic structure implies that sentences about other cases are justifiable by inductive reasoning. We apply this representationalist perspective to the problems of generalization and replication. Using the analytical framework of modal logic, we argue that the modes of reasoning are often not only applied to the context that has been studied empirically, but also on a between-contexts level. Consequently, quantitative researchers mostly adhere to a top-down strategy of generalization, whereas qualitative researchers usually follow a bottom-up strategy of generalization. Depending on which strategy is employed, the role of replication attempts is very different. In deductive reasoning, replication attempts serve as empirical tests of the underlying theory. Therefore, failed replications imply a faulty theory. From an inductive perspective, however, replication attempts serve to explore the scope of the theory. Consequently, failed replications do not question the theory per se, but help to shape its boundary conditions. We conclude that quantitative research may benefit from a bottom-up generalization strategy as it is employed in most qualitative research programs. Inductive reasoning forces us to think about the boundary conditions of our theories and provides a framework for generalization beyond statistical testing. In this perspective, failed replications are just as informative as successful replications, because they help to explore the scope of our theories.

Trust and agency in the context of communication

The communication process is analysed on the basis of the notions of trust and agency. The aim of the paper is to clarify the role played by causality on the one hand and the role played by logical consequences of assumptions about trust in information sources on the other hand. The first part is informal and the second part refers to the logical framework of modal logic though it requires a quite limited background in this area.

Price of privacy

The article proposes a logical framework for reasoning about agents' ability to protect their privacy by hiding certain information from a privacy intruder. It is assumed that the knowledge of the intruder is derived from the observation of pieces of evidence and that there is a cost associated with the elimination of the evidence. The logical framework contains a modal operator labeled by a group of agents and a total budget available to this group. The key contribution of this work is the proposed incorporation of the cost factor into privacy protection reasoning within the standard modal logic framework. The main technical result are the soundness and completeness theorems for the introduced logical system with respect to a formally defined semantics.

Worlds and Propositions Set Free

The authors provide an object-theoretic analysis of two paradoxes in the theory of possible worlds and propositions stemming from Russell and Kaplan. After laying out the paradoxes, the authors provide a brief overview of object theory and point out how syntactic restrictions that prevent object-theoretic versions of the classical paradoxes are justified philosophically. The authors then trace the origins of the Russell paradox to a problematic application of set theory in the definition of worlds. Next the authors show that an object-theoretic analysis of the Kaplan paradox reveals that there is no genuine paradox at all, as the central premise of the paradox is simply a logical falsehood and hence can be rejected on the strongest possible grounds—not only in object theory but for the very framework of propositional modal logic in which Kaplan frames his argument. The authors close by fending off a possible objection that object theory avoids the Russell paradox only by refusing to incorporate set theory and, hence, that the object-theoretic solution is only a consequence of the theory’s weakness.

The Logic of Knowledge and the Flow of Information

In this paper I look at Fred Dretske's account of information and knowledge as developed in Knowledge and The Flow of Information. In particular, I translate Dretske's probabilistic definition of information to a modal logical framework and subsequently use this to explicate the conception of information and its flow which is central to his account, including the notions of channel conditions and relevant alternatives. Some key products of this task are an analysis of the issue of information closure and an investigation into some of the logical properties of Dretske's account of information flow.

A modal framework for modelling abductive reasoning

We present a framework for understanding abduction within modal logic and Kripke semantics; worlds of a Kripke frame will represent possible theories, and a change in theory will be understood as a passage from one world to an adjacent possible world. Further, these steps may agree with the accessibility relation or may ‘backtrack’, accordingly as new information refutes or reinforces our present theory. Our formalism can be used to model not only abduction, but also to talk about the inner structure of theories as well as relations between them, allowing us to interpret many ideas from philosophy of science within the well-understood framework of modal logic.

A modal logic framework for reasoning about comparative distances and topology

We propose and investigate a uniform modal logic framework for reasoning about topology and relative distance in metric and more general distance spaces, thus enabling the comparison and combination of logics from distinct research traditions such as Tarski’s S 4 for topological closure and interior, conditional logics, and logics of comparative similarity. This framework is obtained by decomposing the underlying modal-like operators into first-order quantifier patterns. We then show that quite a powerful and natural fragment of the resulting first-order logic can be captured by one binary operator comparing distances between sets and one unary operator distinguishing between realised and limit distances (i.e., between minimum and infimum). Due to its greater expressive power, this logic turns out to behave quite differently from both S 4 and conditional logics. We provide finite (Hilbert-style) axiomatisations and ExpTime-completeness proofs for the logics of various classes of distance spaces, in particular metric spaces. But we also show that the logic of the real line (and various other important metric spaces) is not recursively enumerable. This result is proved by an encoding of Diophantine equations.

Ultraproducts and possible worlds semantics in institutions

We develop possible worlds (Kripke) semantics at the categorical abstract model theoretic level provided by the so-called ‘institutions’. Our general abstract modal logic framework provides a method for systematic Kripke semantics extensions of logical systems from computing science and logic. We also extend the institution-independent method of ultraproducts of [R. Diaconescu, Institution-independent ultraproducts, Fundamenta Informaticæ 55 (3–4) (2003) 321–348] to possible worlds semantics and prove a fundamental preservation result for abstract modal satisfaction. As a consequence we develop a generic compactness result for possible worlds semantics.

My beliefs about your beliefs: a case study in theory of mind and epistemic logic

We model three examples of beliefs that agents may have about other agents’ beliefs, and provide motivation for this conceptualization from the theory of mind literature. We assume a modal logical framework for modelling degrees of belief by partially ordered preference relations. In this setting, we describe that agents believe that other agents do not distinguish among their beliefs (‘no preferences’), that agents believe that the beliefs of other agents are in part as their own (‘my preferences’), and the special case that agents believe that the beliefs of other agents are exactly as their own (‘preference refinement’). This multi-agent belief interaction is frame characterizable. We provide examples for introspective agents. We investigate which of these forms of belief interaction are preserved under three common forms of belief revision.

Individual Concepts in Modal Predicate Logic

The article deals with the interpretation of propositional attitudes in the framework of modal predicate logic. The first part discusses the classical puzzles arising from the interplay between propositional attitudes, quantifiers and the notion of identity. After comparing different reactions to these puzzles it argues in favor of an analysis in which evaluations of de re attitudes may vary relative to the ways of identifying objects used in the context of use. The second part of the article gives this analysis a precise formalization from a model- and proof-theoretic perspective.

A modal logic framework for multi-agent belief fusion

This article provides a modal logic framework for reasoning about multi-agent belief and its fusion. We propose logics for reasoning about cautiously merged agent beliefs that have different degrees of reliability. These logics are obtained by combining the multi-agent epistemic logic and multi-source reasoning systems. The fusion is cautious in the sense that if an agent's belief is in conflict with those of higher priorities, then his belief is completely discarded from the merged result. We consider two strategies for the cautious merging of beliefs. In the first, called level cutting fusion , if inconsistency occurs at some level, then all beliefs at the lower levels are discarded simultaneously. In the second, called level skipping fusion , only the level at which the inconsistency occurs is skipped. We present the formal semantics and axiomatic systems for these two strategies and discuss some applications of the proposed logical systems. We also develop a tableau proof system for the logics and prove the complexity result for the satisfiability and validity problems of these logics.

Coalgebraic modal logic: soundness, completeness and decidability of local consequence

This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the category of sets. This is achieved through the use of predicate liftings, which generalise atomic propositions and modal operators from Kripke models to arbitrary coalgebras. Predicate liftings also allow us to use induction along the terminal sequence of the underlying endofunctor as a proof principle. This induction principle is systematically exploited to establish soundness, completeness and decidability of the logics. We believe that this induction principle also opens new ways for reasoning about modal logics: Our proof of completeness does not rely on a canonical model construction, and the proof of the finite model property does not use filtrations.

An inferential approach to Information Retrieval and its implementation using a manual thesaurus

Most inferential approaches to Information Retrieval (IR) have been investigated within the probabilistic framework. Although these approaches allow one to cope with the underlying uncertainty of inference in IR, the strict formalism of probability theory often confines our use of knowledge to statistical knowledge alone (e.g. connections between terms based on their co-occurrences). Human-defined knowledge (e.g. manual thesauri) can only be incorporated with difficulty. In this paper, based on a general idea proposed by van Rijsbergen, we first develop an inferential approach within a fuzzy modal logic framework. Differing from previous approaches, the logical component is emphasized and considered as the pillar in our approach. In addition, the flexibility of a fuzzy modal logic framework offers the possibility of incorporating human-defined knowledge in the inference process. After defining the model, we describe a method to incorporate a human-defined thesaurus into inference by taking user relevance feedback into consideration. Experiments on the CACM corpus using a general thesaurus of English, Wordnet, indicate a significant improvement in the system's performance.

INFORMATION RETRIEVAL BY LOGICAL IMAGING

The evaluation of an implication by Imaging is a logical technique developed in the framework of modal logic. Its interpretation in the context of a ‘possible worlds’ semantics is very appealing for ir. In 1989, Van Rijsbergen suggested its use for solving one of the fundamental problems of logical models of IR: the evaluation of the implication d → q (where d and q are respectively a document and a query representation). Since then, others have tried to follow that suggestion proposing models and applications, though without much success. Most of these approaches had as their basic assumption the consideration that ‘a document is a possible world’. We propose instead an approach based on a completely different assumption: ‘a term is a possible world’. This approach enables the exploitation of term‐term relationships which are estimated using an information theoretic measure.

ON MODAL LOGIC INTERPRETATION OF POSSIBILITY THEORY

This paper is a continuation in a series of papers started by Resconi et. al. [8] in which we try to develop interpretations of various uncertainty theories within the framework of standard modal logic. In this paper, we deal with possibility theory. We suggest an interpretation of possibility measures and necessity measures and show that this interpretation is complete for rational-valued measures.

Counterfactual reasoning by (means of) defaults

We show how defaults can be used for counterfactual reasoning. We use a framework of modal logic to reason about both defaults and counterfactuals, in which one can express certainties, possibilities, actualities and (preferred or practical) beliefs in a distinct manner. Firstly, we discuss some properties of our approach in relation to other approaches in the literature.

A logic for reasoning about security

A formal framework called Security Logic ( SL ) is developed for specifying and reasoning about security policies and for verifying that system designs adhere to such policies. Included in this modal logic framework are definitions of knowledge, permission, and obligation . Permission is used to specify secrecy policies and obligation to specify integrity policies. The combination of policies is addressed and examples based on policies from the current literature are given.