Set Of Possible Worlds Research Articles

Hyperintensionality and overfitting

A hyperintensional epistemic logic would take the contents which can be known or believed as more fine-grained than sets of possible worlds. I consider one objection to the idea: Williamson’s Objection from Overfitting. I propose a hyperintensional account of propositions as sets of worlds enriched with topics: what those propositions, and so the attitudes having them as contents, are about. I show that the account captures the conditions under which sentences express the same content; that it can be pervasively applied in formal and mainstream epistemology; and that it is left unscathed by the objection.

The Hyperintensional Variant of Kaplan’s Paradox

David Kaplan famously argued that mainstream semantics for modal logic, which identifies propositions with sets of possible worlds, is affected by a cardinality paradox. Takashi Yagisawa showed that a variant of the same paradox arises when standard possible worlds semantics is extended with impossible worlds to deliver a hyperintensional account of propositions. After introducing the problem, we discuss two general approaches to a possible solution: giving up on sets and giving up on worlds, either in the background semantic framework or in the corresponding conception of propositions. As a result, we conclude that abandoning worlds by embracing a truthmaker-based approach offers a promising way to account for hyperintensional propositions without facing the paradoxical outcome.

A Hyperintensional Logic of Non-prime Evidence

We present a logic of evidence that reduces agents’ epistemic idealisations by combining classical propositional logic with substructural modal logic for formulas in the scope of epistemic modalities. To this aim, we provide a neighborhood semantics of evidence, which provides a modal extension of Fine’s semantics for relevant propositional logic. Possible worlds semantics for classical propositional logic is then obtained by defining the set of possible worlds as a special subset of information states in Fine’s semantics. Finally, we prove that evidence is a hyperintensional and non-prime notion in our logic, and provide a sound and complete axiomatisation of our evidence logic.

Где и как возникают смыслы (материалы круглого стола)

The discussion aims to identify the interpretive mechanisms that provide an interface be­tween text and context (between language and the world, language and culture, language and society). It is a multi-level system of interfaces connected by inversion and recursion relations and operations. We identify this system as a pragmasemantics. It also acts as an interface platform for correlation/transformation of intra-system semantic units and extralinguistic objects within a set of possible worlds. At the same time, this platform is endowed with the autopoietic potential to generate new operating interfaces systems, as well as textualized structures of operations and interpretations (instructions, memory, reframing). The genera­tion of meaning is the result of the interaction of the system platform with the platform-as-context, due to the possibility that one generates and determines the operational potential of the other, acting a recursive loop or Möbius strip. The connection between subjectivity and meaningful semiosis is crucially important due to the key role of textualization and langua­ging. The signified are reproduced by signifiers and thus generate those processes that are structured by interactions and other practices. In complex systems (biological and beyond), the possibility of self-replication arises through recursive self-interpretation. Thus, subjectivi­ty acts as a kind of universal interface that determines the change of contexts, their interac­tion, the generation of possible new ones, ensuring the pro-creativity and pre-adaptability of the system (its stability and development.

Types and Type Theories in Natural Language Analysis

This article reviews the set of possible paths from a semantics based on Simple Type Theories (STTs) toward one based on Rich Type Theories (RTTs) and the motivations behind the move from one to the other. The main elements of this review are threefold. First, it provides a systematic overview of different STTs, including options for what to include as members of the set of basic types, and whether to assume type constructors additional to the one for constructing functional types. Second, this review discusses the main differences between STTs and RTTs, namely, that in RTTs but not in STTs, types are part of the object language. That is, one can refer to and reason with and about types. In turn, this makes available an alternative account of propositions to the one assumed in semantics in the Frege–Church–Montague tradition: Instead of being characterized as sets of possible worlds, propositions can be treated themselves as types, that is, as structured semantic objects. Third and finally, this review provides an outline of some of the main applications of RTTs, including hyperintensionality, quantification, anaphora, polysemy, and modification.

Hyperintensional models for non-congruential modal logics

Abstract In this work, we illustrate applications of a semantic framework for non-congruential modal logic based on hyperintensional models. We start by discussing some philosophical ideas behind the approach; in particular, the difference between the set of possible worlds in which a formula is true (its intension) and the semantic content of a formula (its hyperintension), which is captured in a rigorous way in hyperintensional models. Next, we rigorously specify the approach and provide a fundamental completeness theorem. Moreover, we analyse examples of non-congruential systems that can be semantically characterized within this framework in an elegant and modular way. Finally, we compare the proposed framework with some alternatives available in the literature. In the light of the results obtained, we argue that hyperintensional models constitute a basic, general and unifying semantic framework for (non-congruential) modal logic.

A probabilistic temporal epistemic logic: Decidability

Abstract We study a propositional probabilistic temporal epistemic logic $\textbf {PTEL}$ with both future and past temporal operators, with non-rigid set of agents and the operators for agents’ knowledge and for common knowledge and with probabilities defined on the sets of runs and on the sets of possible worlds. A semantics is given by a class ${\scriptsize{\rm Mod}}$ of Kripke-like models with possible worlds. We prove decidability of $\textbf {PTEL}$ by showing that checking satisfiability of a formula in ${\scriptsize{\rm Mod}}$ is equivalent to checking its satisfiability in a finite set of finitely representable structures. The same procedure can be applied to the class of all synchronous ${\scriptsize{\rm Mod}}$-models. We give an upper complexity bound for the satisfiability problem for ${\scriptsize{\rm Mod}}$.

A Defense of Explanationism against Recent Objections

Abstract In the recent literature on the nature of knowledge, a rivalry has emerged between modalism and explanationism. According to modalism, knowledge requires that our beliefs track the truth across some appropriate set of possible worlds. Modalists tend to focus on two modal conditions: sensitivity and safety. According to explanationism, knowledge requires only that beliefs bear the right sort of explanatory relation to the truth. In slogan form: knowledge is believing something because it's true. In this paper, we aim to vindicate explanationism from some recent objections offered by Gualtiero Piccinini, Dario Mortini, and Kenneth Boyce and Andrew Moon. Together, these authors present five purported counterexamples to the sufficiency of the explanationist analysis for knowledge. In addition, Mortini devises a clever argument that explanationism entails the violation of a plausible closure principle on knowledge. We will argue that explanationism is innocent of all these charges against it, and we hope that the strength of the defense we offer of explanationism is evidence in its favor, and a reason to investigate explanationism further as the long-elusive truth about the nature of knowledge.

Scaling Up Structural Clustering to Large Probabilistic Graphs Using Lyapunov Central Limit Theorem

Structural clustering is one of the most widely used graph clustering frameworks. In this paper, we focus on structural clustering of probabilistic graphs, which comes with significant computational challenges and has, so far, resisted efficient solutions that are able to scale to large graphs, e.g. the state-of-art can only handle graphs with a few million edges. We address the main bottleneck step of probabilistic structural clustering, computing the structural similarity of vertices based on their Jaccard similarity over the set of possible worlds of a given probabilistic graph. The state-of-art used Dynamic Programming, a quadratic run-time algorithm, that does not scale to pairs of vertices of high degree. In this paper we present a novel approach based on Lyapunov Central Limit Theorem. By using a carefully chosen set of random variables we are able to cast the computation of structural similarity to computing a one-tailed area under the Normal Distribution. Our approach has linear runtime as opposed to quadratic, and as such, it scales to much larger inputs. Extensive experiments show that our approach can handle massive graphs at web-scale which the state-of-art cannot.

The Fragmented Mind

<i>The Fragmented Mind</i>

Ontological disagreements, reference, and charity: A challenge for Hirsch's deflationism

AbstractEli Hirsch argues that certain ontological disputes involve a conflict between “equivalent” languages, and that the principle of charity compels each disputant to interpret the other as speaking truly in their own language. For Hirsch, a language's semantics maps sentences (in context) onto sets of possible worlds but assigns no role to reference. I argue that this method leads to an overly uncharitable portrayal of the disputes at issue whereby ontologists who speak “equivalent” languages can only argue about syntax. Lack of charity stems from the fact that this portrayal likely fails to uphold the self‐conception of the disputants – and particularly what I will call the weak self‐conception. As a result, Hirsch's deflationism falls victim of the same principle of charity that informs it.

The Explanationist and the Modalist

AbstractRecent epistemology has witnessed a substantial opposition between two competing approaches to capturing the notion of non-accidentality in the analysis of knowledge: the explanationist and the modalist. According to the latest advocates of the former (e.g., Bogardus and Perrin 2020), S knows that p if and only if S believes that p because p is true. According to champions of the latter (e.g., safety and sensitivity theorists), S knows that p if and only if S's belief that p is true in a relevant set of possible worlds. Because Bogardus and Perrin's explanationism promises to deliver a plausible analysis of knowledge without any need for modal notions, it is an elegant proposal with prima facie appeal. However, such version of explanationism ultimately does not live up to its promises: in this paper, I raise some objections to their explanationist analysis of knowledge while showing that modalism is in much better shape than they think. In particular, I argue that their explanationist condition generates the wrong results in Gettier cases and fake-barn cases. I also offer and defend a novel version of modalism: I introduce a refined safety condition which is shown to successfully handle the same Gettier cases that beset Bogardus and Perrin's version of explanationism. The paper concludes with a reassessment of the explanationist's initial ambition to provide an analysis of knowledge without modal notions. The upshot will be that even if the prospects for such analysis remain dim, our money should be on the modalist, not the explanationist.

Technical Perspective

The paper entitled "Probabilistic Data with Continuous Distributions" overviews recent work on the foundations of infinite probabilistic databases [3, 2]. Prior work on probabilistic databases (PDBs) focused almost exclusively on the finite case: A finite PDB represents a discrete probability distribution over a finite set of possible worlds [4]. In contrast, an infinite PDB models a continuous probability distribution over an infinite set of possible worlds. In both cases, each world is a finite relational database instance. Continuous distributions are essential and commonplace tools for reasoning under uncertainty in practice. Accommodating them in the framework of probabilistic databases brings us closer to applications that naturally rely on both continuous distributions and relational databases.

Multi-Centered Worlds, Limited Accessibility and Ways of Believing

Recent descendants of David Lewis, such as Stephen Torre, Dilip Ninan, and Dirk Kindermann have utilized multi-centered propositions, which are roughly sets of possible worlds centered on a sequence of individuals, to characterize the content of attitudes. In an attempt to explain counterfactual attitudes such as wishing and imagining, Ninan (2012, 2013) developed a more fine-grained characterization of multi-centered propositions than others in the multi-centered camp. While Ninan provides a systematic explanation of the nature of de se attitudes (attitudes one expresses with first-person pronouns like ‘I’ or ‘me’), the nature of other types of indexical attitudes (attitudes one expresses with indexical terms like ‘you,’ ‘today,’ ‘now,’) is not explored. This paper does two things. First, it demonstrates how we can account for indexical attitudes within Ninan’s framework. This, however, has the consequence of generating propositions of limited accessibility of sorts, which in turn conflict with a received and simple model of linguistic communication. According to this simple model, successful communication consists of the transmission of mental content from speaker to hearer. Second, in light of this conflict, the paper proposes an adjustment to Ninan’s theory in order to avoid commitment to propositions of limited accessibility, and thus preserve the shareability of attitudinal content from speaker to hearer.

Algorithmic properties of first-order superintuitionistic logics of finite Kripke frames in restricted languages

Abstract We consider the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate superintuitionistic logics of finite Kripke frames on the logics’ algorithmic properties. By a finite frame we mean a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that positive fragments of many predicate superintuitionistic logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter; this applies to the logics of finite frames of the predicate counterparts of propositional logics lying between the intuitionistic logic and the logic of the weak law of the excluded middle.

Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages

Abstract We study the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate modal logics of finite Kripke frames on the logics’ algorithmic properties. A finite frame is a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that most predicate modal logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter. This applies to the logics of finite frames of the predicate extensions of the sublogics of propositional modal logics $\textbf{GL}$, $\textbf{Grz}$ and $\textbf{KTB}$—among them, $\textbf{K}$, $\textbf{T}$, $\textbf{D}$, $\textbf{KB}$, $\textbf{K4}$ and $\textbf{S4}$.

Truth, Meaning, and Yablo’s Paradox – A Moderate Anti-Realist Approach

Yablo’s Paradox, an infinite-sentence version of the Liar Paradox, aims to show that semantic paradox can emerge even without circularity. I will argue that the lack of meaning/content of the sentences involved is the source of the paradoxical outcome.I will introduce and argue for a Moderate Antirealist (MAR) approach to truth and meaning, built around the twin principles that neither truth nor meaning can outstrip knowability. Accordingly, I will introduce a MAR truth operator that both forges an explicit connection between truth and knowability and distinguishes between truth and factuality. I will also argue that the meaning/content of propositions should be identified not with the set of possible worlds in which the propositions are true/factual, but rather in which they are known.I will show that our MAR framework dissolves Yablo’s Paradox and also confirms our intuition that these sentences are all devoid of content/meaning.

Mentaculus Laws and Metaphysics

The laws of nature are central to our understanding of the world. And while there is often broad agreement about the technical formulations of the laws, there can be sharp disagreement about the metaphysical nature of the laws. For instance, the Newtonian laws of nature can be stated and analyzed by appealing to a set of possible worlds. Yet, some philosophers argue the worlds are mere notational devices, while others take them to be robust, concrete entities in their own right. In this paper, I use a recent view of laws called the Mentaculus as a case study to illustrate the wide variety of metaphysical pictures that can accompany such a view. I conclude that the technical features of the laws -- typically (though not always) given to us by practicing scientists -- are compatible with many different metaphysical foundations.

Probability, coherent belief and coherent belief changes

This paper is about the statics and dynamics of belief states that are represented by pairs consisting of an agent’s credences (represented by a subjective probability measure) and her categorical beliefs (represented by a set of possible worlds). Regarding the static side, we argue that the latter proposition should be coherent with respect to the probability measure and that its probability should reach a certain threshold value. On the dynamic side, we advocate Jeffrey conditionalisation as the principal mode of changing one’s belief state. This updating method fits the idea of the Lockean Thesis better than plain Bayesian conditionalisation, and it affords a flexible method for adding and withdrawing categorical beliefs. We show that it fails to satisfy the traditional principles of Inclusion and Preservation for belief revision and the principle of Recovery for belief withdrawals, as well as the Levi and Harper identities. We take this to be a problem for the latter principles rather than for the idea of coherent belief change.

Uncertainty About Evidence

We develop a logical framework for reasoning about knowledge and evidence in which the agent may be uncertain about how to interpret their evidence. Rather than representing an evidential state as a fixed subset of the state space, our models allow the set of possible worlds that a piece of evidence corresponds to to vary from one possible world to another, and therefore itself be the subject of uncertainty. Such structures can be viewed as (epistemically motivated) generalizations of topological spaces. In this context, there arises a natural distinction between what is actually entailed by the evidence and what the agent knows is entailed by the evidence -- with the latter, in general, being much weaker. We provide a sound and complete axiomatization of the corresponding bi-modal logic of knowledge and evidence entailment, and investigate some natural extensions of this core system, including the addition of a belief modality and its interaction with evidence interpretation and entailment, and the addition of a "knowability" modality interpreted via a (generalized) interior operator.