Viewed from a suitable distance, David Lewis's ontological scheme is simplicity itself. Absolutely everything that exists, according to Lewis, is either a spatiotemporal particular, or a set theoretic construction from such particulars, or a mereological aggregate of such items. Set theoretic constructionalism is not an incidental feature of Lewis's system. The master argument of On the Plurality of Worlds is that a pluriverse composed of infinitely many concrete universes constitutes a “paradise for philosophers. Given his other views, (...) his account of classes has profound implications for general metaphysics, implying as it does an account of the nature of properties and instantiation, propositions and propositional truth, the nature of events, and so on. But Lewis does not consider those implications. The aim of this chapter is to close the circle by bringing Lewis's mature account of sets into contact with his larger ontological scheme. (shrink)

The Newtonian research program consists of the core axioms of the Principia Mathematica, a sequence of force laws and auxiliary hypotheses, and a set of methodological rules. The latter underwent several changes and so it is sometimes claimed that, historically seen, Newton and the Newtonians added methodological rules post constructione in order to further support their research agenda. An argument of Duhem, Feyerabend, and Lakatos aims to provide a theoretical reason why Newton could not have come up with his theory (...) of the Principia in accordance with his own methodology: Since Newton’s starting point, Kepler’s laws, contradict the law of universal gravitation, he could not have applied the so-called method of analysis and synthesis. In this paper, this argument is examined with reference to the Principia’s several editions. Newton’s method is characterized, and necessary general background assumptions of the argument are made explicit. Finally, the argument is criticized based on a contemporary philosophy of science point of view. (shrink)

Nick Bostrom and others have suggested treating decision-making under moral uncertainty as analogous to parliamentary decision-making. The core suggestion of this “parliamentary approach” is that the competing moral theories function like delegates to the parliament, and that these delegates then make decisions by some combination of bargaining and voting. There seems some reason to hope that such an approach might avoid standard objections to existing approaches (for example, the “maximise expected choiceworthiness” (MEC) and “my favourite theory” approaches). However, the parliamentary (...) approach is so far extremely underspecified, making it largely indeterminate how such a model will in fact behave in the respects that those concerned with moral uncertainty care about. This paper explores one way of making it precise. We treat predicaments of moral uncertainty as analogous to bargaining situations alone (setting aside voting), and apply a version of the Nash solution that is standard in bargaining theory. The resulting model does indeed perform in many of the hoped-for ways. However, so also does a version of MEC that employs a structural approach to intertheoretic comparisons. It seems to us an open question which, regarding this version of MEC and the bargaining-theoretic approach, is superior to the other. We identify the key points on which the two differ. (shrink)

Questions concerning the proof-theoretic strength of classical versus nonclassical theories of truth have received some attention recently. A particularly convenient case study concerns classical and nonclassical axiomatizations of fixed-point semantics. It is known that nonclassical axiomatizations in four- or three-valued logics are substantially weaker than their classical counterparts. In this paper we consider the addition of a suitable conditional to First-Degree Entailment—a logic recently studied by Hannes Leitgeb under the label HYPE. We show in particular that, by formulating the (...) theory PKF over HYPE, one obtains a theory that is sound with respect to fixed-point models, while being proof-theoretically on a par with its classical counterpart KF. Moreover, we establish that also its schematic extension—in the sense of Feferman—is as strong as the schematic extension of KF, thus matching the strength of predicative analysis. (shrink)

We consider an approach to propositional synonymy in proof-theoretic semantics that is defined with respect to a bilateral G3-style sequent calculus \(\mathtt{SC2Int}\) for the bi-intuitionistic logic \(\mathtt{2Int}\). A distinctive feature of \(\mathtt{SC2Int}\) is that it makes use of two kind of sequents, one representing proofs, the other representing refutations. The structural rules of \(\mathtt{SC2Int}\), in particular its cut rules, are shown to be admissible. Next, interaction rules are defined that allow transitions from proofs to refutations, and vice versa, mediated (...) through two different negation connectives, the well-known implies-falsity negation and the less well-known coimplies-truth negation of \(\mathtt{2Int}\). By assuming that the interaction rules have no impact on the identity of derivations, the concept of inherited identity between derivations in \(\mathtt{SC2Int}\) is introduced and the notions of positive and negative synonymy of formulas are defined. Several examples are given of distinct formulas that are either positively or negatively synonymous. It is conjectured that the two conditions cannot be satisfied simultaneously. (shrink)

ABSTRACT Topos quantum theory is standardly portrayed as a kind of ‘neo-realist’ reformulation of quantum mechanics.1 1 In this article, I study the extent to which TQT can really be characterized as a realist formulation of the theory, and examine the question of whether the kind of realism that is provided by TQT satisfies the philosophical motivations that are usually associated with the search for a realist reformulation of quantum theory. Specifically, I show that the notion of the quantum state (...) is problematic for those who view TQT as a realist reformulation of quantum theory. 1Introduction 2Topos Quantum Theory 2.1Phase space 2.2Hilbert space 2.3Beyond Hilbert space 2.4Defining realism 2.5The spectral presheaf 2.6The logic of topos quantum theory 3Interpreting States in Topos Quantum Theory 4Interpreting Truth Values and Clopen Subobjects in Topos Quantum Theory 4.1Interpreting the truth values 4.2Interpreting Subcl 5Neo-realism 5.1The covariant approach 6Conclusion. (shrink)

The standard approach to what I call “proof-theoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of proof-theoretic semantics, this paper investigates in detail various notions of proof-theoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used in (...) normalization theory. It is argued that these two sorts of concepts must be kept strictly apart. (shrink)

We present an axiomatic approach for a class of finite, extensive form games of perfect information that makes use of notions like “rationality at a node” and “knowledge at a node.” We distinguish between the game theorist's and the players' own “theory of the game.” The latter is a theory that is sufficient for each player to infer a certain sequence of moves, whereas the former is intended as a justification of such a sequence of moves. While in general the (...) game theorist's theory of the game is not and need not be axiomatized, the players' theory must be an axiomatic one, since we model players as analogous to automatic theorem provers that play the game by inferring (or computing) a sequence of moves. We provide the players with an axiomatic theory sufficient to infer a solution for the game (in our case, the backwards induction equilibrium), and prove its consistency. We then inquire what happens when the theory of the game is augmented with information that a move outside the inferred solution has occurred. We show that a theory that is sufficient for the players to infer a solution and still remains consistent in the face of deviations must be modular. By this we mean that players have distributed knowledge of it. Finally, we show that whenever the theory of the game is group-knowledge (or common knowledge) among the players (i.e., it is the same at each node), a deviation from the solution gives rise to inconsistencies and therefore forces a revision of the theory at later nodes. On the contrary, whenever a theory of the game is modular, a deviation from equilibrium play does not induce a revision of the theory. (shrink)

The notion that manual gestures played an important role in the evolution of human language was strengthened by the discovery of mirror neurons in monkey area F5, the proposed homologue of human Broca's area. This idea is central to the thesis developed by Arbib, and lending further support to a link between motor resonance mechanisms and language/communication development is the case of autism and congenital blindness. We provide an account of how these conditions may relate to the aforementioned theory.

Proof-theoretic semantics is an inferentialist theory of meaning, usually developed in a multiple-assumption and single-conclusion framework. In that framework, this theory seems unable to justify classical logic, so some authors have proposed a multiple-conclusion reformulation to accomplish this goal. In the first part of this paper, the debate originated by this proposal is briefly exposed and used to defend the diverging opinion that proof-theoretic semantics should always endorse a single-assumption and single-conclusion framework. In order to adopt this approach (...) some of its _criteria_ of validity, especially separability, need to be weakened. This choice is evaluated and defended. The main argument in this direction is based on the circular dependences of meaning between multiple assumptions and conjunctions, and between multiple conclusions and disjunctions. In the second part of this paper, some systems that suit the new requirements are proposed for both intuitionistic and classical logic. A proof that they are valid, according to the weakened _criteria_, is sketched. (shrink)

Working in a fixed Grothendieck topos Sh(C, J C ) we generalize \({\mathcal{L}_{{\infty},\omega}}\) to allow our languages and formulas to make explicit reference to Sh(C, J C ). We likewise generalize the notion of model. We then show how to encode these generalized structures by models of a related sentence of \({\mathcal{L}_{{\infty},\omega}}\) in the category of sets and functions. Using this encoding we prove analogs of several results concerning \({\mathcal{L}_{{\infty},\omega}}\) , such as the downward Löwenheim–Skolem theorem, the completeness theorem and (...) Barwise compactness. (shrink)

We introduce a framework for a graph-theoretic analysis of the semantic paradoxes. Similar frameworks have been recently developed for infinitary propositional languages by Cook and Rabern, Rabern, and Macauley. Our focus, however, will be on the language of first-order arithmetic augmented with a primitive truth predicate. Using Leitgeb’s notion of semantic dependence, we assign reference graphs (rfgs) to the sentences of this language and define a notion of paradoxicality in terms of acceptable decorations of rfgs with truth values. It (...) is shown that this notion of paradoxicality coincides with that of Kripke. In order to track down the structural components of an rfg that are responsible for paradoxicality, we show that any decoration can be obtained in a three-stage process: first, the rfg is unfolded into a tree, second, the tree is decorated with truth values (yielding a dependence tree in the sense of Yablo), and third, the decorated tree is re-collapsed onto the rfg. We show that paradoxicality enters the picture only at stage three. Due to this we can isolate two basic patterns necessary for paradoxicality. Moreover, we conjecture a solution to the characterization problem for dangerous rfgs that amounts to the claim that basically the Liar- and the Yablo graph are the only paradoxical rfgs. Furthermore, we develop signed rfgs that allow us to distinguish between ‘positive’ and ‘negative’ reference and obtain more fine-grained versions of our results for unsigned rfgs. (shrink)

Psychological essentialism is a cognitive bias through which human beings conceive the entities around them as having inner essences and basic natures. Social scientists routinely generate flawed inferences because their methods require the truth of psychological essentialism. This article develops set-theoretic analysis as a scientific-constructivist approach that overcomes the bias of psychological essentialism. With this approach, the “sets” of set-theoretic analysis are mental phenomena that establish boundaries and identify similarities and differences among entities whose natural kind composition is (...) not known. The approach is illustrated through a consideration of research on intelligence, race, and poverty in the United States. (shrink)

I thank Judea Pearl for his discussion of my paper and respond to the points he raises. In particular, his attachment to unaugmented directed acyclic graphs has led to a misapprehension of my own proposals. I also discuss the possibilities for developing a non-manipulative understanding of causality.

In this paper we propose and analyze a game-theoretic model of the epistemology of peer disagreement. In this model, the peers' rationality is evaluated in terms of their probability of ending the disagreement with a true belief. We find that different strategies---in particular, one based on the Steadfast View and one based on the Conciliatory View---are rational depending on the truth-sensitivity of the individuals involved in the disagreement. Interestingly, the Steadfast and the Conciliatory Views can even be rational simultaneously (...) in some circumstances. We tentatively provide some reasons to favor the Conciliatory View in such cases. We argue that the game-theoretic perspective is a fruitful one in this debate, and this fruitfulness has not been exhausted by the present paper. (shrink)

I discuss three potential sources of evidence for truth in set theory, coming from set theory’s roles as a branch of mathematics and as a foundation for mathematics as well as from the intrinsic maximality feature of the set concept. I predict that new non first-order axioms will be discovered for which there is evidence of all three types, and that these axioms will have significant first-order consequences which will be regarded as true statements of set theory. The bulk of (...) the paper is concerned with the Hyperuniverse Programme, whose aim is to discover an optimal mathematical principle for expressing the maximality of the set-theoretic universe in height and width. (shrink)

The spacing effect is among the most widely replicated empirical phenomena in the learning sciences, and its relevance to education and training is readily apparent. Yet successful applications of spacing effect research to education and training is rare. Computational modeling can provide the crucial link between a century of accumulated experimental data on the spacing effect and the emerging interest in using that research to enable adaptive instruction. In this paper, we review relevant literature and identify 10 criteria for rigorously (...) evaluating computational models of the spacing effect. Five relate to evaluating the theoretic adequacy of a model, and five relate to evaluating its application potential. We use these criteria to evaluate a novel computational model of the spacing effect called the Predictive Performance Equation. Predictive Performance Equation combines elements of earlier models of learning and memory including the General Performance Equation, Adaptive Control of Thought—Rational, and the New Theory of Disuse, giving rise to a novel computational account of the spacing effect that performs favorably across the complete sets of theoretic and applied criteria. We implemented two other previously published computational models of the spacing effect and compare them to PPE using the theoretic and applied criteria as guides. (shrink)

This paper studies some model-theoretic properties of special groups of finite type. Special groups are a first-order axiomatization of the algebraic theory of quadratic forms, introduced by Dickmann and Miraglia, which is essentially equivalent to abstract Witt rings. More precisely, we consider elementary equivalence, saturation, elementary embeddings, quantifier elimination, stability and Morley rank.

Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can realize these embeddings (...) via a wide variety of forcing notions, while providing control over lower bounds as well. We also give a generalization to class forcing in the context of second-order set theory, and exhibit some further structure in the generic multiverse, such as the existence of exact pairs. (shrink)

We propose an account of dynamic predicates which draws on the notion of force, eliminating reference to events in the linguistic semantics. We treat dynamic predicates as predicates of forces, represented as functions from an initial situation to a final situation that occurs ceteris paribus, that is, if nothing external intervenes. The possibility that opposing forces might intervene to prevent the transition to a given final situation leads us to a novel analysis of non-culminating accomplishment predicates in a variety of (...) languages, including the English progressive. We then apply the force-theoretic framework to the composition of basic Vendlerian eventuality types within a lexical-decomposition syntax. The difference between predicates of forces and predicates of situations is argued to underlie the dynamic/stative contrast, and also to allow for a formal treatment of the difference between be and stay. Consequences for the relationship between language and cognition are discussed. (shrink)

One metaphysical problem about laws is how to find appropriate truthmakers for fully general functional laws. What makes it true, for instance, that an uninstantiated mass would interact with others as prescribed by laws concerning mass? This is the missing value problem. D. M. Armstrong attempted to solve it by appeal to determinable universals. I will offer a trope‐theoretic solution that, while in some ways more metaphysically adventurous than Armstrong's view, avoids commitment to universals and determinables (as different from (...) their determinates). The solution makes use of a special conception of tropes as capable of intrinsic change. It also makes use of a distinction between two ways of having a causal power (a distinction we should make in any case). Existing powers‐based approaches to the problem struggle to avoid the idea that powers mysteriously point beyond themselves. But if tropes are capable of intrinsic change in the way I propose, they can account for the full generality of laws with minimal pointing beyond, and can do so while retaining natures that are credibly intrinsic. (shrink)

Evolutionary game theory and public goods games offer an important framework to understand cooperation during pandemics. From this perspective, the COVID-19 situation can be conceptualized as a dilemma where people who neglect safety precautions act as free riders, because they get to enjoy the benefits of decreased health risk from others’ compliance with policies despite not contributing to or even undermining public safety themselves. At the same time, humans appear to carry a suite of evolved psychological mechanisms aimed at curbing (...) free riding in order to ensure the continued provision of public goods, which can be leveraged to develop more effective measures to promote compliance with regulations. We also highlight factors beyond free riding that reduce compliance rates, such as the emergence of conspiratorial thinking, which seriously undermine the effectiveness of measures to suppress free riding. Together, the current paper outlines the social dynamics that occur in public goods dilemmas involving the spread of infectious disease, highlights the utility and limits of evolutionary game-theoretic approaches for COVID-19 management, and suggests novel directions based on emerging challenges to cooperation. (shrink)

Restall :279–291, 2014) proposes a new, proof-theoretic, logical pluralism. This is in contrast to the model-theoretic pluralism he and Beall proposed in Beall and Restall :475–493, 2000) and in Beall and Restall. What I will show is that Restall has not described the conditions on being admissible to the proof-theoretic logical pluralism in such a way that relevance logic is one of the admissible logics. Though relevance logic is not hard to add formally, one critical component of (...) Restall’s pluralism is that the relevance logic that gets added must have connectives which mean the same thing as the connectives in the already admitted logic. This is what I will show is not possible. (shrink)

On some views, we can be sure that parties to a dispute over the logic of ‘exists’ are not talking past each other if they can characterise ‘exists’ as the only monadic predicate up to logical equivalence obeying a certain set of rules of inference. Otherwise, we ought to be suspicious about the reality of their disagreement. This is what we call a proof- theoretic argument. Pace some critics, who have tried to use proof-theoretic arguments to cast doubts (...) about the reality of disagreements about the logic of ‘exists’, we argue that proof-theoretic arguments can be deployed to establish the reality of several such disagreements. Along the way, we will also utilise this technique to establish similar results about some disagreements over the logic of identity. (shrink)

We study model theoretic tree properties and their associated cardinal invariants. In particular, we obtain a quantitative refinement of Shelah’s theorem for countable theories, show that [Formula: see text] is always witnessed by a formula in a single variable and that weak [Formula: see text] is equivalent to [Formula: see text]. Besides, we give a characterization of [Formula: see text] via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature (...) are indeed [Formula: see text]. (shrink)

To determine what the speaker in a cooperative dialog meant with his assertion, on top of what he explicitly said, it is crucial that we assume that the assertion he gave was optimal. In determining optimal assertions we assume that dialogs are embedded in decision problems (van Rooij 2003) and use backwards induction for calculating them (Benz 2006). In this paper, we show that in terms of our framework we can account for several types of implicatures in a uniform way, (...) suggesting that there is no need for an independent linguistic theory of generalized implicatures. In the final section, we show how we can embed our theory in the framework of signaling games, and how it relates with other game theoretic analyses of implicatures. (shrink)

Abstract A general framework is proposed for accommodating the recent results of studies into ?natural? decision making. A crucial element of this framework is the notion of a ?partial structure?, recently introduced into the semantic approach to scientific theories. It is through the introduction of this element that connections can be made with certain problems regarding inconsistency and rationality in general.

I explore, from a proof-theoretic perspective, the hierarchy of classical and paraconsistent logics introduced by Barrio, Pailos and Szmuc in (Journal o f Philosophical Logic,49, 93-120, 2021). First, I provide sequent rules and axioms for all the logics in the hierarchy, for all inferential levels, and establish soundness and completeness results. Second, I show how to extend those systems with a corresponding hierarchy of validity predicates, each one of which is meant to capture “validity” at a different inferential level. (...) Then, I point out two potential philosophical implications of these results. (i) Since the logics in the hierarchy differ from one another on the rules, I argue that each such logic maintains its own distinct identity (contrary to arguments like the one given by Dicher and Paoli in 2019). (ii) Each validity predicate need not capture “validity” at more than one metainferential level. Hence, there are reasons to deny the thesis (put forward in Barrio, E., Rosenblatt, L. & Tajer, D. (Synthese, 2016)) that the validity predicate introduced in by Beall and Murzi in (Journal o f Philosophy,110(3), 143–165, 2013) has to express facts not only about what follows from what, but also about the metarules, etc. (shrink)