
57240.607903
This paper aims to provide modal foundations for mathematical platonism. I examine Hale and Wright’s (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright’s objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by Hale and Wright and examined in Hale (2013a); examine cardinality issues which arise depending on whether Necessitism is accepted at first and higherorder; and demonstrate how a multidimensional intensional approach to the epistemology of mathematics, augmented with Necessitism, is consistent with Hale and Wright’s notion of there being epistemic entitlement rationally to trust that abstraction principles are true. Epistemic and metaphysical modality may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in explaining our possible knowledge thereof.

527941.60795
A counterpossible conditional is a counterfactual with an impossible antecedent. Common sense delivers the view that some such conditionals are true, and some are false. In recent publications, Timothy Williamson has defended the view that all are true. In this paper we defend the common sense view against Williamson’s objections.

735628.607968
I’m visiting the University of Genoa and talking to two category theorists: Marco Grandis and Giuseppe Rosolini. Grandis works on algebraic topology and higher categories, while Rosolini works on the categorical semantics of programming languages. …

1066118.607983
When thinking about rational agents facing choices, one appealing mathematical model recurs in the literature. From Borges’ story ‘The Garden of Forking Paths’ to a host of technical paradigms, sometimes at war, sometimes at peace, all invoke the picture of a branching tree of finite sequences of events with epistemic indistinguishability relations for agents between these sequences, reflecting their limited powers of observation. Indeed, tree models for computation, with branches standing for process evolutions over time, have long been studied in computer science, cf. [32, 33, 7, 2, 14].

1066146.607996
The intuitive notion of evidence has both semantic and syntactic features. In this paper, we develop an evidence logic for epistemic agents faced with possibly contradictory evidence from different sources. The logic is based on a neighborhood semantics, where a neighborhood N indicates that the agent has reason to believe that the true state of the world lies in N . Further notions of relative plausibility between worlds and beliefs based on the latter ordering are then defined in terms of this evidence structure, yielding our intended models for evidencebased beliefs. In addition, we also consider a second more general flavor, where belief and plausibility are modeled using additional primitive relations, and we prove a representation theorem showing that each such general model is a pmorphic image of an intended one. This semantics invites a number of natural special cases, depending on how uniform we make the evidence sets, and how coherent their total structure. We give a structural study of the resulting ‘uniform’ and ‘flat’ models. Our main result are sound and complete axiomatizations for the logics of all four major model classes with respect to the modal language of evidence, belief and safe belief. We conclude with an outlook toward logics for the dynamics of changing evidence, and the resulting language extensions and connections with logics of plausibility change.

1066233.608009
The literature on the epistemic foundations of game theory uses a variety of mathematical models to formalise talk about the players’ beliefs about the game, beliefs about the rationality of the other players, beliefs about the beliefs of the other players, beliefs about the beliefs about the beliefs of the other players, and so on (see [Bra07] for a recent survey). Examples include Harsanyi’s type spaces ([Har67]), interactive belief structures ([Bra03]), knowledge structures ([Aum76]) plus a variety of logicbased frameworks (see, for example, [Ben01, HM06, Bon02, Boa02, BSZ08]). A recurring issue involves defining a space of all possible beliefs of the players and whether such a space exists. In this paper, we study one such definition: the notion of assumptioncomplete models. This notion was introduced in [Bra03], where it is formulated in terms of “interactive belief models” (which are essentially qualitative versions of type spaces). Assumptioncompleteness is also explored in [BK06], where a number of significant results are found, and connections to modal logic are mentioned. A discussion of that paper, and a syntactic proof of its central result, are to be found in [Pac07].

1066271.608022
A rational belief must be grounded in the evidence available to an agent. However, this relation is delicate, and it raises interesting philosophical and technical issues. Modeling evidence requires richer structures than found in standard epistemic semantics where the accessible worlds aggregate all reliable evidence gathered so far. Even recent more finelygrained plausibility models ordering the epistemic ranges identify too much: belief is indistinguishable from aggregated best evidence. At the opposite extreme, one might model evidence syntactically as “formulas received”, but this seems overly detailed, and we we lose the intuition that evidence can be semantic in nature, zooming in on some actual world.

1066302.60804
A recurring issue in any formal model representing agents’ (changing) informational attitudes is how to account for the fact that the agents are limited in their access to the available inference steps, possible observations and available messages. This may be because the agents are not logically omniscient and so do not have unlimited reasoning ability. But it can also be because the agents are following a predefined protocol that explicitly limits statements available for observation and/or communication. Within the broad literature on epistemic logic, there are a variety of accounts that make precise a notion of an agent’s “limited access” (for example, Awareness Logics, Justification Logics, and Inference Logics). This paper interprets the agents’ access set of formulas as a constraint on the agents’ information gathering process limiting which formulas can be observed.

1066328.608069
Deontic Logic goes back to Ernst Mally’s 1926 work, Grundgesetze des Sollens: Elemente der Logik des Willens [Mally. E.: 1926, Grundgesetze des Sollens: Elemente der Logik des Willens, Leuschner & Lubensky, Graz], where he presented axioms for the notion ‘p ought to be the case’. Some difficulties were found in Mally’s axioms, and the field has much developed. Logic of Knowledge goes back to Hintikka’s work Knowledge and Belief [Hintikka, J.: 1962, Knowledge and Belief: An Introduction to the Logic of the Two Notions, Cornell University Press] in which he proposed formal logics of knowledge and belief.

1066379.608098
We develop a dynamic modal logic that can be used to model scenarios where agents negotiate over the allocation of a finite number of indivisible resources. The logic includes operators to speak about both preferences of individual agents and deals regarding the reallocation of certain resources. We reconstruct a known result regarding the convergence of sequences of mutually beneficial deals to a Pareto optimal allocation of resources, and discuss the relationship between reasoning tasks in our logic and problems in negotiation. For instance, checking whether a given restricted class of deals is sufficient to guarantee convergence to a Pareto optimal allocation for a specific negotiation scenario amounts to a model checking problem; and the problem of identifying conditions on preference relations that would guarantee convergence for a restricted class of deals under all circumstances can be cast as a question in modal logic correspondence theory.

1066500.608116
In this paper we study substantive assumptions in social interaction. By substantive assumptions we mean contingent assumptions about what the players know and believe about each other’s choices and information. We first explain why substantive assumptions are fundamental for the analysis of games and, more generally, social interaction. Then we show that they can be compared formally, and that there exist contexts where no substantive assumptions are being made. Finally we show that the questions raised in this paper are related to a number of issues concerning “large” structures in epistemic game theory.

1066752.60813
We introduce and study a PDLstyle logic for reasoning about protocols, or plans, under imperfect information. Our paper touches on a number of issues surrounding the relationship between an agent’s abilities, available choices, and information in an interactive situation. The main question we address is under what circumstances can the agent commit to a protocol or plan, and what can she achieve by doing so?

1066839.608143
We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some wellknown problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics.

1201415.608206
This essay provides a novel account of iterated epistemic states. The essay argues that states of epistemic determinacy might be secured by countenancing selfknowledge on the model of fixed points in monadic secondorder modal logic, i.e. the modal µcalculus. Despite the epistemic indeterminacy witnessed by the invalidation of modal axiom 4 in the sorites paradox – i.e. the KK principle: φ → φ – an epistemic interpretatation of the Kripke functors of a µautomaton permits the iterations of the transition functions to entrain a principled means by which to account for necessary conditions on selfknowledge. This essay provides a novel account of selfknowledge, which avoids the epistemic indeterminacy witnessed by the invalidation of modal axiom 4 in epistemic logic; i.e. the KK principle: φ → φ. The essay argues, by contrast, that – despite the invalidation of modal axiom 4 on its epistemic interpretation – states of epistemic determinacy might yet be secured by countenancing selfknowledge on the model of fixed points in monadic secondorder modal logic, i.e. the modal µcalculus.

1238509.608234
This paper targets a series of potential issues for the discussion of, and modal resolution to, the alethic paradoxes advanced by Scharp (2013). I aim, then, to provide a novel, epistemicist treatment of the alethic paradoxes. In response to Curry’s paradox, the epistemicist solution that I advance enables the retention of both classical logic and the traditional rules for the alethic predicate: truthelimination and truthintroduction. By availing of epistemic modal logic, the epistemicist approach permits, further, of a descriptively adequate explanation of the indeterminacy that is exhibited by epistemic states concerning liarparadoxical sentences.

1238534.608294
This paper aims to provide a mathematically tractable background against which to model both modal cognitivism and modal expressivism. I argue that epistemic modal algebras comprise a materially adequate fragment of the language of thought, and endeavor to show how such algebras provide the resources necessary to resolve Russell’s paradox of propositions. I demonstrate, then, how modal expressivism can be regimented by modal coalgebraic automata, to which the above epistemic modal algebras are dually isomorphic. I examine, in particular, the virtues unique to the modal expressivist approach here proffered in the setting of the foundations of mathematics, by contrast to competing approaches based upon both the inferentialist approach to conceptindividuation and the codification of speech acts via intensional semantics.

1238554.608317
This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility in the categorytheoretic setting is identifiable with the Kripke functors of modal coalgebraic automata, where the automata model Grothendieck Universes and the functors are further interdefinable with the elementary embeddings of large cardinal axioms. The Kripke functors definable in Grothendieck universes are argued to account for the ontological expansion effected by the elementary embeddings in the category of sets. By characterizing the modal profile of Ωlogical validity, and thus the generic invariance of mathematical truth, modal coalgebraic automata are further capable of capturing the notion of definiteness, in order to yield a noncircular definition of indefinite extensibility.

1238569.60833
This essay aims to redress the contention that epistemic possibility cannot be a guide to the principles of modal metaphysics. I argue that the interaction between the multidimensional intensional framework and intensional plural quantification enables epistemic possibilities to target the haecceitistic properties of individuals. I outline the elements of plural logic, and I specify, then, a multidimensional intensional formula encoding the relation between the epistemic possibility of haecceity comprehension and its metaphysical possibility. I conclude by addressing objections from the indeterminacy of ontological principles relative to the space of epistemic possibilities, and from the consistency of epistemic modal space.

1238596.608343
This essay aims to provide a modal logic for rational intuition. Similarly to treatments of the property of knowledge in epistemic logic, I argue that rational intuition can be codified by a modal operator governed by the axioms of a dynamic provability logic, which augments GL with the modal µcalculus. Via correspondence results between modal logic and firstorder logic, a precise translation can then be provided between the notion of ’intuitionof’, i.e., the cognitive phenomenal properties of thoughts, and the modal operators regimenting the notion of ’intuitionthat’. I argue that intuitionthat can further be shown to entrain conceptual elucidation, by way of figuring as a dynamicinterpretational modality which induces the reinterpretation of both domains of quantification and the intensions of mathematical concepts that are formalizable in monadic first and secondorder formal languages.

1238634.608356
This paper aims to contribute to the analysis of the nature of mathematical modality, and to the applications of the latter to unrestricted quantification and absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of multidimensional intensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the multidimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. I demonstrate, finally, how the duality axioms of the epistemic logic for the semantics can be availed of, in order to defuse the paradox of knowability.

1239031.608368
In Q2, article 3 of the first part of the Summa Theologica, Aquinas argues that we can in fact demonstrate God’s existence, using only our natural reason (without resort to faith). His main argument in favor of this conclusion is an appeal to the authority of St. Paul’s letter to the Romans 1:20. Aquinas considers three objections to his position: 1. The existence of God is an article of faith, revealed by the Scriptures, not a matter of rational proof. 2. We cannot know God’s essence or nature (as Aquinas himself concedes). How can we prove the existence of an utterly unknown thing? 3. Since we cannot see God directly in this life (as, again, Aquinas would concede), we can know God only on the basis of His effects (i.e., creation). However, creation is finite, and God is infinite, and we cannot infer an infinite cause from a finite effect.

1288094.608382
Lean as a Programming Language . . . . . . . . . . . . . . . . . . . . . . . 5 1.2

1288100.608395
Copyright (c) 2016, Jeremy Avigad, Robert Y. Lewis, and Floris van Doorn. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE.

1288109.608408
Lean is an implementation of a logical foundation known as dependent type theory. Specifically, it implements a version of dependent type theory known as the Calculus of Inductive Constructions. The CIC is a formal language with a small and precise set of rules that governs the formation of expressions. In this formal system, moreover, every expression has a type. The type of expression indicates what sort of object the expression denotes. For example, an expression may denote a mathematical object like a natural number, a data type, an assertion, or a proof.

1402859.608422
The thesis of this paper is that we can justify induction deductively relative to one end, and deduction inductively relative to a different end. I will begin by presenting a contemporary variant of Hume (1739; 1748)’s argument for the thesis that we cannot justify the principle of induction. Then I will criticize the responses the resulting problem of induction has received by Carnap (1963; 1968) and Goodman (1954), as well as praise Reichenbach (1938; 1940)’s approach. Some of these authors compare induction to deduction. Haack (1976) compares deduction to induction, and I will critically discuss her argument for the thesis that we cannot justify the principles of deduction next. In concluding I will defend the thesis that we can justify induction deductively relative to one end, and deduction inductively relative to a different end, and that we can do so in a noncircular way. Along the way I will show how we can understand deductive and inductive logic as normative theories, and I will briefly sketch an argument to the effect that there are only hypothetical, but no categorical imperatives.

1724669.60844
Two different programs are in the business of explicating accuracy—the truthlikeness program and the epistemic utility program. Both assume that truth is the goal of inquiry, and that among inquiries that fall short of realizing the goal some get closer to it than others. TL theorists have been searching for an account of the accuracy of propositions. Epistemic utility theorists have been searching for an account of the accuracy of credal states. Both assume we can make cognitive progress in an inquiry even while falling short of the target. I show that the prospects for combining these two programs are bleak. A core accuracy principle, Proximity, that is universally embraced within the Truthlikeness program turns out to be incompatible with a central principle within the Epistemic Utility program, namely Propriety.

1724690.608454
Declaration. The work included here is my own. This thesis is an annotated compilation of published papers, none of which was coauthored. Acknowledgement of assistance received will be found in each paper, and these acknowledgements are also collected together at the end of Chapter 0.

1724746.608469
The determination of “who is a J” within a society is treated as an aggregation of the views of the members of the society regarding this question. Methods, similar to those used in Social Choice theory are applied to axiomatize three criteria for determining who is a J: 1) a J is whoever defines oneself to be a J. 2) a J is whoever a “dictator” determines is a J. 3) a J is whoever an “oligarchy” of individuals agrees is a J.

1738829.608482
We propose a coherence account of the conjunction fallacy applicable to both of its two paradigms (the MA paradigm and the AB paradigm). We compare our account with a recent proposal by Tentori, Crupi and Russo (2013) that attempts to generalize earlier confirmation accounts. Their model works better than its predecessors in some respects, but it exhibits only a shallow form of generality and is unsatisfactory in other ways as well: it is strained, complex, and untestable as it stands. Our coherence account inherits the strength of the confirmation account, but in addition to being applicable to both paradigms, it is natural, simple, and readily testable. It thus constitutes the next natural step for Bayesian theorizing about the conjunction fallacy.

1746220.608499
The principle of plenitude for possible structures (PPS) that I endorsed tells us what structures are instantiated at possible worlds, but not what structures give the entire structure of a possible world, not what worldstructures there are. A possible structure may be a substructure of a worldstructure, instantiated by only a subdomain of the domain of inhabitants of a possible world; or it may be a reduct of a worldstructure, involving only some of the natural properties or relations instantiated at a possible world; or it may be a substructure of a reduct of a worldstructure. A possible structure needn’t be a worldstructure all by itself. For this reason, (PPS) does not provide a complete account of plenitude of worlds when combined with a principle of plenitude for recombinations and a principle of plenitude for worldcontents (such as those in “Principles of Plenitude”). For all that (PPS) says, there could be but one (very large!) worldstructure, with every world corresponding to some arrangement of possibilia within that one structure. In particular, (PPS) will not allow the derivation of various plausible principles of plenitude for worldstructures. For example, (PPS) does not tell us whether substructures of worldstructures are themselves worldstructures, and thus fails to support a principle of solitude according to which any (connected) possible individual can exist all by itself. In this postscript, I first canvas the reasons I had for formulating a principle of plenitude for structures that was noncommittal as to the structure of entire worlds. I then develop a stronger principle that can serve as a principle of plenitude for worldstructures in a complete account of plenitude of worlds. The principle I give is strong enough to entail an appropriate version of the principle of solitude, but not so strong as to entail the existence of gunky worlds. Gunk, I am inclined to believe, is impossible.