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.

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].

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 evidence-based 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 p-morphic 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.

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 logic-based 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 assumption-complete 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). Assumption-completeness 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].

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 finely-grained 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.

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.

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.

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.

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.

We introduce and study a PDL-style 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?

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 well-known 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.

This essay provides a novel account of iterated epistemic states. The essay argues that states of epistemic determinacy might be secured by countenancing self-knowledge on the model of fixed points in monadic second-order modal logic, i.e. the modal µ-calculus. 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 µ-automaton permits the iterations of the transition functions to entrain a principled means by which to account for necessary conditions on self-knowledge. This essay provides a novel account of self-knowledge, 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 self-knowledge on the model of fixed points in monadic second-order modal logic, i.e. the modal µ-calculus.

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: truth-elimination and truth-introduction. 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 liar-paradoxical sentences.

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 concept-individuation and the codification of speech acts via intensional semantics.

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 category-theoretic setting is identifiable with the Kripke functors of modal coalgebraic automata, where the automata model Grothendieck Universes and the functors are further inter-definable 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 Ω-logical 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 non-circular definition of indefinite extensibility.

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 multi-dimensional 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 multi-dimensional 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.

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 µ-calculus. Via correspondence results between modal logic and first-order logic, a precise translation can then be provided between the notion of ’intuition-of’, i.e., the cognitive phenomenal properties of thoughts, and the modal operators regimenting the notion of ’intuition-that’. I argue that intuition-that can further be shown to entrain conceptual elucidation, by way of figuring as a dynamic-interpretational modality which induces the reinterpretation of both domains of quantification and the intensions of mathematical concepts that are formalizable in monadic first- and second-order formal languages.

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 multi-dimensional 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 multi-dimensional 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.

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.

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

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.

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.

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 non-circular 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.

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.

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

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.

We propose a coherence account of the conjunction fallacy applicable to both of its two paradigms (the M-A paradigm and the A-B 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.

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 world-structures there are. A possible structure may be a substructure of a world-structure, instantiated by only a subdomain of the domain of inhabitants of a possible world; or it may be a reduct of a world-structure, 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 world-structure. A possible structure needn’t be a world-structure 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 world-contents (such as those in “Principles of Plenitude”). For all that (PPS) says, there could be but one (very large!) world-structure, 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 world-structures. For example, (PPS) does not tell us whether substructures of world-structures are themselves world-structures, 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 world-structures 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.

The degrees of unsolvability were introduced in the ground-breaking papers of Post [20] and Kleene and Post [7] as an attempt to measure the information content of sets of natural numbers. Kleene and Post were interested in the relative complexity of decision problems arising naturally in mathematics; in particular, they wished to know when a solution to one decision problem contained the information necessary to solve a second decision problem. As decision problems can be coded by sets of natural numbers, this question is equivalent to: Given a computer with access to an oracle which will answer membership questions about a set A, can a program (allowing questions to the oracle) be written which will correctly compute the answers to all membership questions about a set B? If the answer is yes, then we say that B is Turing reducible to A and write B ≤_T A. We say that B ≡T A if B ≤T A and A ≤T B. ≡T is an equivalence relation, and ≤T induces a partial ordering on the corresponding equivalence classes; the poset obtained in this way is called the degrees of unsolvability, and elements of this poset are called degrees.