This paper addresses debates in German idealism that arise in response to the modal shift in logic, proposed by Kant, from a logic of thinking to a logic of experience. With the Kantian logic of experience arises a problem of radical contingency or 'rhapsodic determination' for logic. While Fichte and Hegel attempt to resolve the problem of contingency by constructing rational systems aimed at established the grounds for logic, I show how Schelling brings into view, in a proto-existentialist movement, the (...) way in which the indeterminacy of the will undergirds rational system construction and, thereby, presents a limiting problem for the question of the value of system-building itself. (shrink)

In this paper we study the logical strength of the determinacy of infinite binary games in terms of second order arithmetic. We define new determinacy schemata inspired by the Wadge classes of Polish spaces and show the following equivalences over the system RCA0*, which consists of the axioms of discrete ordered semi‐rings with exponentiation, Δ10 comprehension and Π00 induction, and which is known as a weaker system than the popularbase theory RCA0: 1. Bisep(Δ10, Σ10)‐Det* ↔ WKL0, 2. Bisep(Δ10, (...) Σ20)‐Det* ↔ ATR0 + Σ11 induction, 3. Bisep(Σ10, Σ20)‐Det* ↔ Sep(Σ10, Σ20)‐Det* ↔ Π11‐CA0, 4. Bisep(Δ20, Σ20)‐Det* ↔ Π11‐TR0, where Det* stands for the determinacy of infinite games in the Cantor space (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

We show that each of $\Delta _{3}^{1}-{\rm CA}_{0}+\Sigma _{3}^{1}-{\rm IND}$ and $\Pi _{2}^{1}-{\rm CA}_{0}+\Pi _{3}^{1}-{\rm TI}$ proves $\Delta _{3}^{0}-{\rm Det}$ and that neither $\Sigma _{3}^{1}-{\rm IND}$ nor $\Pi _{3}^{1}-{\rm TI}$ can be dropped. We also show that neither $\Delta _{3}^{1}-{\rm CA}_{0}+\Sigma _{\infty}^{1}-{\rm IND}$ nor $\Pi _{2}^{1}-{\rm CA}_{0}+\Pi _{\infty}^{1}-{\rm TI}$ proves $\Sigma _{3}^{0}-{\rm Det}$. Moreover, we prove that none of $\Delta _{2}^{1}-{\rm CA}_{0}$, $\Sigma _{3}^{1}-{\rm IND}$ and $\Pi _{2}^{1}-{\rm TI}$ is provable in $\Delta _{1}^{1}-{\rm Det}_{0}={\rm ACA}_{0}+\Delta _{1}^{1}-{\rm Det}$.

Many theorists understand human rights as only aiming to secure a minimally decent existence, rather than a positively good or flourishing life. Some of the theoretical considerations that support this minimalist view have been mapped out in the philosophical literature. The aim of this paper is to explain how a relatively neglected theoretical desideratum – namely, determinacy – can be invoked in arguing for human rights minimalism. Most of us want a theory of human rights whose demands can be (...) realized, and which is acceptable to a range of worldviews. But we might also expect our theory to provide determinate answers to questions of scope (i.e. which putative rights are bona fide human rights?) and practical implementation (i.e. what concrete duties are generated by which rights?). A minimalist view of human rights makes it is easier to jointly fulfil all of these desiderata. (shrink)

We analyze the set-theoretic strength of determinacy for levels of the Borel hierarchy of the form$\Sigma _{1 + \alpha + 3}^0 $, forα<ω1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to requireα+ 1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of weak reflection principles, Π1-RAPα, whose consistency strength corresponds exactly to the logical strength of${\rm{\Sigma }}_{1 + (...) \alpha + 3}^0 $determinacy, for$\alpha < \omega _1^{CK} $. This yields a characterization of the levels ofLby or at which winning strategies in these games must be constructed. Whenα= 0, we have the following concise result: The leastθso that all winning strategies in${\rm{\Sigma }}_4^0 $games belong toLθ+1is the least so that$L_\theta \models {\rm{``}}{\cal P}\left$exists, and all wellfounded trees are ranked”. (shrink)

Schmidt’s game and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games, $\mathsf {AD}_{\mathbb R}$, which is a much stronger axiom than that asserting all integer games are determined, $\mathsf {AD}$. (...) One of our main results is a general theorem which under the hypothesis $\mathsf {AD}$ implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt’s $(\alpha,\beta,\rho )$ game on $\mathbb R$ is determined from $\mathsf {AD}$ alone, but on $\mathbb R^n$ for $n \geq 3$ we show that $\mathsf {AD}$ does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt’s $(\alpha, \beta, \rho )$ game on $\mathbb {R}$ has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt’s game. These results highlight the obstacles in obtaining the determinacy of Schmidt’s game from $\mathsf {AD}$. (shrink)

DuBose, D.A., Determinacy and the sharp function on the reals, Annals of Pure and Applied Logic 55 237–263. We characterize in terms of determinacy, the existence of the least inner model of “every real has a sharp”. We let 1 be the sharp function on the reals and define two classes of sets, * and *+, which lie strictly between β<ω2 an d Δ. We show that the determinacy of * follows from L[#1] xvR; “every real has (...) a sharp”; and we show that the existence of indiscernibles for L[#1] is equivalent to a slightly determinacy hypothesis, the determinacy of *+. (shrink)

Competent speakers of natural languages can borrow reference from one another. You can arrange for your utterances of ‘Kirksville’ to refer to the same thing as my utterances of ‘Kirksville’. We can then talk about the same thing when we discuss Kirksville. In cases like this, you borrow “ aboutness ” from me by borrowing reference. Now suppose I wish to initiate a line of reasoning applicable to any prime number. I might signal my intention by saying, “Let p be (...) any prime.” In this context, I will be using the term ‘p’ to reason about the primes. Although ‘p’ helps me secure the aboutness of my discourse, it may seem wrong to say that ‘p’ refers to anything. Be that as it may, this paper explores what mathematical discourse would be like if mathematicians were able to borrow freely from one another not just the reference of terms that clearly refer, but, more generally, the sort of aboutness present in a line of reasoning leading up to a universal generalization. The paper also gives reasons for believing that aboutness of this sort really is freely transferable. A key implication will be that the concept “set of natural numbers” suffers from no mathematically significant indeterminacy that can be coherently discussed. (shrink)

We show, assuming weak large cardinals, that in the context of games of length \ with moves coming from a proper class, clopen determinacy is strictly weaker than open determinacy. The proof amounts to an analysis of a certain level of L that exists under large cardinal assumptions weaker than an inaccessible. Our argument is sufficiently general to give a family of determinacy separation results applying in any setting where the universal class is sufficiently closed; e.g., in (...) third, seventh, or \\)th order arithmetic. We also prove bounds on the strength of Borel determinacy for proper class games. These results answer questions of Gitman and Hamkins. (shrink)

This chapter analyzes the different ways to describe brain behaviour with the goal to provide a basis for an informed discussion of the nature of decisions and actions that humans perform in their lives. The chapter is organized as follows. Section 2 outlines a number of concepts exhibiting how many subtle details and distinctions lie behind the broad notions of determinacy and stochasticity. These details are necessary for a discussion, in Section 3, of particular aspects relevant for the characterization (...) of brain states and their dynamics. The descriptions of brain behaviour currently provided by neuroscience depend on the level and context of the descriptions. There is no clear-cut evidence for ultimately determinate or ultimately stochastic brain behaviour. As a consequence, there is no solid neurobiological basis to argue either in favour of or against any fundamental determination or openness of human decisions and actions. (shrink)

We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by realtime 1-counter Büchi automata is equivalent to the determinacy of analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge ofω-languages accepted by 1-counter Büchi automata is equivalent to the analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a (...) 1-counter Büchi automatonand a Büchi automatonsuch that: There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge gameW,L()); There exists a model of ZFC in which the Wadge gameW,L()) is not determined. Moreover these are the only two possibilities, i.e. there are no models of ZFC in which Player 1 has a winning strategy in the Wadge gameW,L()). (shrink)

The idea that legal theories seek not only to explain but to evaluate the moral justification of particular areas of law is quite familiar. Yet little attention has been paid to the minimal criteria of adequacy for justificatory legal theories. Whereas many theories claim to identify the moral grounds that justify a particular area of law, such as contracts or torts, none of them explains how its justification determines the outcomes of adjudication governed by the law in that area. In (...) this brief Essay for the William and Mary Law Review Symposium on Law and Morality, I argue that a particular area of law can be justified only by identifying moral reasons that fully determine the results of adjudication. No matter how compelling the moral reasons a legal theory identifies, and how tight the fit between those reasons and the structure and content of the legal rules governing a judicial decision, a legal theory fails to justify a particular area of law if the reasoning it identifies falls short of fully determining the results in the judicial decisions governed by that law. Though this bold claim may seem unrealistic, I argue that legal theories can satisfy this determinacy requirement by identifying determinate but inconclusive reasoning that explains outcomes in adjudication. While such reasoning may prove to be erroneous, that does not undermine its justificatory force. (shrink)

The paper discusses some ways in which the phenomenon of borderline cases may be thought to bear on the traditional philosophical idea that certain domains of facts are fully open to our view. The discussion focusses on a very influential argument (due to Tim Williamson) to the effect that, roughly, no such domains of luminous facts exist. Many commentators have felt that the vagueness unavoidably inherent in the description of the facts that are best candidates for being luminous plays an (...) illicit role in the argument. The paper investigates this issue by centring around the idea that vagueness brings with itself borderline cases, and that these in turn generate absence of a fact of the matter and hence epistemically benign lack of knowledge. It is argued that, given the possibility of absence of a fact of the matter, the idea of luminosity should be reformulated using the notion of determinacy, and that the resulting reformulation is not immediately subject to the original anti-luminosity argument. However, it is shown that the specific understanding of determinacy required by this strategy validates a new argument against the reformulated version of luminosity. Moreover, reflection on the connection between mistake and absence of a fact of the matter offers another argument against such version, with the surprising upshot that, granting the soundness of the original anti-luminosity argument, not even the determinacy of a certain fact would guarantee its knowability. (shrink)

This paper investigates formal logics for reasoning about determinacy and independence. Propositional Dependence Logic D and Propositional Independence Logic I are recently developed logical systems, based on team semantics, that provide a framework for such reasoning tasks. We introduce two new logics L_D and L_I, based on Kripke semantics, and propose them as alternatives for D and I, respectively. We analyse the relative expressive powers of these four logics and discuss the way these systems relate to natural language. We (...) argue that L_D and L_I naturally resolve a range of interpretational problems that arise in D and I.We also obtain sound and complete axiomatizations for L_D and L_I . (shrink)

Few arguments against intentional states in animals have stood the test of time. But one objection by Stich and Davidson has never been rebutted. In my reconstruction it runs: Ascribing beliefs to animals is vacuous, unless something counts as an animal believing one specific “content” rather than another; Nothing counts as an animal believing one specific content rather than another, because of their lack of language; Ergo: Ascribing beliefs to animals is vacuous. Several attempts to block the argument challenge the (...) first premise, notably the appeals to “naked” belief ascriptions and alternative representational formats. This essay defends the first premise and instead challenges the second premise. There are non-linguistic “modes of presentation”; these can be determined by attributing to animals specific needs and capacities—a “ hermeneutic ethology” based on lessons from the debate about radical translation/interpretation in the human case. On that basis we can narrow down content by exclusion. What remains is an “imponderability of the mental” which does not rule out attributions of intentional states to animals. (shrink)

Assume V = L ⊨ ZF + DC + Θ > ω2 + μ is a normal fine measure on.

For several partial sharp functions # on the reals, we characterize in terms of determinacy, the existence of indiscernibles for several inner models of “# exists for every real r”. Let #10=1#10 be the identity function on the reals. Inductively define the partial sharp function, β#1γ+1, on the reals so that #1γ+1 =1#1γ+1 codes indiscernibles for L [#11, #12,…, #1γ] and #1γ+1=#1γ+1). We sho w that the existence of β#1γ follows from the determinacy of *Σ01)*+ games . Part (...) I proves the converse. (shrink)

We characterize, in terms of determinacy, the existence of the least inner model of "every object of type k has a sharp." For k ∈ ω, we define two classes of sets, (Π 0 k ) * and (Π 0 k ) * + , which lie strictly between $\bigcup_{\beta and Δ(ω 2 -Π 1 1 ). Let ♯ k be the (partial) sharp function on objects of type k. We show that the determinancy of (Π 0 k ) (...) * follows from $L \lbrack\ sharp_k \rbrack \models "\text{every object of type} k \text{has a sharp},$ and we show that the existence of indiscernibles for L[ ♯ k ] is equivalent to a slightly stronger determinacy hypothesis, the determinacy of (Π 0 k ) * +. (shrink)

We characterize in terms of determinacy, the existence of the least inner model of “every real has a sharp”. We let #1 be the sharp function on the reals and define two classes of sets, * and *+, which lie strictly between β<ω2- and Δ. We show that the determinacy of * follows from L[#1] “every reak has a sharp”; and we show that the existence of indiscernibles for L[#1] is equivalent to a slightly stronger determinacy hypothesis, (...) the determinacy of *+. (shrink)

A skeptical worry known as ‘the indeterminacy of computation’ animates much recent philosophical reflection on the computational identity of physical systems. On the one hand, computational explanation seems to require that physical computing systems fall under a single, unique computational description at a time. On the other, if a physical system falls under any computational description, it seems to fall under many simultaneously. Absent some principled reason to take just one of these descriptions in particular as relevant for computational explanation, (...) widespread failure of computational explanation would appear to follow. This paper advances a new solution to the indeterminacy of computation. Very roughly, I argue that the computational identity of a physical system is determinate relative to a contextually specified way of regarding that system computationally—known as a labelling scheme. When a system simultaneously implements multiple computations, it does so relative to different labelling schemes. But relative to a fixed labelling scheme, a physical system has a unique computational identity. I argue that this relativistic conception of computational identity vindicates computational explanation in the face of simultaneous implementation. (shrink)

Assume ZF + AD +V=L and letκ< Θ be an uncountable cardinal. We show thatκis Jónsson, and that if cof = ω thenκis Rowbottom. We also establish some other partition properties.

This paper distinguishes between definiteness and determinacy. Definiteness is seen as a morphological category which, in English, marks a uniqueness presupposition, while determinacy consists in denoting an individual. Definite descriptions are argued to be fundamentally predicative, presupposing uniqueness but not existence, and to acquire existential import through general type-shifting operations that apply not only to definites, but also indefinites and possessives. Through these shifts, argumental definite descriptions may become either determinate or indeterminate. The latter option is observed in (...) examples like ‘Anna didn’t give the only invited talk at the conference’, which, on its indeterminate reading, implies that there is nothing in the extension of ‘only invited talk at the conference’. The paper also offers a resolution of the issue of whether possessives are inherently indefinite or definite, suggesting that, like indefinites, they do not mark definiteness lexically, but like definites, they typically yield determinate readings due to a general preference for the shifting operation that produces them. (shrink)

Let o(κ) denote the Mitchell order of κ. We show how to reduce long games which run to the first ordinal admissible in the play, to iteration games on models with a cardinal κ so that (1) κ is a limit of Woodin cardinals: and (2) o(κ) = κ⁺⁺. We use the reduction to derive several optimal determinacy results on games which run to the first admissible in the play.

A major challenge in the philosophy of mathematics is to explain how mathematical language can pick out unique structures and acquire determinate content. In recent work, Button and Walsh have introduced a view they call ‘internalism’, according to which mathematical content is explained by internal categoricity results formulated and proven in second-order logic. In this paper, we critically examine the internalist response to the challenge and discuss the philosophical significance of internal categoricity results. Surprisingly, as we argue, while internalism arguably (...) explains how we pick out unique mathematical structures, this does not suffice to account for the determinacy of mathematical discourse. (shrink)

We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example: Theorem A. $\mathrm{D}\mathrm{e}\mathrm{t}\text{\hspace{0.17em}}({\mathrm{\Pi }}_{1}^{1}-\mathrm{I}\mathrm{N}\mathrm{D})$ ⇒ there exists an inner model with a strong cardinal. Theorem B. Det(AQI) ⇒ there exist type-1 mice and hence inner models with proper classes of strong cardinals. where ${\mathrm{\Pi }}_{1}^{1}-\mathrm{I}\mathrm{N}\mathrm{D}\phantom{\rule{0ex}{0ex}}$ (AQI) is the pointclass of boldface ${\mathrm{\Pi }}_{1}^{1}$ -inductive (respectively arithmetically quasi-inductive) sets of reals.

Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first-order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non-standard interpretations that confronts higher-order logics on their more traditional, set-based semantics. We challenge both claims. Our challenge is based on a Henkin-style semantics for plural (...) logic that does not resort to sets or set-like objects to interpret plural variables, but adopts the view that a plural variable has many objects as its values. Using this semantics, we also articulate a generalized notion of ontological commitment which enables us to develop some ideas of earlier critics of the alleged ontological innocence of plural logic. (shrink)

This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics. From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate, motivate two important constraints on attempts to meet our challenge, and then use these constraints to develop an argument against determinacy and discuss (...) a particularly popular approach to resolving indeterminacy, before offering some brief closing reflections. We believe our discussion poses a serious challenge for most philosophical theories of mathematics, since it puts considerable pressure on all views that accept a non-trivial amount of determinacy for even basic arithmetic. (shrink)

My thanks to the Editors of Philosophy & Public Affairs for very helpful comments on an earlier draft. I also had the benefit of an exchange with Christopher McMahon. 1. Christopher McMahon, “The Indeterminacy of Republican Policy,” Philosophy & Public Affairs 33 (2005): 67–93, at p. 89. All parenthetical references in the text are to this article.

In constructing the three-valued logic, Jan Łukasiewicz was highly inspirited by the Aristotelian idea of logical contingency. Nevertheless, we can construct a four-valued logic for explicating the Stoic idea of logical determinacy. In this system, we have the following truth values: 0 (‘possibly false), 1 (‘necessarily false’), 2 (‘possibly true’), 3 (‘necessarily true’), where the designated truth value is represented by the two values: 2 and 3.

Comparativism maintains that physical quantities are ultimately relational in character. For example, an object’s having 1 kg rest mass depends on the relations it stands in to other objects in the universe. Comparativism, its advocates allege, reveals that quantities are not metaphysically mysterious: Quantities are reducible to familiar relations holding among physical objects. Modal accounts of intrinsicality—such as Lewis’s duplication account or Langton and Lewis’s combinatorial account—are popular accounts preserving many of our core intuitions regarding which properties are intrinsic. I (...) argue that to endorse both comparativism and a modal account of intrinsicality, we must reject the plausible thesis that determinable properties are instantiated solely in virtue of their determinates. I call this ‘the determinacy tension’ and I suggest approaches for dissolving it. (shrink)

In recent years, a number of authors have defended the coherence and philosophical utility of the notion of metaphysical indeterminacy. Concurrently, the idea that reality can be stratified into more or less fundamental ‘levels’ has gained significant traction in the literature. Here, I examine the relationship between these two notions. Specifically, I consider the question of what metaphysical determinacy at one level of reality tells us about the possibility of metaphysical determinacy at other more or less fundamental levels. (...) Towards this end, I propose a novel conception of the way in which fundamental states of affairs determine derivative states of affairs in the presence of indeterminacy and construct a corresponding formal model of multilevel systems that demonstrates the compatibility of determinacy at the fundamental level with indeterminacy at higher levels, thereby rebutting Barnes' suggestion that indeterminacy at any level of reality implies indeterminacy at the fundamental level. (shrink)

We prove the determinacy of all Δ 1 1 games on arbitrary trees, and we use this result and the assumption that a measurable cardinal exists to demonstrate the determinacy of all games on ω ω that belong both to – Π 1 1 and to its dual.

DETERMINACY is the claim that covert shifts in visual attention sometimes affect the determinacy of visual content (capital letters will distinguish the claim from the familiar word, 'determinacy'). Representationalism is the claim that visual phenomenology supervenes on visual representational content. Both claims are popular among contemporary philosophers of mind, and DETERMINACY has been employed in defense of representationalism. I claim that existing arguments in favor of DETERMINACY are inconclusive. As a result, DETERMINACY-based arguments in (...) support of representationalism are not strong ones. (shrink)

The notion of individuation has both a semantic and an ontological face. More exactly, the claim can be defended that individuation has a proprietary linguistic or conceptual aspect as distinct from an ontological one. Ontologists such as W. V. Quine would credit neither the possibility of such a divergence nor the intelligibility of its proposal. The ontology of a language, for Quine, is inseparable from its individuative resources, mechanisms such as identity, pluralization, pronouns, and so forth. I aim here to (...) pick apart the ontological from the semantic aspect of individuation at one place where they interact, thereby to show that the distinction, far from being a mere abstraction lacking explanatory leverage, is vital for achieving an adequate characterization of the representational powers of language. I begin by attempting to put the issue into some kind of perspective, a didactic move prudent in virtue of the brassiness of the subsequent tacks. (shrink)

We show under that every set of reals is I‐regular for any σ‐ideal I on the Baire space such that is proper. This answers the question of Khomskii [7, Question 2.6.5]. We also show that the same conclusion holds under if we additionally assume that the set of Borel codes for I‐positive sets is. If we do not assume, the notion of properness becomes obscure as pointed out by Asperó and Karagila [1]. Using the notion of strong properness similar to (...) the one introduced by Bagaria and Bosch [2], we show under without using that every set of reals is I‐regular for any σ‐ideal I on the Baire space such that is strongly proper assuming every set of reals is ∞‐Borel and there is no ω1‐sequence of distinct reals. In particular, the same conclusion holds in a Solovay model. (shrink)

From the hypothesis that all Turing closed games are determined we prove: (1) the Continuum Hypothesis and (2) every subset of ℵ1 is constructible from a real.