Across languages, we find indefinites that trigger modal inferences. This article contributes to a semantic typology of these items by contrasting Spanish algún with indefinites like German irgendein or Italian uno qualsiasi. While irgendein-type indefinites trigger a Free Choice effect (Kratzer and Shimoyama 2002; Chierchia 2006), algún simply signals that at least two individuals in its domain are possibilities. Additionally, algún, but not irgendein, can convey that the speaker does not know how many individuals satisfy the existential claim in the (...) world of evaluation. We contend that the two types of indefinites impose different constraints on their domain of quantification: irgendein and its kin are domain wideners (Kratzer and Shimoyama 2002), whereas algún is an ‘anti-singleton’ indefinite (its domain cannot be restricted to a singleton). This, together with the fact that algún does not require uniqueness, allows us to derive the contrast between irgendein and algún by using the pragmatic reasoning presented by Kratzer and Shimoyama. (shrink)

Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic interest of (...) this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that ‘indefinite extensibility’ arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch. (shrink)

This paper investigates the truth conditions of sentences containing indefinite noun phrases, focusing on occurrences in attitude reports, and, in particular, a puzzle case due to Walter Edelberg. It is argued that indefinites semantically contribute the (thought-)object they denote, in a manner analogous to attributive definite descriptions. While there is an existential reading of attitude reports containing indefinites, it is argued that the existential quantifier is contributed by the de re interpretation of the indefinite (as the de re (...) reading adds existential quantification to the interpretation of definites on Kaplan’s analysis). (shrink)

I argue that we can and should extend Tarski's model-theoretic criterion of logicality to cover indefinite expressions like Hilbert's ɛ operator, Russell's indefinite description operator η, and abstraction operators like 'the number of'. I draw on this extension to discuss the logical status of both abstraction operators and abstraction principles.

In this chapter I will look at the semantic analysis of indefinites in English, and their treatment in the framework of Dynamic Logic, choice functions (epsilon terms), and Kit Fine’s arbitrary objects. In the end I will make some comparative remarks about the latter and the account in terms of Skolem functions that Fine criticized, and propose an alternative framework (team semantics).

The principle of sufficient reason threatens modal collapse. Some have suggested that by appealing to the indefinite extensibility of contingent truth, the threat is neutralized. This paper argues that this is not so. If the indefinite extensibility of contingent truth is developed in an analogous fashion to the most promising models of the indefinite extensibility of the concept set, plausible principles permit the derivation of modal collapse.

Indefinite probability statements can be analysed in terms of statements which attribute probability to propositions. Therefore, there is no need to find a special place in probability theory for them; once we have an adequate account of statements that straightforwardly attribute probability to propositions, we will automatically have an adequate account of indefinite probability statements.

The paper proposes a novel solution to the problem of scope posed by natural language indefinites that captures both the difference in scopal freedom between indefinites and bona fide quantifiers and the syntactic sensitivity that the scope of indefinites does nevertheless exhibit. Following the main insight of choice functional approaches, we connect the special scopal properties of indefinites to the fact that their semantics can be stated in terms of choosing a suitable witness. This is in contrast to bona fide (...) quantifiers, the semantics of which crucially involves relations between sets of entities. We provide empirical arguments that this insight should not be captured by adding choice/Skolem functions to classical first-order logic, but in a semantics that follows Independence-Friendly Logic, in which scopal relations involving existentials are part of the recursive definition of truth and satisfaction. These scopal relations are resolved automatically as part of the interpretation of existentials. Additional support for this approach is provided by dependent indefinites, a cross-linguistically common class of special indefinites that can be straightforwardly analyzed in our semantic framework. (shrink)

Indefinite singular (IS) noun phrases are a familiar but under-theorized feature of moral discourse (such as in "A virtuous person prioritizes self-improvement"). While widely used in definitions and generalizations about character, their semantic and normative significance has gone largely unexamined. This paper argues that IS moral generalizations serve a dual function in moral thought: a definitional role in articulating principles (e.g., "An act is right iff it maximizes happiness") and a metalinguistic role in negotiating moral concepts (e.g., "A good (...) person is generous"). We propose that many IS moral generalizations operate not merely as descriptive statements, but as normative proposals that shape how moral terms should be used. We distinguish these uses from standard characterizing generalizations and show that IS constructions allow for moral concepts---such as virtue, goodness, and rightness---to be dynamically defined, contested, and taught. This reveals a novel perspective on how language contributes to the evolution and application of moral norms. (shrink)

In recent years, many people writing on set theory have invoked the notion of an indefinitely extensible concept. The notion, it is usually claimed, plays an important role in solving the paradoxes of absolute infinity. It is not clear, however, how the notion should be formulated in a coherent way, since it appears to run into a number of problems concerning, for example, unrestricted quantification. In fact, the notion makes perfectly good sense if one endorses a dialetheic solution to the (...) paradoxes. It then morphs from a supposed solution to the paradoxes into a diagnosis of their structure. In this paper I show how. (shrink)

We look at recent accounts of the indefinite extensibility of the concept set and compare them with a certain linguistic model of indefinite extensibility. We suggest that the linguistic model has much to recommend over alternative accounts of indefinite extensibility, and we defend it against three prima facie objections.

Fictions can leave certain features of their fictional worlds _indefinite_. For instance, a novel, film, play, or painting which does not declare whether a certain character is right-handed or left-handed leaves his handedness indefinite. The view that indefiniteness is a consequence of _incompleteness_ in a fictional world is presented. A logic for indefiniteness is developed—articulated using a third truth value, _indeterminate_—in which a disjunction can be true without having a true disjunct. Whether incompleteness is not just a feature of (...) fictional worlds but also of the actual world is considered, taking superposition (found in quantum mechanics) as an example of incompleteness. It is shown that there is nothing objectionable in this (and thus nothing objectionable in taking fictional worlds to be incomplete either). Finally, the preferred account is compared with an alternative which treats indefiniteness in terms of the representation of more than one world. (shrink)

Dummett’s notion of indefinite extensibility is influential but obscure. The notion figures centrally in an alternative Dummettian argument for intuitionistic logic and anti-realism, distinct from his more famous, meaning-theoretic arguments to the same effect. Drawing on ideas from Dummett, a precise analysis of indefinite extensibility is proposed. This analysis is used to reconstruct the poorly understood alternative argument. The plausibility of the resulting argument is assessed.

One of the big questions about indefinites is why they can escape scope islands. In the recent approach of Brasoveanu and Farkas :1–55, 2011) scopal relations with syntactically dominating quantifiers are hard wired into the semantic definition of the existential quantifier, which immediately explains why the semantic scope of indefinites may exceed their syntactic scope. In this paper, I argue for the revival of an alternative approach which places the explanatory burden on the idea that indefinites are essentially referential expressions, (...) similar to definites, and not plain existential quantifiers. I propose one fully explicit variant of such theories and argue that it comes with a number of conceptual and empirical advantages over competing theories. (shrink)

The Monist’s call for papers for this issue ended: “if formalism is true, then it must be possible in principle to mechanize meaning in a conscious thinking and language-using machine; if intentionalism is true, no such project is intelligible”. We use the Grelling-Nelson paradox to show that natural language is indefinitely extensible, which has two important consequences: it cannot be formalized and model theoretic semantics, standard for formal languages, is not suitable for it. We also point out that object-object mapping (...) theories of semantics, the usual account for the possibility of non intentional semantics, doesn’t seem able to account for the indefinitely extensible productivity of natural language. (shrink)

It is argued that an indefinite statement as introduced by Aristotle in De Int. 7 refers to a universal which may partly partake in contradictory accidental predicates together. This fact is mirrored on the semantic level by ascribing truth to some degree to both parts of a contradiction. Accordingly, Aristotle should be interpreted as saying in De Int. 9 that the statement that a certain individual object will be F at some time in its contingent future is to be (...) taken to be true to some degree. This is because an individual object cannot yet, with respect to its contingent future, be regarded as factual but only as - time-independently - exemplifying a universal. In this context, fundamental connections become apparent between indefinite statements on the one hand and Aristotelian modal logic, statistics and theory of science on the other. (shrink)

In this article, I consider Descartes’ enigmatic claim that we must assert that the material world is indefinite rather than infinite. The focus here is on the discussion of this claim in Descartes’ late correspondence with More. One puzzle that emerges from this correspondence is that Descartes insists to More that we are not in a position to deny the indefinite universe has limits, while at the same time indicating that we conceive a contradiction in the notion that (...) the universe has a limit. I reject one attempt to resolve this apparent conflict which appeals to Descartes’ admission to More that divine omnipotence requires that God can create a vacuum in nature, and focus instead on his response to More’s claim that this omnipotence requires the possibility of a completion of the division of matter that results in atoms. Finally, I distinguish Descartes’ indefinite from two other kinds of infinity with which it might well be confused. The first is an essentially incomplete ‘potential infinity’ that Descartes discusses in the Third Meditation, and the second is the sort of quantitative infinity that Leibniz – contrary to Descartes – denies can constitute a completed whole. (shrink)

Across languages, we find indefinites that overtly mark a speaker’s lack of knowledge: they signal that the speaker is unable to give any further information about who or what satisfies her existential claim (Becker, 1999; Haspelmath, 1997). From now on, we will refer to the marking of the speaker’s lack of knowledge as an epistemic effect and to the indefinites that induce epistemic effects as epistemic indefinites.

This essay clarifies quantifier variance and uses it to provide a theory of indefinite extensibility that I call the variance theory of indefinite extensibility. The indefinite extensibility response to the set-theoretic paradoxes sees each argument for paradox as a demonstration that we have come to a different and more expansive understanding of ‘all sets’. But indefinite extensibility is philosophically puzzling: extant accounts are either metasemantically suspect in requiring mysterious mechanisms of domain expansion, or metaphysically suspect in (...) requiring nonstandard assumptions about mathematical objects. Happily, the view of quantifier meanings that underwrites quantifier variance can be used to provide an account of indefinite extensibility that is both metasemantically and metaphysically satisfying. Section 1 introduces the puzzle of indefinite extensibility; section 2 develops and clarifies the metasemantics of quantifier variance; section 3 solves section 1's puzzle of indefinite extensibility by applying section 2's account of quantifier meanings; and section 4 compares the theory developed in section 3 to several other theories in the literature. (shrink)

The purpose of this paper is to show that the mathematics of quantum mechanics (QM) is the vector (Hilbert) space version of the mathematics of partitions at the set level. Since partitions are the math tool to describe indefiniteness and definiteness, this shows how the reality so well described by QM is a non-classical reality featuring the objective indefiniteness of superposition states. The lattice of partitions gives a skeletal model of quantum reality with the partition versions of pure states, non-classical (...) mixed states (i.e., including a superposition state), and the completely distinguished classical state that satisfies the partition logic version of the Principle of Identity of Indistinguishables. Both the key notions of quantum states and quantum observables are respectively the (density) matrix versions of partitions and vector space version of a numerical attributes. The operation of projective measurement given by the Lüders mixture operation is the vector space version of the partition join operation between the partition prefiguring the density matrix of the state being measured and the partition prefiguring the observable being measured. All this adds specific key concepts and structure to Abner Shimony’s idea of a literal understanding of QM to form what might be called the "Objective Indefiniteness Interpretation" of QM. (shrink)

This paper stems from previous works of the authors in which a new measure of the simplicity of an explanation based on its degree of arbitrariness is proposed: The more the explanation is arbitray, the less appealing it is, with explanations having no arbitrariness – called constrained – being the preferred ones. In previous works, as commonly done in the literature of abductive logic programming, a set of hypotheses is not an explanation, unless it is definite, i.e. it explains all (...) the data belonging to the observation. In this paper, we follow a different perspective and define the concept of indefinite constrained explanations, i.e. constrained explanations that are not definite, but admit some indefiniteness. An indefinite constrained explanation captures the intuition of the existence of an explanation (indefinite explanation) that would best explain the given evidence, while not making arbitrary choices (constrained explanation). The main contribution of the paper is the study of abduction in indefinite deductive theories: specifically, the paper studies the framework of abductive logic programming extended with integrity constraints in the setting in which both the initial knowledge base and the abductive explanations are indefinite (may contain occurrences of null values) and the domain is possibly infinite. Furthermore, the paper discusses the complexity of problems concerning indefinite (constrained) explanations. (shrink)

Discourse dynamics, pragmatics, and indefinites Content Type Journal Article Pages 1-30 DOI 10.1007/s11098-012-9882-y Authors Karen S. Lewis, Department of Philosophy, University of Southern California, Los Angeles, CA, USA Journal Philosophical Studies Online ISSN 1573-0883 Print ISSN 0031-8116.

The paper proves a conjecture of Solomon Feferman concerning the indefiniteness of the continuum hypothesis relative to a semi-intuitionistic set theory.

The prototypes of definiteness and indefiniteness in English are the definite article the and the indefinite article a/an, and singular noun phrases (NPs)1 determined by them. That being the case it is not to be predicted that the concepts, whatever their content, will extend satisfactorily to other determiners or NP types. However it has become standard to extend these notions. Of the two categories definites have received rather more attention, and more than one researcher has characterized the category of (...) definite NPs by enumerating NP types. Westerståhl (1985), who is concerned only with determiners in the paper cited, gives a very short list: demonstrative NPs, possessive NPs, and definite descriptions. Prince (1992) lists proper names and personal pronouns, as well as NPs with the, a demonstrative, or a possessive NP as determiner. She notes, in addition, that “certain quantifiers (e.g. all, every) have been argued to be definite” (Prince 1992: 299). This list, with the quantifiers added, agrees with that given by Birner & Ward (1998, 114). Ariel (1988, 1990) adds null anaphoric NPs. (shrink)

ConclusionIn conclusion, I shall indicate one consequence of (3.4). The major resultof work on infinitely iterated Prisoner's Dilemma games is that there existcooperative equilibria in such games. I have suggested above that myaccount of finitely, but indefinitely, iterated Prisoner's Dilemma gamesreflects the nature of genuine iterated Prisoner's Dilemmas more accu-rately than accounts involving infinite iterations. If my suggestion iscorrect, then one consequence of (3.4) - of there being only uncooperativeequilibria in finitely, but indefinitely, iterated games - is to call intoquestion (...) the significance of the existence of cooperative equilibria ininfinitely iterated Prisoner's Dilemma games. (shrink)

In this paper I argue against grounding being necessarily well-founded, and provide some reasons to think it's actually not well-founded.

The-predicativism is the view that proper names (henceforth, names) are common count nouns with a denominative use in all their literal occurrences. A core tenet of this view is that bare singular names in (apparent) argument position of a predicate function as the descriptive component of a covert incomplete definite description with a definite article as a phonologically null determiner. This article presents two closely related counterexamples to the-predicativism. The argument will be made that indefinite singular names (ISNs) – (...) that is, singular names preceded by indefinites, with or without fronted-modifiers – generate a significant asymmetry between names and common count nouns. (shrink)

The view that mathematical objects are indefinite in nature is presented and defended, hi the first section, Field's argument for fictionalism, given in response to Benacerraf's problem of identification, is closely examined, and it is contended that platonists can solve the problem equally well if they take the view that mathematical objects are indefinite. In the second section, two general arguments against the intelligibility of objectual indefiniteness are shown erroneous, hi the final section, the view is compared to (...) mathematical structuralism, and it is shown that a version of structuralism should be understood as embracing the same view. (shrink)

The interpretation of indefinite noun phrases poses a dilemma that goes to the heart of the relationship between semantic content and truth. On the one hand, it is argued, indefinites are referring expressions, and contribute their referents to the utterance’s semantic content (the proposition assigned to the situated utterance, by linguistic convention, as its communicated content). On the other hand, it is apparent that the truth conditions of utterances containing indefinites are not object-dependent, but existential; the referents of indefinites, (...) if they have any, must be ignored in the calculation of truth value. These two claims are consistent only if we give up a central assumption of truth-conditional semantics: that the semantic content of an utterance, in conjunction with a circumstance of evaluation, determines its truth value. (shrink)

Mathematics is indefinite with respect to topics. This is illustrated by the Modern Mathematics Movement, which brought about revolutionary changes to the mathematics curriculum. These changes were facilitated by an incoherent combination of political preoccupations, military interests, economic priorities, and intrinsic mathematical ideas. That mathematics is indefinite with respect to applications is illustrated by the radical change in research related to algorithms. The Entscheidungsproblem is a specific metamathematical problem, whose solution presupposed that the notion of algorithm was clearly (...) specified. This specification was done by means of the Turing machine, a conception tremendously important in the development of computing. That mathematics is indefinite with respect to applications is also illustrated by the development of modern cryptography, which refers to cryptography based on computing. In the development of modern cryptography, number theory plays a huge role. Results from apparently harmless pure number theory have acquired massive economic and also military interests. (shrink)

I argue that indefinite descriptions are referring terms. This is not the ambiguity thesis: that sometimes they are referring terms and sometimes something else, such as quantifiers. No. On my view they are always referring terms; and never quantifiers. I defend this thesis by modifying the standard conception of what a referring term is: a modification that needs to be made anyway, irrespective of the treatment of indefinites. I derive this approach from my speech-act theoretic semantics. The basic thought (...) is that referring terms have as their meanings speech-acts of a certain kind called proto-referring acts. These are acts in which speakers advertise or present intentions to denote, where denotation is a word-world relation, and advertising an intention is acting as if one has intentions, where it is open whether one has them or not, or whether the referring term used denotes or not. I show how this works for proper names. The meaning of a proper name is the speech-act proto-referring act type defined by a certain referential tree. This gives us the basis for an account of proper name meaning irrespective of denotation: a uniform treatment of full and empty names. Applied to indefinites, we can capture cases where speakers perform proto-acts—in which they advertise an intention to denote something—where they intend to denote, but others where they do not, but they still perform the proto-act: advertising an intention to denote. Two cases are ‘Fred saw a hippo’ and ‘Jane did not see a hippo’. In both cases U performs the same proto-act, represent it as Rpro, but possesses the advertised intention in the first case, but not in the second. In the first case, Rpro gains referential content—its descriptive content is expanded to include seen by Fred, whereas in its second token use it has no content augmentation or sentential determination of reference. In the second case ‘a hippo’ is an empty referring term, just as ‘Pegasus’ is empty in ‘Fred did not see Pegasus; he does not exist’. But in both sentences ‘a hippo’ functions in the same basic way: in both cases the basic proto-referring act Rpro is performed. Using this approach, I show how definite descriptions can be construed as indefinites with added meaning. (shrink)

It is widely assumed that the scope of indefinites is island insensitive, i.e., that, generally, an indefinite inside of a syntactic island, such as an adjunct clause, is capable of taking scope outside of that island. This paper challenges this assumption by studying the scope behaviour of the Spanish plural indefinite algunos ’). It presents an experimental study that shows that the scope of algunos is not free and depends on its syntactic environment, at least in the dialect (...) of Spanish studied here. The paper discusses some of the implications of the study for current theories of indefinite scope: it points out the problems that choice functions and singleton indefinites have with the Spanish data, and it also discusses the implications for Schwarzschild's solution to the so-called ‘Donald Duck’ problem. (shrink)

In a recent volume of this journal John Carroll argued that there exist only uncooperative equilibria in indefinitely repeated prisoner's dilemma games. We show that this claim depends on modeling such games as finitely but indefinitely repeated games, which reduce simply to finitely repeated games. We propose an alternative general model of probabilistically indefinitely repeated games, and discuss the appropriateness of each of these models of indefinitely repeated games.

This book examines the semantics and pragmatics of determiners or pronouns that signal ignorance on the part of the speaker, such as Spanish algún. It offers novel empirical observations and important theoretical insights on epistemic indefinites and related topics such as modal free relatives, modified numerals, and epistemic modals.

In this paper I provide a decompositional analysis of three kinds of plural indefinites in two related languages, European Spanish and Brazilian Portuguese. The three indefinites studied are bare plurals, the unos (Spanish)/uns (Portuguese) type, and the algunos (Spanish)/alguns (Portuguese) type. The paper concentrates on four properties: semantic plurality, positive polarity, partitivity, and event distribution. The logic underlying the analysis is that of compositionality, applied at the subword level: as items become bigger in form (with the addition of morphemes), they (...) also acquire more semantic properties. The paper proposes the “indefinite hierarchy", which establishes a set of components for languages to build their indefinites with, in a particular order. (shrink)

Descartes' philosophy contains an intriguing notion of the infinite, a concept labeled by the philosopher as indefinite. Even though Descartes clearly defined this term on several occasions in the correspondence with his contemporaries, as well as in his Principles of Philosophy, numerous problems about its meaning have arisen over the years. Most commentators reject the view that the indefinite could mean a real thing and, instead, identify it with an Aristotelian potential infinite. In the first part of this (...) article, I show why there is no numerical infinity in Cartesian mathematics, as such a concept would be inconsistent with the main fundamental attribute of numbers: to be comparable with each other. In the second part, I analyze the indefinite in the context of Descartes' mathematical physics. It is my contention that, even with no trace of infinite in his mathematics, Descartes does refer to an actual indefinite because of its application to the material world within the system of his physics. This fact underlines a discrepancy between his mathematics and physics of the infinite, but does not lead a difficulty in his mathematical physics. Thus, in Descartes' physics, the indefinite refers to an actual dimension of the world rather than to an Aristotelian mathematical potential infinity. In fact, Descartes establishes the reality and limitlessness of the extension of the cosmos and, by extension, the actual nature of his indefinite world. This indefinite has a physical dimension, even if it is not measurable. La filosofía de Descartes contiene una noción intrigante de lo infinito, un concepto nombrado por el filósofo como indefinido. Aunque en varias ocasiones Descartes definió claramente este término en su correspondencia con sus contemporáneos y en sus Principios de filosofía, han surgido muchos problemas acerca de su significado a lo largo de los años. La mayoría de comentaristas rechaza la idea de que indefinido podría significar una cosa real y, en cambio, la identifica con un infinito potencial aristotélico. En la primera parte de este artículo muestro por qué no hay infinito numérico en las matemáticas cartesianas, en la medida en que tal concepto sería inconsistente con el principal atributo fundamental de los números: ser comparables entre sí. En la segunda parte analizo lo indefinido en el contexto de la física matemática de Descartes. Mi argumento es que, aunque no hay rastro de infinito en sus matemáticas, Descartes se refiere a un indefinido real a causa de sus aplicaciones al mundo material dentro del sistema de su física. Este hecho subraya una discrepancia entre sus matemáticas y su física de lo infinito, pero no implica ninguna dificultad en su física matemática. Así pues, en la física de Descartes, lo indefinido se refiere a una dimensión real del mundo más que a una infinitud potencial matemática aristotélica. De hecho, Descartes establece la realidad e infinitud de la extensión del cosmos y, por extensión, la naturaleza real de su mundo indefinido. Esta indefinición tiene una dimensión física aunque no sea medible. (shrink)

In the war on terrorism, the imprisonment of suspected terrorists by the United States has raised a host of issues,1 among them that of indefinite detention. Over the years, there has been a great...

This chapter argues that the Dummett (2001) position is well-motivated. The position restricts the knowability principle to atomic statements, and defines intuitionistic truth inductively from there. An instructive account of Dummett's development in (1990) and (1996) is offered. There Dummett attempts to clarify the notion of indefinite extensibility of such concepts as set, natural number, and real number, and argues that only intuitionistic logic can illuminate a proper understanding of the notion. If this is correct, then the Dummett (2001) (...) theory of truth is well-motivated, and so, we have a principled solution to the knowability paradox. (shrink)

A critical philosophy of mathematics considers mathematics as being indefinite. Mathematical constructions are not heading towards definite formats; they are tentative and always open to change. In this chapter I concentrate on showing mathematics as being indefinite with respect to concepts and proofs. I illustrate this with reference to the changing roles of the notion of infinitesimal. Before the nineteenth century the term infinitesimal was used liberally, but in the nineteenth century it became a concern to ensure rigour (...) in mathematical reasoning, and to get the infinitesimals under control. The notion of function is indefinite; in fact, it has not always been part of the mathematical vocabulary. Neither Newton nor Leibniz operated with the notion of function. The very word function was first used by Bernoulli, and Euler made some further clarifications of the notion. The notion of infinity has, in its own way, brought about huge controversies in mathematics. The notion is indefinite, and this indefiniteness has a huge impact on what to consider as a valid mathematical proof. (shrink)

This paper aims to show that intuitionistic Kripke models are a powerful tool for interpreting Kant’s ‘Critical Philosophy’. Part I reviews some old work of mine that applies these models to provide a reading of Kant’s second antinomy about the divisibility of matter and to answer several attacks on Kant’s antinomies. But it also points out three shortcomings of that original application. First, the reading fails to account for Kant’s second antinomy claim that matter is divisible ‘ad infinitum’ and his (...) first antinomy claim that empirical space extends only ‘ad indefinitum’. Secondly, in conformity with the ‘assertability’ heuristics for Kripke models, it attributes an assertability semantics to Kant. In fact, it attributes a pair of assertability semantics to Kant, but neither of these accords with modern assertabilism. And finally, one must wonder the propriety of using contemporary assertability and contemporary formal logic to save an eighteenth-century text. Part II goes historical to solve the problems about assertability. It outlines the Descartes–Leibniz dialectic about the infinite/indefinite distinction and demonstrates how each position reflects larger philosophical positions. It then demonstrates that Kant’s two versions of assertability and his version of the infinite/indefinite dichotomy emerge naturally from his attempts (pre-critical and then critical) to resolve a tension in Leibniz’s philosophy. In light of this demonstration, Part III adapts the Kripke model interpretation so that it does reflect that dichotomy. It then refines the Kripke semantics in order to model some of Kant’s signature critical doctrines, and it shows how Kant’s critical version of the infinite/indefinite distinction differs from all those of his predecessors and from his own pre-critical version. It also addresses the question of using contemporary logical machinery to interpret Kant. Part IV turns the tables and uses what we have learned about Kant and modern semantics in order to address a pair of issues in the contemporary critical foundations of logic. (shrink)

In his thought-provoking, carefully crafted, and impressively researched article, Don Scheid makes a compelling argument for an indefinite detention paradigm for “mega-terrorists.” Scheid's proposa...

The colonisation of mathematics is engraved in the version of the history of mathematics that presents it as a Western achievement. As part of the process of colonisation, Eurocentrism developed into an all-dominating perspective on the world, and the white-supremacy ideology became part of the modern outlook. The European colonisation of mathematics was accomplished by cutting away the non-Greek roots of mathematics, and by describing Greek culture as European. The Eurocentric presentation of the history of mathematics is, however, a myth (...) that has transformed into an unquestionable given, repeated in many mathematics textbooks, including brief sketches of the history of mathematics. Ethnomathematical studies have contributed profoundly to the recognition of the multiplicities of the enculturation of mathematics. For a critical philosophy of mathematics, it is crucial to acknowledge this multiplicity and to recognise mathematics as being indefinite with respect to culture. (shrink)

Of all the cases made against classical logic, Michael Dummett's is the most deeply considered. Issuing from a systematic and original conception of the discipline, it constitutes one of the most distinctive achievements of twentieth century British philosophy. Although Dummett builds on the work of Brouwer and Heyting, he provides the case against classical logic with a new, explicit and general foundation in the philosophy of language. Dummett's central arguments, widely celebrated if not widely endorsed, concern the implications of the (...) relation between meaning and use for both the inference rules that govern logical connectives and the relation between truth and its recognition. It is less often noted that Dummett has a further argument against classical logic, one based on the semantic and set-theoretic paradoxes. That is the topic of this paper. (shrink)

This paper claims that indefinite descriptions, singular and plural, have different scope properties than genuine quantifiers. This claim is based on their distinct behavior in island constructions: while indefinites in islands can have intermediate (and maximal) scope readings, quantifiers cannot. Further, the simplest in situ interpretation strategy for indefinites results in incorrect truth conditions for intermediate (and maximal) scope readings. I introduce a mechanism which “auto-matically” preserves the restriction on free variables corresponding to indefinites, in a way which allows (...) the restriction to be carried up in the course of semantic interpretation and to be used at the level where the variable is quantified. This mechanism is a realization of the “indefinites-as-free-variables” proposal of Lewis, Kamp, and Heim, but emphasizes the role of the restriction. I then show that distributive readings of plural indefinites also display island-escaping behavior and argue for an independent, island-insensitive distribution mechanism. (shrink)

Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets which are category-theoretically dual to (...) one another and which are developed in two dual mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is "follow the math" by showing how the mathematics of set partitions can be transported in a natural way to complex vector spaces where it yields the mathematical machinery of QM. And then we show how the machinery of QM can be transported the other way down to set-like vector spaces over ℤ₂ yielding a rather fulsome "toy" or pedagogical model of "quantum mechanics over sets." In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics. (shrink)

The formal semantics that we have proposed for definite and indefinite descriptions analyzes them both as variable-binding operators and as referring terms. It is the referential analysis which makes it possible to account for the facts outlined in Section 2, e.g. for the purely ‘instrumental’ role of the descriptive content; for the appearance of unusually wide scope readings relative to other quantifiers, higher predicates, and island boundaries; for the fact that the island-escaping readings are always equivalent to maximally wide (...) scope quantifiers; and for the appearance of violations of the identity conditions on variables in deleted constituents. We would emphasize that this is not a random collection of observations. They cohere naturally with each other, and with facts about other phrases that are unambigously referential.We conceded at the outset of this paper that the referential use of an indefinite noun phrase does not, by itself, motivate the postulation of a referential interpretation. Our argument has been that the behavior of indefinites in complex sentences cannot be economically described, and certainly cannot be explained, unless a referential interpretation is assumed. It could be accounted for in pragmatic terms only if the whole theory of scope relations and of conditions on deletion could be eliminated from the semantics and incorporated into a purely pragmatic theory. But this seems unlikely. (shrink)

Contemporary discussions do not always clearly distinguish two different forms of vagueness. Sometimes focus is on the imprecision of predicates, and sometimes the indefiniteness of statements. The two are intimately related, of course. A predicate is imprecise if there are instances to which it neither definitely applies nor definitely does not apply, instances of which it is neither definitely true nor definitely false. However, indefinite statements will occur in everyday discourse only if speakers in fact apply imprecise predicates to (...) such indefinite instances. (What makes an instance indefinite is, it should be clear, predicate-relative.) The basic issue in the present inquiry is whether this indefiniteness ever really occurs; the basic question is, Why should it ever occur? (shrink)