It is widely held that there is a problem of talking about or otherwise representing things that not exist. But what exactly is this problem? This paper presents a formulation of the problem in terms of the conflict between the fact that there are truths about non-existent things and the fact that truths must be answerable to reality, how things are. Given this, the problem of singular negative existential statements is no longer the central or most difficult aspect of the (...) problem of non-existence, despite what some philosophers say. (shrink)

In this paper I am concerned with an analysis of negative existential sentences that contain proper names only by using negative or neutral free logic. I will compare different versions of neutral free logic with the standard system of negative free logic (Burge, Sainsbury) and aim to defend my version of neutral free logic that I have labeled non-standard neutral free logic.

Famously, Saul Kripke proposes that there are contingent a priori truths, and has offered a number of examples to illustrate his claim. The most well-known example involves the standard meter bar in Paris. Purportedly, a certain agent knows a priori that the bar is one meter long. However, in response to a long-standing objection to such examples - the "existential complaint" - generally only modified examples having a conditional form are now considered candidates for the contingent a priori. Gareth Evans (...) argues that these conditionals must be understood free-logically, and on this basis argues against Keith Donnellan's analysis of the contingent a priori. I show Evans' argument mistaken. I also take issue with the existential complaint, and suggest a way of understanding Kripke's original examples that is not subject to it. My approach focuses the debate in its right place, and allows us to take Kripke's original examples seriously. (shrink)

Indexicals are unique among expressions in that they depend for their literal content upon extra-semantic features of the contexts in which they are uttered. Taking this peculiarity of indexicals into account yields solutions to variants of Frege's Puzzle involving objects of attitude-bearing of an indexical nature. If names are indexicals, then the classical versions of Frege's Puzzle can be solved in the same way. Taking names to be indexicals also yields solutions to tougher, more recently-discovered puzzles such as Kripke's well-known (...) case involving Paderewski. We argue that names are in fact rigidly designating indexicals. We also argue that fully developed, the direct reference theory's best strategy for solving the puzzles amounts to the adoption of the indexical theory of names – a move that we argue should be thought of as a natural development of the direct reference theory, and not as antagonistic to it. (shrink)

A model-theoretic approach to the semantics of set-theoretic discourse.

What are the meanings of empty names such as ‘Vulcan,’ ‘Pegasus,’ and ‘Santa Claus’ in such sentences as ‘Vulcan is the tenth planet,’ ‘Pegasus flies,’ and especially ‘Santa Claus does not exist’?Our view, developed in Adams et al., consists of a direct-reference account of the meaning of empty names in combination with a pragmatic-implicature account of why we have certain intuitions that seem to conflict with a direct-reference account.

Descriptions are phrases of the form ‘an F’, ‘the F’, ‘Fs’, ‘the Fs’ and NP's F (e.g. ‘John's mother’). They can be indefinite (e.g., ‘an F’ and ‘Fs’), definite (e.g. ‘the F’ and ‘the Fs’), singular (e.g., ‘an F’, ‘the F’) or plural (e.g., ‘the Fs’, ‘Fs’). In English plural indefinite descriptions lack an article and are for that reason also known as ‘bare plurals’. How to account for the semantics and pragmatics of descriptions has been one of the central (...) topics in philosophy for centuries. This entry focuses on the historical and contemporary philosophical debate about Bertrand Russell’s theory of descriptions and the theories that developed as responses to this theory. (shrink)

The truth of a statement depends on the world in two ways: what the statement says is true if the world is as the statement says it is; on the other hand, what the expressions in the statement mean depends on what the world is like (for instance, on what conventions are in place). Each of these two kinds of dependence of truth on the world corresponds to one of the dimensions on the two-dimensional semantic framework, developed in the 1970’ (...) in the work of Evans, Kaplan, Kripke and Stalnaker. The introduction provides a systematic overview of the framework, the ideas of its earlier originators, recent developments and criticism. Finally, it gives a brief overview over the contributions to the volume. (shrink)

What are the prospects for a linguistic approach to ontology? Given that it seems that there are true subject-predicate sentences containing empty names, traditional linguistic approaches to ontology appear to be flawed. I argue that in order to determine what there is, we need to determine which sentences ascribe properties (and relations) to objects, and that there does not appear to be any formal criterion for this. This view is then committed to giving an account of what predicates do in (...) sentences when they do not ascribe properties. I sketch an approach to the varieties of predication. (shrink)

A popular claim in recent philosophy of mind and action is that events only exist once they are over. This has been taken to have the consequence that many temporal phenomena cannot be understood ‘from the inside’, as they are unfolding, purely in terms of events. However, as I argue here, the claim that events exist only when over is incoherent. I consider two ways of understanding the claim and the notion of existence it involves: one that ties existence to (...) the logic of quantification, and another that assimilates the existence of an event to that of a continuant. The first of these, I argue, cannot be the right way to understand the target claim, as there are serious logical obstacles to regarding this notion of existence as temporally variable. The second, on the other hand, leaves it ultimately mysterious what the existence of an event could amount to. I close with the suggestion that the undeniable temporal asymmetries which parties to the debate are interested in are properly understood in perspectival, not ontological, terms. (shrink)

Confused terms appear to signify more than one entity. Carnap maintained that any putative name that is associated with more than one object in a relevant universe of discourse fails to be a genuine name. Although many philosophers have agreed with Carnap, they have not always agreed among themselves about the truth-values of atomic sentences containing such terms. Some hold that such atomic sentences are always false, and others claim they are always truth-valueless. Field maintained that confused terms can still (...) refer, albeit partially, and offered a supervaluational account of their semantic properties on which some atomic sentences with confused terms can be true. After outlining many of the most important theoretical considerations for and against various semantic theories for such terms, we report the results of a study designed to investigate which of these accounts best accords with the truth-value judgments of ordinary language users about sentences containing these terms. We found that naïve participants view confused names as capable of successfully referring to one or more objects. Thus, semantic theories that judge them to involve total reference failure do not comport well with patterns of ordinary usage. (shrink)

A long-standing theme in discussion of perception and thought has been that our primary cognitive contact with individual objects and events in the world derives from our perceptual contact with them. When I look at a duck in front of me, I am not merely presented with the fact that there is at least one duck in the area, rather I seem to be presented withthisthing (as one might put it from my perspective) in front of me, which looks to (...) me to be a duck. Furthermore, such a perception would seem to put me in a position not merely to make the existential judgment that there is some duck or other present, but rather to make a singular, demonstrative judgment, that that is a duck. My grounds for an existential judgment in this case derives from my apprehension of the demonstrative thought and not vice versa. (shrink)

Ted Sider has famously argued that existence, in the unrestricted sense of ontology, cannot be vague, as long as vagueness is modeled by means of precisifications. The first section of Chapter 9 exposes some controversial assumptions underlying Sider’s alleged reductio of vague existence. The upshot of the discussion is that, although existence cannot be vague, it can be super-vague, i.e. higher-order vague, for all orders. The second section develops and defends a novel framework, dubbed negative supervaluationary semantics, which makes room (...) for the possibility of super-vague existence. (shrink)

This article studies seriously actualistic quantified modal logics. A key component of the language is an abstraction operator by means of which predicates can be created out of complex formulas. This facilitates proof of a uniform substitution theorem: if a sentence is logically true, then any sentence that results from substituting a predicate abstract for each occurrence of a simple predicate abstract is also logically true. This solves a problem identified by Kripke early in the modern semantic study of quantified (...) modal logic. A tableau proof system is presented and proved sound and complete with respect to logical truth. The main focus is on seriously actualistic T, an extension of T, but the results established hold also for systems based on other propositional modal logics. Following Menzel it is shown that the formal language studied also supports an actualistic account of truth simpliciter. (shrink)

In this extended critical discussion of 'Kant's Modal Metaphysics' by Nicholas Stang (OUP 2016), I focus on one central issue from the first chapter of the book: Stang’s account of Kant’s doctrine that existence is not a real predicate. In §2 I outline some background. In §§3-4 I present and then elaborate on Stang’s interpretation of Kant’s view that existence is not a real predicate. For Stang, the question of whether existence is a real predicate amounts to the question: ‘could (...) there be non-actual possibilia?’ (p.35). Kant’s view, according to Stang, is that there could not, and that the very notion of non-actual or ‘mere’ possibilia is incoherent. In §5 I take a close look at Stang’s master argument that Kant’s Leibnizian predecessors are committed to the claim that existence is a real predicate, and thus to mere possibilia. I argue that it involves substantial logical commitments that the Leibnizian could reject. I also suggest that it is danger of proving too much. In §6 I explore two closely related logical commitments that Stang’s reading implicitly imposes on Kant, namely a negative universal free logic and a quantified modal logic that invalidates the Converse Barcan Formula. I suggest that each can seem to involve Kant himself in commitment to mere possibilia. (shrink)

R.M. Sainsbury's account of reference has many compelling and attractive features. But it has the undesirable consequence that sentences of the form "x is thinking about y" can never be true when y is replaced by a non-referring term. Of the two obvious ways to deal with this problem within Sainsbury's framework, I reject one and endorse the other. This endorsement is also within the spirit of Sainsbury's account of reference. /// La explicación que ofrece R.M. Sainsbury de la referencia (...) tiene muchas características convincentes y atractivas, pero tiene la consecuencia indeseable de que oraciones de la forma "x está pensando acerca de y" nunca pueden ser verdaderas cuando se reemplaza y con un término no referencial. De las dos maneras obvias de tratar este problema dentro del marco teórico de Sainsbury, rechazo una y acepto la otra. Aceptar esta última también cae dentro del espíritu de la explicación de la referencia ofrecida por Sainsbury. (shrink)

I critically discuss some aspects of Mark Sainsbury's Reference without Referents, from an otherwise sympathetic viewpoint. My objections focus on the adequacy of the truth-conditional framework that Sainsbury presupposes. I argue that, as semantic theories, truth-conditional accounts are both too ambitious, and too austere to be fully explanatory, and that both problems have consequences for an account of reference. The latter problem has to do with the difficulties to capture in a truth-conditional framework the descriptive contribution of indexicals and, in (...) my view, proper names. The former has to do with the non-semantic contribution of context to the determination of truth-conditions in general and reference in particular. /// En este artículo discuto críticamente algunos aspectos del libro de Mark Sainsbury Reference without Referents, desde una perspectiva por lo demás próxima. Mis objeciones se centran en la adecuación del marco que Sainsbury presupone: la semántica de condiciones de verdad. Argumento que tales propuestas son, en tanto que teorías semánticas, a la vez demasiado ambiciosas y demasiado austeras para ser genuinamente explicativas, y que ambos problemas tienen consecuencias en lo que respecta a la explicación de la referencia. El segundo problema tiene que ver con las dificultades para representar la contribución descriptiva de los deícticos y, en mi opinión, de los nombres propios también en el marco de condiciones de verdad. El primero tiene que ver con la contribución no semántica del contexto a la determinación de las condiciones de verdad en general y la referencia en particular. (shrink)

When are two tokens of a name tokens of the same name? According to this paper, the answer is a matter of the historical connections between the tokens. For each name, there is a unique originating event, and subsequent tokens are tokens of that name only if they derive in an appropriate way from that originating event. The conditions for a token being a token of a given name are distinct from the conditions for preservation of the reference of a (...) name. Hence a name may change its reference. Defending the theory requires considerations about the identity of acts, and about empty names and disagreement. The upshot is a causal theory of the identity of names, but not what would normally be counted as a causal theory of the reference of names. Although reference is often transmitted causally, what determines semantic reference is conventionalized speaker-reference. (shrink)

The result of combining classical quantificational logic with modal logic proves necessitism – the claim that necessarily everything is necessarily identical to something. This problem is reflected in the purely quantificational theory by theorems such as ∃x t=x; it is a theorem, for example, that something is identical to Timothy Williamson. The standard way to avoid these consequences is to weaken the theory of quantification to a certain kind of free logic. However, it has often been noted that in order (...) to specify the truth conditions of certain sentences involving constants or variables that don’t denote, one has to apparently quantify over things that are not identical to anything. In this paper I defend a contingentist, non-Meinongian metaphysics within a positive free logic. I argue that although certain names and free variables do not actually refer to anything, in each case there might have been something they actually refer to, allowing one to interpret the contingentist claims without quantifying over mere possibilia. (shrink)

The theory of object-dependent thought is summarized and the central motivation for it is sketched, namely, that thought content is constitutively truth conditional. Consequences for the epistemology of thought are briefly noted. Rival conceptions of singular thought are sketched.

Les théoriciens sémantistes de la fiction cherchent typiquement à expliquer nos relations sémantiques au fictionnel dans le contexte plus général des théories de la référence, privilégiant une explication de la sémantique sur le psychologique. Dans cet article, nous défendons une dépendance inverse. Par l’éclaircissement de nos relations psychologiques au fictionnel, nous trouverons un guide pour savoir comment développer une sémantique de la fiction. S’ensuivra une esquisse de la sémantique.

The logic of fiction has been a stand-alone research programme only since the early 1970s.1 It is a fair question as to why in the first place fictional discourse would have drawn the interest of professional logicians. It is a question admitting of different answers. One is that, since fictional names are “empty”, fiction is a primary datum for any logician seeking a suitably comprehensive logic of denotation. Another answer arises from the so-called incompleteness problem, exemplified by the fact (or (...) apparent fact) that some fictional sentences – think of “Sherlock Holmes’ mother was nick-named ‘Polly’” − are neither true nor false. These are sentences to command the attention of logicians who work on non-bivalent logics. A further spur to logical engagement is the supposed fictionality of certain kinds of ideal models in science and certain classes of mathematical objects. No doubt, there are other features of fictional discourse that provide the logician with a natural entré, but perhaps it would also be correct to say that the fiction’s biggest draw for logicians is that our quite common beliefs about the fictional constitute what Nicholas Rescher calls “aporetic clusters”, so named after the Latinized Greek aporos for “impassable”.2 An aporetic cluster is a set of claims such that.. (shrink)

The thesis of existentialism holds that if a proposition p exists and predicates something of an object a, then in any world where a does not exist, p does not exist either. If “possibly, p” entails “in some possible world, the proposition that p exists and is true,” then existentialism is prima facie incompatible with the truth of claims like “possibly, the Eiffel Tower does not exist.” In order to avoid this claim, a distinction between two kinds of world-indexed truth (...) –and two associated kinds of modality –is needed. This paper embodies an attempt to develop a full account of just such a distinction. (shrink)

L’article publié ici se propose d’emprunter une voie qui n’avait pas été empruntée dans les explorations précédentes de l’auteur. En effet, on verra qu’il s’agit ici de surmonter les difficultés auxquelles sont confrontées les théories réalistes de la fiction et en particulier la théorie artefactuelle dont Amie Thomasson est l’auteur. La question principale s’édicte en ces termes : s’il y a des personnages de fiction, comment se fait-il qu’il nous soit naturel de dire que tel ou tel personnage n’existe pas (...) ? Comment rendre compte des affirmations de non-existence des personnages de fiction, en particulier lorsqu’on est un réaliste de la fiction ? L’auteur confronte trois manières de traiter ces affirmations : l’approche par la quantification restreinte et l’approche linguistique et enfin l’approche des conditions d’application, point de vue auquel l’auteur se rallie. La première approche consiste à faire des restrictions spatiales ou catégorielles afin de cerner le champ de la référence. La seconde approche, qui est celle que les théoriciens de la référence directe emploient majoritairement, consiste à passer par le métalangage, c’est-à-dire de passer à un niveau sémantique supérieur. La dernière approche, celle par les conditions d’application, est construite sur les vertus des deux premières approches. Enfin, l’auteur élargira l’apport de cette dernière approche aux débats métaphysiques. (shrink)

The aim of this paper is to assess the relative merits of two accounts of the semantics of proper names. The enterprise is of particular interest because the theories are very similar in fundamental respects. In particular, they can agree on three major features of names: names are rigid designators; different co-extensive names can have different cognitive significance; empty proper names can be meaningful. Neither theory by itself offers complete explanations of all three features. But each theory is consistent with (...) them and goes some way towards explaining them. (shrink)

The aim of this paper is to explore the possibility that, at least, some metaphysical debates are ‘metalinguistic negotiations’. I will take the dispute between the dominant approaches of realism and the anti-realism ones about the ontological status of scientific models as a case-study. I will argue that such a debate may be better understood as a disagreement, at bottom normatively, motivated, insofar as a normative and non-factual question may be involved in it: how the relevant piece of language ought (...) to be used. Even though I will generally assess the prospects for a broadly deflationist approach, I shall outline a sense in which the dispute can be recast as ‘minimally substantive’. (shrink)

Logic ought to guide our thinking. It is better, more rational, more intelligent to think logically than to think illogically. Illogical thought leads to bad judgment and error. In any case, if logic had no role to play as a guide to thought, why should we bother with it?The somewhat naïve opinions of the previous paragraph are subject to attack from many sides. It may be objected that an activity does not count as thinking at all unless it is at (...) least minimally logical, so logic is constitutive of thought rather than a guide to it. Or it may be objected that whereas logic describes a system of timeless relations between propositions, thinking is a dynamic process involving revisions, and so could not use a merely static guide. Or again the objection may be that there is no such thing as logic, only a whole variety of different logics, not all of which could possibly be good guides. (shrink)

We outline a pragmatic-normative understanding of logic as a discipline that is completely anchored in the sphere of action, rules, means and ends: We characterize inferring as a speech act which is in need of regulation and we connect inferential rules with consequence relations. Furthermore, we present a scenario which illustrates how one actually assesses or can in principle assess the quality of logical rules with respect to justificatory questions. Finally, we speculate on the origin of logical rules as a (...) means of supporting our practice of inferring. (shrink)

In this paper, I will defend a new compositional semantics for single-domain free logic. This semantics makes use of a distinction between the semantic value of a singular term and its semantic referent. The semantic value of a singular term is conceived of as a set that either contains the semantic referent or no element at all. The semantic referent is the object that the term designates. Before I will introduce this new semantics for single-domain predicate and an S5-type modal (...) logic in detail, I will present five related problems of compositionality of the standard semantics for these logical systems. After that I will show in detail how my new proposed alternative semantics can be used to solve nearly all of the outlined five problems. (shrink)

Sainsbury 2005 and 2009 offers a theory of empty names that purports to account for the content and truth-value of all utterances involving them. The goal is to do this while offering a homogenous semantic treatment: both empty and nonempty names make the same kind of contribution to truth-values. The account is based on a new theory of reference that purports to be an alternative among nondescriptivist accounts. According to the new theory, there is reference even without referents. In this (...) paper I argue that the theory does not offer a homogeneous semantics for names and that, thus, it fails to offer a more satisfactory alternative to the theories already available. I conclude by briefly describing a way in which a theory could in fact offer such homogeneous semantics for names via a cognitive theory of empty names. (shrink)

In this paper, I will show that typical formal semantic reconstructions of the rigidity of proper names neglect the important aspect that the rigidity of names is determined by our ordinary use of a name relative to the actual world. This fact was clearly pointed out by Kripke, but overlooked by the subsequent discussion concerning this topic. Based on this diagnosis, I will distinguish three different actualized notions of rigidity. Firstly, I will introduce two different new varieties of known versions (...) of rigidity; namely, actualized persistent and actualized obstinate rigidity. Secondly, I will make use of the tools provided by free logic and introduce a new and overlooked version of rigidity, which I will call actually restricted rigidity. Against this background, I will argue that we have different options to formally model the rigidity of proper names in natural languages. Which option we choose, mainly depends on our philosophical background assumptions. For someone who believes in non-existent objects and the possibility of naming such objects, it is the best choice to hold that proper names are actually obstinately rigid designators. For someone who rejects non-existent objects and the possibility of naming such objects, actually restricted rigidity is a better choice than actualized persistent rigidity. (shrink)

Name der Zeitschrift: Deutsche Zeitschrift für Philosophie Jahrgang: 65 Heft: 4 Seiten: 657-688.

In this paper I aim to defend a version of the view that ‘exist’ expresses primarily a property of individual objects, a property that each of them has. In the first section, I will distinguish the three main types of rival conceptions concerning the semantic status of ‘exist’ that will define the subsequent discussion. In the second section it will be shown that the best explanation of our overall use of ‘exist’ in natural language requires the treatment of ‘exist’ as (...) a predicate that can be applied to individual objects. The third section will be concerned with the problem of the semantic analysis of negative singular existential sentences that contain proper names. I will briefly defend a specific solution to this problem that makes use of a single-domain free logic. In the following sections, I will try to defend this view of existence in more detail. That is, I will deal with three different specific challenges that prima facie seem to speak against the proposed view and for some rival alternatives; and I will show how these challenges can be met. (shrink)

In section III of Pryor 2006a, I argued against the view that the mere fact that a thought- type is hyper-reliable directly gives one justification to believe a thought of that type. A close alternative says that our merely appreciating that the thought-type is hyper-reliable directly gives us that justification.

IPC, the intuitionistic predicate calculus, has the property(i) Vc(A c /x) xA.Furthermore, for certain important , IPC has the converse property (ii) xA Vc(A c /x). (i) may be given up in various ways, corresponding to different philosophic intuitions and yielding different systems of intuitionistic free logic. The present paper proves the strong completeness of several of these with respect to Kripke style semantics. It also shows that giving up (i) need not force us to abandon the analogue of (ii).

The paper presents a new theory of perceptual demonstrative thought, the property-dependent theory. It argues that the theory is superior to both the object-dependent theory (Evans, McDowell) and the object-independent theory (Burge).

We present a plural logic that is as expressively strong as it can be without sacrificing axiomatisability, axiomatise it, and use it to chart the expressive limits set by axiomatisability. To the standard apparatus of quantification using singular variables our object-language adds plural variables, a predicate expressing inclusion (is/are/is one of/are among), and a plural definite description operator. Axiomatisability demands that plural variables only occur free, but they have a surprisingly important role. Plural description is not eliminable in favour of (...) quantification; on the contrary, quantification is definable in terms of it. Predicates and functors (function signs) can take plural as well as singular terms as arguments, and both many-valued and single-valued functions are expressible. The system accommodates collective as well as distributive predicates, and the condition for a predicate to be distributive is definable within it; similarly for functors. An essential part of the project is to demonstrate the soundness and completeness of the calculus with respect to a semantics that does without set-theoretic domains and in which the use of settheoretic extensions of predicates and functors is replaced by the sui generis relations and functions for which the extensions were at best artificial surrogates. Our metalanguage is designed to solve the difficulties involved in talking plurally about individuals and about the semantic values of plural items. (shrink)

This article presents a sequent calculus for a negative free logic with identity, called N . The main theorem (in part 1) is the admissibility of the Cut-rule. The second part of this essay is devoted to proofs of soundness, compactness and completeness of N relative to a standard semantics for negative free logic.

The aim of this paper is to show what sorts of logics are required by externalist and internalist accounts of the meanings of natural kind nouns. These logics give us a new perspective from which to evaluate the respective positions in the externalist-internalist debate about the meanings of such nouns. The two main claims of the paper are the following: first, that adequate logics for internalism and externalism about natural kind nouns are second-order logics; second, that an internalist second-order logic (...) is a free logic—a second order logic free of existential commitments for natural kind nouns, while an externalist second-order logic is not free of existential commitments for natural kind nouns—it is existentially committed. (shrink)

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, proper (...) classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so class-valued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the IMPS Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems. (shrink)

Some philosophers, notably Professors Quine and Geach, have stressed the analogies they see between pronouns of the vernacular and the bound variables of quantification theory. Geach, indeed, once maintained that ‘for a philosophical theory of reference, then, it is all one whether we consider bound variables or pronouns of the vernacular'. This slightly overstates Geach's positition since he recognizes that some pronouns of ordinary language do function differently from bound variables; he calls such pronouns ‘pronouns of laziness'. Geach's characterisation of (...) pronouns of laziness has varied from time to time, but the general idea should be clear from a paradigm example: A man who sometimes beats his wife has more sense than one who always gives in to her.The pronouns ‘one’ and ‘her’ go proxy for a noun or a noun phrase in the sense that the pronoun is replaceable in paraphrase by simple repetition of its antecedent. (shrink)

The essay outlines the character of free logic, and motivation for its construction and development. It details some technical achievements of high philosophical interest, but urges that the role of existence assumptions in logic is still not fully understood, that unresolved old problems, both technical and philosophical, abound, and presents some new problems of considerable philosophical import in free logic.

Partial functions are ubiquitous in both mathematics and computer science. Therefore, it is imperative that the underlying logical formalism for a general-purpose mechanized mathematics system provide strong support for reasoning about partial functions. Unfortunately, the common logical formalisms — first-order logic, type theory, and set theory — are usually only adequate for reasoning about partial functionsin theory. However, the approach to partial functions traditionally employed by mathematicians is quite adequatein practice. This paper shows how the traditional approach to partial functions (...) can be formalized in a range of formalisms that includes first-order logic, simple type theory, and Von-Neumann—Bernays—Gödel set theory. It argues that these new formalisms allow one to directly reason about partial functions; are based on natural, well-understood, familiar principles; and can be effectively implemented in mechanized mathematics systems. (shrink)

A sentence containing a number of definite descriptions, each lying within the scope of its predecessor, is naturally read as asserting the uniqueness of a sequence of objects satisfying the descriptions. The project of providing a general uniform procedure for eliminating embedded definite descriptions that gets this and other logical forms right is impeded by several puzzles.