By playing a crucial role in settling open issues in the philosophical debate about logical consequence, logical evidence has become the holy grail of inquirers investigating the domain of logic. However, despite its indispensable role in this endeavor, logical evidence has retained an aura of mystery. Indeed, there seems to be a great disharmony in conceiving the correct nature and scope of logical evidence among philosophers. In this paper, I examine four widespread conceptions of logical evidence to argue that all (...) should be reconsidered. First, I argue that logical apriorists are more tolerant of logical evidence than empiricists. Second, I argue that evidence for logic should not be read out of natural language. Third, I argue that if logical intuitions are to count as logical evidence, then their evidential content must not be propositional. Finally, I argue that the empiricist proposal of treating experts' judgments as evidence suffers from the same problems as the rationalist conception. (shrink)

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover at most (...) one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

Most literature on vagueness deals with the phenomenon as applied to predicates. On the contrary, even the idea of vague connectives seems to be taken as an oxymoron. The goal of this article is to propose an understanding of vague logical connectives based on vague quantifiers. The main idea is that the phenomenon of vagueness translates to connectives in terms of the property of Abnormality. I also argue that Prior’s Tonk can, according to this approach, be considered a vague connective. (...) In order to do so, I provide a sound and complete interpretation for it, based on Strict-Tolerant semantics, in which Tonk has a non-normal truth-table. (shrink)

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for (...) number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)

Existential claims are widely held to be grounded in their true instances. However, this principle is shown to be problematic by arguments due to Kit Fine. Stephan Krämer has given an especially simple form of such an argument using propositional quantifiers. This note shows that even if a schematic principle of existential grounds for propositional quantifiers has to be restricted, this does not immediately apply to a corresponding non-schematic principle in higher-order logic.

In a series of papers Ladyman and Presnell raise an interesting challenge of providing a pre-mathematical justification for homotopy type theory. In response, they propose what they claim to be an informal semantics for homotopy type theory where types and terms are regarded as mathematical concepts. The aim of this paper is to raise some issues which need to be resolved for the successful development of their types-as-concepts interpretation.

Although idealism was widely defended in the history of philosophy, it is nowadays almost universally considered a non-starter. This holds in particular for a strong form of idealism, which asserts that not just minds or the mental in general, but our human minds in particular are metaphysically central to reality. Such a view seems to be excessively anthropocentric and contrary to what we by now know about our place in the universe. Nonetheless, there is reason to think that such a (...) strong form of idealism is indeed correct. In this paper, I will present an argument for idealism of this kind through considerations about a harmony between our thought and reality. The central argument in favour of idealism will come from a possibly unexpected source: we can see that a strong form of idealism is true simply from considerations about our language alone. I shall argue that thinking about how we represent reality allows us to conclude that idealism is true, and thus that reality must be a certain way. But no argument of this kind seems to allow for a metaphysical conclusion like idealism, since considerations about our language alone only show how we represent reality, not how reality is. And thus idealism can’t possibly follow, since it concerns how reality is, not just how we represent it to be. A good part of the second half of the paper is devoted to showing how such an argument is possible after all, and that it really does establish idealism. (shrink)

This thesis advances and defends a novel two-category ontology of objects and dimensions, latterly conceived as respects of comparability. The proposed 'dimensionist' ontology is set out and brought to bear on discussions of determinables and determinates, the problem of universals, fact ontologies, and nomic governance. Dimensionism is argued to fare well in comparison to a range of rival ontological accounts of property possession. A metametaphysical framework is set out to undergird the discussion, which draws on both realist and pragmatist resources.

Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ($AC$) or ‘Frege Constraint’ ($FC$), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how$AC$generalizes Frege’s views while$FC$comes closer to his original conceptions. Different authors diverge on the interpretation of$FC$and on whether it (...) applies to definitions of both natural and real numbers. Our aim is to trace the origins of$FC$and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish$AC$from$FC$(§2). We discuss six rationales which may motivate the adoption of different instances of$AC$and$FC$(§3). We turn to the possible interpretations of$FC$(§4), and advance a Semantic$FC$(§4.1), arguing that while it suits Frege’s definition of natural numbers (4.1.1), it cannot reasonably be imposed on definitions of real numbers (§4.1.2), for reasons only partly similar to those offered by Crispin Wright (§4.1.3). We then rehearse a recent exchange between Bob Hale and Vadim Batitzky to shed light on Frege’s conception of real numbers and magnitudes (§4.2). We argue that an Architectonic version of$FC$is indeed faithful to Frege’s definition of real numbers, and compatible with his views on natural ones. Finally, we consider how attributing different instances of$FC$to Frege and appreciating the role of the Architectonic$FC$can provide a more perspicuous understanding of his foundational program, by questioning common pictures of his logicism (§5). (shrink)

There are two ways of thinking about the natural numbers: as ordinal numbers or as cardinal numbers. It is, moreover, well-known that the cardinal numbers can be defined in terms of the ordinal numbers. Some philosophies of mathematics have taken this as a reason to hold the ordinal numbers as (metaphysically) fundamental. By discussing structuralism and neo-logicism we argue that one can empirically distinguish between accounts that endorse this fundamentality claim and those that do not. In particular, we argue that (...) if the ordinal numbers are metaphysically fundamental then it follows that one cannot acquire cardinal number concepts without appeal to ordinal notions. On the other hand, without this fundamentality thesis that would be possible. This allows for an empirical test to see which account best describes our actual mathematical practices. We then, finally, discuss some empirical data that suggests that we can acquire cardinal number concepts without using ordinal notions. However, there are some important gaps left open by this data that we point to as areas for future empirical research. (shrink)

Mereological universalists and nihilists disagree on the conditions for composition. In this paper, we show how this debate is a function of one’s chosen semantics for plural quantifiers. Debating mereologists have failed to appreciate this point because of the complexity of the debate and extraneous theoretical commitments. We eliminate this by framing the debate between universalists and nihilists in a formal model where these two theses about composition are contradictory. The examination of the two theories in the model brings clarity (...) to a debate in which opponents frequently talk past one another. With the two views stated precisely, our investigation reveals the dependence of the mereologists’ ontological commitments on the semantics of plural quantifiers. Though we discuss the debate with respect to a simplified and idealized model, the insights provided will make more complex debates on composition more productive and deflationist criticisms of the debate less substantial. (shrink)

InThe Language Animal, Charles Taylor’s struggle to provide a theoretical framework for his narration of the self finally becomes obvious. About 30 years after he wrote his great and fascinatingSources of the Self, Taylor closes the gap between the self as a radical being-in-the-world and its analytical premises. Even if the main topic of Taylor’s new book may seem to be only a comparison of what he calls ‘HHH-theory’ and ‘HLC-theory,’ there are two other authors, the combination of whose ideas (...) clarifies not only his approach to language but also to his concept of ‘reality’ as such: Gottlob Frege and Ludwig Wittgenstein.DansThe Language Animal, les efforts que doit déployer Charles Taylor pour fournir un point de vue théorique permettant de fonder son récit du soi deviennent finalement apparents. Quelque trente ans après son grand et fascinantLes sources du moi, il comble l’écart qui subsistait entre le soi en tant qu’être au monde radical et ses prémisses analytiques. Même si le principal sujet du nouveau livre de Taylor semble se résumer à une comparaison entre ce qu’il appelle la théorie HHH et la théorie HLC, on y retrouve deux auteurs dont les idées, combinées, permettent de clarifier son approche du langage et du concept de «réalité» : Gottlob Frege et Ludwig Wittgenstein. (shrink)

Callard (2007) argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: why a mathematical object does not affect other concrete objects and other mathematical objects, what counts as a mathematical object, and how (...) we can have knowledge about an unchanging object. (shrink)

The problem analysed in this paper is whether we can gain knowledge by using valid inferences, and how we can explain this process from a model-theoretic perspective. According to the paradox of inference (Cohen & Nagel 1936/1998, 173), it is logically impossible for an inference to be both valid and its conclusion to possess novelty with respect to the premises. I argue in this paper that valid inference has an epistemic significance, i.e., it can be used by an agent to (...) enlarge his knowledge, and this significance can be accounted in model-theoretic terms. I will argue first that the paradox is based on an equivocation, namely, it arises because logical containment, i.e., logical implication, is identified with epistemological containment, i.e., the knowledge of the premises entails the knowledge of the conclusion. Second, I will argue that a truth-conditional theory of meaning has the necessary resources to explain the epistemic significance of valid inferences. I will explain this epistemic significance starting from Carnap’s semantic theory of meaning and Tarski’s notion of satisfaction. In this way I will counter (Prawitz 2012b)’s claim that a truth-conditional theory of meaning is not able to account the legitimacy of valid inferences, i.e., their epistemic significance. (shrink)

Arguing first that the best way to understand what a continuant is is as something that primarily has its properties at a time rather than atemporally, the paper then defends the idea that there are occurrent continuants. These are things that were, are, or will be happening—like the ongoing process of someone reading or my writing this paper, for instance. A recently popular philosophical view of process is as something that is referred to with mass nouns and not count nouns. (...) This has mistakenly encouraged the view that the only way to think of process is as the stuff of events, and has obscured the possibility of thinking of processes as individual continuants. (shrink)

Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...) foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett’s program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+ε without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett’s contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett’s framework is improved by these results as they show that questions of realism and anti-realism are not an “all or nothing” matter, but that there are plausibly metaphysical stances between the poles of anti-realism and realism, because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as “is smart,” which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong logic. (shrink)

This paper discusses three approaches to the semantics of event nominalizations and adverbial modification: the Davidsonian account, the Kimian account, and the truthmaker account. It argues that a combination of all three accounts is needed for the semantics of the full range of event, trope, and state nominalizations in English.

Cet ouvrage est consacre a l'etude de la distinction linguistique entre noms massifs (lait, mobilier, desordre, amour...) et noms comptables (chat, equipe, combat, chose...). Les premiers sont normalement invariables, tandis que les seconds s'emploient librement au singulier et au pluriel. Apres avoir etabli qu'il s'agit bien d'une distinction morpho-syntaxique, l'ouvrage discute la possibilite de caracteriser semantiquement cette distinction. Les recherches existantes ne tiennent compte, essentiellement, que des noms s'appliquant au domaine materiel. Ce travail, au contraire, examine en detail aussi bien (...) les noms concrets qu'abstraits. Il propose de redefinir les proprietes souvent attribuees aux noms massifs (references distributive et cumulative), grace, notamment, a une nouvelle notion de partie. Concernant les noms comptables, il montre qu'aucune propriete purement semantique n'est necessaire pour qu'un nom soit de ce type. L'ouvrage se termine par l'analyse des conversions (emplois massifs des noms comptables, et vice versa), et du role qu'y jouent les facteurs pragmatiques. (shrink)

On the one hand they seem to be quite obviously truth conditionally equivalent, but on the other hand they seem to be about different things. Whereas (1) is about Jupiter and its moons, (2) is about numbers. In particular, the word ‘four’ appears in (1) in the position of an adjective or determiner, whereas it seems to be a name for a number in (2). Furthermore, (2) appears to be an identity statement claiming that what two number terms stand for (...) is the same thing. Several authors have proposed answers to this puzzle, including Frege, by either pinning the issue on the nature of numbers, pretense, fictionalism, or the nature of reference, but all these proposals agreed that (2) is an identity statement and that in it ‘four’ is a name for an object. In [Hofweber, 2007] I argued that (2) is not an identity statement and that ‘four’ in (2) is not a name for an object. The argument for it was this: When we look at what role (2) has in actual communication we can see that (2) has a focus effect on a discourse that arises from the syntax without special intonation. (shrink)

In Grundgesetze, Vol. II, §91, Frege argues that ‘it is applicability alone which elevates arithmetic from a game to the rank of a science’. Many view this as an in nuce statement of the indispensability argument later championed by Quine. Garavaso has questioned this attribution. I argue that even though Frege's applicability argument is not a version of ia, it facilitates acceptance of suitable formulations of ia. The prospects for making the empiricist ia compatible with a rationalist Fregean framework appear (...) thus much less dim than expected. Nonetheless, those arguing for such compatibility eventually face an hardly surmountable dilemma. (shrink)

(First paragraph.) Different views on the relation between phenomenal reality, the world as we consciously experience it, and noumenal reality, the world as it is independent from an experiencing subject, have different implications for a collection of interrelated issues of meaning and reality including aspects of metaphysics, the philosophy of language, and philosophical methodology. Exploring some of these implications, this paper compares and brings together analytic, continental, and Buddhist approaches, focusing on relevant aspects of the philosophy of Donald Davidson, Jacques (...) Derrida, Dharmakīrti, and Dōgen. Prima facie, these philosophers have little in common, and indeed the differences are vast. Even in case of the two Western thinkers there is a fundamental difference between Davidson’s anti-dualist identification of phenomenal, experienced reality with the noumenal, real, external world on the one hand, and the bracketing or elimination of noumenal reality at the base of Derrida’s thought on the other, which lead to radically different ideas with regards to (the possibility and nature of) objectivity and our linguistic access to the real/external/noumenal world. Nevertheless, there are important similarities between Dharmakīrti’s theory of apoha and Davidson’s and Derrida’s theories of triangulation and différance respectively, and these similarities can be exploited to bridge some of the differences and attempt a constructive engagement. (shrink)

Ontology is the study of what there is, what kinds of things make up reality. Ontology seems to be a very difficult, rather speculative discipline. However, it is trivial to conclude that there are properties, propositions and numbers, starting from only necessarily true or analytic premises. This gives rise to a puzzle about how hard ontological questions are, and relates to a puzzle about how important they are. And it produces the ontologyobjectivity dilemma: either (certain) ontological questions can be trivially (...) answered using only uncontroversial premises, or the uncertainties of ontology are really a threat to the truth of basically everything we say or believe. The main aim of this dissertation is to resolve these puzzles and to shed some light on the discipline of ontology. I defend a view inspired by Carnap’s internal-external distinction about what there is, but one according to which both internal and external questions are fully factual and meaningful. In particular, I argue that the trivial arguments are valid, but they do not answer any ontological questions. Furthermore, I propose an account of the function of our talk about properties, propositions and natural numbers. According to this account our talk about them has no ontological presuppositions for its literal and objective truth. This avoids the ontology-objectivity dilemma, and solves the puzzles about ontology. To do this I look at quantification and noun phrases in general, and at their relation to ontology. I argue that quantifiers are semantically underspecified in a certain respect, and play two different roles in communication. I discuss the relation between syntactic form and information structure, the function of certain non-referential, non-quantificational noun phrases, the uses of bare number determiners, and how arithmetic truths are learned and taught. The more metaphysical issues discussed include: inexpressible properties, logicism about arithmetic, nominalism, Carnap’s view about ontology, the problem of universals, the relationship between ontology and objectivity, different projects within ontology, non-existent.... (shrink)

The concept of indiscernibility in a structure is analysed with the aim of emphasizing that in asserting that two objects are indiscernible, it is useful to consider these objects as members of (the domain of) a structure. A case for this usefulness is presented by examining the consequences of this view to the philosophical discussion on identity and indiscernibility in quantum theory.

Ontology is the philosophical discipline that tries to find out what there is: what entities make up reality, what is the stuff the world is made from? Thus, ontology is part of metaphysics, and in fact it seems to be about half of all of metaphysics. It tries to establish what (kinds of) things there are, the other half tries to find out what the (general) properties of these things are and what (general) relations they have to each other. Settling (...) questions in ontology would bring with it major progress in metaphysics. And it would bring with it major progress in a variety of areas in philosophy outside of metaphysics as well. Many philosophical debates outside of metaphysics are quite directly influenced by how things turn out in ontology. Whether or not there are certain entities will give rise to quite different answers in various philosophical debates. I would like to very briefly describe three debates where ontological questions play a central role. These debates are either directly from metaphysics or from other areas of philosophy and they will be of central importance throughout this paper. (shrink)

In his groundbreaking Grundlagen, Frege (1884) pointed out that number words like ‘four’ occur in ordinary language in two quite different ways and that this gives rise to a philosophical puzzle. On the one hand ‘four’ occurs as an adjective, which is to say that it occurs grammatically in sentences in a position that is commonly occupied by adjectives. Frege’s example was (1) Jupiter has four moons, where the occurrence of ‘four’ seems to be just like that of ‘green’ in (...) (2) Jupiter has green moons. On the other hand, ‘four’ occurs as a singular term, which is to say that it occurs in a position that is commonly occupied by paradigmatic cases of singular terms, like proper names: (3) The number of moons of Jupiter is four. Here ‘four’ seems to be just like ‘Wagner’ in (4) The composer of Tannhäuser is Wagner, and both of these statements seem to be identity statements, ones with which we claim that what two singular terms stand for is identical. But that number words can occur both as singular terms and as adjectives is puzzling. Usually adjectives cannot occur in a position occupied by a singular term, and the other way round, without resulting in ungrammaticality and nonsense. To give just one example, it would be ungrammatical to replace ‘four’ with ‘the number of moons of Jupiter’ in (1): (5) *Jupiter has the number of moons of Jupiter moons. This ungrammaticality results even though ‘four’ and ‘the number of moons of Jupiter’ are both apparently singular terms standing for the same object in (3). So, how can it be that number words can occur both as singular terms and as adjectives, while other adjectives cannot occur as singular terms, and other singular terms cannot occur as adjectives? (shrink)

Nominalizations are expressions that are particularly challenging philosophically in that they help form singular terms that seem to refer to abstract or derived objects often considered controversial. The three standard views about the semantics of nominalizations are [1] that they map mere meanings onto objects, [2] that they refer to implicit arguments, and [3] that they introduce new objects, in virtue of their compositional semantics. In the second case, nominalizations do not add anything new but pick up objects that would (...) be present anyway in the semantic structure of a corresponding sentence without a nominalization. In the first and third case, nominalizations in a sense ‘create’ new objects’, enriching the ontology on the basis of the meaning of expressions. I will argue that there is a fourth kind of nominalization which requires a quite different treatment. These are nominalizations that introduce ‘new’ objects, but only partially characterize them. Such nominalizations generally refer to events or tropes. I will explore an account according on which such nominalizations refer to truth makers. (shrink)

For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice (...) are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all. (shrink)

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions cast in set-theoretic terms and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, (...) set theory has proceeded in the opposite direction, from a web of intensions to a theory of extensionpar excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical structures and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression ofmathematicalmoves, whatever and sometimes in spite of what has been claimed on its behalf.What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by current set theory. The whole transfinite landscape can be viewed as the result of Cantor's attempt to articulate and solve the Continuum Problem. (shrink)

Ernst Friedrich Ferdinand Zermelo transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two (...) decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details. (shrink)

Kit Fine (2007) outlines an account of semantic coordination, an account motivated by the role of semantic coordination in cognition. Actually, Fine outlines two accounts of semantic coordination, one in terms of co-reference and another in terms of synonymy. I argue, first, that Fine's two accounts are not equivalent, with one being logically stronger than the other, but second and more importantly, that neither account is correct. I outline an alternative account of semantic coordination – the epistemic conception of semantic (...) coordination – that links semantic and epistemic aspects of cognition more directly. The most surprising result of my alternative account of semantic coordination is that semantic coordination is a relation that can hold between elements of thought and language of different semantic types. I close the paper by briefly outlining some applications of the idea that semantic coordination is a relation that can hold across semantic types. (shrink)

Despite their recent development, geo-ontologies represent a complicated conundrum for the different experts involved in their design. Computer scientists use ontologies for describing the meaning of data and their semantics in order to make information resources built for humans understandable also for artificial agents. Geographers pursue conceptualizations that describe the domain of interest in a way that should be accessible, informative, and complete for their final recipients. In this context, philosophers are not required to sketch the historical background of ontology. (...) Rather, they should offer conceptual solutions for carving nature at the joints and choosing how best to categorize and classify the different entities belonging to the geographical domain in question. Accordingly, the aim of this paper is to combine assumptions and requirements coming from these different areas of research in order to show what different categories might potentially complete the current domain of geo-ontologies. (shrink)

What is the role of the environment, and of the information it provides, in cognition? More specifically, may there be a role for certain artefacts to play in this context? These are questions that motivate "4E" theories of cognition (as being embodied, embedded, extended, enactive). In his take on that family of views, Hajo Greif first defends and refines a concept of information as primarily natural, environmentally embedded in character, which had been eclipsed by information-processing views of cognition. He continues (...) with an inquiry into the cognitive bearing of some artefacts that are sometimes referred to as 'intelligent environments'. Without necessarily having much to do with Artificial Intelligence, such artefacts may ultimately modify our informational environments. -/- With respect to human cognition, the most notable effect of digital computers is not that they might be able, or become able, to think but that they alter the way we perceive, think and act. (shrink)

Contemporary semantic theories can be classified along two dimensions: (i) the way and time-course in which contextual factors influence sentence truth-conditions; and (ii) whether and to what extent comprehension involves sensory, motor and emotional processes. In order to explore this theoretical space, our ERP study investigates the time-course of the interaction between the lexically specified telic component of a noun (the function of the object to which the noun refers to, e.g., a funnel is generally used to pour liquids into (...) containers) and an ad-hoc affordance contextually induced by the situation described in the discourse. We found that, if preceded by a neutral discourse context, a verb incongruent with the noun’s telic component as in “She uses the funnel to hang her coat” elicited an enhanced N400 compared to a congruent verb as in “She uses the funnel to pour water into a container”. However, if the situation introduced in the preceding discourse induced a new function for the object as an ad-hoc affordance (e.g., the funnel is glued to the wall and the agent wants to hang the coat), we observed a crossing-over regarding the direction of the N400 effect: comparing the ad-hoc affordance-inducing context with the neutral context, the N400 for the incongruent verb was significantly reduced, whereas the N400 for the congruent verb was significantly enhanced. We explain these results as a consequence of the incorporation of the contextually triggered ad-hoc affordance into the meaning of the noun. Combining these results with an analysis of semantic similarity values between test sentences and contexts, we argue that one possibility is that the incorporation of an ad-hoc affordance may be explained on the basis of the mental simulation of concurrent motor information. (shrink)

The paper critically scrutinizes the widespread idea that Russell subscribes to a "Universalist Conception of Logic." Various glosses on this somewhat under-explained slogan are considered, and their fit with Russell's texts and logical practice examined. The results of this investigation are, for the most part, unfavorable to the Universalist interpretation.

Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the (...) potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover. (shrink)

This paper provides an externalist account of talk and thought that clearly distinguishes the two. It is argued that linguistic meanings and concepts track different phenomena and have different explanatory roles. The distinction, understood along the lines proposed, brings theoretical gains in a cluster of related areas. It provides an account of meaning change which accommodates the phenomenon of contested meanings and the possibility of substantive disagreement across theoretical divides, and it explains the nature and value of conceptual engineering in (...) a way that addresses recent prominent concerns. (shrink)

ABSTRACT A review of the book by Gottfried Gabriel and Sven Schlotter Frege und die kontinentalen Ursprünge der analytischen Philosophie (Münster, Mentis, 2017).

A feature of Frege's philosophy of arithmetic that has elicited a great deal of attention in the recent secondary literature is his contention that numbers are ‘self‐subsistent’ objects. The considerable interest in this thesis among the contemporary philosophy of mathematics community stands in marked contrast to Kreisel's folk‐lore observation that the central problem in the philosophy of mathematics is not the existence of mathematical objects, but the objectivity of mathematics. Although Frege was undoubtedly concerned with both questions, a goal of (...) the present paper is to argue that his success in securing the objectivity of arithmetic depends on a less contentious commitment to numbers as objects than either he or his critics have supposed. As such, this paper is an articulation and defense of both Frege's analysis of arithmetic and Kreisel's observation. (shrink)

The article attempts to reconsider the relationship between Leibniz’s and Kant’s philosophy of geometry on the one hand and the nineteenth century debate on the foundation of geometry on the other. The author argues that the examples used by Leibniz and Kant to explain the peculiarity of the geometrical way of thinking are actually special cases of what the Jewish-German mathematician Felix Hausdorff called “transformation principle”, the very same principle that thinkers such as Helmholtz or Poincaré applied in a more (...) general form in their celebrated philosophical writings about geometry. The first two parts of the article try to show that Leibniz’s and Kant’s philosophies of geometry, despite their differences, appear to be preoccupied with the common problem of the impossibility to grasp conceptually the intuitive difference between two figures (such as a figure and its scaled, displaced or mirrored copy). In the third part, it is argued that from the perspective of Hausdorff’s philosophical-geometrical reflections, this very same problem seems to find a more radical application in Helmholtz’s or Poincaré’s thought experiments on the impossibility of distinguishing distorted copies of our universe from the original one. I draw the conclusion that in Hausdorff’s philosophical work, which has received scholarly attention only recently, one can find not only an original attempt to frame these classical arguments from a set-theoretical point of view, but also the possibility of considering the history of philosophy of geometry from an uncommon perspective, where especially the significance of Kant’s infamous appeal to “intuition” can be judged by more appropriate standards. (shrink)

We attempt to defend the species-as-individuals hypothesis by examining the logical role played by the binomials (e.g., "Homo sapiens," "Pinus ponderosa") in biological discourse about species. Those who contend that the binomials can be properly understood as functioning in biological theory as singular terms opt for an objectual account of species and view species as individuals. Those who contend that the binomials can in principle be eliminated from biological theory in favor of predicate expressions opt for a predicative account of (...) species and view species as kinds. We contend that biologists' talk about species is talk about species as individuals, and we conclude that the most plausible account of species is an objectual account. (shrink)

Wright claims that “the epistemology of good abstraction principles should be assimilated to that of basic principles of logical inference”. In this paper I follow Wright’s recommendation, but I consider a different epistemology of logic, namely anti-exceptionalism. Anti-exceptionalism’s main contention is that logic is not a priori, and that the choice between rival logics should be based on abductive criteria such as simplicity, adequacy to the data, strength, fruitfulness, and consistency. This paper’s goal is to lay down the foundations for (...) an application of the anti-exceptionalist methodology to abstraction principles. In Sect. 1 I outline my strategy. I show that anti-exceptionalism has a bearing on the so-called Bad Company problem (Sect. 2), and explore some versions of anti-exceptionalism about abstraction (Sect. 3). I then consider new criteria for choosing between rival abstraction principles, and, in particular, between consistent revisions of Frege’s Basic Law V (Sect. 4). I finally consider how the abstractionist can compare between competing criteria (Sect. 5), and argue that anti-exceptionalists should be pluralists about abstraction (Sect. 6). (shrink)

We tend to talk about parts in the same way in which we talk about whole objects. Yet a part is not something to be included in an inventory of the world over and above the whole to which it belongs, and a whole is not something to be included in an inventory over and above its own parts. This paper is an attempt to clarify a way of dealing with this tension which may be labeled the Minimalist View: an (...) element in the field of a part‐whole relation is to be included in an inventory of the world if, and only if, it does not overlap any other element that is itself included in the inventory. As it turns out, a clarification of this view involves both a defense of mereological extensionality and an account of the topological distinction between detached and undetached parts. (shrink)

In this paper two concepts of psychologism in logic are outlined: the one which Frege and Husserl fought against and the new psychologism, or cognitivism, which underlies a cognitive turn in contemporary logic. Four issues such cognitively oriented logic should be interested in are indicated. They concern: new fields opened for logical analysis, new methods and tools needed to address these fields, neural basis of logical reasoning, and an educational problem: how to teach such logic? Several challenging questions, which arise (...) in the context of these issues, are listed. (shrink)

In this paper I discuss Frege's distinction between objects and concepts and suggest a solution of Frege's paradox of the concept horse. The expression ''the concept horse'' is not eliminated and the concept is not identified with its extension, but the concept is identified with the sense of the corresponding predicate. This solution fits better into a fregean ontology and philosophy of language than alternative solutions and allows for a general answer to the question why Frege's system is infected with (...) Russell's paradox. Russell's paradox is caused by the reification of a concept. Certain problems of modern set theory seem to have a similar cause.Eine weithin sichtbare Warnungstafel muss aufgerichtet. (shrink)

The doctrine that meanings are entitieswith a determinate and independent reality is often believed tohave been undermined by Quine's thought experiment of radicaltranslation, which results in an argument for the indeterminacy oftranslation. This paper argues to the contrary. Starting fromQuine's assumption that the meanings of observation sentences arestimulus meanings, i.e., set-theoretical constructions of neuronalstates uniquely determined by inter-subjectively observable facts,the paper shows that this meaning assignment, up to isomorphism,is uniquely extendable to all expressions that occur inobservation sentences. To do so, (...) a theorem recently proven byHodges is used. To derive the conclusion, one only has to assumethat languages are compositional, abide by a generalized contextprinciple and by what I call the category principle. Theseassumptions originating in Frege and Husserl are coherent withQuine's overall position. It is concluded that Quine'snaturalistic approach does not justify scepticism with regard tomeaning, but should rather result in a view that affiliatessemantics with neuroscience. (shrink)

After the publication of Begriffsschrift, a conflict erupted between Frege and Schröder regarding their respective logical systems which emerged around the Leibnizian notions of lingua characterica and calculus ratiocinator. Both of them claimed their own logic to be a better realisation of Leibniz’s ideal language and considered the rival system a mere calculus ratiocinator. Inspired by this polemic, van Heijenoort (1967b) distinguished two conceptions of logic—logic as language and logic as calculus—and presented them as opposing views, but did not explain (...) Frege’s and Schröder’s conceptions of the fulfilment of Leibniz’s scientific ideal. -/- In this paper I explain the reasons for Frege’s and Schröder’s mutual accusations of having created a mere calculus ratiocinator. On the one hand, Schröder’s construction of the algebra of relatives fits with a project for the reduction of any mathematical concept to the notion of relative. From this stance I argue that he deemed the formal system of Begriffsschrift incapable of such a reduction. On the other hand, first I argue that Frege took Boolean logic to be an abstract logical theory inadequate for the rendering of specific content; then I claim that the language of Begriffsschrift did not constitute a complete lingua characterica by itself, more being seen by Frege as a tool that could be applied to scientific disciplines. Accordingly, I argue that Frege’s project of constructing a lingua characterica was not tied to his later logicist programme. (shrink)

In this dissertation, I argue that we should be pluralists about truth and in turn, eliminativists about the property Truth. Traditional deflationists were right to suspect that there is no such property as Truth. Yet there is a plurality of pluralities of properties which enjoy defining features that Truth would have, were it to exist. So although, in this sense, truth is plural, Truth is non-existent. The resulting account of truth is indebted to deflationism as the provenance of the suspicion (...) that Truth doesn't exist. But it would be hasty to simply classify the account as deflationary. Each of the 'truth-like' properties that it recognizes is highly substantive--that is, complex and explanatorily potent. So we should deflate Truth by recognizing that it doesn't exist, but we should also recognize that one of the most vital tasks in truth theory is to discover the essences of the many truth-like properties. My aim here is to do precisely this. (shrink)