The Caesar problem arises for abstractionist views, which seek to secure reference for terms such as ‘the number of Xs’ or #X by stipulating the content of ‘unmixed’ identity contexts like ‘#X = #Y’. Frege objects that this stipulation says nothing about ‘mixed’ contexts such as ‘# X = Julius Caesar’. This article defends a neglected response to the Caesar problem: the content of mixed contexts is just as open to stipulation as that of unmixed contexts.

In ontological debates, realists typically argue for their view via one of two approaches. The _Quinean approach_ employs naturalistic arguments that say our scientific practices give us reason to affirm the existence of a kind of entity. The _Fregean approach_ employs linguistic arguments that say we should affirm the existence of a kind of entity because our discourse contains reference to those entities. These two approaches are often seen as distinct, with _indispensability arguments_ typically associated with the former, but not (...) the latter, approach. This paper argues for a connection between the two approaches on the grounds that the typical arguments of the Fregean approach can be reformulated as indispensability arguments. This connection is significant in at least two ways. First, it implies that indispensability arguments provide a common framework within which to compare the Quinean and Fregean approaches, which allows for a more precise delineation of the two approaches. Second, it implies the possibility of analogical relations that allow proponents and opponents of each approach to draw upon the ideas that have been developed regarding the other. (shrink)

This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...) rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism, via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction. (shrink)

Logic is about reasoning, or so the story goes. This thesis looks at the concept of logic, what it is, and what claims of correctness of logics amount to. The concept of logic is not a settled matter, and has not been throughout the history of it as a notion. Tools from conceptual analysis aid in this historical venture. Once the unsettledness of logic is established we see the repercussions in current debates in the philosophy of logic. Much of the (...) battle over the ‘one true logic’ is conceptually talking past each other. The theory of logics-as-formalizations is presented as a conceptually open theory of logic which is Carnapian in flavour and grounding. Rudolf Carnap’s notions surrounding ‘external’ and ‘pseudo-questions’ about linguistic frameworks apply to formalizations, thus logics, as well. An account of what formalizations are, a more structured sub-set of modelling, is given to ground the claim that logics are formalizations. Finally, a novel account of correctness, the COFE framework, is developed which allows the notions of logical monism, pluralism and nihilism to be more precisely formulated than they currently are in the discourse. (shrink)

Gottlob Frege takes logic to be the science of truth throughout his career. However, the mature Frege makes remarks which seem to go against the idea that logic is the science of truth. This paper shows that we can explain away this tension in the mature Frege’s conception of logic if we accept that truth is an object, that is, the truth-vale True qua the reference of a sentence, for Frege. Even though the main thesis of this paper is a (...) conditional, I provide reasons why we ought to consider the possibility seriously that truth is an object for Frege. I also explain how we can make sense of Frege’s conception of logic as the science of truth while accepting truth is an object. (shrink)

Is logic normative for reasoning? In the wake of work by Gilbert Harman and John MacFarlane, this question has been reduced to: are there any adequate bridge principles which link logical facts to normative constraints on reasoning? Hitherto, defenders of the normativity of logic have exclusively focussed on identifying adequate validity bridge principles: principles linking validity facts—facts of the form 'gamma entails phi'—to normative constraints on reasoning. This paper argues for two claims. First, for the time being at least, Harman’s (...) challenge cannot be surmounted by articulating validity bridge principles. Second, Harman’s challenge can be met by articulating invalidity bridge principles: principles linking invalidity facts of the form 'gamma does not entail phi' to normative constraints on reasoning. In doing so, I provide a novel defence of the normativity of logic. (shrink)

Frege claims that the laws of logic are characterized by their “generality,” but it is hard to see how this could identify a special feature of those laws. I argue that we must understand this talk of generality in normative terms, but that what Frege says provides a normative demarcation of the logical laws only once we connect it with his thinking about truth and science. He means to be identifying the laws of logic as those that appear in every (...) one of the scientific systems whose construction is the ultimate aim of science, and in which all truths have a place. Though an account of logic in terms of scientific systems might seem hopelessly antiquated, I argue that it is not: a basically Fregean account of the nature of logic still looks quite promising. (shrink)

The translation of both ‘bedeuten’ and ‘Bedeutung’ in Frege's works remains sufficiently problematic that some contemporary authors prefer to leave these words untranslated. Here a case is made for returning to Russell's initial choice of ‘to indicate’ and ‘indication’ as better alternatives than the more usual ‘meaning’, ‘reference’, or ‘denotation’. It is argued that this choice has the philosophical payoff that Frege's controversial doctrines concerning the semantic values of sentences and predicative expressions are rendered far more comprehensible by it, and (...) that this translational strategy fulfills the desiderata of offering a translation which is acceptable both before and after Frege introduced the distinction between sense and reference or, as this paper would have it, between the sense of an expression and what it indicates. (shrink)

Some writers object to logical pluralism on the grounds that logic is normative. The rough idea is that the relation of logical consequence has consequences for what we ought to think and h...

Mathematical platonism is the view that abstract mathematical objects exist. Ontological pluralism is the view that there are many modes of existence. This paper examines the prospects for...

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)

Neo-Fregeans such as Bob Hale and Crispin Wright seek a foundation of mathematics based on abstraction principles. These are sentences involving a relation called the abstraction relation. It is usually assumed that abstraction relations must be equivalence relations, so reflexive, symmetric and transitive. In this article I argue that abstraction relations need not be reflexive. I furthermore give an application of non-reflexive abstraction relations to restricted abstraction principles.

This paper argues that Carnap both did not view and should not have viewed Frege's project in the foundations of mathematics as misguided metaphysics. The reason for this is that Frege's project was to give an explication of number in a very Carnapian sense — something that was not lost on Carnap. Furthermore, Frege gives pragmatic justification for the basic features of his system, especially where there are ontological considerations. It will be argued that even on the question of the (...) independent existence of abstract objects, Frege and Carnap held remarkably similar views. I close with a discussion of why, despite all this, Frege would not accept the principle of tolerance. (shrink)

This paper offers a new interpretation for Wittgenstein`s treatment of mathematical identities. As it is widely known, Wittgenstein`s mature philosophy of mathematics includes a general rejection of abstract objects. On the other hand, the traditional interpretation of mathematical identities involves precisely the idea of a single abstract object – usually a number –named by both sides of an equation.

It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church's “lambda calculus” for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903–1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the lambda calculus. Russell also anticipated Schönfinkel's combinatory logic approach of treating multiargument functions as functions having other functions as value. (...) Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and “value-ranges”. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Russell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the lambda calculus. (shrink)

In the past few years, a number of philosophers ; Horgan and Tienson 2002; Pitt 2004) have maintained the following three theses: there is a distinctive sort of phenomenology characteristic of conscious thought, as opposed to other sorts of conscious mental states; different conscious thoughts have different phenomenologies; and thoughts with the same phenomenology have the same intentional content. The last of these three claims is open to at least two different interpretations. It might mean that the phenomenology of a (...) thought expresses its intentional content, where intentional content is understood as propositional, and propositions are understood as mind-and language-independent abstract entities. And it might mean that the phenomenology of a thought is its intentional content—that is, that the phenomenology of a thought, like the phenomenology of a sensation, constitutes its content. The second sort of view is a kind of psychologism. Psychologistic views hold that one or another sort of thing—numbers, sentences, propositions, etc.—that we can think or know about is in fact a kind of mental thing. Since Frege, psychologism has been in bad repute among analytic philosophers. It is widely held that Frege showed that such views are untenable, since, among other things, they subjectivize what is in fact objective, and, hence, relativize such things as consistency and truth to the peculiarities of human psychology. The purpose of this paper is to explore the consequences of the thesis that intentional mental content is phenomenological and to try to reach a conclusion about whether it yields a tenable view of mind, thought and meaning. I believe the thesis is not so obviously wrong as it will strike many philosophers of mind and language. In fact, it can be defended against the standard objections to psychologism, and it can provide the basis for a novel and interesting account of mentality. (shrink)

Given the plethora of competing logical theories of validity available, it’s understandable that there has been a marked increase in interest in logical epistemology within the literature. If we are to choose between these logical theories, we require a good understanding of the suitable criteria we ought to judge according to. However, so far there’s been a lack of appreciation of how logical practice could support an epistemology of logic. This paper aims to correct that error, by arguing for a (...) practice-based approach to logical epistemology. By looking at the types of evidence logicians actually appeal to in attempting to support their theories, we can provide a more detailed and realistic picture of logical epistemology. To demonstrate the fruitfulness of a practice-based approach, we look to a particular case of logical argumentation—the dialetheist’s arguments based upon the self-referential paradoxes—and show that the evidence appealed to support a particular theory of logical epistemology, logical abductivism. (shrink)

Despite its importance for early analytic philosophy, Gottlob Frege's account of existence statements, according to which they classify concepts, has been thought to succumb to a number of well-worn criticisms. This article does two things. First, it argues that, by remaining faithful to the letter of Frege's claim that concepts are functions, the Fregean account can be saved from many of the standard criticisms. Second, it examines the problem that Frege's account fails to generalize to cases which involve definite descriptions (...) and proper names. To deal with this the proffered analysis deviates from the letter of Frege's views, while remaining within its spirit. It proposes, in opposition to Frege, that expressions which grammatically look like singular terms should not always be read as referring to objects, but are sometimes best analysed as indicating functions. (shrink)

According to a certain conception of language, any sentence can, when used on an occasion, have any of indefinitely many truth-conditions. Such a conception of language gives us reason to think that the question of whether the notion of truth has a distinctive content cannot be settled by looking solely at the predication of truth. By focusing on the predicate ‘true’ when trying to determine the significance of the notion of truth, we may have been looking in the wrong place.

Embodied approaches to cognition see abstract thought and language as grounded in interactions between mind, body, and world. A particularly important challenge for embodied approaches to cognition is mathematics, perhaps the most abstract domain of human knowledge. Conceptual metaphor theory, a branch of cognitive linguistics, describes how abstract mathematical concepts are grounded in concrete physical representations. In this paper, we consider the implications of this research for the metaphysics and epistemology of mathematics. In the case of metaphysics, we argue that (...) embodied mathematics is neutral in the sense of being compatible with all existing accounts of what mathematical entities really are. However, embodied mathematics may be able to revive an older position known as psychologism and overcome the difficulties it faces. In the case of epistemology, we argue that the evidence collected in the embodied mathematics literature is inconclusive: It does not show that abstract mathematical thinking is constituted by metaphor; it may simply show that abstract thinking is facilitated by metaphor. Our arguments suggest that closer interaction between the philosophy and cognitive science of mathematics could yield a more precise, empirically informed account of what mathematics is and how we come to have knowledge of it. (shrink)

In this paper, it is examined how, if at all, the logical laws can be normative for human reasoning, wherein the notion of normativity is analyzed primarily with respect to Wittgenstein’s philosophy. During the ancient and the medieval periods, logic was being considered in terms of discourse and dialogical practice, but since Descartes and especially Kant, it has been treated as a system of laws with which the process of individual human reasoning has been compared. Therefore, normativity can be investigated (...) in the private sphere and in the public sphere. Wittgenstein discussed both aspects of normativity: in his early philosophy, the focus is on the laws of logic that are primarily normative for the state of affairs in the world, while in his later works the emphasis is on a social aspect of normativity, which is derived from the adopted rules that have been applied in a certain community. Taken that way, logic is certainly normative in the public sphere, but the more difficult issue is whether logic is normative for thinking, regarding the difference between the logical laws and those of thought. (shrink)

In this essay, I critically discuss Dale Jacquette's new English translation of Frege's work Die Grundlagen der Arithmetik as well as his Introduction and Critical Commentary (Frege, G. 2007. The Foundations of Arithmetic. A Logical-Mathematical Investigation into the Concept of Number . Translated with an Introduction and Critical Commentary by Dale Jacquette. New York: Longman. xxxii + 112 pp.). I begin with a short assessment of Frege's book. In sections 2 and 3, I examine several claims that Jacquette makes in (...) his Introduction and Critical Commentary and put matters in the right perspective. In sections 4-7, I analyse errors and shortcomings in Jacquette's (and Austin's) translation(s) and show how they can be avoided. In this context, I consider several issues of interest for Frege's logic and philosophy of arithmetic. I conclude with general remarks. (shrink)

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

In recent years, neologicists have demonstrated that Hume's principle, based on the one-to-one correspondence relation, suffices to construct the natural numbers. This formal work is shown to be relevant for empirical research on mathematical cognition. I give a hypothetical account of how nonnumerate societies may acquire arithmetical knowledge on the basis of the one-to-one correspondence relation only, whereby the acquisition of number concepts need not rely on enumeration (the stable-order principle). The existing empirical data on the role of the one-to-one (...) correspondence relation for numerical abilities is assessed and additional empirical tests are proposed. In the final part, it is argued that the fact that the successor relation and the one-to-one correspondence relation can play independent roles in number concept acquisition may be a complication for testing the Whorfian hypothesis. (shrink)

Why does Frege claim that logical axioms are ‘self‐evident,’ to be recognized as true ‘independently of other truths,’ and then offer arguments for those axioms? I argue that he thinks the arguments provide us with the justification that we need for accepting the axioms and that this is compatible with his remarks about self‐evidence. This compatibility depends on philosophical considerations connected with the ‘critical method’: an interesting approach to the justification of axioms endorsed by leading philosophers at the time.

In the explanations of logical laws and inference rules of the mature version of Begriffsschrift in Grundgesetze, Frege uses the predicate “… is the True.” Scholars like Greimann maintain that this predicate is a metalinguistic truth-predicate for Frege. This paper examines an argument for this claim that is based on the “nominal reading” of Frege’s conception of sentences—the claim that for Frege a sentence “p” is equivalent to a nonsentential phrase like “the truth-value of the thought that p.” In particular, (...) this paper attempts to establish two points concerning this argument based on the nominal reading. First, the argument implies a claim about the nature of assertion which Frege repeatedly denies in his mature works. Secondly, the nominal reading on which the argument depends is false. A sentence “p” is not equivalent to a nonsentential phrase like “the truth-value of the thought that p” for Frege. Our discussion will lead to an important lesson about Frege’s conception of sentences and of assertion. (shrink)

In this essay, I distinguish two ways of depsychologizing psychology: ‘anti-psychologism’ and ‘non-psychologism’. Both positions are responses to the Fregean sharp distinction between the logical and the psychological. But where anti-psychologism, which I find in John McDowell, attempts to overcome the sharp distinction by arguing that psychological states and their expressions are apt to be articulated into judgments, Stanley Cavell's non-psychologism, a powerful and neglected alternative, wants to overcome the sharp distinction by abandoning judgment as the paradigm expression of thought (...) and communication. (shrink)

This paper identifies a tension in Frege’s philosophy and offers a diagnosis of its origins. Frege’s Context Principle can be used to dissolve the problem of propositional unity. However, Frege’s official response to the problem does not invoke the Context Principle, but the distinction between ‘saturated’ and ‘unsaturated’ propositional constituents. I argue that such a response involves assumptions that clash with the Context Principle. I suggest, however, that this tension is not generated by deep-seated philosophical commitments, but by Frege’s occasional (...) attempt to take a dubious shortcut in the justification of his conception of propositional structure. (shrink)

This essay examines Frege's reaction to Russell's Paradox and his views about the grounding of existence claims in mathematics. It is argued that Frege's strict requirements on existential proofs would rule out the attempt to ground arithmetic in. It is hoped that this discussion will help to clarify the ways in which Frege's position is both coherent and significantly different from the neo-logicist position on the issues of: what's required for proofs of existence; the connection between models, consistency, and existence; (...) and the prospects for a logical grounding of arithmetic in the wake of the paradox. (shrink)

The paper analyses Frege's approach to the identity conditions for the entity labelled by him as Sinn. It starts with a brief characterization of the main principles of Frege's semantics and lists his remarks on the identity conditions for Sinn. They are subject to a detailed scrutiny, and it is shown that, with the exception of the criterion of intersubstitutability in oratio obliqua, all other criteria have to be discarded. Finally, by comparing Frege's views on Sinn with Carnap's method of (...) extension and intension and the method of intensional isomorphism, it is proved that these methods do not provide a criterion for the identity of Frege's Sinn, even for extensional contexts, that the concept of intension does not coincide, as stated by Carnap, in these contexts, with Frege's concept of Sinn, and that Carnap's claim that in oratio obliqua Frege's semantics leads to an infinite hierarchy of Sinn entities can be questioned at least hypothetically in the light of certain new historical facts. (shrink)

According to Øystein Linnebo's account of abstractionism, abstraction principles, received as Fregean criteria of identity, can be used to reduce facts about singular reference to objects such as directions and numbers to facts that do not involve such objects. In this article, first I show how the resources of Linnebo's metasemantics successfully handle Dummett's challenge against the referentiality of the singular terms formed by abstraction principles. Then, I argue that Linnebo's metasemantic commitments do not provide us with tools for dispelling (...) the threat of a version of referential indeterminacy, according to which nothing in our use of a singular term, even when it is guided by an associated criterion of identity, could determine which particular object it refers to. I end by examining the bearing of the indeterminacy challenge to Linnebo's treatment of Frege's Caesar Problem: in the absence of an argument against the indeterminacy of reference, it is unclear how numerical expressions could qualify as genuine singular terms. (shrink)

Simon Blackburn has shown that there is an analogy between the problem of moral motivation in ethics (how can moral reasons move us?) and the problem of what we might call the power of logical reasons (how can logical reasons move us, what is the force of the 'logical must?'). In this paper, I explore further the parallel between the internalism problem in ethics and the problem of the power of logical reasons, and defend a version of psychologism about reasons, (...) although not one of the Humean form. I discuss two forms of cognitivism: (i) a pure cognitivism and 'hard' realism modelled after Dancy's parallel conception in ethics: when we grasp logical reasons we grasp facts, which are directly known to us; (ii) a Kantian form of cognitivism, based on the idea that compulsion by reason goes with the capacity of reflection. I argue that (i) is implausible, and that (ii) fails to meet the internalist requirement. One would then seem to be left with what Dancy calls, about practical reasoning, psychologism about reasons. I sketch what might be such a psychologism. But it fails to account for the objectivity of logical reasons. I argue that the relevant state here is a form of dispositional knowledge of logical reasons. (shrink)

According to the Explanatory Problem with Frege's Platonism about Thoughts, the sharp separation between the psychological and the logical on which Frege famously insists is too sharp, leaving Frege no resources to show how it could be legitimate to invoke logical laws in an explanation of our activities of thinking. I argue that there is room in Frege's philosophy for such justificatory explanations. To see how, we need first to understand correctly the lesson of Frege's attack on psychologism as fundamentally (...) marking a contrast between justification and explanation, and, second, we must take Frege to be committed to the idea that the laws of truth are normatively constitutive for the process of thinking. (shrink)

At several points in his later writings Wittgenstein discusses imaginary forms of life and ways of thinking that appear queer or alien from our point of view; concepts so different from ours that those who think from within them seem to be alternatives to us. In this paper I argue that reflection on the notions of difference and possibility in play here shows that imaginary cases of alien conceptual schemes or forms of life such as those considered by Wittgenstein are (...) not all cases of concepts that are entirely unintelligible for us; rather they may represent possible, albeit distant, ways of thinking for us. Such cases serve to aid imaginative reflection on our own case(s). By making us appreciate the possibility of the strange, they help us better to appreciate the arbitrariness of the familiar. I end the paper by considering what the implications of this reading of Wittgenstein’s position might be for Donald Davidson’s rejection of conceptual relativism. (shrink)

This paper engages the question ‘Does the consistency of a set of axioms entail the existence of a model in which they are satisfied?’ within the frame of the Frege-Hilbert controversy. The question is related historically to the formulation, proof and reception of Gödel’s Completeness Theorem. Tools from mathematical logic are then used to argue that there are precise senses in which Frege was correct to maintain that demonstrating consistency is as difficult as it can be, but also in which (...) Hilbert was correct to maintain that demonstrating existence given consistency is as easy as it can be. (shrink)

I attempt to explain Frege's handling of the Julius Caesar issue in terms of his more general philosophical commitments. These only became fully explicit in his middle-period writings, but his earlier moves are best explained, I suggest, if we suppose them to be implicit in his earlier thinking. These commitments conditionally justify Frege in rejecting Hume's Principle as either a definition or axiom but in accepting Axiom V. However, the general epistemological picture they constitute has serious problems in accounting for (...) how knowledge is possible at all of such propositions as that Julius Caesar is not a number. (shrink)

It is easy to think of Frege as having offered two unintentionally discordant criteria for the identity of senses—one tied to the truth conditions of sentences, and one meant to capture relations of cognitive discriminability. This reading, however, is doubly mistaken; the discord between these two ways of thinking of senses has a Fregean resolution, but neither the resolution nor either of the original two pictures affords a genuine criterion for the identity of senses.

Let me start with the enigmatic dictum of Blaise Pascal: ‘l'homme passe l'homme’ – ‘man goes beyond himself’; ‘humanity transcends itself’. What does this mean? On one plausible interpretation, Pascal is adverting to that strange restlessness of the human spirit which so many philosophers have pondered on, from Augustine before him, to Kierkegaard and many subsequent writers since. To be human is to recognize that we are, in a certain sense, incomplete beings. We are on a journey to a horizon (...) that always seems to recede from view. Unlike all the other animals, who need nothing further for their thriving and flourishing once the appropriate environmental conditions are provided, human beings, even when all their needs are catered for – physical, biological, social, cultural – and even when they enjoy a maximally secure and enriching environment, still have a certain resistance to resting content with existence defined within a given set of parameters. They still have the restless drive to reach forward to something more. (shrink)

It is argued that the finitist interpretation of wittgenstein fails to take seriously his claim that philosophy is a descriptive activity. Wittgenstein's concentration on relatively simple mathematical examples is not to be explained in terms of finitism, But rather in terms of the fact that with them the central philosophical task of a clear 'ubersicht' of its subject matter is more tractable than with more complex mathematics. Other aspects of wittgenstein's philosophy of mathematics are touched on: his view that mathematical (...) propositions are 'grammatical propositions' (and so, Strictly speaking, Not genuine propositions at all) and his view that in mathematics 'everything is algorithm, Nothing meaning'. His views on consistency and his anti-Foundationalism are linked with this central thesis. (shrink)

I shall compare two views of legal concepts: as nodes in inferential nets and as categories in an ontology (a conceptual architecture). Firstly, I shall introduce the inferential approach, consider its implications, and distinguish the mere possession of an inferentially defined concept from the belief in the concept’s applicability, which also involves the acceptance of the concept’s constitutive inferences. For making this distinction, the inferential and eliminative analysis of legal concepts proposed by Alf Ross will be connected to the views (...) on theoretical concepts in science advanced by Frank Ramsey and Rudolf Carnap. Consequently, the mere comprehension of a legal concept will be distinguished from the application of the concept to a particular legal system, since application presupposes a doctrinal commitment, namely, the belief that the inferences constituting the concept hold in that system. Then, I shall consider how concepts can be characterised by defining the corresponding terms and placing them within an ontology. Finally, I shall argue that there is a tension between the inferential and the ontological approach, but that both need to be taken into account, to capture the meaning and the cognitive function of legal concepts. (shrink)

Extensionalism, as I understand it here, is the view that physical reality consists exclusively of extensional entities. On this view, intensional entitities must either be eliminated in favor of an ontology of extensional entities, or be reduced to such an ontology, or otherwise be admitted as non-physical. In this paper I argue that extensionalism is a misguided philosophical doctrine. First, I argue that intensional phenomena are not confined to the realm of language and thought. Rather, the ontology of such phenomena (...) is intimately entwined with the ontology of properties. After providing some evidence to the popularity of extensionalism in contemporary analytic philosophy, I investigate the motivating reasons behind it. Considering several explanations, I argue that the main motivating reason is rooted in the identification of matter with extension, an identification which is one of the hallmarks of the mechanistic conception of nature inherited from the founding fathers of our modern scientific outlook. I then argue that such a conception is not only at odds with a robust ontology of properties but is also at odds with our best contemporary physics. Rather than vindicating extensionalism contemporary science undermines the position, and the lesson to be drawn from this surprising fact is that extensionalism needs no longer be espoused as a regulative ideal of naturalistic philosophy. I conclude by showing that the ontological approach to intensional phenomena advocated throughout the paper also gains support from an examination of the historical context within which ‘intension’ was first introduced as a semantic notion. (shrink)

ABSTRACT Is it possible to effect singular reference to mathematical objects in the abstractionist framework? I will argue that even if mathematical expressions pass the relevant syntactic and inferential tests to qualify as singular terms, that does not mean that their semantic function is to refer to a particular object. I will defend two arguments leading to this claim: the permutation argument for the referential indeterminacy of mathematical terms, and the argument from the semantic idleness of the terms introduced by (...) abstraction principles. (shrink)

The investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity of second and higher-order logic. However, the epistemological significance of such investigations has not received much attention in the contemporary foundational debate.We discuss Girard’s normalization proof for second order type theory or System F and compare it with two faulty consistency arguments: the one given by Frege for the logical system of the Grundgesetze and (...) the one given by Martin-Löf for the intuitionistic type theory with a type of all types.The comparison suggests that the question of the circularity of second order logic cannot be reduced to Russell’s and Poincaré’s 1906 “vicious circle” diagnosis. Rather, it reveals a bunch of mathematical and logical ideas hidden behind the hazardous idea of impredicative quantification, constituting a vast domain for foundational research. (shrink)