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)

Two different conceptions of language dominate philosophical reflection on the nature of human language and of human linguistic powers. The first is the conception of language as a calculus of meaning, and of understanding as computational interpretation. This conception is rooted in the exigencies of function-theoretic logic. The notions pivotal to this conception are truth, truth-condition, sense and force, naming and describing (representation), and theory of meaning for natural languages. The alternative conception is an anthropological one, which conceives of language (...) above all as a form of human communicative behaviour, constituted by human practices and manifest in human action. The notions pivotal to this conception are practice, language-game, use and rule of use as given by explanations of meaning, understanding and criteria of understanding. The fundamental principles that inform each of these conceptions are explained. The radical flaws of calculus conceptions of language are laid bare. (shrink)

Conceptual primitivism is the view that truth is among our most basic and fundamental concepts. It cannot be defined, analyzed, or reduced into concepts that are more fundamental. Primitivism is opposed to both traditional attempts at defining truth (in terms of correspondence, coherence, or utility) and deflationary theories that argue that the notion of truth is exhausted by means of the truth schema. Though primitivism might be thought of as a view of last resort, I believe that the view is (...) independently attractive, and can be argued for directly. In this paper I offer what I take to be the strongest argument in favor of conceptual primitivism, which relies upon the Fregean doctrine of the omnipresence of truth. (shrink)

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.

This paper explores the relationship between Hume's Prinicple and Basic Law V, investigating the question whether we really do need to suppose that, already in Die Grundlagen, Frege intended that HP should be justified by its derivation from Law V.

Influenced by G. E. Moore, Russell broke with Idealism towards the end of 1898; but in later years he characterized his meeting Peano in August 1900 as ?the most important event? in ?the most important year in my intellectual life?. While Russell discovered his paradox during his post-Peano period, the question arises whether he was already committed, during his pre-Peano Moorean period, to assumptions from which his paradox may be derived. Peter Hylton has argued that the pre-Peano Russell was thus (...) vulnerable to (at least one version of) Russell's paradox and hence that the paradox exposes a pre-existing difficulty in Russell's Moorean philosophy. Contrary to Hylton, I argue that the Moorean Russell adhered to views which insulated him against the paradox. Further, I argue that Russell became vulnerable to his paradox as a result of changes in his Moorean position occasioned, first, by his acceptance of Cantor's theory of the transfinite, and, second, by his correspondence with Frege. I conclude with some general comments regarding Russell's acceptance of naïve set theory. (shrink)

This paper examines some of the most important metaphors and analogies that epistemologists have used to discuss the structure and validity of knowledge. After reviewing foundational, coherentist, and other metaphors for knowledge, we discuss the metaphilosophical significance of the prevalence of such metaphors. We argue that they support a view of philosophy as akin to science rather than poetry or rhetoric. Keywords: epistemology, metaphor, analogy, metaphilosophy, foundations, coherence.

One of the most striking differences between Frege's Begriffsschrift (logical system) and standard contemporary systems of logic is the inclusion in the former of the judgement stroke: a symbol which marks those propositions which are being asserted , that is, which are being used to express judgements . There has been considerable controversy regarding both the exact purpose of the judgement stroke, and whether a system of logic should include such a symbol. This paper explains the intended role of the (...) judgement stroke in a way that renders it readily comprehensible why Frege insisted that this symbol was an essential part of his logical system. The key point here is that Frege viewed logic as the study of inference relations amongst acts of judgement , rather than – as in the typical contemporary view – of consequence relations amongst certain objects (propositions or well-formed formulae). The paper also explains why Frege's use of the judgement stroke is not in conflict with his avowed anti-psychologism, and why Wittgenstein's criticism of the judgement stroke as 'logically quite meaningless' is unfounded. The key point here is that while the judgement stroke has no content , its use in logic and mathematics is subject to a very stringent norm of assertion. (shrink)

Barnes, A. and P. Thagard Empathy and analogy. Dialogue: Canadian Philosophical Review, 36: 705-720. HTML Croft, D., & Thagard, P.. Dynamic imagery: A computational model of motion and visual analogy. In L. Magnani and N. Nersessian, Model-based reasoning: Science, technology, values. New York: Kluwer/Plenum, 259-274. PDF only. HTML description of program and code for DIVA.

George Boolos has argued that the iterative conception of set justifies most, but not all, the ZFC axioms, and that a second conception of set, the Frege-von Neumann conception (FN), justifies the remaining axioms. This article challenges Boolos's claim that FN does better than the iterative conception at justifying the axioms in question.

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)

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The categorical duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The hyperintensional profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal and hyperintensional profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely (...) logical. Second, the foregoing provides a hyperintensional account of the interpretation of mathematical and metamathematical vocabulary. (shrink)

Philosophers debate about which logical system, if any, is the One True Logic. This involves a disagreement concerning the sufficient conditions that may single out the correct logic among various candidates. This paper discusses whether there are necessary conditions for the correct logic; that is, I discuss whether there are features such that if a logic is correct, then it has those features, although having them might not be sufficient to single out the correct logic. Traditional rationalist arguments suggest that (...) the necessary conditions of thought are necessary and sufficient conditions singling out the correct logical and mathematical theories. In the contemporary debate, Chalmers advocates a view along this line. Jago, analogously, suggests that the necessary conditions for thought—or, as he calls them, our basic epistemic expectations—single out a family of logical and mathematical theories. Warren and Williamson, on the other hand, argue that there are no necessary conditions of thought. I argue that there are necessary conditions for thought, and these are necessary but not sufficient conditions to be the correct logic; indeed, these are features that all logics—correct or incorrect—share. No view we can understand is ruled out by the necessary conditions for thought, but we cannot understand quite any view. Human linguistic and conceptual abilities are genetically constrained, and these constraints are our best guide to the boundaries of logic. Arguing for this, I tackle two dogmas of modern rationalism: namely, the view that the biological constraints of human cognition have no bearing on the boundaries of the epistemic space, and the view that the boundaries of thought coincide with the boundaries of language. (shrink)

¿Qué es la mente humana? ¿Es un aspecto de la naturaleza? ¿O acaso la trasciende? ¿Se puede explicar enteramente al hombre en un lenguaje científico-natural? Desde que Aristóteles definió al hombre como un "animal racional", el pensamiento occidental ha erigido una distinción ontológica entre hombre y naturaleza. Sin embargo, el darwinismo nos ha legado una verdad ineludible: que somos organismos animales, cuyas capacidades conceptuales no escapan a los procesos propios de la naturaleza. ¿Se puede, entonces, pensar al hombre como un (...) ser mental, y al mismo tiempo como un organismo natural? Para ello, sin duda, será necesario reformular los conceptos tradicionales de "mente" y de "naturaleza" en busca de una reconciliación conceptual. El siguiente volumen recorre campos como la fenomenología trascendental, la lógica, la biología, la semiótica, la lingüística y la psicología con el fin de superar este dualismo tan antiguo como la filosofía. (shrink)

This chapter focuses on picture theory, which is sometimes spoken of as a theory of the proposition. By a proposition, Wittgenstein like Frege means something that determines its sense by means of a correlation between the mode of combination of its constituent symbols and the structure of its sense. It has been an orthodoxy amongst Tractatus interpreters, and continues to be such in the wider philosophical community, that Wittgenstein follows the Russell in offering a correspondence theory of truth. The expression (...) of a Tractarian fact provides that fact's mode of combination, and from here there can be no further question as to the ontological types of its constituents. Tractarian facts, it is however added, are not truth conditions but items whose existence explains the obtaining of truth conditions: a Tractarian atomic fact is not a truth condition but a truthmaker. (shrink)

El excepcionalismo de la lógica asume que la lógica tiene un estatus epistemológico privilegiado, es normativa y no es revisable. El antiexcepcionalismo, en cambio, rechaza ese carácter excepcional. Recientemente, se ha sugerido que el problema de la adopción de reglas lógicas es transversal a la discusión que se da entre excepcionalistas y antiexcepcionalistas. En este trabajo defiendo la tesis contraria. En particular, sostengo que, si el problema de la adopción de reglas es un problema central para la lógica como disciplina, (...) el antiexcepcionalismo debe ser incorrecto. En caso contrario, debe ser incorrecta la postura excepcionalista. Evalúo, además, consecuencias directas del análisis presente sobre la discusión en torno al monismo y el pluralismo de la lógica. (shrink)

Wittgenstein seeks to throw light on our concept of understanding by looking at how misunderstandings arise and what kinds of failure they involve. He discerns a peculiar sort of misunderstanding in the writings of the social anthropologist James Frazer. In Frazer’s hands, the anthropological project of enabling us to understand human behavior seems to yield the result that there are certain forms of human behavior that simply cannot be understood. The source of Frazer’s misunderstanding, according to Wittgenstein, is that he (...) places narrow requirements on what could count as meaningful, prior to and independently of his encounter with the subjects of his interpretation. Frazer, like some of the philosophers Wittgenstein addresses in his other works, succumbs to nonsense in his very attempt to draw the limits of sense. My aim in this chapter is to clarify the connections between Wittgenstein’s criticism of Frazer and his criticism of his fellow philosophers, in particular of Frege. The materials I draw on stem from various periods in Wittgenstein’s career, and they reveal, in my mind, an important continuity in Wittgenstein’s thought: addressing misunderstanding, in Wittgenstein’s view, is fundamentally an ethical problem, not a theoretical one. (shrink)

The historical consensus is that logical evidence is special. Whereas empirical evidence is used to support theories within both the natural and social sciences, logic answers solely to a priori evidence. Further, unlike other areas of research that rely upon a priori evidence, such as mathematics, logical evidence is basic. While we can assume the validity of certain inferences in order to establish truths within mathematics and test scientifi c theories, logicians cannot use results from mathematics or the empirical sciences (...) without seemingly begging the question. Appeals to rational intuition and analyticity in order to account for logical knowledge are symptomatic of these commitments to the apriority and basicness of logical evidence. This chapter argues that these historically prevalent accounts of logical evidence are mistaken, and that if we take logical practice seriously we fi nd that logical evidence is rather unexceptional, sharing many similarities to the types of evidence appealed to within other research areas. (shrink)

In his early essay, “Must We Mean What We Say”, Cavell argues that the claims of ordinary language philosophers regarding “what we say when” are not empirical generalizations about a given group of speakers but are rather to be understood as measuring the limits of what counts as a coherent act of thinking and speaking. Cavell’s charge against the skeptic about the external world is that he seeks to think and speak beyond these limits. In this paper I compare Cavell’s (...) response to the skeptic to Davidson’s. Both base their responses on a broadly Kantian approach that appeals to the conditions under which thinking or speaking about objects is possible. On this approach the skeptic isn’t giving a false answer to an intelligible question, but rather, the question to which the skeptic is giving an answer is shown to be in some way unintelligible. But while Davidson’s critique of the skeptic is based on the conditions of ascribing meaning to one’s words, and contents to one’s beliefs, Cavell’s critique is based on the failure of the skeptic to mean the words he uses in the way that he needs. This difference expresses an underlying disagreement about the meaning of “meaning”: for Davidson the world comes into view through the meaning of our words and concepts, through the contents of our beliefs; for Cavell, the world comes into view through the agreement in “criteria” that are a condition of applying words and concepts to the world. This difference illuminates what Cavell calls “the truth of skepticism”: the idea that “my relation to the world and to others in general is not one of knowing”. (shrink)

This paper studies the notions of conceptual grounding and conceptual explanation (which includes the notion of mathematical explanation), with an aim of clarifying the links between them. On the one hand, it analyses complex examples of these two notions that bring to the fore features that are easily overlooked otherwise. On the other hand, it provides a formal framework for modeling both conceptual grounding and conceptual explanation, based on the concept of proof. Inspiration and analogies are drawn with the recent (...) research in metaphysics on the pair metaphysical grounding–metaphysical explanation, and especially with the literature in philosophy of science on the pair causality-causal explanation. (shrink)

I explore the possibility of a structuralist interpretation of homotopy type theory (HoTT) as a foundation for mathematics. There are two main aspects to HoTT's structuralist credentials. First, it builds on categorical set theory (CST), of which the best-known variant is Lawvere's ETCS. I argue that CST has merit as a structuralist foundation, in that it ascribes only structural properties to typical mathematical objects. However, I also argue that this success depends on the adoption of a strict typing system which (...) undermines the metaphysical seriousness of this structuralism. Homotopy type theory adds to CST a distinctive theory of identity between sets, which arguably allows its objects to be seen as ante rem structures. I examine the prospects for such a view, and address many other interpretive problems as they arise. (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)

This chapter intervenes in recent debates in Kant scholarship about the possibility of a general logical alien. Such an alien is a thinker whose laws of thinking violate ours. She is third-personal as she is radically unlike us. Proponents of the constitutive reading of Kant’s conception of general logic accordingly suggest that Kant rules out the possibility of such an alien as unthinkable. I add to this an often-overlooked element in Kant’s thinking: there is reason to think that he grants—and (...) in fact presupposes—the possibility of a transcendental logical alien. Such an alien is a knower whose laws of experience purport to violate ours. She is first-personal as she is radically like us. In other words, she is us, insofar as we are alienated from ourselves and our experience. I go on to draw an analogy between her, a dogmatist, and another transcendental alien, an evil agent. Just as a dogmatist is alienated from her (our) experiential laws, an evil agent is alienated from her (our) moral law. These forms of theoretical and practical self-conceit require self-knowledge in the form of a critique of speculative or practical reason. In bringing this point out, I aim to shift from the question of whether logical laws constitute our thinking to the question of whether grasping our experiential and moral laws as our laws constitutes our reason. (shrink)

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)

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)

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)

This paper aims to answer the question of whether or not Frege's solution limited to value-ranges and truth-values proposed to resolve the "problem of indeterminacy of reference" in section 10 of Grundgesetze is a violation of his principle of complete determination, which states that a predicate must be defined to apply for all objects in general. Closely related to this doubt is the common allegation that Frege was unable to solve a persistent version of the Caesar problem for value-ranges. It (...) is argued that, in Frege’s standards of reducing arithmetic to logic, his solution to the indeterminacy does not give rise to any sort of Caesar problem in the book. (shrink)

This paper concerns the three great modal dichotomies: (i) the necessary/contingent dichotomy; (ii) the a priori/empirical dichotomy; and (iii) the analytic/synthetic dichotomy. These can be combined to produce a tri-dichotomy of eight modal categories. The question as to which of the eight categories house statements and which do not is a pivotal battleground in the history of analytic philosophy, with key protagonists including Descartes, Hume, Kant, Kripke, Putnam and Kaplan. All parties to the debate have accepted that some categories are (...) void. This paper defends the contrary view that all eight categories house statements—a position I dub ‘octopropositionalism’. Examples of statements belonging to all eight categories are given. (shrink)

The aim of this paper is to provide a dynamic interpretation of Kant’s logical hylomorphism. Firstly, various types of the logical hylomorphism will be illustrated. Secondly, I propose to reevaluate Kant’s constitutivity thesis about logic. Finally, I focus on the design of logical norms as specific kinds of artefacts.

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for two-dimensional (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

In his late fragment, ‘Sources of Knowledge of Mathematics and Natural Sciences’ Frege laments the tendency to confuse functions with objects and says, ‘It is here that the tendency of language by its use of the definite article to stamp as an object what is a function and hence a non-object, proves itself to be the source of inaccurate and misleading expressions and also of errors of thought. Probably most of the impurities that contaminate the logical source of knowledge have (...) their origins in this.’ This paper applies Frege’s logical insights to a series of influential modal arguments for the existence of souls or minds, in order to demonstrate that they hinge on the error of treating functions as objects. Related arguments have also resulted in the claim that identity statements are necessary, and that apparent identities ‘really’ involve relations of constitution. Here it is argued that once we recognise that all these arguments involve the assimilation of objects and concepts (which Frege identifies with functions) they can be seen to be defective. (shrink)

Neo-Fregean approaches to set theory, following Frege, have it that sets are the extensions of concepts, where concepts are the values of second-order variables. The idea is that, given a second-order entity $X$, there may be an object $\varepsilon X$, which is the extension of X. Other writers have also claimed a similar relationship between second-order logic and set theory, where sets arise from pluralities. This paper considers two interpretations of second-order logic—as being either extensional or intensional—and whether either is (...) more appropriate for this approach to the foundations of set theory. Although there seems to be a case for the extensional interpretation resulting from modal considerations, I show how there is no obstacle to starting with an intensional second-order logic. I do so by showing how the $\varepsilon$ operator can have the effect of “extensionalizing” intensional second-order entities. (shrink)

The paper offers a new account of Spinoza's conception of “substance”, the fundamental building block of reality. It shows that it can be demonstrated apriori within Spinoza's metaphysical framework that (i) contrary to Idealist readings, for Spinoza there can be no substance that is not determined or modified by some other entity produced by substance; and that (ii) there can be no substance (and hence no being) that is not a thinking substance.

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

ArgumentThis paper builds on a neglected philosophical idea,Evidenz. Max Weber used it in his discussion ofVerstehen, as the goal of understanding either action or such things as logic. It was formulated differently by Franz Brentano, but with a novel twist: thatanyonewho understood something would see the thing to be understood as self-evident, not something dependent on inference, argument, or reasoning. The only way one could take something as evident in this sense is by being able to treat other people as (...) having the same responses – by empathy with them, in the weak sense of following their thought. Brentano's philosophical claim is that without some stopping point at what is self-evident, justifications fall into infinite regress. This is radically opposed to much of conventional philosophy. The usual solutions to the regress problem rely on problematic claims about the supposed hidden transcendental structure behind reasoning. In contrast, empathy is a genuine natural phenomenon and a better explanation for the actual phenomenon of making sense of the reasoning of others. What is evident to all who are capable of understanding is an empirically-defined subset of this class. (shrink)

The paper discusses the emergence of Frege's puzzle and the introduction of the celebrated distinction between sense and reference in the context of Frege's logicist project. The main aim of the paper is to show that not logicism per se is mainly responsible for this introduction, but Frege's constant struggle against formalism. Thus, the paper enlarges the historical context, and provides a reconstruction of Frege's philosophical development from this broader perspective.

Wittgenstein responds in his Notes on Logic to a discussion of Russell's Principles of Mathematics concerning assertion. Russell writes: "It is plain that, if I may be allowed to use the word assertion in a non-psychological sense, the proposition "p implies q" asserts an implication, though it does not assert p or q. The p and the q which enter into this proposition are not strictly the same as the p or the q which are separate propositions." (PoM p35) Wittgenstein (...) replies: "Assertion is merely psychological. In not-p, p is exactly the same as if it stands alone; this point is absolutely fundamental." (NB p95) Wittgenstein's response is intriguing, not least because of the centrality to his Tractatus of the idea that a proposition says something. This paper will examine that idea, distinguishing it from 'merely psychological' assertion, and explore in this context how we should understand the occurrence of a Tractarian proposition within another. (shrink)

I have three main objectives in this essay. First, in chapter 2, I shall put forward and justify what I call worldlessness, by which I mean the following: All truths (as well as falsehoods) are wholly independent of any circumstances, not only time and place but also possible worlds. It follows from this view that whatever is actually true must be taken as true with respect to every possible world, which means that all truths are (in a sense) necessary. However, (...) the account I shall propound is different from what is known in the trade as necessitarianism, i.e. the view that there is only one possible world, viz. the actual one, for the doctrine of the worldlessness of truth values, despite its commitment to the necessity of truths and falsehoods, is quite compatible with the idea of there being other possible worlds. Another important issue in chapter 2, explored in particular in section 2.12, is the claim that there is no real change in the world. Secondly, in chapter 3 I consider the eminent traditional argument for determinism, deriving from Aristotle, namely, logical determinism, i.e. determinism justified by an appeal to the logical principle of bivalence (that all proper statements, including those concerning the future, are either true or false). In this connection I try to show that, (i), the formulation of the conclusion of this argument as "Whatever will happen will happen of necessity" is implausible, at least from the modern point of view, (ii), the formulation as "Whatever will happen will happen inevitably" is more to the point, and (iii), on the basis of the worldless and timeless aspect advocated in chapter 2, this latter formulation is quite harmless, essentially amounting to the trivial statement, "Whatever will happen will happen". Thirdly, in chapter 4 I study theological determinism, or determinism that arises from God's supposed providential control over everything that happens. In this connection, I shall survey some historical accounts of the relation between human free will and determinism (not only theological but also causal determinism); the philosophers the views of whom I shall attend to include Chrysippus, St. Augustine, Boethius and Aquinas. I shall in particular consider G.W. Leibniz' theodicean aspirations, viz. his solution to the problem of evil and, especially, his compatibilist attempts to reconcile human free will with the strictly deterministic flow of actual events. I think it is important to try to explicate Leibniz' ingenious account of these matters, since it seems that it has not been fully appreciated in the literature, not even by contemporary Leibniz scholars (such as B. Mates, R.C. Sleigh, C. Wilson, R.M. Adams and D. Rutherford). In providing the Leibnizian compatibilist solution of the problem of determinism and freedom in chapter 4, I shall utilize the approach of chapter 2. (shrink)

Are Fregean thoughts compositionally complex and composed of senses? We argue that, in Begriffsschrift, Frege took 'conceptual contents' to be unstructured, but that he quickly moved away from this position, holding just two years later that conceptual contents divide of themselves into 'function' and 'argument'. This second position is shown to be unstable, however, by Frege's famous substitution puzzle. For Frege, the crucial question the puzzle raises is why "The Morning Star is a planet" and "The Evening Star is a (...) planet" have different contents, but his second position predicts that they should have the same content. Frege's response to this antinomy is of course to distinguish sense from reference, but what has not previously been noticed is that this response also requires thoughts to be compositionally complex, composed of senses. That, however, raises the question just how thoughts are composed from senses. We reconstruct a Fregean answer, one that turns on an insistence that this question must be understood as semantic rather than metaphysical. It is not a question about the intrinsic nature of residents of the third realm but a question about how thoughts are expressed by sentences. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

I argue here that Frege’s eventual view on the relation between sentences and the thoughts they express is that, ideally, a sentence expresses exactly one thought, and a thought is expressed by exactly one (canonical) sentence. This may clash with some mainstream views of Frege, for it has the consequence of de-emphasizing the philosophical significance of the question of how it is possible for someone to regard one sentence as true yet regard another sentence that expresses the same thought as (...) false. This account of Frege was developed by taking a long-range look at his writings over the course of his life. (shrink)

A widely held view about Fregean Sense has it that the determination of a sign's referent by the sign's sense is achieved viasatisfaction: the sense specifies a condition (or set of conditions) and the referent is that entity, if any, which uniquely satisfies that (set of) condition(s). This is usually held in conjunction with the claim that the sense is existentially and qualitatively independent of the referent: if the referent did not exist, or did not uniquely satisfy the sense, the (...) sense would still exist and would still specify the same condition(s) that it actually does (and might determine a different referent than its actual one). Proponents of this view give several reasons for holding it. I describe these reasons and argue that they are not convincing. More generally, I try to show that the notion of satisfaction has no useful application within Frege's system. I then suggest an alternative account of the determination of a referent by a sense that I think is truer to Frege and more illuminating. Compared to the satisfaction view, my account construes determination as a more naturalistic and epistemically real relation between speakers and things in the world. (shrink)