Given the old conception of the relation greater than, the proposition that the whole is greater than the part is an immediate consequence. But being greater in this sense is not incompatible with being equal in the sense of one-one correspondence. Some who failed to recognize this formulated invalid arguments against the possibility of infinite quantities. Others who did realize that the relations of equal and greater when so defined are compatible, concluded that such relations are not appropriately taken as (...) quantitative relations, at least, not in general. If suitable quantitative relations were to be defined, there was a possibility of retaining either the traditional definition of greater and finding another concept of equality, or of retaining the concept of equality as one-one correspondence and defining greater than so that it is incompatible with equality in this sense. The former alternative was implicitly taken by Bolzano, who never succeeded in defining suitable quantitative relations. The latter alternative was taken by Cantor, and is the basis of his great success in constructing a mathematical theory of the transfinite. (shrink)

We seek means of distinguishing logical knowledge from other kinds of knowledge, especially mathematics. The attempt is restricted to classical two-valued logic and assumes that the basic notion in logic is the proposition. First, we explain the distinction between the parts and the moments of a whole, and theories of ?sortal terms?, two theories that will feature prominently. Second, we propose that logic comprises four ?momental sectors?: the propositional and the functional calculi, the calculus of asserted propositions, and rules for (...) (in)valid deduction, inference or substitution. Third, we elaborate on two neglected features of logic: the various modes of negating some part(s) of a proposition R, not only its ?external? negation not-R; and the assertion of R in the pair of propositions ?it is (un)true that R? belonging to the neglected logic of asserted propositions, which is usually left unstated. We also address the overlooked task of testing the asserted truth-value of R. Fourth, we locate logic among other foundational studies: set theory and other theories of collections, metamathematics, axiomatisation, definitions, model theory, and abstract and operator algebras. Fifth, we test this characterisation in two important contexts: the formulation of some logical paradoxes, especially the propositional ones; and indirect proof-methods, especially that by contradiction. The outcomes differ for asserted propositions from those for unasserted ones. Finally, we reflect upon self-referring self-reference, and on the relationships between logical and mathematical knowledge. A subject index is appended. (shrink)

Recent work has defended “Euclidean” theories of set size, in which Cantor’s Principle (two sets have equally many elements if and only if there is a one-to-one correspondence between them) is abandoned in favor of the Part-Whole Principle (if A is a proper subset of B then A is smaller than B). It has also been suggested that Gödel’s argument for the unique correctness of Cantor’s Principle is inadequate. Here we see from simple examples, not that Euclidean theories of set (...) size are wrong, but that they must be either very weak and narrow or largely arbitrary and misleading. (shrink)

The paper discusses Bernard Bolzano’s epistemological approach to believing and knowing with regard to the epistemic requirements of an axiomatic model of science. It relates Bolzano’s notions of believing, knowing and evaluation to notions of infallibility, immediacy and foundational truth. If axiomatic systems require their foundational truths to be infallibly known, this knowledge involves both evaluation of the infallibility of the asserted truth and evaluation of its being foundational. The twofold attempt to examine one’s assertions and to do so by (...) searching for the objective grounds of the truths asserted lies at the heart of Bolzano’s notion of knowledge. However, the explanatory task of searching for grounds requires methods that cannot warrant infallibility. Hence, its constitutive role in a conception of knowledge seems to imply the fallibility of such knowledge. I argue that the explanatory task contained in Bolzanian knowing involves a high degree of epistemic virtues, and that it is only through some salient virtue that the credit of infallibility can distinguish Bolzanian knowing from a high degree of Bolzanian believing. (shrink)

This paper defends the view that Newtonian forces are real, symmetrical and non-causal relations. First, I argue that Newtonian forces are real; second, that they are relations; third, that they are symmetrical relations; fourth, that they are not species of causation. The overall picture is anti-Humean to the extent that it defends the existence of forces as external relations irreducible to spatio-temporal ones, but is still compatible with Humean approaches to causation (and others) since it denies that forces are a (...) species of causation. (shrink)

Everyday reasoning is replete with arguments which, though not logically valid, nonetheless harbor a measure of credibility in their own right. Here the claim that such arguments force us to acknowledge material validity, in addition to logical validity, is advanced, and criteria that attempt to unpack this concept are examined in detail. Of special concern is the effort to model these criteria on explications of logical validity that rely on notions of substitutivity and logical form. It is argued, however, that (...) such a parallel is not easily located and that it is uncertain that a construal of material validity can be fashioned after traditional accounts of logical validity. Attention is also given to the topics of enthymemes and to the proper domain of logic. (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)

Several accounts of logical truth are compared and shown to define distinct concepts. Nevertheless, conditions are given under which they happen to declare exactly the same sentences logically true. These conditions involve the variety of objects in the domain, the richness of the language, and the logical resources available. It is argued that the class of sentences declared logically true by each of the accounts depends on particularities of the actual world.

Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicist's conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no principles of (...) inference specific to a given local topic. Poincaré, a Kantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical reasoning (proof). In this essay, I try to isolate and clarify this idea and to describe the mathematical epistemology which underlies it. Central to this epistemology (which is basically Kantian in orientation, and closely similar to that advocated by Brouwer) is a principle of epistemic conservation which says that knowledge of a given type cannot be extended by means of an inference unless that inference itself constitutes knowledge belonging to the given type. (shrink)

Hume argued that inductive inferences do not have rational justification. My aim is to reject Hume’s argument. The discussion is partly motivated by an analogy with Carroll’s Paradox, which concerns deductive inferences. A first radically externalist reply to Hume (defended by Dauer and Van Cleve) is that justified inductive inferences do not require the subject to know that nature is uniform, though the uniformity of nature is a necessary condition for having the justification. But then the subject does not have (...) reasons for believing what she believes. I defend a moderate externalist account that seeks to partly accommodate that objection to the radical externalist proposal. It is based on an extension of Peacocke’s theory of concepts: possession conditions for predicative concepts standing for natural properties include (fallible) dispositions to project them to new cases in accordance with inductive inferential patterns. (shrink)

'Ontological dependence' is a term of philosophical jargon which stands for a rich family of properties and relations, often taken to be among the most fundamental ontological properties and relations. Notions of ontological dependence are usually thought of as 'carving reality at its ontological joints', and as marking certain forms of ontological 'non-self-sufficiency'. The use of notions of dependence goes back as far as Aristotle's characterization of substances, and these notions are still widely used to characterize other concepts and to (...) formulate metaphysical claims. This paper first gives an overview of the varieties of these notions, and then discusses some of their main applications. (shrink)

The paper proposes a strategy for understanding metaphysical grounding in deflationary terms and, more generally, proposes a form of methodological deflationism with respect to the notions of ground. The idea is to define a deflationary is grounded in-predicate by appeal to the two-place non-causal connective ‘because’ and a deflationary truth predicate. To this end, we discuss the explanatory role of the truth-predicate in non-causal explanations and develop a theory of truth for the language of the ‘because’-connective. We argue that at (...) least from a logical perspective our deflationary notion of ground is up to the task. (shrink)

Provides a comprehensive overview and introduction to the Routledge Handbook of Propositions.

In this short paper we consider Linnebo's thin/thick dichotomy: first, we show that it does not overlap with the very common one between abstract/concrete objects; second, on the basis of some difficulties with the distinction, we propose, as a possible way out, to move from thin/thick objects to thin/thick concepts.

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitmate” in virtue of being “more basic” or “more fundamental”. This paper addresses two related issues. First, (...) we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistomology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally “legitimate”. (shrink)

Husserl’s early picture of explanation in the sciences has never been completely provided. This lack represents an oversight, which we here redress. In contrast to currently accepted interpretations, we demonstrate that Husserl does not adhere to the much maligned deductive-nomological (DN) model of scientific explanation. Instead, via a close reading of early Husserlian texts, we reveal that he presents a unificationist account of scientific explanation. By doing so, we disclose that Husserl’s philosophy of scientific explanation is no mere anachronism. It (...) is, instead, tenable and relevant. We discuss how Husserl and other contemporary thinkers draw theoretical inspiration from the same source—namely, Bernard Bolzano. Husserl’s theory of scientific explanation shares a common language and discusses the same themes as, for example, Phillip Kitcher and Kit Fine. To advance our novel reading, we discuss Husserl’s investigations of grounding, inter-lawful explanation, intra-mathematical explanation, and scientific unification. (shrink)

In this article, I use Edmund Gettier’s Ten Coins hypothetical scenario to illustrate some reasoning errors in the use of definite descriptions. The Gettier problem, central as it is to modern epistemology, is first and foremost an argument, which Gettier (Analysis 23(6):121–123, 1963) constructs to prove a contrary conclusion to a widely held view in epistemology. Whereas the epistemological claims in the case have been extensively analysed conceptually, the strategies and tools from other philosophical disciplines such as analytic philosophy of (...) language, logic and argumentation that Gettier deploys in the case have scarcely received any attention. This work abstracts from the epistemological content and examines Gettier’s handling of the definite description involved, and how that affects the cogency of his argument. (shrink)

Say that two sentences are ground-theoretically equivalent iff they are interchangeable salva veritate in grounding contexts. Notoriously, ground-theoretic equivalence is a hyperintensional matter: even logically equivalent sentences may fail to be interchangeable in grounding contexts. Still, there seem to be some substantive, general principles of ground-theoretic equivalence. For example, it seems plausible that any sentences of the form A∧B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \wedge B$$\end{document} and B∧A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B (...) \wedge A$$\end{document} are ground-theoretically equivalent. What, then, are in general the conditions for two sentences to stand in the relation of ground-theoretic equivalence, and what are the logical features of that relation? This paper develops and defends an answer to these questions based on the mode-ified truthmaker theory of content presented in my recent paper ‘Towards a theory of ground-theoretic content’ :785–814, 2018). (shrink)

In Modal Logic as Metaphysics, Timothy Williamson claims that the possibilism-actualism (P-A) distinction is badly muddled. In its place, he introduces a necessitism-contingentism (N-C) distinction that he claims is free of the confusions that purportedly plague the P-A distinction. In this paper I argue first that the P-A distinction, properly understood, is historically well-grounded and entirely coherent. I then look at the two arguments Williamson levels at the P-A distinction and find them wanting and show, moreover, that, when the N-C (...) distinction is broadened (as per Williamson himself) so as to enable necessitists to fend off contingentist objections, the P-A distinction can be faithfully reconstructed in terms of the N-C distinction. However, Williamson’s critique does point to a genuine shortcoming in the common formulation of the P-A distinction. I propose a new definition of the distinction in terms of essential properties that avoids this shortcoming. (shrink)

This paper defends the view that Newtonian forces are real symmetrical and non-causal relations. In the first part, I argue that Newtonian forces are real; in the second part, that they are relations; in the third part, that they are symmetrical relations; in the fourth part, that they are not causal relations, (but causal relata) by which I mean that they are not species of causation. The overall picture is anti-humean to the extent that it defends the existence of forces, (...) irreducible to spatio-temporal relations, but is still compatible with humean approaches to causation (and others) since it denies that forces are species of causation. (shrink)

In recent years, metaphysics has undergone what some describe as a revolution: it has become standard to understand a vast array of questions as questions about grounding, a metaphysical notion of determination. Why should we believe in grounding, though? Supporters of the revolution often gesture at what I call the Argument from Explanatoriness: the notion of grounding is somehow indispensable to a metaphysical type of explanation. I challenge this argument and along the way develop a “reactionary” view, according to which (...) there is no interesting sense in which the notion of grounding is explanatorily indispensable. I begin with a distinction between two conceptions of grounding, a distinction which extant critiques of the revolution have usually failed to take into consideration: grounding qua that which underlies metaphysical explanation and grounding qua metaphysical explanation itself. Accordingly, I distinguish between two versions of the Argument from Explanatoriness: the Unexplained Explanations Version for the first conception of grounding, and the Expressive Power Version for the second. The paper’s conclusion is that no version of the Argument from Explanatoriness is successful. (shrink)

Husserlian Essentialism is the view, maintained byEdmundHusserl throughout his career, that necessary truths obtain because essentialist truths obtain. In this thesis I have two goals. First, to reconstruct and flesh out Husserlian Essentialism and its connections with surrounding areas of Husserl's philosophy in full detail – something which has not been done yet. Second, to assess the theoretical solidity of the view. As regards the second point, after having presented Husserlian Essentialism in the first two chapters, I raise a serious (...) problem for it in Chapter 3. In the remainder of the thesis I endeavour to solve the problem. In order to do so, I propose to amend both Husserl's theory of essence and his theory of predication. The bulk of the emendation consists in working out an account of essence and an account of predication that do not presuppose, or in any way imply, the claims that: 1) for a universal to be in the essence of an object, either the object or one of its parts must instantiate the universal; 2) for a universal to be truly predicated of an object, either the object or one of its parts must instantiate the universal. These claims, notice, apart from being what gets Husserl in trouble, are well entrenched not only in Husserl's, but in most theories of essence and predication. It is thus interesting to see what an alternative option may be – even regardless of the Husserlian setting in which I work it out. (shrink)

I examine the theory of consequentia of the medieval logician, John Buridan. Buridan advocates a strict commitment to what we now call proposition-tokens as the bearers of truth-value. The analysis of Buridan's theory shows that, within a token-based semantics, amendments to the usual notions of inference and consequence are made necessary, since pragmatic elements disrupt the semantic behaviour of propositions. In my reconstruction of Buridan's theory, I use some of the apparatus of modern two-dimensional semantics, such as two-dimensional matrices and (...) the distinction between the context of formation and the context of evaluation of utterances. (shrink)

The well-known argument of Frederick Fitch, purporting to show that verificationism (= Truth implies knowability) entails the absurd conclusion that all the truths are known, has been disarmed by Dorothy Edgington''s suggestion that the proper formulation of verificationism presupposes that we make use of anactuality operator along with the standardly invoked epistemic and modal operators. According to her interpretation of verificationism, the actual truth of a proposition implies that it could be known in some possible situation that the proposition holds (...) in theactual situation. Thus, suppose that our object language contains the operatorA — it is actually the case that ... — with the following truth condition: vA iff w0, wherew 0 stands for the designated world of the model — the actual world. Then we can formalize the verificationist claim as follows. (shrink)

Despite attending Husserl’s classes, participating in the discussions of the Göttingen phenomenological circle, and writing prolifically on phenomenology, Eugen Enyvvari seems to have been virtually ignored by phenomenological scholarship. I use an array of unpublished sources and a survey of his juvenilia to reconstruct Enyvvari’s biography and intellectual formation, including his confrontation with Melchior Palagyi’s critique of Husserl and Bolzano. Based on both his reports and records from the Göttingen University Archives, I attempt to establish the influences to which he (...) could have been exposed in Göttingen. I rely on a careful micro-analysis of the development of Husserl’s notion of noematical meaning at the time of Enyvvari’s stay in Göttingen in order to asses Enyvvari’s specific contribution to phenomenology and his significance from the point of view of the general historiography of phenomenology. (shrink)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

The prospects of a causal interpretation of probability are examined. Various accounts both from the history of scientific method and from recent developments in the tradition of the method of arbitrary functions, in particular by Strevens, Rosenthal, and Abrams, are briefly introduced and assessed. I then present a specific account of causal probability with the following features: First, the link between causal probability and a particular account of induction and causation is established, namely eliminative induction and the related difference-making account (...) of causation in the tradition of Bacon, Herschel, and Mill. Second, it is shown how a causal approach is useful beyond applications of the method of arbitrary functions and is able to deal with various shades of both ontic and epistemic probabilities. Furthermore, I clarify the notion of causal symmetry as a central element of an objective version of the principle of indifference and relate probabilistic independence to causal irrelevance. (shrink)

This book presents novel formalizations of three of the most important medieval logical theories: supposition, consequence and obligations. In an additional fourth part, an in-depth analysis of the concept of formalization is presented - a crucial concept in the current logical panorama, which as such receives surprisingly little attention.Although formalizations of medieval logical theories have been proposed earlier in the literature, the formalizations presented here are all based on innovative vantage points: supposition theories as algorithmic hermeneutics, theories of consequence analyzed (...) with tools borrowed from model-theory and two-dimensional semantics, and obligations as logical games. For this reason, this is perhaps the first time that these medieval logical theories are made fully accessible to the modern philosopher and logician who wishes to obtain a better grasp of them, but who has always been held back by the lack of appropriate ‘translations' into modern terms.Moreover, the book offers a reflection on the very nature of logic, a reflection that is prompted by the comparisons between medieval and modern logic, their similarities and dissimilarities. It is thus a contribution not only to the history of logic, but also to the philosophy of logic, the philosophy of language and semantics.The analysis of medieval logic is also relevant for the modern philosopher and logician in that, being the unifying methodology used across all disciplines at that time, logic really provided unity to science. It thus presents a unified model of scientific investigation, where logic plays the aggregating role. (shrink)

Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is to provide an arena (...) for exploring relations and interactions between mathematical fields, their relative strengths, etc. Given the different goals, there is little point to determining a single foundation for all of mathematics. (shrink)

This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ?say? something), and they have subject matter (they are ?about? something). The asymmetry between establishing validity and establishing invalidity has long been noted: validity is established through an information-processing procedure exhibiting a step-by-step deduction of the conclusion from the premise-set. Invalidity is established by (...) exhibiting a countermodel satisfying the premises but not the conclusion. The process of establishing validity focuses on information content; the process of establishing invalidity focuses on subject matter. Corcoran's information-theoretic concept of logical consequence corresponds to the former. Tarski's model-theoretic concept of logical consequence formulated in his famous 1936 no-countermodels definition corresponds to the latter. Both are found to be indispensable for understanding the rationale of the deductive method and each complements the other. This study discusses the ontic question of the nature of logical consequence and the epistemic question of the human capabilities presupposed by practical applications of these two concepts as they make validity and invalidity accessible to human knowledge. (shrink)

John Cook Wilson (1849–1915) was Wykeham Professor of Logic at New College, Oxford and the founder of ‘Oxford Realism’, a philosophical movement that flourished at Oxford during the first decades of the 20th century. Although trained as a classicist and a mathematician, his most important contribution was to the theory of knowledge, where he argued that knowledge is factive and not definable in terms of belief, and he criticized ‘hybrid’ and ‘externalist’ accounts. He also argued for direct realism in perception, (...) criticizing both empiricism and idealism, and argued for a moderate nominalist view of universals as being in rebus and only ‘apprehended’ by their particulars. His influence helped swaying Oxford away from idealism and, through figures such as H. A. Prichard, Gilbert Ryle, or J. L. Austin, his ideas were also to some extent at the origin of ‘moral intuitionism’ and ‘ordinary language philosophy’ which defined much of Oxford philosophy until the second half of the twentieth-century. Nevertheless, his name and legacy were all but forgotten for generations after World War II. Still, his views on knowledge are with us today, being in part at work in the writings of philosophers as diverse as John McDowell, Charles Travis, and Timothy Williamson. (shrink)

The aim of this paper is to provide a definition of the the notion of complete and immediate formal grounding through the concepts of derivability and complexity. It will be shown that this definition yields a subtle and precise analysis of the concept of grounding in several paradigmatic cases.

Quine’s general approach is to treat ontology as a matter of what a theory says there is. This turns ontology into a question of which existential statements are consequences of that theory. This approach is contrasted favourably with the view that takes ontological commitment as a relation to things. However within the broadly Quinean approach we can distinguish different accounts, differing as to the nature of the consequence relation best suited for determining those consequences. It is suggested that Quine’s own (...) narrowly formal account fails. Then a consideration of the necessitation approach championed by Jackson and Lewis shows that it does not do justice to the role of acknowledging consequences in determining rationality. I suggest that an approach which puts a priori consequence as the key relation does a better job. The task of spelling out the nature of a priori consequence is sketched, along with reasons to doubt the adequacy of the double indexing approach to analysing the a priori. The sorts of relations we can stand in to theories which allow us to inherit ontological commitments are touched on with a number of important philosophical strategies for introducing belief-like attitudes which nevertheless avoid ontological commitment. (shrink)

In a series of publications beginning in the 1980s, John Etchemendy has argued that the standard semantical account of logical consequence, due in its essentials to Alfred Tarski, is fundamentally mistaken. He argues that, while Tarski's definition requires us to classify the terms of a language as logical or non-logical, no such division is guaranteed to deliver the correct extension of our pre-theoretical or intuitive consequence relation. In addition, and perhaps more importantly, Tarski's account is claimed to be incapable of (...) explaining an essential modal/epistemological feature of consequence, namely, its necessity and apriority. Bernard Bolzano (1781-1848) is widely recognized as having anticipated Tarski's definition in his Wissenschaftslehre (or Theory of Science ) of 1837. Because of the similarities between his account and Tarski's, Etchemendy's arguments have also been extended to cover Bolzano. The purpose of this article is to consider Bolzano's theory in the light of these criticisms. We argue that, due to important differences between Bolzano's and Tarski's theories, Etchemendy's objections do not apply immediately to Bolzano's account of consequence. Moreover, Bolzano's writings contain the elements of a detailed philosophical response to Etchemendy. (shrink)

G. F. Stout is famous as an early twentieth century proselyte for abstract particulars, or tropes as they are now often called. He advanced his version of trope theory to avoid the excesses of nominalism on the one hand and realism on the other. But his arguments for tropes have been widely misconceived as metaphysical, e.g. by Armstrong. In this paper, I argue that Stout’s fundamental arguments for tropes were ideological and epistemological rather than metaphysical. He moulded his scheme to (...) fit what is actually given to us in perception, arguing that our epistemic practices would break down in an environment where only universals were given to us. (shrink)

This thesis is on the history and philosophy of logic and semantics. Logic can be described as the ‘science of reasoning’, as it deals primarily with correct patterns of reasoning. However, logic as a discipline has undergone dramatic changes in the last two centuries: while for ancient and medieval philosophers it belonged essentially to the realm of language studies, it has currently become a sub-branch of mathematics. This thesis attempts to establish a dialogue between the modern and the medieval traditions (...) in logic, by means of ‘translations’ of the medieval logical theories into the modern framework of symbolic logic, i.e. formalizations. One of its conclusions is that, when properly understood within their own framework, the interest of medieval logical theories for modern investigations go beyond mere historical interest, but that a thorough conceptual analysis of such theories must be undertaken in order to avoid conceptual misprojections. While such translations of medieval into modern logic have been attempted before, the approach presented here is innovative in that attention is paid to the similarities as well as to the dissimilarities between the two traditions, and to what can be learned from the medieval masters for modern investigations in logic and semantics. (shrink)

This paper analyses the features of Kant’s transcendental philosophy, which Kant himself described as transcendental idealism. On the one hand, Kant’s transcendentalism rests on the distinction between things in themselves and appearances. On the other hand, our ‘mode of cognition’ [Critique, B25] cognition is representative in that is based on representations — subjective and objective ones. A synthesis of the above considerations suggests that Kant’s transcendentalism rests on the conceptual triad “[objective] object — appearance — and [mental] representation “. Kant’s (...) transcendentalism is impossible without the ‘premise’ of appearance. The correct interpretation of Kant’s transcendental philosophy should keep the double difference of appearance both from thing–in–itself and representation. In this transcendental triad, the Kantian appearance has an intermediate status since it is located between objective things and subjective representations. However, the conceptual status of the appearance needs to be clarified, since Kant himself does not give a clear answer to this question, and at present there are several interpretations, differing primarily in the definition of the concept of the appearance. For me, appearance can be correlated with objective–objectual representation. It would be unwise to identify appearance with thing in itself, which was characteristic of pre–Kantian philosophy, or appearance with representation, which was the case in phenomenalist interpretations of transcendental idealism à la Berkley. The Kantian appearance, as emphasised in BXXVII of his Critique, is an appearance of an object, which — although implicitly — suggests a semantic relationship of reference. In this case, the appearance is not an object, but just ‘a designation of an object’ [Critique, B235]. Appearance is impossible without what appears in it. This paper puts forward a number of arguments in favour of the objectual status of Kant’s concept of appearance. (shrink)

I characterize the deductivist ideal of justification and, following to a great extent Toulmin’s work The Uses of Argument, I try to explain why this ideal is erroneous. Then I offer an alternative model of justification capable of making our claims to knowledge about substantial matters sound and reasonable. This model of justification will be based on a conception of justification as the result of good argumentation, and on a model of argumentation which is a pragmatic linguistic reconstruction of Toulmin’s (...) model of argument. (shrink)

In this paper I argue that Bolzano's concept of deducibility and Tarski's concept of logical consequence differ with respect to their philosophical intent. I distinguish between epistemic and ontic approaches to logic, and argue that Bolzano's deducibility presupposes an epistemic approach, while Tarski's logical consequence presupposes an ontic approach.

The standard semantic definition of consequence with respect to a selected set X of symbols, in terms of truth preservation under replacement (Bolzano) or reinterpretation (Tarski) of symbols outside X, yields a function mapping X to a consequence relation ⇒x. We investigate a function going in the other direction, thus extracting the constants of a given consequence relation, and we show that this function (a) retrieves the usual logical constants from the usual logical consequence relations, and (b) is an inverse (...) to—more precisely, forms a Galois connection with—the Bolzano-Tarski function. (shrink)

The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.

In a series of papers published in the sixties and seventies, Jaakko Hintikka, drawing upon Kant’s conception, defines an argument to be analytic whenever it does not introduce new individuals into the discussion and argues that there exists a class of arguments in polyadic first-order logic that are to be synthetic according to this sense. His work has been utterly overlooked in the literature. In this paper, I claim that the value of Hintikka’s contribution has been obscured by his formalisation (...) of the original definition. Therefore, I provide (i) a brief reconstruction of the historical framework of the problem and the revolutionary import of Hintikka’s contribution, (ii) a clarification of the most complicated steps of Hintikka’s elaboration of his insight, (iii) a criticism of several features that play a fundamental role in Hintikka’s formalisation and (iv) a selection from Hintikka’s own material of some valuable suggestions towards a clear and workable formalisation. As for the pars construens, I isolate in the approach of depth-bounded first-order logics (D'Agostino et al. 2021) an alternative formalisation of the notion of syntheticity as the introduction of new individuals in the reasoning, and I show that it is not affected by the same difficulties as Hintikka’s proposal. In so doing, I hope to have contributed to the realisation of the project of rehabilitating Kant’s analytic–synthetic distinction in the context of modern first-order logic with the purpose of showing, against the logical empiricist movement, that logic is not analytic. (shrink)