According to non-conceptualist interpretations, Kant held that the application of concepts is not necessary for perceptual experience. Some have motivated non-conceptualism by noting the affinities between Kant's account of perception and contemporary relational theories of perception. In this paper I argue (i) that non-conceptualism cannot provide an account of the Transcendental Deduction and thus ought to be rejected; and (ii) that this has no bearing on the issue of whether Kant endorsed a relational account of perceptual experience.

In this paper, I propose a new nonconceptual reading of the B-Deduction. As Hanna correctly remarks :399–415, 2011: 405), the word “cognition” has in both editions of the first Critique a wide sense, meaning nonconceptual cognition, and a narrow meaning, in Kant’s own words “an objective perception”. To be sure, Kant assumes the first meaning to account for why the Deduction is unavoidable. And if we take this meaning as a premise of the B-Deduction, then there is (...) a gap in the argument since the categories are certainly not conditions for non-conceptual cognition. Still, I believe it is not this wide meaning but rather the narrow one that figures in any premise of the B-Deduction. Thus, in the reading that I am proposing, categories are not conditions for representing something, or even conditions for representing something objectively. Instead, they are conditions for the recognition that what we represent through the senses exists mind-independently. In the first step of the B-Deduction, this cognition in the narrow sense takes the form of the propositional thinking that the nonconceptually represented object of the sensible intuition exists objectively. In contrast, in the second step of the B-Deduction, this cognition in the narrow sense takes the form of the apprehension of what our human senses represent nonconceptually as existing objectively. (shrink)

A normalization procedure is given for classical natural deduction with the standard rule of indirect proof applied to arbitrary formulas. For normal derivability and the subformula property, it is sufficient to permute down instances of indirect proof whenever they have been used for concluding a major premiss of an elimination rule. The result applies even to natural deduction for classical modal logic.

A running commentary is offered on the first half of Frege’s Grundlagen der Arithmetik, §17, and suggests that Frege anticipated the method of demonstration used by Paul Bernays for the Deduction Theorem.

In this book Catarina Dutilh Novaes—the accomplished philosopher of logic and historian of logic—takes a wide-angle view of the field and argues that the great.

The framework of natural deduction is extended by permitting rules as assumptions which may be discharged in the course of a derivation. this leads to the concept of rules of higher levels and to a general schema for introduction and elimination rules for arbitrary n-ary sentential operators. with respect to this schema, (functional) completeness "or", "if..then" and absurdity is proved.

Gentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents (...) have at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933. (shrink)

Natural deduction is the type of logic most familiar to current philosophers, and indeed is all that many modern philosophers know about logic. Yet natural deduction is a fairly recent innovation in logic, dating from Gentzen and Jaśkowski in 1934. This article traces the development of natural deduction from the view that these founders embraced to the widespread acceptance of the method in the 1960s. I focus especially on the different choices made by writers of elementary textbooks—the (...) standard conduits of the method to a generation of philosophers—with an eye to determining what the ‘essential characteristics’ of natural deduction are. (shrink)

Anderson and Belnap presented indexed Fitch-style natural deduction systems for the relevant logics R, E, and T. This work was extended by Brady to cover a range of relevant logics. In this paper I present indexed tree natural deduction systems for the Anderson–Belnap–Brady systems and show how to translate proofs in one format into proofs in the other, which establishes the adequacy of the tree systems.

This paper examines Hobbes’s criticisms of Robert Boyle’s air-pump experiments in light of Hobbes’s account in _De Corpore_ and _De Homine_ of the relationship of natural philosophy to geometry. I argue that Hobbes’s criticisms rely upon his understanding of what counts as “true physics.” Instead of seeing Hobbes as defending natural philosophy as “a causal enterprise … [that] as such, secured total and irrevocable assent,” 1 I argue that, in his disagreement with Boyle, Hobbes relied upon his understanding of natural (...) philosophy as a mixed mathematical science. In a mixed mathematical science one can mix facts from experience with causal principles borrowed from geometry. Hobbes’s harsh criticisms of Boyle’s philosophy, especially in the _Dialogus Physicus, sive De natura aeris_, should thus be understood as Hobbes advancing his view of the proper relationship of natural philosophy to geometry in terms of mixing principles from geometry with facts from experience. Understood in this light, Hobbes need not be taken to reject or diminish the importance of experiment/experience; nor should Hobbes’s criticisms in _Dialogus Physicus_ be understood as rejecting experimenting as ignoble and not befitting a philosopher. Instead, Hobbes’s viewpoint is that experiment/experience must be understood within its proper place – it establishes the ‘that’ for a mixed mathematical science explanation. (shrink)

Filosofia, História, Psicologia, Ciências Cognitivas e Práticas Matemáticas: este é o horizonte investigativo muito seminal no qual o livro The Dialogical Roots of Deduction se encontra situado. Catarina Dutilh-Novaes, por meio dessa obra, buscou defender a tese de que a dedução tem raízes dialógicas. Mais que isso, a lógica estaria ligada às práticas humanas, interações sociais, cultura e sociedade. Não estando, assim, vinculada com algum âmbito de formas platônicas, de representações da mente ou do sujeito transcendental. A lógica tem, (...) nesta perspectiva dialógica, um fator humano incontornável. O livro mostra o fator profundamente social da dedução como situadas nas práticas humanas. Neste cenário, a normatividade da lógica, por exemplo, surge de algumas dinâmicas sociais especializadas e não de fenômenos mentais privados. A sua abrangente investigação do conceito e de práticas dedutivas nos revela perspectivas variadas para se defender que a dedução tem raízes dialógicas e que elas ainda estão presentes hoje em teorias e nas práticas dedutivas. (shrink)

We examine Charles S. Peirce's mature views on the logic of science, especially as contained in his later and still mostly unpublished writings. We focus on two main issues. The first concerns Peirce's late conception of retroduction. Peirce conceived inquiry as performed in three stages, which correspond to three classes of inferences: abduction or retroduction, deduction, and induction. The question of the logical form of retroduction, of its logical justification, and of its methodology stands out as the three major (...) threads in his later writings. The other issue concerns the second stage of scientific inquiry, deduction. According to Peirce's later formulation, deduction is divided not only into two kinds but also into two sub-stages: logical analysis and mathematical reasoning, where the latter is either corollarial or theorematic. Save for the inductive stage, which we do not address here, these points cover the essentials of Peirce's latest thinking on the l.. (shrink)

It seems beyond doubt that a thinker can come to know a conclusion by deducing it from premisses that he knows already, but philosophers have found it puzzling how a thinker could acquire knowledge in this way. Assuming a broadly externalist conception of knowledge, I explain why judgements competently deduced from known premisses are themselves knowledgeable. Assuming an exclusionary conception of judgeable content, I further explain how such judgements can be informative. (According to the exclusionary conception, which I develop from (...) some remarks in Ramsey, a judgement's content is given by the hitherto live possibilities that it excludes or rules out.) I propose that the value of logic lies in its allowing us to combine different sources of knowledge, so that we can learn things that we could not learn from those sources individually. I conclude by arguing that while single-conclusion logics possess that value, multiple-conclusion logics do not. (shrink)

Gentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents (...) have at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933. (shrink)

We give in this paper a short semantical proof of the strong normalization for full propositional classical natural deduction. This proof is an adaptation of reducibility candidates introduced by J.-Y. Girard and simplified to the classical case by M. Parigot.

In the wake of the massive criticism of Kant’s deduction of the categories in the first Critique, Fichte starts providing what he takes an improved version of such a deduction to be. This article aims at investigating the transformation he thereby introduces into the Kantian thought. I will do so mainly with respect to the deduction’s architectonical dimension, i.e. by investigating the role of the deduction for the Wissenschaftslehre as a whole. Concretely, I will defend the (...) following theses: (1) By identifying the deduction of the categories and the metaphysical system, Fichte moves away both from Kant’s conception of a system and from that of a deduction. (2) It is the unique character of Fichte’s conception of a first principle of a system that allows for an identification between system and deduction. (shrink)

I defend a novel interpretation of Kant's conceptualism regarding the contents of our perceptual experiences. Conceptualist interpreters agree that Kant's Deduction aims to prove that intuitions require the categories for their spatiality and temporality. But conceptualists disagree as to which features of space and time make intuitions require the categories. Interpreters have cited the singularity, unity, infinity, and homogeneity of space and time. But this is incompatible with Kant's Aesthetic, which aims to prove that these same features qualify space (...) and time as intuitions, not concepts. On my interpretation, the feature is objectivity. Space and time are objective, in that they ground our judgments in geometry and mechanics. (shrink)

The two major options on which the current debate on the interpretation of quantum mechanics relies, namely realism and empiricism, are far from being exhaustive. There is at least one more position available, which is metaphysically as agnostic as empiricism, but which shares with realism a committment to considering the structure of theories as highly significant. The latter position has been named transcendentalism after Kant. In this paper, a generalized version of Kant's method is used. This yields a reasoning that (...) one is entitled to call a transcendental deduction of some major formal features of quantum mechanics. (shrink)

This paper contains a critical analysis of the interpretation of Kant's second edition version of the Transcendental Deduction offered by Béatrice Longuenesse in her recent book: Kant and the Capacity to Judge. Though agreeing with much of Longuenesse's analysis of the logical function of judgment, I question the way in which she tends to assign them the objectifying role traditionally given to the categories. More particularly, by way of defending my own interpretation of the Deduction against some of (...) her criticisms, I argue that Longuenesse fails to show how either part of the two-part proof may be plausibly thought to have established the necessity of the categories (as opposed to the logical functions). Finally, I question certain aspects of her 'radical' interpretation of the famous footnote at B160-1, where Kant distinguishes between 'form of intuition' and 'formal intuition'. (shrink)

Gentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents (...) have at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933. (shrink)

This paper is about the nature of the relationship between (1) the doctrine of Non-Conceptualism about mental content, (2) Kant's Transcendental Idealism, and (3) the Transcendental Deduction of the Pure Concepts of the Understanding, or Categories, in the B (1787) edition of the Critique of Pure Reason, i.e., the B Deduction. Correspondingly, the main thesis of the paper is this: (1) and (2) yield serious problems for (3), yet, in exploring these two serious problems for the B (...) class='Hi'>Deduction, we also discover some deeply important and perhaps surprising philosophical facts about Kant's theory of cognition and his metaphysics. (shrink)

One recent trend in Kant scholarship has been to read Kant as undertaking a project in philosophical semantics, as opposed to, say, epistemology, or transcendental metaphysics. This trend has evolved almost concurrently with a debate in contemporary philosophy of mind about the nature of concepts and their content. Inferentialism is the view that the content of our concepts is essentially inferentially articulated, that is, that the content of a concept consists entirely, or in essential part, in the role that that (...) concept plays in a system of inferences. By contrast, relationalism is the view that this content is fixed by a mental or linguistic item's standing in a certain relation to its object. The historical picture of Kant and the contemporary debate about concepts intersect in so far as contemporary inferentialists about conceptual content often cite Immanuel Kant not only as one of the founding fathers of a tradition that leads more or less straightforwardly to contemporary inferentialism, but also as the philosopher who first saw the fatal flaws in any attempt to articulate the content of our concepts relationally. (shrink)

Deduction has frequently been condemned as a useless intellectual instrument because of its tautological character. To thoroughgoing opponents of rationalism, the pretensions of deduction are on the same level with those of induction. Both presume to yield more knowledge from the fact that we have some knowledge; and this is an impossible paradox, which not even so powerful an opponent of the “Philosophy of Experience” as Bradley could resolve to his own satisfaction. I wish in this brief note (...) to indicate how this semblance of paradox may very easily be removed without having to deny, as has often been thought necessary, the tautologicality of deduction. (shrink)