The thesis of this paper is that we can justify induction deductively relative to one end, and deduction inductively relative to a different end. I will begin by presenting a contemporary variant of Hume (1739; 1748)'s argument for the thesis that we cannot justify the principle of induction. Then I will criticize the responses the resulting problem of induction has received by Carnap (1963; 1968) and Goodman (1954), as well as praise Reichenbach (1938; 1940)'s approach. -/- Some of these (...) authors compare induction to deduction. Haack (1976) compares deduction to induction, and I will critically discuss her argument for the thesis that we cannot justify the principles of deduction next. In concluding I will defend the thesis that we can justify induction deductively relative to one end, and deduction inductively relative to a different end, and that we can do so in a non-circular way. Along the way I will show how we can understand deductive and inductive logic as normative theories, and I will briefly sketch an argument to the effect that there are only hypothetical, but no categorical imperatives. (shrink)

In this article, I consider critical arguments levelled against central elements of my view, expounded in my book Kant’s Deduction and Apperception (Schulting 2012b; KDA), that the categories are derived a priori from the principle of apperception, the ‘I think’. This view goes back to a much earlier, and more famous attempt by Klaus Reich, first proposed in 1932 (see Reich 2001), to argue that the functions of thought are ultimately and a priori derivable from the objective unity of (...) apperception. Reich looked to textual sources outside the Deduction for support, while I argued that TD itself provides supporting grounds for this view, or at least for the derivation of the categories from apperception. This has not been a popular view among Kantians, and gathering from the criticisms against my take on it, one may safely assume it is not going to be the standard view any time soon (see Dyck 2014 and Stephenson 2014; though Quarfood 2014 is much more positive). While responding to my critics, I go over some of the main planks of my interpretation of the so-called ‘first step’ of the B-Deduction, which was delineated in much greater detail in KDA. Among other things, against Corey Dyck I maintain that the analytic unity of consciousness is crucially important to the argument of §16 of the Deduction and I argue that the categories are more intimately related to the functions of judgement than some interpreters, including Dyck, make them out to be. Secondly, I argue that the progressive argument of TD should not be construed as a transcendental argument against the sceptic, as so many Anglophone readers of TD (still) do. To construe TD as aimed at the sceptic is to underestimate Kant’s epistemic confidence and to miss the real point of the Critical project, which should be seen more in the context of rationalism, namely: justifying the use of the pure concepts of the understanding by showing that they are only objectively valid in conjunction with empirical intuitions of objects. In this context, I criticise a standard reading of the reciprocity between the subject and object of experience and critically consider construals of a supposed gap in Kant’s argument. I argue why, in his critique of KDA, Andrew Stephenson is mistaken in thinking that showing that the categories apply to the objects of experience is not entailed by showing that the categories are instantiated in the experience of objects. Thirdly, I defend my claim that the derivation of the categories is a proper deduction, by answering the critique of a level confusion in my argument. This concerns a methodological point about the way TD proceeds, and why it involves self-consciousness. In the last part of the article, I respond to the incisive question, raised by Marcel Quarfood, whether it is at all possible to derive the category of contingency from within the first-person perspective. In formulating an answer, I point out that we can only have a negative concept of contingency, which at the same time shows the limits of the transcendental-subjective perspective. Ironically, on account of Kant’s radical subjectivism about the possibility of knowledge we are at the same time barred from accessing what is truly merely subjectively valid. This shows that Kant’s radical subjectivism is not a psychological subjectivism. (shrink)

In this paper, I discuss the debate on Kant and nonconceptual content in the context of the main argument of the B-Deduction. Kantian conceptualists (Bowman 2011; Griffith 2012; Gomes 2014) have responded to the recent nonconceptualist offensive, with reference to A89ff./B122ff. (§13)—which, confusingly, the nonconceptualists also cite as evidence for their contrary reading—by arguing that the nonconceptualist view conflicts with the central goal of TD, namely, to argue that all intuitions are subject to the categories. I contend that the (...) conceptualist reading of A89ff./B122ff. is unfounded, but also that the nonconceptualists are wrong to believe that intuitions as such refer strictly to objects independently of the functions of the understanding, and that they are mistaken about the relation between figurative synthesis and intellectual synthesis. I argue that Kant is a conceptualist, albeit not in the sense that standard conceptualists assume. Perceptual knowledge is always judgemental, though without this resulting in the standard conceptualist claim that, necessarily, all intuitions or all perceptions per se stand under the categories (strong conceptualism). I endorse the nonconceptualist view that, for Kant, perception per se, i.e. any mere or ‘blind’ intuition of objects (i.e. objects as indeterminate appearances) short of perceptual knowledge, does not necessarily stand under the categories. Perception is not yet perceptual knowledge. In this context, I point out the common failure in the literature on TD, both of the conceptualist and nonconceptualist stripe, to take account of the modal nature of Kant’s argument for the relation between intuition and concept insofar as cognition should arise from it. (shrink)

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.

This presentation of Aristotle's natural deduction system supplements earlier presentations and gives more historical evidence. Some fine-tunings resulted from conversations with Timothy Smiley, Charles Kahn, Josiah Gould, John Kearns,John Glanvillle, and William Parry.The criticism of Aristotle's theory of propositions found at the end of this 1974 presentation was retracted in Corcoran's 2009 HPL article "Aristotle's demonstrative logic".

For deductive reasoning to be justified, it must be guaranteed to preserve truth from premises to conclusion; and for it to be useful to us, it must be capable of informing us of something. How can we capture this notion of information content, whilst respecting the fact that the content of the premises, if true, already secures the truth of the conclusion? This is the problem I address here. I begin by considering and rejecting several accounts of informational content. I (...) then develop an account on which informational contents are indeterminate in their membership. This allows there to be cases in which it is indeterminate whether a given deduction is informative. Nevertheless, on the picture I present, there are determinate cases of informative (and determinate cases of uninformative) inferences. I argue that the model I offer is the best way for an account of content to respect the meaning of the logical constants and the inference rules associated with them without collapsing into a classical picture of content, unable to account for informative deductive inferences. (shrink)

One of the strongest motivations for conceptualist readings of Kant is the belief that the Transcendental Deduction is incompatible with nonconceptualism. In this article, I argue that this belief is simply false: the Deduction and nonconceptualism are compatible at both an exegetical and a philosophical level. Placing particular emphasis on the case of non-human animals, I discuss in detail how and why my reading diverges from those of Ginsborg, Allais, Gomes and others. I suggest ultimately that it is (...) only by embracing nonconceptualism that we can fully recognise the delicate calibration of the trap which the Critique sets for Hume. (shrink)

Varieties of natural deduction systems are introduced for Wansing’s paraconsistent non-commutative substructural logic, called a constructive sequential propositional logic (COSPL), and its fragments. Normalization, strong normalization and Church-Rosser theorems are proved for these systems. These results include some new results on full Lambek logic (FL) and its fragments, because FL is a fragment of COSPL.

Critical Notice of Kant's Transcendental Deduction, by Henry Allison. Oxford: Oxford University Press, 2015. Pp. Xv + 477.

I give an argument against nonconceptualist readings of Kants claim that intuitions and concepts constitute two distinct kinds of representation than is assumed by proponents of nonconceptualist readings. I present such an interpretation and outline the alternative reading of the Deduction that results.

In my reply to the respective critiques by Corey Dyck, Marcel Quarfood and Andrew Stephenson of my book Kant’s Deduction and Apperception: Explaining the Categories (Palgrave 2012), I go over some of the main planks of my interpretation of the first step of the B-Deduction. In response to Dyck, I explain that there are several reasons why I believe that the deduction of the categories must indeed be seen as a logical derivation from the unity of apperception, (...) and also why this view of the Transcendental Deduction does not make the Metaphysical Deduction redundant. Furthermore, I maintain that, pace Dyck, the analytic unity of consciousness is crucially central to the argument of § 16 of the Deduction. Lastly, I argue that the categories are more intimately related to the functions of judgement than Dyck makes them out to be. In response to Stephenson, I argue that the progressive argument of the Transcendental Deduction should not be construed as a transcendental argument against the sceptic. Secondly, I criticise his reading of the reciprocity between the subject and object of experience. In this context, I also point out that Stephenson misconstrues my reading of the supposed gap in Kant’s reasoning, and argue that his fourfold gap is not pertinent to Kant. Thirdly, I defend my claim that the derivation of the categories is a proper deduction, by answering Stephenson’s critique of a level confusion in my argument and pointing out why he is mistaken to think that showing that the categories are applied to the objects of experience is not entailed by showing that the categories are instantiated in the experience of objects. In the last part of the paper, I respond to Quarfood’s question whether it is at all possible to derive the category of contingency from within the first-person perspective. I attempt to formulate an answer, while pointing out that we can only have a negative concept of contingency. (shrink)

It is tempting to think that multi premise closure creates a special class of paradoxes having to do with the accumulation of risks, and that these paradoxes could be escaped by rejecting the principle, while still retaining single premise closure. I argue that single premise deduction is also susceptible to risks. I show that what I take to be the strongest argument for rejecting multi premise closure is also an argument for rejecting single premise closure. Because of the symmetry (...) between the principles, they come as a package: either both will have to be rejected or both will have to be revised. (shrink)

It is commonly claimed that the conclusion of a valid deductive argument is contained in its premises and says nothing new. In 'Deduction and Novelty,' in The Reasoner 5 (4), pp. 56-57, I refuted that claim. In The Reasoner, 8 (3), pp. 24-25, David McBride criticised my refutation. I show that McBride’s arguments are unsound.

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a (...) way that restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction-and cut-free sequent calculus equivalent to the Hubert system for basic modal logic shows the standard failure argument untenable by proving the underivability of DA from A. (shrink)

This essay partly builds on and partly criticizes a striking idea of Dieter Henrich. Henrich argues that Kant's distinction in the first Critique between the question of fact (quid facti) and the question of law (quid juris) provides clues to the argumentative structure of a philosophical "Deduction". Henrich suggests that the unity of apperception plays a role analogous to a legal factum. By contrast, I argue, first, that the question of fact in the first Critique is settled by the (...) Metaphysical Deduction, which establishes the purity of origin of the Categories, and, second, that in the second Critique, the relevant factum is the Fact of Reason, which amounts to the fact that the Moral Law is pure in origin. (shrink)

In this paper, I argue that in the sense of greatest epistemological concern for Kant, empirical cognition is “rational sensory discrimination”: the identification or differentiation of sensory objects from each other, occurring through a capacity to become aware of and express judgments. With this account of empirical cognition, I show how the transcendental deduction of the first Critique is most plausibly read as having as its fundamental assumption the thesis that we have empirical cognition, and I provide evidence that (...) Kant understood Hume as granting this assumption. (shrink)

James Van Cleve has argued that Kant’s Transcendental Deduction of the categories shows, at most, that we must apply the categories to experience. And this falls short of Kant’s aim, which is to show that they must so apply. In this discussion I argue that once we have noted the differences between the first and second editions of the Deduction, this objection is less telling. But Van Cleve’s objection can help illuminate the structure of the B Deduction, (...) and it suggests an interesting reason why the rewriting might have been thought necessary. (shrink)

In the transcendental deduction, the central argument of the Critique of Pure Reason, Kant seeks to secure the objective validity of our basic categories of thought. He distinguishes objective and subjective sides of this argument. The latter side, the subjective deduction, is normally understood as an investigation of our cognitive faculties. It is identified with Kant’s account of a threefold synthesis involved in our cognition of objects of experience, and it is said to precede and ground Kant’s proof (...) of the validity of the categories in the objective deduction. I challenge this standard reading of the subjective deduction, arguing, first, that there is little textual evidence for it, and, second, that it encourages a problematic conception of how the deduction works. In its place, I present a new reading of the subjective deduction. Rather than being a broad investigation of our cognitive faculties, it should be seen as addressing a specific worry that arises in the course of the objective deduction. The latter establishes the need for a necessary connection between our capacities for thinking and being given objects, but Kant acknowledges that his readers might struggle to comprehend how these seemingly independent capacities are coordinated. Even worse, they might well believe that in asserting this necessary connection, Kant’s position amounts to an implausible subjective idealism. The subjective deduction ismeant to allay these concerns by showing that they rest on a misunderstanding of the relation between these faculties. This new reading of the subjective deduction offers a better fit with Kant’s text. It also has broader implications, for it reveals the more philosophically plausible account of our relation to the world as thinkers that Kant is defending – an account that is largely obscured by the standard reading of the subjective deduction. (shrink)

It is often taken for granted by writers who propose--and, for that matter, by writers who oppose--'justifications' of inductions, that deduction either does not need, or can readily be provided with, justification. The purpose of this paper is to argue that, contrary to this common opinion, problems analogous to those which, notoriously, arise in the attempt to justify induction, also arise in the attempt to justify deduction.

The definitions of ‘deduction’ found in virtually every introductory logic textbook would encourage us to believe that the inductive/deductive distinction is a distinction among kinds of arguments and that the extension of ‘deduction’ is a determinate class of arguments. In this paper, we argue that that this approach is mistaken. Specifically, we defend the claim that typical definitions of ‘deduction’ operative in attempts to get at the induction/deduction distinction are either too narrow or insufficiently precise. We (...) conclude by presenting a deflationary understanding of the inductive/deductive distinction; in our view, its content is nothing over and above the answers to two fundamental sorts of questions central to critical thinking. (shrink)

Gentzen’s and Jaśkowski’s formulations of natural deduction are logically equivalent in the normal sense of those words. However, Gentzen’s formulation more straightforwardly lends itself both to a normalization theorem and to a theory of “meaning” for connectives . The present paper investigates cases where Jaskowski’s formulation seems better suited. These cases range from the phenomenology and epistemology of proof construction to the ways to incorporate novel logical connectives into the language. We close with a demonstration of this latter aspect (...) by considering a Sheffer function for intuitionistic logic. (shrink)

Building on the work of Peter Hinst and Geo Siegwart, we develop a pragmatised natural deduction calculus, i.e. a natural deduction calculus that incorporates illocutionary operators at the formal level, and prove its adequacy. In contrast to other linear calculi of natural deduction, derivations in this calculus are sequences of object-language sentences which do not require graphical or other means of commentary in order to keep track of assumptions or to indicate subproofs. (Translation of our German paper (...) "Ein Redehandlungskalkül. Ein pragmatisierter Kalkül des natürlichen Schließens nebst Metatheorie"; online available at http://philpapers.org/rec/CORERE.). (shrink)

We argue that the need for commentary in commonly used linear calculi of natural deduction is connected to the “deletion” of illocutionary expressions that express the role of propositions as reasons, assumptions, or inferred propositions. We first analyze the formalization of an informal proof in some common calculi which do not formalize natural language illocutionary expressions, and show that in these calculi the formalizations of the example proof rely on commentary devices that have no counterpart in the original proof. (...) We then present a linear natural deduction calculus that makes use of formal illocutionary expressions in such a way that unique readability for derivations is guaranteed – thus showing that formalizing illocutionary expressions can eliminate the need for commentary. (shrink)

This is a companion paper to Braüner where a natural deduction system for propositional hybrid logic is given. In the present paper we generalize the system to the first-order case. Our natural deduction system for first-order hybrid logic can be extended with additional inference rules corresponding to conditions on the accessibility relations and the quantifier domains expressed by so-called geometric theories. We prove soundness and completeness and we prove a normalisation theorem. Moreover, we give an axiom system first-order (...) hybrid logic. (shrink)

Considering the instability of nonlinear dynamics, the deductive inference rule Modus ponens itself is not enough to guarantee the validity of reasoning sequences in the real physical world, and similar results cannot necessarily be obtained from similar causes. Some kind of stability hypothesis should be added in order to draw meaningful conclusions. Hence, the uncertainty of deductive inference appears to be like that of inductive inference, and the asymmetry between deduction and induction becomes unrecognizable such as to undermine the (...) basis for the fundamental cleavage between analytic truth and synthetic truth, as W. V. O. Quine pointed out. Induction is not inferior to deduction from a pragmatic point of view. (shrink)

I take up Kant's remarks about a " transcendental deduction" of the "concepts of space and time". I argue for the need to make a clearer assessment of the philosophical resources of the Aesthetic in order to account for this transcendental deduction. Special attention needs to be given to the fact that the central task of the Aesthetic is simply the "exposition" of these concepts. The Metaphysical Exposition reflects upon facts about our usage to reveal our commitment to (...) the idea that these concepts refer to pure intuitions. But the legitimacy of these concepts still hangs in the balance: these concepts may turn out to refer to nothing real at all. The subsequent Transcendental Exposition addresses this issue. The objective validity of the concepts of space and time, and hence their transcendental deduction, hinges on careful treatment of this last point. (shrink)

This article argues that many (often Anglophone) interpreters of the Deduction have mistakenly identified Kant's aim as vindicating ordinary knowledge of objects and as refuting Hume's (alleged) skepticism about such knowledge. Instead, the article contends that Kant's aims were primarily negative. His primary mission (in the Deduction) was not to justify application of the categories to experience, but to show that any use beyond the domain of experience could not be justified. To do this, he needed to show (...) that their proper use in attaining metaphysical knowledge was restricted to (actual and possible) experience. The central theoretical claims of previous metaphysics were thus rendered void. Kant was not out to save ordinary knowledge from the skeptic, at least not originally, since he did not think such knowledge was in danger. Rather, he wanted to sustain the skeptical claim that we cannot justify metaphysical claims about things in themselves, hence that we cannot gain metaphysical knowledge about the existence of God, the immortality of the soul, the beginning of time, and the least parts of matter. He wanted to achieve this result in a more theoretically satisfying manner than had skeptical authors. Doing so would also allow him to curb certain forms of skeptical empiricism, by showing that we cannot disprove, or rule out as unintelligible, human freedom, an afterlife, or divine providence. The article ends by considering the retrospective rereadings of the Deduction by Hermann Cohen and P. F. Strawson, concluding that these rereadings force Kant's text into an alien mold, thereby diminishing its philosophical value as a work with its own distinctive aims and methods. (shrink)

In this paper two different natural deduction systems forhybrid logic are compared and contrasted.One of the systems was originally given by the author of the presentpaper whereasthe other system under consideration is a modifiedversion of a natural deductionsystem given by Jerry Seligman.We give translations in both directions between the systems,and moreover, we devise a set of reduction rules forthe latter system bytranslation of already known reduction rules for the former system.

I argue for a novel, non-subjectivist interpretation of Kant’s transcendental idealism. Kant’s idealism is often interpreted as specifying how we must experience objects or how objects must appear to us. I argue to the contrary by appealing to Kant’s Transcendental Deduction. Kant’s Deduction is the proof that the categories are not merely subjectively necessary conditions we need for our cognition, but objectively valid conditions necessary for objects to be appearances. My interpretation centres on two claims. First, Kant’s method (...) of self-knowledge consists in his determining what makes our cognitive faculty finite in contrast to God’s infinite cognitive faculty. Second, Kant’s limitation of our knowledge to appearances consists in his developing an account according to which appearances and our finite cognitive faculty are conceived of in terms of each other and in contrast to noumena in the positive sense and God’s infinite cognitive faculty. (shrink)

The relation between logic and thought has long been controversial, but has recently influenced theorizing about the nature of mental processes in cognitive science. One prominent tradition argues that to explain the systematicity of thought we must posit syntactically structured representations inside the cognitive system which can be operated upon by structure sensitive rules similar to those employed in systems of natural deduction. I have argued elsewhere that the systematicity of human thought might better be explained as resulting from (...) the fact that we have learned natural languages which are themselves syntactically structured. According to this view, symbols of natural language are external to the cognitive processing system and what the cognitive system must learn to do is produce and comprehend such symbols. In this paper I pursue that idea by arguing that ability in natural deduction itself may rely on pattern recognition abilities that enable us to operate on external symbols rather than encodings of rules that might be applied to internal representations. To support this suggestion, I present a series of experiments with connectionist networks that have been trained to construct simple natural deductions in sentential logic. These networks not only succeed in reconstructing the derivations on which they have been trained, but in constructing new derivations that are only similar to the ones on which they have been trained. (shrink)

A truth-preservation fallacy is using the concept of truth-preservation where some other concept is needed. For example, in certain contexts saying that consequences can be deduced from premises using truth-preserving deduction rules is a fallacy if it suggests that all truth-preserving rules are consequence-preserving. The arithmetic additive-associativity rule that yields 6 = (3 + (2 + 1)) from 6 = ((3 + 2) + 1) is truth-preserving but not consequence-preserving. As noted in James Gasser’s dissertation, Leibniz has been criticized (...) for using that rule in attempting to show that arithmetic equations are consequences of definitions. -/- A system of deductions is truth-preserving if each of its deductions having true premises has a true conclusion—and consequence-preserving if, for any given set of sentences, each deduction having premises that are consequences of that set has a conclusion that is a consequence of that set. Consequence-preserving amounts to: in each of its deductions the conclusion is a consequence of the premises. The same definitions apply to deduction rules considered as systems of deductions. Every consequence-preserving system is truth-preserving. It is not as well-known that the converse fails: not every truth-preserving system is consequence-preserving. Likewise for rules: not every truth-preserving rule is consequence-preserving. There are many famous examples. In ordinary first-order Peano-Arithmetic, the induction rule yields the conclusion ‘every number x is such that: x is zero or x is a successor’—which is not a consequence of the null set—from two tautological premises, which are consequences of the null set, of course. The arithmetic induction rule is truth-preserving but not consequence-preserving. Truth-preserving rules that are not consequence-preserving are non-logical or extra-logical rules. Such rules are unacceptable to persons espousing traditional truth-and-consequence conceptions of demonstration: a demonstration shows its conclusion is true by showing that its conclusion is a consequence of premises already known to be true. The 1965 Preface in Benson Mates (1972, vii) contains the first occurrence of truth-preservation fallacies in the book. (shrink)

The paper studies the extension of harmony and stability, major themes in proof-theoretic semantics, from single-conclusion natural-deduction systems to multiple -conclusions natural-deduction, independently of classical logic. An extension of the method of obtaining harmoniously-induced general elimination rules from given introduction rules is suggested, taking into account sub-structurality. Finally, the reductions and expansions of the multiple -conclusions natural-deduction representation of classical logic are formulated.

This paper raises obvious questions undermining any residual confidence in Mates work and revealing our embarrassing ignorance of true nature of Stoic deduction. It was inspired by the challenging exploratory work of JOSIAH GOULD.

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)

Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT —a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has a (...) contextual DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case. (shrink)

In this paper, I claim that Kant’s subjective deduction in the first edition of the KrV is to be understood in terms of an investigation of the fundamental force(s) (Grundkraft) of the soul, an investigation essential to Wolffian psychology and much debated throughout Germany in the second half of the 1700’s. I argue that the subjective deduction is indeed presented by means of the exposition of the three-fold syntheses but only insofar as these syntheses are employed as pointers (...) towards each of three original sources of cognition. Only taken as such can we do justice not only to the argument’s unique concern as distinct from that of the objective deduction, but also to its dependence upon the justificatory results of that deduction which was after all Kant’s “primary concern”. (shrink)

In this paper, it is argued that only in the section on dialectic in the Critique of Judgment does Kant reach a definitive and conclusive version of deduction, after discovering the concept of the supersensible. In the section on the deduction of pure aesthetic judgments, Kant does not satisfactorily explain the critical distinction between the sensible nature of humanity and the supersensible nature of human reason presupposed in the concept of universal communicability. While the concept of the supersensible (...) illustrates this distinction, it is only through this concept that Kant that can justify the specific possibility of claiming subjective validity in taste. The priority of the solution found in the dialectic is illustrated not only by a comparative analysis of the two sections, but also by a historical reconstruction of the process of the formation of the work, which shows that the first formulation of the concept of validity coincides with the use of the concept of the supersensible. (shrink)

Deductive inference is usually regarded as being "tautological" or "analytical": the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of first-order logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view. We propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of growing (...) computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of "depth" or "informativeness" of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure "intelim logic", which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is "analytic" in a particularly strict sense, in that it rules out any use of "virtual information", which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed. (shrink)

In this paper we will show Peirce’s distinction between deduction, induction and abduction. The aim of the paper is to show how Peirce changed his views on the subject, from an understanding of deduction, induction and hypotheses as types of reasoning to understanding them as stages of inquiry very tightly connected. In order to get a better understanding of Peirce’s originality on this, we show Peirce’s distinctions between qualitative and quantitative induction and between theorematical and corollarial deduction, (...) passing then to the distinction between mathematics and logic. In the end, we propose a sketch of a comparison between Peirce and Whitehead concerning the two thinkers’ view of mathematics, hoping that this could point to further inquiries. (shrink)

Gentzen's “Untersuchungen” [1] gave a translation from natural deduction to sequent calculus with the property that normal derivations may translate into derivations with cuts. Prawitz in [8] gave a translation that instead produced cut-free derivations. It is shown that by writing all elimination rules in the manner of disjunction elimination, with an arbitrary consequence, an isomorphic translation between normal derivations and cut-free derivations is achieved. The standard elimination rules do not permit a full normal form, which explains the cuts (...) in Gentzen's translation. Likewise, it is shown that Prawitz' translation contains an implicit process of cut elimination. (shrink)

Kant’s claim in the Subjective Deduction that we have multiple fundamental mental powers appears to be susceptible to some a priori metaphysical arguments made against multiple fundamental mental powers by Christian Wolff who held that these powers would violate the unity of thought and entail that the soul is an extended composite. I argue, however, that in the Second Paralogism and his lectures on metaphysics, Kant provides arguments that overcome these objections by showing that it is possible that a (...) composite could ground the unity of thought, that properties are powers and therefore the soul could possess multiple powers, and the soul is a thing in itself so it cannot be an extended composite. These arguments lend additional support to the attribution of multiple mental powers to us in the Subjective Deduction. (shrink)

This paper provides a finer analysis of the well-known form of the Local Deduction Theorem in contraction-free logics . An infinite hierarchy of its natural strengthenings is introduced and studied. The main results are the separation of its initial four members and the subsequent collapse of the hierarchy.

In this paper we present labelled sequent calculi and labelled natural deduction calculi for the counterfactual logics CK + {ID, MP}. As for the sequent calculi we prove, in a semantic manner, that the cut-rule is admissible. As for the natural deduction calculi we prove, in a purely syntactic way, the normalization theorem. Finally, we demonstrate that both calculi are sound and complete with respect to Nute semantics [12] and that the natural deduction calculi can be effectively (...) transformed into the sequent calculi. (shrink)

Harold Hodes in [1] introduces an extension of first-order modal logic featuring a backtracking operator, and provides a possible worlds semantics, according to which the operator is a kind of device for ‘world travel’; he does not provide a proof theory. In this paper, I provide a natural deduction system for modal logic featuring this operator, and argue that the system can be motivated in terms of a reading of the backtracking operator whereby it serves to indicate modal scope. (...) I prove soundness and completeness theorems with respect to Hodes’ semantics, as well as semantics with fewer restrictions on the accessibility relation. (shrink)

Le « cogito, ergo sum » cartésien apparaît depuis quarante ans comme « inférence et performance » (J. Hintikka). Mais de quelle inférence s'agit-il précisément ? Pour le savoir, cet article poursuit deux objectifs : d'abord, montrer que la question pertinente à laquelle il s'agit de répondre ne concerne pas la relation logique interne qui lie le cogito au sum, et qui est une intuition, mais celle, externe, qui lie le « cogito, ergo sum » tout entier au « quicquid (...) cogitat, est ». Ensuite, montrer que cette dernière relation est tout à la fois une induction et une déduction. (shrink)

In this paper, I will present a Fitch–style natural deduction proof theory for modal paralogics (modal logics with gaps and/or gluts for negation). Besides the standard classical subproofs, the presented proof theory also contains modal subproofs, which express what would follow from a hypothesis, in case it would be true in some arbitrary world.

This paper studies, with techniques ofAlgebraic Logic, the effects of putting a bound on the cardinality of the set of side formulas in the Deduction Theorem, viewed as a Gentzen-style rule, and of adding additional assumptions inside the formulas present in Modus Ponens, viewed as a Hilbert-style rule. As a result, a denumerable collection of new Gentzen systems and two new sentential logics have been isolated. These logics are weaker than the positive implicative logic. We have determined their algebraic (...) models and the relationships between them, and have classified them according to several standard criteria of Abstract Algebraic Logic. One of the logics is protoalgebraic but neither equivalential nor weakly algebraizable, a rare situation where very few natural examples were hitherto known. In passing we have found new, alternative presentations of positive implicative logic, both in Hilbert style and in Gentzen style, and have characterized it in terms of the restricted Deduction Theorem: it is the weakest logic satisfying Modus Ponens and the Deduction Theorem restricted to at most 2 side formulas. The algebraic part of the work has lead to the class of quasi-Hilbert algebras, a quasi-variety of implicative algebras introduced by Pla and Verdú in 1980, which is larger than the variety of Hilbert algebras. Its algebraic properties reflect those of the corresponding logics and Gentzen systems. (shrink)

Many believe that Goodman’s new riddle of induction proves the impossibility of a purely syntactical theory of confirmation. After discussing and rejecting Jackson’s solution to Goodman’s paradox, I formulate the “new riddle of deduction,” in analogy to the new riddle of induction. Since it is generally agreed that deductive validity can be defined syntactically, the new riddle of induction equally does not show that inductive validity cannot be defined syntactically. I further rely on the analogy between induction and (...) class='Hi'>deduction in order to explain why some predicates, such as “grue,” are unprojectible. (shrink)

In this paper we deal with two types of reasoning: induction, and deduction First, we present a unified computational model of deductive reasoning through models, where deduction occurs in five phases: Construction, Integration, Conclusion, Falsification, and Response. Second, we make an attempt, to analyze induction through the same phases. Our aim is an explorative evaluation of the mental processes possibly shared by deductive and inductive reasoning.