Deductive reasoning is the kind of reasoning in which, roughly, the truth of the input propositions (the premises) logically guarantees the truth of the output proposition (the conclusion), provided that no mistake has been made in the reasoning. The premises may be propositions that the reasoner believes or assumptions that the reasoner is exploring. Deductive reasoning contrasts with inductive reasoning, the kind of reasoning in which the truth of the premises need not guarantee the truth of the conclusion.

Cette étude montre comment le météorologue Edward Lorenz, dans deux articles de 1963 et 1964, explore les propriétés des systèmes chaotiques par des allers-retours entre une déduction mathématique (basée sur la théorie des systèmes dynamiques) et une étude des solutions numériques du système dit « de Lorenz » dans un régime d’instabilité. This study aims at showing how the metereologist Edward Lorenz, in two papers of 1963 and 1964, explores the properties of chaotic systems thanks to the interplay between a (...) mathematical deduction (based on dynamical systems theory) and a study of the mathematical solutions of the Lorenz system in an instability regime. (shrink)

Duncan Pritchard recently proposed a Wittgensteinian solution to closure-based skepticism. According to Wittgenstein, all epistemic systems assume certain truths. The notions that we are not disembodied brains, that the Earth has existed for a long time and that one’s name is such-and-such all function as “hinge commitments.” Pritchard views a hinge commitment as a positive propositional attitude that is not a belief. Because closure principles concern only knowledge-apt beliefs, they do not apply to hinge commitments. Thus, from the fact that (...) a subject knows that he is sitting in a room, and the fact that the subject’s sitting in a room entails his bodily existence, it does not follow that the subject also knows that he is not an envatted brain. This paper rejects Pritchard’s non-belief reading of hinge commitments. I start by showing that the non-belief reading fails to solve the skeptical paradox because the reasons that Pritchard uses to support the non-belief reading do not exempt hinge propositions from closure principles. I then proceed to argue that the non-belief reading is false as it claims that hinge commitments, unlike ordinary beliefs, are rationally unresponsive—with the help of a scenario in which a subject’s experience is internally chaotic, we can safely conclude that the hinge commitment that one is not systematically mistaken about the world is equally responsive to one’s evidential situations. (shrink)

I discuss three elements of Dennis Schulting’s new book on the transcendental deduction of the pure concepts of the understanding, or categories. First, that Schulting gives a detailed account of the role of each individual category. Second, Schulting’s insistence that the categories nevertheless apply ‘en bloc’. Third, Schulting’s defence of Kant’s so-called reciprocity thesis that subjective unity of consciousness and objectivity in the sense of cognition’s objective purport are necessary conditions for the possibility of one another. I endorse these (...) fascinating but unfashionable claims and sketch my own version of what they amount to, which is quite different to Schulting’s own construal. I point to some fundamental limitations and problems for Schulting’s position and argue that his project needs to be reshaped or at least reconceived in the face of them. Even if Schulting’s argument is sound, it does not provide a deduction, properly speaking, of the categories. (shrink)

Deductive Cogency holds that the set of propositions towards which one has, or is prepared to have, a given type of propositional attitude should be consistent and closed under logical consequence. While there are many propositional attitudes that are not subject to this requirement, e.g. hoping and imagining, it is at least prima facie plausible that Deductive Cogency applies to the doxastic attitude involved in propositional knowledge, viz. belief. However, this thought is undermined by the well-known preface paradox, leading a (...) number of philosophers to conclude that Deductive Cogency has at best a very limited role to play in our epistemic lives. I argue here that Deductive Cogency is still an important epistemic requirement, albeit not as a requirement on belief. Instead, building on a distinction between belief and acceptance introduced by Jonathan Cohen and recent developments in the epistemology of understanding, I propose that Deductive Cogency applies to the attitude of treating propositions as given in the context of attempting to understand a given phenomenon. I then argue that this simultaneously accounts for the plausibility of the considerations in favor of Deductive Cogency and avoids the problematic consequences of the preface paradox. (shrink)

This book stands at the intersection of two topics: the decidability and computational complexity of hybrid logics, and the deductive systems designed for them. Hybrid logics are here divided into two groups: standard hybrid logics involving nominals as expressions of a separate sort, and non-standard hybrid logics, which do not involve nominals but whose expressive power matches the expressive power of binder-free standard hybrid logics.The original results of this book are split into two parts. This division reflects the division of (...) the book itself. The first type of results concern model-theoretic and complexity properties of hybrid logics. Since hybrid logics which we call standard are quite well investigated, the efforts focused on hybrid logics referred to as non-standard in this book. Non-standard hybrid logics are understood as modal logics with global counting operators ) whose expressive power matches the expressive power of binder-free standard hybrid logics. The relevant results comprise: 1. Establishing a sound and complete axiomatization for the modal logic K with global counting operators ), which can be easily extended onto other frame classes, 2. Establishing tight complexity bounds, namely NExpTime-completeness for the modal logic with global counting operators defined over the classes of arbitrary, reflexive, symmetric, serial and transitive frames ), MT), MD), MB), MK4) with numerical subscripts coded in binary. Establishing the exponential-size model property for this logic defined over the classes of Euclidean and equivalential frames ), MS5).Results of the second type consist of designing concrete deductive systems for standard and non-standard hybrid logics. More precisely, they include: 1. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for a rich class of binder-free standard hybrid logics. An interesting feature of indicated calculi is the nonbranching character of the rule, 2. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for non-standard hybrid logics. The internalization technique applied to a tableau calculus for the modal logic with global counting operators is novel in the literature, 3. Devising the first hybrid algorithm involving an inequality solver for modal logics with global counting operators. Transferring the arithmetical part of reasoning to an inequality solver turned out to be sufficient in ensuring termination.The book is directed to philosophers and logicians working with modal and hybrid logics, as well as to computer scientists interested in deductive systems and decision procedures for logics. Extensive fragments of the first part of the book can also serve as an introduction to hybrid logics for wider audience interested in logic.The content of the book is situated in the areas of formal logic and theoretical computer science with some elements of the theory of computational complexity. (shrink)

In focusing on the systematic deduction of the categories from a principle, Schulting takes up anew the controversial project of the eminent German Kant scholar Klaus Reich, whose monograph “The Completeness of Kant's Table of Judgments” made the case that the logical functions of judgement can all be derived from the objective unity of apperception and can be shown to link up with one another systematically. -/- Common opinion among Kantians today has it that Kant did not mean to (...) derive the functions of judgement, and accordingly the categories, from the principle of apperception. Schulting challenges this standard view and aims to resuscitate the main motivation behind Reich’s project. He argues, in agreement with Reich’s main thesis about the derivability of the functions of judgement, that Kant indeed does mean to derive, in full a priori fashion, the categories from the principle of apperception. -/- Schulting also shows that, given the general assumptions of the Critical philosophy, Kant's derivation is successful and that absent an account of the derivation of the categories from apperception, the B-Deduction cannot really be understood. -/- New edition. First published 2012 as „Kant’s Deduction and Apperception. Explaining the Categories" (Palgrave Macmillan). (shrink)

In this study on deduction, the authors argue that people reason by imagining the relevant state of affairs, ie building an internal model of it, formulating a tentative conclusion based on this model and then searching for alternative models.

This volume examines the notion of an analytic proof as a natural deduction, suggesting that the proof's value may be understood as its normal form--a concept with significant implications to proof-theoretic semantics.

We present a sound and complete Fitch-style natural deduction system for an S5 modal logic containing an actuality operator, a diagonal necessity operator, and a diagonal possibility operator. The logic is two-dimensional, where we evaluate sentences with respect to both an actual world (first dimension) and a world of evaluation (second dimension). The diagonal necessity operator behaves as a quantifier over every point on the diagonal between actual worlds and worlds of evaluation, while the diagonal possibility quantifies over some (...) point on the diagonal. Thus, they are just like the epistemic operators for apriority and its dual. We take this extension of Fitch’s familiar derivation system to be a very natural one, since the new rules and labeled lines hereby introduced preserve the structure of Fitch’s own rules for the modal case. (shrink)

In this paper I focus on two contrasting concepts of deduction and induction that have appeared in introductory (formal) logic texts over the past 75 years or so. According to the one, deductive and inductive arguments are defined solely by reference to what arguers claim about the relation between the premises and the conclusions. According to the other, they are defined solely by reference to that relation itself. Arguing that these definitions have defects that are due to their simplicity, (...) I develop definitions that remove these defects by assigning a combination of roles to both arguers’ claims concerning the premises/conclusion relation and the relation itself. Along the way I also present and briefly defend definitions of both deductive and inductive validity that are significantly different from the norm. (shrink)

In philosophy, an argument consists of a set of statements called premises that serve as grounds for affirming another statement called the conclusion. Philosophers typically distinguish arguments in natural languages (such as English) into two fundamentally different types: deductive and inductive. Each type of argument is said to have characteristics that categorically distinguish it from the other type. The two types of argument are also said to be subject to differing evaluative standards. Pointing to paradigmatic examples of each type of (...) argument helps to clarify their key differences. The distinction between the two types of argument may hardly seem worthy of philosophical reflection, as evidenced by the fact that their differences are usually presented as straightforward, such as in many introductory philosophy textbooks. Nonetheless, the question of how best to distinguish deductive from inductive arguments, and indeed whether there is a coherent categorical distinction between them at all, turns out to be considerably more problematic than commonly recognized. This article identifies and discusses a range of different proposals for marking categorical differences between deductive and inductive arguments while highlighting the problems and limitations attending each. Consideration is also given to the ways in which one might do without a distinction between two types of argument by focusing solely on the application of evaluative standards to arguments instead. (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 print edition of Read's account of logical thought includes the original publication's diagrams and tables. In this excellent book, Read commences by offering an overview of past attitudes and definitions of logic. Individual chapters consider the various means by which logical processes are conceived and developed in the mind. Philosophical arguments, spatial reasoning and mathematical forms of logic are discussed in great depth, with illustrations appended where deemed necessary. Read, an academic and philosopher, employs his decades long experience of (...) teaching students about the logical disciplines to write a detailed and comprehensive accounts of logic's aspects. Throughout this work, he attempts to retain a clarity of expression which makes the concepts discussed understandable for readers previously unversed in the types of logical expression. Read also sets out the principles behind setting up a hypothesis, and highlights the common logical fallacies which crop up in human reasoning. Extensive attention is given to the process of syllogism, by which sound conclusions are drawn from the premises offered. Throughout the book, Read is attentive in defining and explaining the esoteric terminology native to logic. As well as allowing the reader swift comprehension of the various subtopics, this explanatory style confers improved flow to the book. At the conclusion we are offered many questions corresponding to each chapter: these are intended to test the reader's understanding, and enforce the mental habits of utilising logical and critical thought. First published in 1898 and subject to several editions with revisions and refinements thereafter, Logic Deductive and Inductive retains a good reputation among academics and logicians. Its successful balance between striking a concise introductory tone with an embrace of the core elements of the subject has served to preserve its popularity into the modern day. (shrink)

Probabilistic models have started to replace classical logic as the standard reference paradigm in human deductive reasoning. Mental probability logic emphasizes general principles where human reasoning deviates from classical logic, but agrees with a probabilistic approach (like nonmonotonicity or the conditional event interpretation of conditionals). -/- This contribution consists of two parts. In the first part we discuss general features of reasoning systems including consequence relations, how uncertainty may enter argument forms, probability intervals, and probabilistic informativeness. These concepts are of (...) central importance for the psychological task analysis. In the second part we report new experimental data on the paradoxes of the material conditional, the probabilistic modus ponens, the complement task, and data on the probabilistic truth table task. The results of the experiments provide evidence for the hypothesis that people represent indicative conditionals by conditional probability assertions. (shrink)

The paper explores a deductive-nomological account of metaphysical explanation: some truths metaphysically explain, or ground, another truth just in case the laws of metaphysics determine the latter truth on the basis of the former. I develop and motivate a specific conception of metaphysical laws, on which they are general rules that regulate the existence and features of derivative entities. I propose an analysis of the notion of ‘determination via the laws’, based on a restricted form of logical entailment. I argue (...) that the DN-account of ground can be defended against the well-known objections to the DN-approach to scientific explanation. The goal of the paper is to show that the DN-account of metaphysical explanation is a well-motivated and defensible theory. (shrink)

The paper introduces a new type of rules into Natural Deduction, elimination rules by composition. Elimination rules by composition replace usual elimination rules in the style of disjunction elimination and give a more direct treatment of additive disjunction, multiplicative conjunction, existence quantifier and possibility modality. Elimination rules by composition have an enormous impact on proof-structures of deductions: they do not produce segments, deduction trees remain binary branching, there is no vacuous discharge, there is only few need of permutations. (...) This new type of rules fits especially to substructural issues, so it is shown for Lambek Calculus, i.e. intuitionistic non-commutative linear logic and to its extensions by structural rules like permutation, weakening and contraction. Natural deduction formulated with elimination rules by composition from a complexity perspective is superior to other calculi. (shrink)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', `heuristic' or (...) `pragmatic', but that there must be an element of what it is rational to believe on the evidence, that is, of non-deductive logic. (shrink)

ABSTRACTThe new paradigm in the psychology of reasoning redirects the investigation of deduction conceptually and methodologically because the premises and the conclusion of the inferences are assumed to be uncertain. A probabilistic counterpart of the concept of logical validity and a method to assess whether individuals comply with it must be defined. Conceptually, we used de Finetti's coherence as a normative framework to assess individuals' performance. Methodologically, we presented inference schemas whose premises had various levels of probability that contained (...) non-numerical expressions and, as a control, sure levels. Depending on the inference schemas, from 60% to 80% of the participants produced coherent conclusions when the premises were uncertain. The data also show that except for schemas involving conjunction, performance was consistently lower with certain than uncertain premises, the rate of conjunction fallacy was consistently low (not exceeding 20%,.. (shrink)

This chapter describes the main accounts of deductive competence, which explain what is computed in carrying out deductions. It argues that people have a modicum of competence, which is useful in daily life and a prerequisite for acquiring logical expertise. It outlines the three main sorts of theory of deductive performance, which explain how people make deductions: They rely on factual knowledge, formal rules, or mental models. It reviews recent experimental studies of deductive reasoning in order to help readers to (...) assess these theories of performance. (shrink)

A simple summary of the difference between induction and deduction.

It has been the dominant view that probabilistic explanations of particular facts must be inductive in character. I argue here that this view is mistaken, and that the aim of probabilistic explanation is not to demonstrate that the explanandum fact was nomically expectable, but to give an account of the chance mechanism(s) responsible for it. To this end, a deductive-nomological model of probabilistic explanation is developed and defended. Such a model has application only when the probabilities occurring in covering laws (...) can be interpreted as measures of objective chance, expressing the strength of physical propensities. Unlike inductive models of probabilistic explanation, this deductive model stands in no need of troublesome requirements of maximal specificity or epistemic relativization. (shrink)

The present article illustrates a conflict between the claim that rational belief sets are closed under deductive consequences, and a very inclusive claim about the factors that are sufficient to determine whether it is rational to believe respective propositions. Inasmuch as it is implausible to hold that the factors listed here are insufficient to determine whether it is rational to believe respective propositions, we have good reason to deny that rational belief sets are closed under deductive consequences.

The discussion on mathematical explanation has inherited the same sense of dissatisfaction that philosophers of science expressed, in the context of scientific explanation, towards the deductive-nomological model. This model is regarded as unable to cover cases of bona fide mathematical explanations and, furthermore, it is largely ignored in the relevant literature. Surprisingly, the reasons for this ostracism are not sufficiently manifest. In this paper I explore a possible extension of the model to the case of mathematical explanations and I claim (...) that there are at least two reasons to judge the deductive-nomological picture of explanation as inadequate in that context. (shrink)

In this book, Alison Laywine considers the mystery of the Transcendental Deduction in Immanuel Kant's Critique of Pure Reason. What is it supposed to accomplish and how? Laywine argues that Kant's peculiar adaptation of his early account of a world is the key to this mystery. Collecting evidence from the Critique and other writings by Kant--in order to identify what he took himself to be doing on his own terms--she holds that Kant deliberately adapted elements of his early metaphysics (...) both to set the agenda of the Deduction and to carry it out. Laywine highlights how the most important metaphysical element in Kant's thought was his early account of a world; a world is not just the sum-total of all substances created by God, but a whole unified by the universal laws of community prescribed by God that externally relate any given substance to all others. From this conception of a world, Kant then extracted a way to conceive key elements in the Deduction: experience as the whole of all possible appearances unified by the universal laws human understanding gives to nature. (shrink)

The distinction between the syntactic and the semantic approach to scientific theories emerged in formal philosophy of science. The semantic approach is commonly considered more advanced and more successful than the syntactic one, but the transition from the one approach to the other was not brought about without any loss. In essence, it is the formal analysis of atomic propositions and the analysis of deductive reasoning that dropped out of consideration in at least some of the elaborated versions of the (...) semantic approach. In structuralist theory of science, as founded by Sneed and Stegmüller, the focus is on global propositions concerning the question of whether or not certain empirical systems satisfy a set-theoretic predicate that encodes the axioms of a scientific theory. Hence, an analysis of deductive reasoning from atomic premisses with the help of a given theory is missing. The objective of the present paper is to develop a deductive system on the basis of the structuralist framework. This system comes with a novel formulation of empirical propositions in structuralism. (shrink)

The paper discusses the origin of dark matter and dark energy from the concepts of time and the totality in the final analysis. Though both seem to be rather philosophical, nonetheless they are postulated axiomatically and interpreted physically, and the corresponding philosophical transcendentalism serves heuristically. The exposition of the article means to outline the “forest for the trees”, however, in an absolutely rigorous mathematical way, which to be explicated in detail in a future paper. The “two deductions” are two successive (...) stage of a single conclusion mentioned above. The concept of “transcendental invariance” meaning ontologically and physically interpreting the mathematical equivalence of the axiom of choice and the well-ordering “theorem” is utilized again. Then, time arrow is a corollary from that transcendental invariance, and in turn, it implies quantum information conservation as the Noether correlate of the linear “increase of time” after time arrow. Quantum information conservation implies a few fundamental corollaries such as the “conservation of energy conservation” in quantum mechanics from reasons quite different from those in classical mechanics and physics as well as the “absence of hidden variables” (versus Einstein’s conjecture) in it. However, the paper is concentrated only into the inference of another corollary from quantum information conservation, namely, dark matter and dark energy being due to entanglement, and thus and in the final analysis, to the conservation of quantum information, however observed experimentally only on the “cognitive screen” of “Mach’s principle” in Einstein’s general relativity. therefore excluding any other source of gravitational field than mass and gravity. Then, if quantum information by itself would generate a certain nonzero gravitational field, it will be depicted on the same screen as certain masses and energies distributed in space-time, and most presumably, observable as those dark energy and dark matter predominating in the universe as about 96% of its energy and matter quite unexpectedly for physics and the scientific worldview nowadays. Besides on the cognitive screen of general relativity, entanglement is available necessarily on still one “cognitive screen” (namely, that of quantum mechanics), being furthermore “flat”. Most probably, that projection is confinement, a mysterious and ad hoc added interaction along with the fundamental tree ones of the Standard model being even inconsistent to them conceptually, as far as it need differ the local space from the global space being definable only as a relation between them (similar to entanglement). So, entanglement is able to link the gravity of general relativity to the confinement of the Standard model as its projections of the “cognitive screens” of those two fundamental physical theories. (shrink)

Deductive Irrationality examines and critiques economic rationalism by assessing the work of influential political philosophers and economic theorists such as Thomas Hobbes, John Locke, Adam Smith, Friedrich Hayek, Gunnar Myrdal, and John F. Muth. It is one of the first serious attempts to investigate the dominant sub-fields in economic theory through the lens of political philosophy.

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)

Deductive inference seems to reveal semantic connections between their premise(s) and conclusion that were there all along. This looks inconsistent with Wittgenstein's later views on meaning. The paper argues that W's treatment of aspects suggests a Wittgensteinian treatment of deduction that accommodates the troublesome phenomenon without conceding its force.

Curry's paradox, sometimes described as a general version of the better known Russell's paradox, has intrigued logicians for some time. This paper examines the paradox in a natural deduction setting and critically examines some proposed restrictions to the logic by Fitch and Prawitz. We then offer a tentative counterexample to a conjecture by Tennant proposing a criterion for what is to count as a genuine paradox.

Hypothetico-deductive (H-D) confirmation builds on the idea that confirming evidence consists of successful predictions that deductively follow from the hypothesis under test. This article reviews scope, history and recent development of the venerable H-D account: First, we motivate the approach and clarify its relationship to Bayesian confirmation theory. Second, we explain and discuss the tacking paradoxes which exploit the fact that H-D confirmation gives no account of evidential relevance. Third, we review several recent proposals that aim at a sounder and (...) more comprehensive formulation of H-D confirmation. Finally, we conclude that the reputation of hypothetico-deductive confirmation as outdated and hopeless is undeserved: not only can the technical problems be addressed satisfactorily, the hypothetico-deductive method is also highly relevant for scientific practice. (shrink)

This paper defends the view that the classification of an argument as being deductive ought to rest exclusively upon psychological considerations; specifically, upon whether the argument's author holds certain beliefs. This account is justified on theoretical and pedagogical grounds, and situated within a general taxonomy of competing proposals. Epistemological difficulties involved in the application of psychological definitions are recognized but claimed to be ineliminable from the praetice of argumentation. The paper concludes by discussing embryonic arguments where the author's relevant beliefs (...) are not sufficiently fine-grained so as to accord the argument deductive or inductive status. (shrink)

This essay presents deductive arguments to an introductory-level audience via a discussion of Aristotle's three types of rhetoric, the goals of and differences between deductive and non-deductive arguments, and the major features of deductive arguments (e.g., validity and soundness).

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)

The structure of derivations in natural deduction is analyzed through isomorphism with a suitable sequent calculus, with twelve hidden convertibilities revealed in usual natural deduction. A general formulation of conjunction and implication elimination rules is given, analogous to disjunction elimination. Normalization through permutative conversions now applies in all cases. Derivations in normal form have all major premisses of elimination rules as assumptions. Conversion in any order terminates.Through the condition that in a cut-free derivation of the sequent Γ⇒C, no (...) inactive weakening or contraction formulas remain in Γ, a correspondence with the formal derivability relation of natural deduction is obtained: All formulas of Γ become open assumptions in natural deduction, through an inductively defined translation. Weakenings are interpreted as vacuous discharges, and contractions as multiple discharges. In the other direction, non-normal derivations translate into derivations with cuts having the cut formula principal either in both premisses or in the right premiss only. (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)

This paper presents an outline of a new theory of relevant deduction which arose from the purpose of solving paradoxes in various fields of analytic philosophy. In distinction to relevance logics, this approach does not replace classical logic by a new one, but distinguishes between relevance and validity. It is argued that irrelevant arguments are, although formally valid, nonsensical and even harmful in practical applications. The basic idea is this: a valid deduction is relevant iff no subformula of (...) the conclusion is replaceable on some of its occurrences by any other formula salva validitate of the deduction. The paper first motivates the approach by showing that four paradoxes seemingly very distant from each other have a common source. Then the exact definition of relevant deduction is given and its logical properties are investigated. An extension to relevance of premises is discussed. Finally the paper presents an overview of its applications in philosophy of science, ethics, cognitive psychology and artificial intelligence. (shrink)

Hybrid deduction-refutation systems are presented for four first degree entailment based logics. The hybrid systems are shown to deductively and refutationally sound and complete with respect to their logics. The proofs of completeness are presented in a uniform way. This paper builds on work in [6], where Goranko presented a deductively and refutationally sound and complete hybrid system for classical logic.

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)

The fifth “deduction” in Plato’s Parmenides concerns the consequences that follow for a one from the hypothesis that it is not. I argue that the subject of this hypothesis is, effectively, any Form, considered just insofar as it is one Form. The hypothesis, I further argue, does not concern any essential aspect of a Form, but rather posits its contingent non-instantation. The motion this deduction attributes to its one is a special type of motion: motion into and out (...) of instantiation. (shrink)

This chapter offers a concise and elementary introduction to fundamental concepts in deductive and deontic reasoning.

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions (...) of the connectives in question. (shrink)