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.

A number of single- and dual-process theories provide competing explanations as to how reasoners evaluate conditional arguments. Some of these theories are typically linked to different instructions—namely deductive and inductive instructions. To assess whether responses under both instructions can be explained by a single process, or if they reflect two modes of conditional reasoning, we re-analysed four experiments that used both deductive and inductive instructions for conditional inference tasks. Our re-analysis provided evidence consistent with a single process. In two new (...) experiments we established a double dissociation of deductive and inductive instructions when validity and plausibility of conditional problems were pitted against each other. This indicates that at least two processes contribute to conditional reasoning. We conclude that single-process theories of conditional reasoning cannot explain the observed results. Theories that postulate at least two processes are needed to account for our findings. (shrink)

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)

In the Critique of Practical Reason, Kant presents the moral law as the sole ‘fact of pure reason’ that neither needs nor admits of a deduction to establish its authority. This claim may come as a surprise to many readers of his earlier Groundwork of the Metaphysics of Morals. In the last section of the Groundwork, Kant seemed to offer a sketch of just such a ‘deduction of the supreme principle of morality’ . Although notoriously obscure, this sketch (...) shows that Kant hoped to base the moral law in the freedom that rational agents can claim as members of the ‘intelligible world’ that transcendental idealism makes available to us. In contrast, the second Critique abandons all aspirations of deriving morality from more basic notions of freedom and practical rationality. (shrink)

The ampliativity of deduction has been defended in several ways—such as the semi-decidability of the theories, the surprise of unexpected consequences, the need of new individuals in deduction, or ampliative inference as deduction with suppressed premises (Dummett, Frege. Philosophy of mathematics. Duckworth, London, 1991; Hintikka, Logic, language-games and information. Oxford University Press, Oxford, 1973; Musgrave, Imre Lakatos and theories of scientific change. Kluwer, Boston, 1989; Rota, Indiscrete thoughts. Birkhäuser, Boston, 1997). These lines of defensive arguments fail if (...) we characterize ampliativity in a strong way, that is, in terms of ‘novelty’ stricto sensu, as an inferential process producing genuinely new pieces of information in the conclusion of a reasoning (Cellucci, Rethinking logic. Springer, Dordrecht, 2013; Ippoliti, Heuristic reasoning. Springer, Basel, 2015). Nevertheless, it is possible to defend a weaker stance of the ampliativity of deduction and its usefulness. This requires interpreting novelty more loosely, that is, as the inferential production of new relations among known entities. This stance maintains that, in this sense, deduction can provide us with new knowledge. Thus, a typical deductive activity, as re-proving theorems, can be thought as heuristic and ampliative because it establishes new relations between known elements (e.g. between given axioms or properties and the statement of the theorem). I argue that also this line of defence is defective for it does not account for the necessity and usefulness of reproving theorems. In particular, a foundationalist, axiomatic-deductive viewpoint does not fully account for the usefulness and the weak ampliative power of deduction. (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)

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.

I provide a general introduction to the notion at issue in this volume, i.e. deduction, and to some akin notions like inference and reasoning. I also argue that logic is, or should be concerned with three attitudes: an archaeological attitude, a nomological attitude, and an aetiological attitude. Then, I sum up the content of the contributions in this volume.

The idea that there are three types of argumentation, deduction, induction, and a third type variously called hypothesis, abduction, or retroduction, first appeared in an 1867 paper by Charles S. Peirce, “On the Natural Classification of Arguments”. According to Peirce’s tripartite division of argumentation, induction is not merely any form of argument that fails to be deductive, but argumentation that generalizes from a sample. In later writings Peirce broadened his notion to mean any testing of hypotheses through observation—as Peirce (...) said, “trying how things will act”. If we understand an observation to be a sample of experience, as Peirce did in 1901 (R L 409:3... (shrink)

Dogmatism holds that an experience or seeming that p can provide prima facie immediate justification for believing p in virtue of its phenomenology. Dogmatism about perceptual justification has appealed primarily to proponents of representational theories of perceptual experience. Call dogmatism that takes perceptual experience to be representational "representational phenomenal dogmatism." As we show, phenomenal seemings play a crucial role in dogmatism of this kind. Despite its conventional appeal to representational theorists, dogmatism is not by definition committed to any particular view (...) of perceptual experience. Naive realists and disjunctivists who hold that perceptual experience is a perceptual relation of direct acquaintance can also endorse dogmatism. Indeed, we argue that they ought to do so. Otherwise, they cannot maintain that relationalism about perceptual experience has an epistemic advantage compared to the representational view. We then show that even if we grant that relationalism has this advantage, only dogmatism that takes perceptual experience to be representational can be extended to account for non-deductive inferential justification. As an account of the latter is required to avoid succumbing to skepticism, relationalism, we argue, does not have the epistemic advantage its defenders claim it has. (shrink)

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)

We outline Karl Popper’s theory of deduction, which he developed in the 1940s. In his theory it is assumed that a consequence relation is given or otherwise constructed by postulation. Logical operations, which may be available in this consequence relation, are then characterized by means of relational definitions, and logical operators are introduced as names for these operations by means of inferential definitions. Using logically structured sentences thus introduced, the inference laws for them are immediately obtained from the inferential (...) definitions. (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)

This Is A New Release Of The Original 1905 Edition.

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)

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)

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)

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)

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)

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)

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.

This chapter contains sections titled: The Textual Status of Hegel's “Historical Deduction” The Place of the “Historical Deduction” within the Argumentative Task of the Lectures ' Introduction The Three “Common Ideas of Art” and the Emergence of the Standpoint of the “Historical Deduction” From Kant to Schiller to Schlegel: The Third Critique, the Culture of Reflectivity, and the Rise of the Concept of the Beautiful The Problem of History and the Narrative Structure of Hegel's Philosophy of Art.

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 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)

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)

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 concerns the connection between speech act theory, especially the theory of assertion, and deduction, especially Natural Deduction.From a very abstract point of view, an assertion of a content p can be described as the ascription of the property of being p to the actual index, or point of evaluation. This is the abstract characterization of assertoric force. Let’s assume that the actual index is a possible world, namely the actual world. Thus, the conclusion of a closed (...) argument, as an act, is an assertion, and thereby characterized as the ascription of the conclusion content as a property to the actual world.The question that will concern us in this talk is how this idea extends to the status of other acts in the practice of Natural Deduction. In these terms, what is the force of an inference that depends on one or more open assumptions? What is the force of the assumption itself? What is the force of an assertion that depends on an entire derivation? Do we need to ascribe a force to the derivation as a whole? Is there a coherent complete theory of act forces of Natural Deduction along these lines? What is the relation of such a theory to the theory of validity of an argument? (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)

We show that in quantum logic of closed subspaces of Hilbert space one cannot substitute quantum operations for classical (standard Hilbert space) ones and treat them as primitive operations. We consider two possible ways of such a substitution and arrive at operation algebras that are not lattices what proves the claim. We devise algorithms and programs which write down any two-variable expression in an orthomodular lattice by means of classical and quantum operations in an identical form. Our results show that (...) lattice structure and classical operations uniquely determine quantum logic underlying Hilbert space. As a consequence of our result, recent proposals for a deduction theorem with quantum operations in an orthomodular lattice as well as a, substitution of quantum operations for the usual standard Hilbert space ones in quantum logic prove to be misleading. Quantum computer quantum logic is also discussed. (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.

In this paper, I shall examine attempts to furnish formal models for deductive scientific explanation. All such attempts have had certain defects. The most serious of these defects is to be found in the fact that the extant models seem to be formally restrictive in ways that do not allow any obvious generalization of their conditions which will encompass the full range of all those scientific explanations which must be considered plausible candidates for translation into deductive models.

The notion of local deduction theorem (which generalizes on the known instances of indeterminate deduction theorems, e.g. for the infinitely-valued ukasiewicz logic C ) is defined. It is then shown that a given finitary non-pathological logic C admits the local deduction theorem iff the class Matr(C) of all matrices validating C has the C-filter extension property (Theorem II.1).

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)

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)

From the second half of the last century, there has been a widespread view in the Anglophone world that Kant’s transcendental deduction (aka TD) aims to vindicate our common-sense view of the world as composed of public and objective particulars against some unqualified forms of skepticism. This widespread assumption has raised serious doubt not only about the success of TD but also about the very nature of its argument in both editions of the Critique. Yet, if there is a (...) connection between TD and global skepticism, the intriguing question is: Who is this skeptic? According to Strawson, “the skeptic” is a hypothesis of a purely sense-datum experience. In contrast, the fact that TD turns on the key notion of self-consciousness has induced several other scholars to assume that the skeptic is no none but a Cartesian external-world skeptic. None of those readings find textual support or are compatible with the very structure of the first Critique. The question is: Does this mean that TD aims only to undermine empiricism, as Guyer suggests? I do not believe so. I am pretty convinced that TD addresses a peculiar form of global skepticism, namely “Hume’s challenge to reason.” Assuming that we cognize (erkennen) and experience (erfahren) appearances as objects as a requirement of Newtonian physics, TD aims to provide a justification of the principle that nature is uniform that is superior to Hume’s justification. (shrink)

We present a natural deduction system for dual-intuitionistic logic. Its distinctive feature is that it is a single-premise multiple-conclusions system. Its relationships with the natural deduction systems for intuitionistic and classical logic are discussed.

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)

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)

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)

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 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)

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)