The paper focuses on the concepts of truth, truth-making and truth-preservation and their role in defining deductive validity as analysed by the late-medieval nominalist scholar Martin Le Maistre in his Tractatus consequentiarum. This treatise, examined from the point of view of fourteenth-century British and Parisian influences, can be characterised as a critical adoption of the previous logical tradition and as the analysis of validity in term of truth-preservation. Part of this analysis is a study of self-referential phenomena, (...) in particular, of self-referential inferences which are addressed in terms of a Bradwardinian implicit-meaning analysis of self-reference by Le Maistre. Also, his analysis of “consequentia formalis” summarises the fourteenth-century development of the discussion and compares alternative approaches towards formality. (shrink)

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)

The purpose of this study is to provide an account of the fallaciousness of begging the question without thereby indicting as fallacious all otherwise acceptable deductively valid reasoning. The solution that we suggest exploits the intuition that all good arguments are weakly circular. The fallaciousness of begging the question is not that the reasoning is circular simpliciter. Rather, begging the question is a fallacy because the conclusion relies on an undischarged assumption that the audience cannot accept without further argumentation. In (...) the face of such an argument the arguer might just as well have merely asserted the conclusion. (shrink)

David Miller claims that every valid deductive argument begs the question. Other philosophers and logicians have made similar claims. I show that the claim is false. Its appeal depends on the existence of logical terminology, particularly concerning what a proposition 'contains' or its 'logical content,' that is best understood as metaphoric and that, given its aptness to mislead, would be better eschewed. I show how the terminology appears to derive from early modern theories of the nature of mind, ideas (...) and reasoning that have since been rejected. (shrink)

This chapter contains sections titled: Validity and Necessity The Validity Paradox Gödel Arithmetizing the Validity Paradox The Validity Paradox in S5 Validity, Necessity, and Deductive Inference.

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)

I express my dissatisfaction with the common ways to treat the semantic paradoxes. Not only do they give rise to revenge paradoxes, they ignore the wisdom contained in the ordinary reaction to paradoxes. I instead propose an account that vindicates the ordinary reaction to paradox by putting the blame on us philosophers. It is the wrong conception of what a valid inference is, one that is central to “the ideal of deductive logic” that gives rise to the problem. The (...) solution outlined gives us a new way to accept defeat in light of the paradoxes: the arguments that lead to them are based on valid forms of reasoning, but their conclusions are nonetheless rationally rejected. (shrink)

Like Kant's Critique of Pure Reason, we now have both an A- and a substantially revised B-edition of Henry Allison's commentary to and defence of that work: Kant's Transcendental Idealism: An Interpretation and Defense . Unlike Kant's Critique, however, Allison's first edition is unlikely to persist as an occasional rival to the second, since reviewers rightly consider the second to have superseded the landmark scholarship of the first. Nevertheless, before Allison's first edition is entirely supplanted, I would like to save (...) a neglected insight from its treatment of the B-deduction that is not retained in the second edition. (shrink)

Critics of marcus singer's deduction of the generalization argument from the principle of consequences and the generalization principle ("generalization in ethics," page 66) insist that his use of "everyone" in that deduction is ambiguous, I.E., "everyone" is used both collectively and distributively, And that the deduction is invalid. In this paper, I provide a valid deduction of the generalization argument from those premises which avoids this difficulty entirely. I argue that the conclusion so deduced is logically and morally equivalent to (...) singer's. (shrink)

It is often claimed that the conclusion of a deductively valid argument is contained in its premises. Popper refuted this claim when he showed that an empirical theory can be expected always to have logical consequences that transcend the current understanding of the theory. This implies that no formalisation of an empirical theory will enable the derivation of all its logical consequences. I call this result ‘Popper-incompleteness.’ This result appears to be consistent with the view of deductive reasoning as (...) a process of unfurling the content of the premises; but I suggest that the result about validity impugns this theory of reasoning. (shrink)

A new deduction system for deciding validity for the minimal decidable normal modal logic K is presented in this article. Modal logics could be very helpful in modelling dynamic and reactive systems such as bio-inspired systems and process algebras. In fact, recently the Connectionist Modal Logics has been presented, which combines the strengths of modal logics and neural networks. Thus, modal logic K is the basis for these approaches. Soundness, completeness and the fact that the system itself is a (...) decision procedure are proved in this article. The main advantages of this approach are: first, the system is deterministic, i.e. it generates one proof tree for a given formula; second, the system is a validity-checker, hence it generates a proof of a formula ; and third, the language of deduction and the language of a logic coincide. Some of these advantages are compared with other classical approaches. (shrink)

The traditional picture of logic takes it for granted that "valid arguments have a fundamental epistemic significance", but neither model theory nor traditional proof theory dealing with formal system has been able to give an account of this significance. Since valid arguments as usually understood do not in general have any epistemic significance, the problem is to explain how and why we can nevertheless use them sometimes to acquire knowledge. It is suggested that we should distinguish between arguments and acts (...) of inferences and that we have to reconsider the latter notion to arrive at the desired explanation. More precisely, the notions should be developed so that the following relationship holds: one gets in possession of a ground for a conclusion by inferring it from premisses for which one already has grounds, provided that the inference in question is valid. The paper proposes explications of the concepts of ground and deductively valid inference so that this relationship holds as a conceptual truth. Logical validity of inference is seen as a special case of deductive validity, but does not add anything as far as epistemic significance is concerned—it resides already in the deductively valid inferences. (shrink)

The fallibility of deduction is the thesis that a thoughtful speaker-reasoner can wrongly believe that an inference is deductively valid. The author presents an argument to the effect that the fallibility of deduction is incompatible with the widespread view that deduction is epistemically unfruitful (the conclusion is contained in the premises, and the transition from premises to conclusion never extends knowledge). If the fallibility of deduction is a fact, the argument presented is a refutation of the doctrine of the unfruitfulness (...) of deduction. But its relevance goes further, because the argument reveals a connection between two different problems: a solution to the problem of fruitfulness (how can deductive inferences increase our knowledge?) is a starting point for a solution to the problem of fallibility (how can deductive inferences be wrong?). To solve the problem of fruitfulness, Michael Dummett proposed some general ideas concerning meaning and inference. Starting from these meaning-theoretical ideas, one can develop an explanation of the fallibility of deduction. The general explanatory strategy does not depend on the choice between a theory of meaning centered on truth, verification, inferential role, or on other key notions. (shrink)

This paper explores an idea of Stoic descent that is largely neglected nowadays, the idea that an argument is valid when the conditional formed by the conjunction of its premises as antecedent and its conclusion as consequent is true. As it will be argued, once some basic features of our naıve understanding of validity are properly spelled out, and a suitable account of conditionals is adopted, the equivalence between valid arguments and true conditionals makes perfect sense. The account of (...) validity outlined here, which displays one coherent way to articulate the Stoic intuition, accords with standard formal treatments of deductive validity and encompasses an independently grounded characterization of inductive validity. (shrink)

Until quite recently, the epistemology of logical laws has not been much discussed and neither has how one can be justified in claiming that a particular inference is valid. The transfer of warrant from premises to conclusion in modus ponens will be examined in the paper through assessing Paul Boghossian's inferentialist proposal of assuming 'blind reasoning'. It will be argued that merely being justified in inferring according to a logical law a priori is worthless unless one can also be justified (...) in claiming that the inference is valid. An alternative to Boghossian's solution will be presented which addresses this need as well as the needs Boghossian had identified previously. (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.

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)

We investigate the philosophical significance of the existence of different semantic systems with respect to which a given deductive system is sound and complete. Our case study will be Corcoran's deductive system D for Aristotelian syllogistic and some of the different semantic systems for syllogistic that have been proposed in the literature. We shall prove that they are not equivalent, in spite of D being sound and complete with respect to each of them. Beyond the specific case of (...) syllogistic, the goal is to offer a general discussion of the relations between informal notions—in this case, an informal notion of deductive validity—and logical apparatuses such as deductive systems and (model-theoretic or other) semantic systems that aim at offering technical, formal accounts of informal notions. Specifically, we will be interested in Kreisel's famous 'squeezing argument'; we shall ask ourselves what a plurality of semantic systems (understood as classes of mathematical structures) may entail for the cogency of specific applications of the squeezing argument. More generally, the analysis brings to the fore the need for criteria of adequacy for semantic systems based on mathematical structures. Without such criteria, the idea that the gap between informal and technical accounts of validity can be bridged is put under pressure. (shrink)

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).

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)

Byrne has demonstrated that although subjects can make deductively valid inferences of the modus ponens and modus tollens forms, these valid inferences can be suppressed by presenting an appropriate additional premise “If R then Q” with the original conditional “If P then Q”. This suppression effect challenges the assumption of all syntactic theories of conditional reasoning that formal rules of inference such as modus ponens is part of mental logic. This paper argues that both the syntactic and the mental model (...) accounts of the suppression effect are inadequate because they fail to give a principled account of the critical interpretive component involved in reasoning. In contrast, the relative salience model proposed in this study emphasized the centrality of the interpretative processes with the critical component being the relative salience of premises as judged by subjects on the basis of their prior knowledge activated in particular problem situations. Using 120 undergraduates and 120 policemen as subjects, predictions from the model were tested and confirmed in a suppression paradigm and evidence of convergent validity for the construct of salience were obtained. The results cannot be reconciled with either the syntactic view or the mental model view that have dominated theories of conditional reasoning. (shrink)

According to Michael Dummett, deductive inference stands in need of justification which must be provided by the theory of meaning for natural language. Such a theory, he insists, should deliver an explanation for the two essential features of deduction: validity and fruitfulness. Dummett claims that only a molecularist theory of meaning could offer the desired justification. In this paper, I will consider and criticize his solution to the problem of the justification of deduction: the so-called molecular verificationist explanation. (...) My aim here will be to show that Dummett's solution does not succeed in reconciling the conflicting demands of the respective explanations of validity and fruitfulness. (shrink)

Extant semantic theories for languages containing vague expressions violate intuition by delivering the same verdict on two principles of classical propositional logic: the law of noncontradiction and the law of excluded middle. Supervaluational treatments render both valid; many-Valued treatments, Neither. The core of this paper presents a natural deduction system, Sound and complete with respect to a 'mixed' semantics which validates the law of noncontradiction but not the law of excluded middle.

Inference to the best explanation (IBE) is considered to be the main means of discovery and justification of scientific hypotheses and theories. The article investigates this inference and its relationship to the main kinds of inference: deduction, induction and abduction. IBE has an abductive inference mechanism, but, contrary to abduction, infers not a possible, but a true conclusion. IBE is an inductive inference, because it is underdetermined by the rules of deduction and by evidence. The article claims that despite its (...) abductive and inductive features which demonstrate that it is not and cannot be deductive inference, IBE nevertheless makes pretense to an absolute truth of its inference, i.e. claims for an almost deductive validity. (shrink)

A philosophical argument in ordinary language is made the basis for a series of deductive logic exercises. Problems of translating the reasoning and alternative symbolizations are discussed to help guide students toward accurate charitable formalizations. Finally, the inference is critically evaluated in light of its deductive validity.

. This paper continues a systematic approach to build natural deduction calculi and corresponding proof procedures for non-classical logics. Our attention is now paid to the framework of paraconsistent logics. These logics are used, in particular, for reasoning about systems where paradoxes do not lead to the `deductive explosion', i.e., where formulae of the type `A follows from false', for any A, are not valid. We formulate the natural deduction system for the logic PCont, explain its main concepts, define (...) a proof searching technique and illustrate it by examples. The presentation is accompanied by demonstrating the correctness of these developments. (shrink)

This is a dialogue between Lisa and Max on Dag Prawitz's work concerning the concept of deductive validity. Lisa first explains Prawitz's criticisms of the presently prevailing non‐epistemic analyses of validity. Then Lisa describes three different ways in which Prawitz attempted to develop an epistemic concept of validity. Max asks questions for clarification, raises some objections and compares Prawitz's three approaches with other lines of thought. Two inference rules are specially discussed: disjunction introduction and ex contradictione (...) quodlibet. Max and Lisa view Prawitz's contribution as part of a variegated, ongoing research pursuing an explication of the concept of validity, which began with the Socratic and Platonic distinction between merely persuasive arguments and good arguments and with Aristotle's definition of syllogism. Throughout the history of philosophy, logicians have aimed at clarifying a pre‐theoretic idea of good deduction and have proposed, criticized, refined, adjusted precise theoretical concepts of validity. Prawitz has criticized current views about validity, has advanced new proposals and revised them. His work is an important chapter of the long pursuit of the concept of validity. (shrink)

Hybrid deduction–refutation systems are deductive systems intended to derive both valid and non-valid, i.e., semantically refutable, formulae of a given logical system, by employing together separate derivability operators for each of these and combining ‘hybrid derivation rules’ that involve both deduction and refutation. The goal of this paper is to develop a basic theory and ‘meta-proof’ theory of hybrid deduction–refutation systems. I then illustrate the concept on a hybrid derivation system of natural deduction for classical propositional logic, for which (...) I show soundness and completeness for both deductions and refutations. (shrink)

The principle that epistemic justification is necessarily transmitted to all the known logical consequences of a justified belief continues to attract critical attention. That attention is not misplaced. If the Transmission Principle is valid, anyone who thinks that a given belief is justified must defend the view that every known consequence of the belief is also justification of the conclusion in an obviously valid argument. Once created, the gap is hard to fill, whatever the circumstances. Reflection principle is modified, the (...) possibility of deductive justification is threatened. If some known consequence fails to be justified, the failure may extend to every known consequence. To reject Transmission is to insert a logical gap between the justification of a premise and the justification of the conclusion in an obviously valid argument. Once created, the gap is hard to fill, whatever the circumstances. Reflection on the Transmission Principle therefore usefully brings us face to face with the following dilemma: accept the principle and hand Cartesian scepticism a powerful weapon, or modify the principle and risk undermining deductive justification. (shrink)

If ever there were scientific procedures that seemed immune to history, induction and deduction would be them. Their validity seemingly unimpinged by the vicissitudes of history, they appear a proper subject of discussion for philosophers and scientists, but not for historians. Historians pride themselves on demonstrating that the internal logic of theory is historical — a response to particular conditions. Breakdowns in logic present historians with opportune moments for historicization, for showing how theoreticians’ efforts to respond to their situation (...) made them move inconsistently. When it comes to induction and deduction, however, they seem so universal and basic — the logic and methodology of science unthinkable without them — that any effort to demonstrate their historically bound character risks relativizing science and, some would argue, undermine rational exchange. Can the historian participate, qua historian, in the conversation on induction and deduction without unraveling science and undermining rational exchange? (shrink)

ABSTRACT Real-time graphical interval logic is a modal logic for reasoning about time in which the basic modality is the interval. The logic differs from other logics in that it has a natural intuitive graphical representation that resembles the timing diagrams drawn by system designers. We have developed an automted deduction system for the logic, which includes a theorem prover and a user interface. The theorem prover checks the validity of proofs in the logic and produces counterexamples to invalid (...) proofs. The user interface includes a graphical editor that enables the user to create graphical formulas on a workstation display, and a database and proof manager that tracks proof dependencies and allows graphical formulas to be stored and retrieved. In this paper we describe the logic, the automated deduction system, and an application to robotics. (shrink)

We investigate how the perceived uncertainty of a conditional affects a person's choice of conclusion. We use a novel procedure to introduce uncertainty by manipulating the conditional probability of the consequent given the antecedent. In Experiment 1, we show first that subjects reduce their choice of valid conclusions when a conditional is followed by an additional premise that makes the major premise uncertain. In this we replicate Byrne. These subjects choose, instead, a qualified conclusion expressing uncertainty. If subjects are given (...) a third statement that qualifies the likelihood of the additional premise, then the uncertainty of the conclusions they choose is systematically related to the suggested uncertainty. Experiment 2 confirms these observations in problems that omit the additional premise and qualify the first premise directly. Experiment 3 shows that the qualifying statement also affects the perceived probability of the consequent given the antecedent of the conditional. Experiment 4 investigates the effect of suggested uncertainty on the fallacies and shows that increases in uncertainty reduce the number of certain conclusions that are chosen while affirming the consequent but have no effect on denying the antecedent. We discuss our results in terms of rule theories and mental models and conclude that the latter give the most natural account of our results. (shrink)

In section 1, I develop epistemic communism, my view of the function of epistemically evaluative terms such as ‘rational’. The function is to support the coordination of our belief-forming rules, which in turn supports the reliable acquisition of beliefs through testimony. This view is motivated by the existence of valid inferences that we hesitate to call rational. I defend the view against the worry that it fails to account for a function of evaluations within first-personal deliberation. In the rest of (...) the paper, I then argue, on the basis of epistemic communism, for a view about rationality itself. I set up the argument in section 2 by saying what a theory of rational deduction is supposed to do. I claim that such a theory would provide a necessary, sufficient, and explanatorily unifying condition for being a rational rule for inferring deductive consequences. I argue in section 3 that, given epistemic communism and the conventionality that it entails, there is no such theory. Nothing explains why certain rules for deductive reasoning are rational. (shrink)

Seldom does human reasoning fit the standards of deduction. Yet logicians have tended to use the strict standards of deductive validity for assessing all inferences. _Within Reason_ develops instead a way of assessing arguments and inferences that is directly appropriate to the non-deductive forms people regularly use. It uses analogy, and argument from analogy, to provide a thread that unites various forms: raising objections, inductions of various sorts, arguments to explanation, and arguments to action. The discussion is (...) developed progressively, at each stage building on the skills learned previously. And at each stage the book suggetss ways of assessing the strength of reasoning. Most of the examples are taken from the great texts of the Western tradition. In the exercises that form an integral part of each chapter an even broader range of writing is presented, offering instructors an extraordinary variety of texts at various levels ot assign to their students. The thesis implicit in the book is that induction, arguments to explanation and arguments to action are all sophisticated forms of arguments from analogy. Burbidge is concerned not simply to warn against the misuse of non-deductive reasoning but also to develop a more positive theory that allows us both to better udnerstand the workings of non-deductive reasoning, and to better employ it. (shrink)

The paper deals with two versions of the fragment with unit, tensor, linear implication and storage operator of intuitionistic linear logic. The first version, ILL, appears in a paper by Benton, Bierman, Hyland and de Paiva; the second one, ILL+, is described in this paper. ILL has a contraction rule and an introduction rule !I for the exponential; in ILL+, instead of a contraction rule, multiple occurrences of labels for assumptions are permitted under certain conditions; moreover, there is a different (...) introduction rule for the exponential, !I+, which is closer in spirit to the necessitation rule for the normalizable version of S4 discussed by Prawitz in his monograph “Natural Deduction”.It is relatively easy to adapt Prawitz's treatment of natural deduction for intuitionistic logic to ILL+; in particular one can formulate a notion of strong validity permitting a proof of strong normalization.The conversion rules for ILL explicitly mentioned in the paper by Benton et al. do not suffice for normal forms with subformula property, but we can show that this can be remedied by addition of a special permutation conversion plus some “satellite” permutation conversions.Some discussion of the categorical models which might correspond to ILL+ is given. (shrink)

Havstad (2022) argues that the argument from inductive risk for the claim that non-epistemic values have a legitimate role to play in the internal stages of science is deductively valid. She also defends its premises and thus soundness. This is, as far as we are aware, the best reconstruction of the argument from inductive risk in the existing literature. However, there is a small flaw in this reconstruction of the argument from inductive risk which appears to render the argument invalid. (...) This flaw is superficial, and a small amendment to it rescues the claim of validity. (shrink)

Corcoran, John. 1974. Aristotelian Syllogisms: Valid arguments or true generalized conditionals?, Mind 83, 278–81. MR0532928 (58 #27178) This tightly-written and self-contained four-page paper must be studied and not just skimmed. It meticulously analyses quotations from Aristotle and Lukasiewicz to establish that Aristotle was using indirect deductions—as required by the natural-deduction interpretation—and not indirect proofs—as required by the axiomatic interpretation. Lukasiewicz was explicit and clear about the subtle fact that Aristotle’s practice could not be construed as correctly performed indirect proof. Lukasiewicz (...) evidence is presented fully; it is irrefutable. But, instead of considering the possibility that Aristotle’s discourses were not intended to express indirect proofs of universalized conditions presupposing axiomatic premises, Lukasiewicz came to the amazing conclusion that Aristotle did not understand indirect proof. This paper builds on the admirable Lukasiewicz scholarship to establish a conclusion diametrically opposed to the one Lukasiewicz asserted. This paper points out that if Aristotle had not been establishing an underlying logic but he was instead axiomatizing a theory of terms, Aristotelians could not claim for Aristotle the title Founder of Logic. People who take Euclid, Peano, and Zermelo to have first axiomatized geometry, arithmetic, and set theory, respectively, do not think that this qualifies them for the titles Founder of Geometry, Founder of Arithmetic, and Founder of Set Theory, respectively. Such people do think that this would qualify the three for the titles Founder of Axiomatic Geometry, Founder of Axiomatic Arithmetic, and Founder of Axiomatic Set Theory, respectively: titles that carry no honors in logic. Being the Founder of Axiomatic Term Theory would likewise carry no honors in the field of logic. CORRECTION: The paper inadvertently implied that Euclid was the first to axiomatize geometry. In order to understand Aristotle’s Analytics as a treatise on demonstration it helps to realize that axiomatized geometry was studied in Plato’s Academy when Aristotle was a student. Euclid was the author of the only ancient axiomatized geometry now available. (shrink)

This book provides philosophers and logicians with a broad spectrum of views on contemporary research on the problem of deduction, its justification and explanation. The variety of distinct approaches exemplified by the single chapters allows for a dialogue between perspectives that, usually, barely communicate with each other. The contributions concern (in a possibly intertwined way) three major perspectives in logic: philosophical, historical, formal. The philosophical perspective has to do with the relationship between deductive validity and truth, and questions (...) the alleged conclusiveness of deduction and its epistemic contribution. It also discusses the role of linguistic acts in deductive practice, and provides a cognitive-didactic contribution on how we may learn through deduction. In the historical perspective, the contributions discuss the ideas of some major historical figures, such as Bolzano, Girard, Gödel, and Peano. Finally, in the formal perspective, the mathematics of deduction is dealt with mainly from an intuitionistic-constructivist or proof-theoretic point of view, with focus on “ecumenic” or internalistic approaches to logical validity, on the nature and identity of proofs, and on dialogical setups. Chapter [14] is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com. (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).

The standard deduction theorem or introduction rule for implication, for classical logic is also valid for intuitionistic logic, but just as with predicate logic, other rules of inference have to be restricted if the theorem is to hold for weaker implicational logics.In this paper we look in detail at special cases of the Gentzen rule for and show that various subsets of these in effect constitute deduction theorems determining all the theorems of many well known as well as not well (...) known implicational logics. In particular systems of rules are given which are equivalent to the relevance logics E,R, T, P-W and P-W-I. (shrink)

One of the most debated questions in psychology and cognitive science is the nature and the functioning of the mental processes involved in deductive reasoning. However, all existing theories refer to a specific deductive domain, like syllogistic, propositional or relational reasoning.Our goal is to unify the main types of deductive reasoning into a single set of basic procedures. In particular, we bring together the microtheories developed from a mental models perspective in a single theory, for which we (...) provide a formal foundation. We validate the theory through a computational model (UNICORE) which allows fine‐grained predictions of subjects' performance in different reasoning domains.The performance of the model is tested against the performance of experimental subjects—as reported in the relevant literature—in the three areas of syllogistic, relational and propositional reasoning. The computational model proves to be a satisfactory artificial subject, reproducing both correct and erroneous performance of the human subjects. Moreover, we introduce a developmental trend in the program, in order to simulate the performance of subjects of different ages, ranging from children (3–6) to adolescents (8–12) to adults (>21). The simulation model performs similarly to the subjects of different ages.Our conclusion is that the validity of the mental model approach is confirmed for the deductive reasoning domain, and that it is possible to devise a unique mechanism able to deal with the specific subareas. The proposed computational model (UNICORE) represents such a unifying structure. (shrink)

In this paper, we propose a computational interpretation of the generalized Kreisel–Putnam rule, also known as the generalized Harrop rule or simply the Split rule, in the style of BHK semantics. We will achieve this by exploiting the Curry–Howard correspondence between formulas and types. First, we inspect the inferential behavior of the Split rule in the setting of a natural deduction system for intuitionistic propositional logic. This will guide our process of formulating an appropriate program that would capture the corresponding (...) computational content of the typed Split rule. Our investigation can also be reframed as an effort to answer the following question: is the Split rule constructively valid in the sense of BHK semantics? Our answer is positive for the Split rule as well as for its newly proposed general version called the S rule. (shrink)