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)

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to be shown either that the given proposition follows or that it does not follow.* Using a formal deduction to show that a conclusion follows and using a countermodel to show that a conclusion does not follow are both traditional practices recognized by Aristotle and used down through the history of logic. (...) These practices presuppose, respectively, a criterion of validity and a criterion of invalidity each of which has been extended and refined by modern logicians: deductions are studied in formal syntax (proof theory) and coun¬termodels are studied in formal semantics (model theory). The purpose of this paper is to compare these two criteria to the corresponding criteria employed in Boole’s first logical work, The Mathematical Analysis of Logic (1847). In particular, this paper presents a detailed study of the relevant metalogical passages and an analysis of Boole’s symbolic derivations. It is well known, of course, that Boole’s logical analysis of compound terms (involving ‘not’, ‘and’, ‘or’, ‘except’, etc.) contributed to the enlargement of the class of propositions and arguments formally treatable in logic. The present study shows, in addition, that Boole made significant contributions to the study of deduc¬tive reasoning. He identified the role of logical axioms (as opposed to inference rules) in formal deductions, he conceived of the idea of an axiomatic deductive sys¬tem (which yields logical truths by itself and which yields consequences when ap¬plied to arbitrary premises). Nevertheless, surprisingly, Boole’s attempt to imple¬ment his idea of an axiomatic deductive system involved striking omissions: Boole does not use his own formal deductions to establish validity. Boole does give symbolic derivations, several of which are vitiated by “Boole’s Solutions Fallacy”: the fallacy of supposing that a solution to an equation is necessarily a logical consequence of the equation. This fallacy seems to have led Boole to confuse equational calculi (i.e., methods for gen-erating solutions) with deduction procedures (i.e., methods for generating consequences). The methodological confusion is closely related to the fact, shown in detail below, that Boole had adopted an unsound criterion of validity. It is also shown that Boole totally ignored the countermodel criterion of invalid¬ity. Careful examination of the text does not reveal with certainty a test for invalidity which was adopted by Boole. However, we have isolated a test that he seems to use in this way and we show that this test is ineffectual in the sense that it does not serve to identify invalid arguments. We go beyond the simple goal stated above. Besides comparing Boole’s earliest criteria of validity and invalidity with those traditionally (and still generally) employed, this paper also investigates the framework and details of THE MATHEMATICAL ANALYSIS OF LOGIC. (shrink)

A first-order language with a defined identity predicate is proposed whose apparatus for atomic predication is sensitive to grammatical categories of natural language. Subatomic natural deduction systems are defined for this naturalistic first-order language. These systems contain subatomic systems which govern the inferential relations which obtain between naturalistic atomic sentences and between their possibly composite components. As a main result it is shown that normal derivations in the defined systems enjoy the subexpression property which subsumes the subformula property with (...) respect to atomic and identity formulae as a special case. The systems admit a proof-theoretic semantics which does not only apply to logically compound but also to atomic and identity formulae—as well as to their components. The potential of the defined systems for a meticulous first-order analysis of natural inferences whose validity crucially depends on expressions of some of the aforementioned categories is demonstrated. (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)

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.

Few of Kant’s distinctions have generated as much puzzlement and criticism as the one he draws in the Prolegomena between judgments of experience, which he describes as objectively and universally valid, and judgments of perception, which he says are merely subjectively valid. Yet the distinction between objective and subjective validity is central to Kant’s account of experience and plays a key role in his Transcendental Deduction of the categories. In this paper, I reject a standard interpretation of the (...) distinction, according to which judgments of perception are merely subjectively valid because they are made without sufficient investigation. In its place, I argue that for Kant, judgments of perception are merely subjectively valid because they merely report sequences of perceptions had by a subject without claiming that what is represented by the perceptions is connected in the objects the perceptions are of. Whereas the interpretation I criticize undercuts Kant’s strategy in the Deduction, I argue, my interpretation illuminates it. (shrink)

. 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)

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)

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)

Minimalism is the view that our concept of truth is constituted by our disposition to accept instances of the truth schema ‘The proposition that p is true if and only if p’. The generalisation problem is the challenge to account for universal generalisations concerning logical truths such as ‘Every proposition of the form 〈if p, then p〉 is true’. This paper argues that such generalisations can be deduced using a single example of the logical truth in question and a single (...) corresponding instance of the truth schema, employing the logical method of reasoning with arbitrary instances of universal and existential generalisations. Suggesting an inferentialist construal of Minimalism, the paper introduces conditional and general acceptance dispositions, distinguishes inferential meaning constitution from implicit definition, highlights the inferential nature of acceptance of instances of the truth schema, sketches a suitable account of structured propositions, compares higher-order with first-order means of quantification, and argues that the conception of truth Minimalism attributes to ordinary speakers is essentially inferential. It finally applies the deductive strategy to generalisations concerning logical validity as well as more complex logical truths. (shrink)

(1) If a deductive argument is valid, then the conclusion is not novel. (2) If the conclusion of an argument is not novel, the argument is not useful. So, (3) if a deductive argument is valid, it is not useful. This conclusion, (3), is unacceptable. Since the argument is valid, we must reject at least one premise. So, should we reject (1) or (2)? This puzzle is usually known as the ‘scandal of deduction’. Analytic philosophers have tried to reject (...) (1) but have assumed premise (2). I argue here that Aristotle would deny (2). Aristotle thinks that at least some deductive arguments are useful, even though they present no new conclusions. Thus, Aristotle’s view contrasts with analytic philosophers of logic, who assume that all useful deductive arguments present novel conclusions. I don’t claim that Aristotle ‘solves’ the problem: it was never posed in Aristotle’s time. Rather, I suggest that Aristotle does not face the problem because he assumes deductions can be useful, without presenting novel conclusions. Aristotle’s view of deduction tames the scandal. (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)

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 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).

The problem of Kant’s Neglected Alternative is that while his Aesthetic provides an argument that space and time are empirically real – in applying to all appearances – its argument seems to fall short of the conclusion that space and time are transcendentally ideal, in not applying to any things in themselves. By considering an overlooked passage in which Kant explains why his Transcendental Deduction is ‘unavoidably necessary’, I argue that it is not solely in his Aesthetic but more (...) so in his Deduction where he intends to provide his argument for the transcendental ideality of space and time. His Deduction shows that space and time do not have a valid application to any things in themselves by arguing that the categories do have a valid application to everything in space and time, but that the categories do not have a valid application to any things in themselves. (shrink)

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)

Legal terms, such as “ownership,”“contract,”“validity,”“negligence,” are used as middle terms in legal deduction. The paper distinguishes two problems regarding this use. One is the logical function of terms for deduction within a normative system. Specific problems dealt with in this connection are meaning, definition, and economy of expression. The other problem connected with middle terms is the “moulding” and possible manipulation of the meaning of legal terms, for arriving at desired conclusions in a given scheme of inference. (...) It is indicated how the moulding of contested legal concepts, if not restricted, will obscure the ratio of legal rules. This problem is relevant, inter alia, to arguments ex analogia in the law. (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)

Aristotle’s Prior Analytics identifies deductions simpliciter with inferential necessity, so that a deduced conclusion is necessarily inferred from some premises. Modern logical reconstructions claim that inferential necessity in Aristotle corresponds to logical validity. However, this logical reconstruction fails on two accounts. First, logical validity does not highlight Aristotle’s distinction between inferential necessity and predicative necessity, meaning that the inferential necessity of a deduction is not of the same kind as the predicative necessity of a non‑deductive argument. Second, (...) logical validity does not explain the relevance of Aristotle’s distinction between complete and incomplete deductions. Logicians speak of complete deduction by adding the term “obvious” or “transparent” to logical validity, and then criticize Aristotle’s view for being unclear. However, Aristotle’s position is not confronted with this difficulty. There is nothing to add to inferential necessity, which already means complete deducibility, as opposed to incomplete deducibility, deemed to be potentially complete. Accordingly, the Prior Analytics reduces the incomplete deductions to the complete deductions in order to prove the potential, inferential necessity of the incomplete deductions. Logical validity would have been faithful to Aristotle’s text, if it had been possible to coin a notion of potential validity, distinct from both validity and invalidity. (shrink)

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

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)

How do people make deductions? The orthodox view in psychology is that they use formal rules of inference like those of a “natural deduction” system.Deductionargues that their logical competence depends, not on formal rules, but on mental models. They construct models of the situation described by the premises, using their linguistic knowledge and their general knowledge. They try to formulate a conclusion based on these models that maintains semantic information, that expresses it parsimoniously, and that makes explicit something not (...) directly stated by any premise. They then test the validity of the conclusion by searching for alternative models that might refute the conclusion. The theory also resolves long-standing puzzles about reasoning, including how nonmonotonic reasoning occurs in daily life. The book reports experiments on all the main domains of deduction, including inferences based on prepositional connectives such as “if” and “or,” inferences based on relations such as “in the same place as,” inferences based on quantifiers such as “none,” “any,” and “only,” and metalogical inferences based on assertions about the true and the false. Where the two theories make opposite predictions, the results confirm the model theory and run counter to the formal rule theories. Without exception, all of the experiments corroborate the two main predictions of the model theory: inferences requiring only one model are easier than those requiring multiple models, and erroneous conclusions are usually the result of constructing only one of the possible models of the premises. (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)

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)

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)

This article identifies and formalizes the logical features of analogous terms that justify their use in deduction. After a survey of doctrines in Aristotle, Aquinas, and Cajetan, the criteria of “analogy of proper proportionality” are symbolized in first-order predicate logic. A common genus justifies use of a common term, but does not provide the inferential link required for deduction. Rather, the respective differentiae foster this link through their identical proportion. A natural-language argument by analogy is formalized so as (...) to exhibit these criteria, thereby showing the validity of analogical deduction. (shrink)

Any theory of explanation must account for the explanatory successes of statistical scientific theories. This should not be done by endorsing determinism. These considerations have been taken as sufficient ground for rejecting the demand on explanations to be deductive. The arguments for doing so, in Coffa (1974) and Salmon (1977, 1984, 1988), are, however, not persuasive. Deductivism is a viable position. Considering that doubts can be raised against the explanatory validity of probabilistic causal relations and the intuitive plausibility of (...) deductivism, it is also a recommendable position, though elaboration is needed in accounting for some of the uses of statistical theories in explanations. (shrink)

Major recent interpretations of Kant's first "critique" (wolff, Strawson, Bennett) have taken his transcendental deduction to be an argument from the fact of consciousness to the existence of an objective world. I argue that it is unclear such an argument can succeed and there are overwhelming reasons to believe kant understood his deduction as having a very different form, namely as moving from the premise that there is empirical knowledge to the conclusion that there are universally valid pure (...) categories. Detailed support for this contention is offered in an analysis of the second edition version of the deduction. (shrink)

I examine the division of labor between the Metaphysical Deduction (MD) and the Transcendental Deduction (TD). Against a common reading, I argue that the MD is insufficient to prove the a priori origin of the categories. For both Kant and his main opponent, namely Hume, the question of whether the categories have an a priori origin in the pure understanding is inseparable from the question of whether they have objective validity. Since the MD does not establish the (...) objective validity of the categories, it cannot establish their a priori origin either. The MD is nevertheless an indispensable part of Kant's project because it lays the argumentative groundwork for the proof structure of the TD and because it provides the systematic plan for the future metaphysics of experience. (shrink)

John Corcoran’s natural deduction system for Aristotle’s syllogistic is reconsidered.Though Corcoran is no doubt right in interpreting Aristotle as viewing syllogisms as arguments and in rejecting Lukasiewicz’s treatment in terms of conditional sentences, it is argued that Corcoran is wrong in thinking that the only alternative is to construe Barbara and Celarent as deduction rules in a natural deduction system.An alternative is presented that is technically more elegant and equally compatible with the texts.The abstract role assigned by (...) tradition and Lukasiewicz to Barbara and Celarent is retained.The two ‘ perfect syllogisms’ serve as ‘basic elements’ in the construction of an inductively defined set of valid syllogisms.The proposal departs from Lukasiewicz, and follows Corcoran, however, in construing the construction as one in natural deduction.The result is a sequent system with fewer rules and in which Barbara and Celarent serve as basic deductions.To compare the theory to Corcoran’s, his original is reformulated in current terms and generalized.It is shown to be equivalent to the proposed sequent system, and several variations are discussed.For all systems mentioned, a method of Henkin‐style completeness proofs is given that is more direct and intuitive than Corcoran’s original. (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)

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

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)

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 Justification of Deduction is the title of one of Michael Dummett’s essays. It names also an important theme in his writings to which he returned in the book The Logical Basis of Metaphysics. In the essay he distinguishes different levels of justification of increasing philosophical depth. At the third and deepest level, the focus is on explaining deduction rather than on justifying it. The task is to explain how deduction can be both legitimate and useful in (...) giving us knowledge. I suggest that it can be described as essentially being the task to say what it is that gives a deduction its epistemic force. It is a fact that deduction has such force, consisting in its capacity to provide grounds for assertions and thereby extend our knowledge, but it is a fact that has to be explained. What is it that gives a deduction this capacity? This task is more challenging than is usually assumed. Obviously, it is not the validity of its inferences, as this is usually understood, which gives a deduction its epistemic force. Truth conditional theory of meaning does not seem to have any satisfactory solution to offer, and I argue that nor have inferential theories of meaning, which take the meaning of sentences to be determined by inference rules accepted in a language. In the last part of the paper, I sketch a different approach to the problem. The main idea is here to give the concept of inference a richer content, so that to perform an inference is not only to make a speech act in which a conclusion is claimed to be supported by a number of premisses, but is in addition to operate on the grounds for the premisses with the aim of getting a ground for the conclusion. I suggest that it is thanks to such operations that deductions provide grounds for their final conclusions. (shrink)

One semantic and two syntactic decision procedures are given for determining the validity of Aristotelian assertoric and apodeictic syllogisms. Results are obtained by using the Aristotelian deductions that necessarily have an even number of premises.

This thesis challenges a general prejudice against visualization in the history of logic and mathematics, by providing a semantic analysis of two graphical representation systems--a traditional Venn diagram representation system and an extension of it. While Venn diagrams have been used to solve problems in set theory and to test the validity of syllogisms in logic, they have not been considered valid proofs but heuristic tools for finding valid formal proofs. ;I present Venn diagrams which have been used in (...) logic as a formal system of representation equipped with its own syntax and semantics. I also show that Venn-I, with the rules of transformation that I specify, is sound and complete. The soundness of Venn-I refutes a worry that mathematicians and logicians have about graphical representation: Fallacies often have arisen from the misuse of diagrams in mathematical proofs, especially in geometry. However, the validity of the transformation rules assures us that the correct application of these rules will not lead to fallacies. ;I extend Venn-I so that this new system has a general way to convey disjunctive information. A new formation rule is introduced and new transformation rules are added. I call this extended system Venn-II. The semantics of Venn-II is formalized as an extension of the semantics of Venn-I. The soundness and completeness of Venn-II are also proven. I then show that Venn-II and a monadic first-order language are equivalent in their expressiveness. ;The analysis of Venn-I and Venn-II leads to interesting issues which have their analogues in other deductive systems. I conclude the thesis with a discussion of some fundamental differences between graphical systems and linguistic systems. (shrink)

Neuroscience has studied deductive reasoning over the last 20 years under the assumption that deductive inferences are not only de jure but also de facto distinct from other forms of inference. The objective of this research is to verify if logically valid deductions leave any cerebral electrical trait that is distinct from the trait left by non-valid deductions. 23 subjects with an average age of 20.35 years were registered with MEG and placed into a two conditions paradigm (100 trials for (...) each condition) which each presented the exact same relational complexity (same variables and content) but had distinct logical complexity. Both conditions show the same electromagnetic components (P3, N4) in the early temporal window (250–525 ms) and P6 in the late temporal window (500–775 ms). The significant activity in both valid and invalid conditions is found in sensors from medial prefrontal regions, probably corresponding to the ACC or to the medial prefrontal cortex. The amplitude and intensity of valid deductions is significantly lower in both temporal windows (p = 0.0003). The reaction time was 54.37% slower in the valid condition. Validity leaves a minimal but measurable hypoactive electrical trait in brain processing. The minor electrical demand is attributable to the recursive and automatable character of valid deductions, suggesting a physical indicator of computational deductive properties. It is hypothesized that all valid deductions are recursive and hypoactive. (shrink)

In the Organon Aristotle describes some deductive schemata in which inconsistencies do not entail the trivialization of the logical theory involved. This thesis is corroborated by three different theoretical topics by him discussed, which are presented in this paper. We analyse inference schema used by Aristotle in the Protrepticus and the method of indirect demonstration for categorical syllogisms. Both methods exemplify as Aristotle employs classical reductio ad absurdum strategies. Following, we discuss valid syllogisms from opposite premises (contrary and contradictory) studied (...) by the Stagerian in the Analytica Priora (B15). According to him, the following syllogisms are valid from opposite premises, in which small Latin letters stand for terms such as subject and predicate, and capital Latin letters stand for the categorical propositions such as in the traditional notation: (i) in the second figure, Eba,Aba ` Eaa (Cesare), Aba, Eba ` Eaa (Camestres), Eba, I ba ` Oaa (Festino), and Aba,Oba ` Oaa (Baroco); (ii) in the third one, Eab,Aab ` Oaa (Felapton), Oab,Aab ` Oaa (Bocardo) and Eab, Iab ` Oaa (Ferison). Finally, we discuss the passage from the Analytica Posteriora (A11) in which Aristotle states that the Principle of Non-Contradiction is not generally presupposed in all demonstrations (scientific syllogisms), but only in those in which the conclusion must be proved from the Principle; the Stagerian states that if a syllogism of the first figure has the major term consistent, the other terms of the demonstration can be each one separately inconsistent. These results allow us to propose an interpretation of his deductive theory as a broad sense paraconsistent theory. Firstly, we proceed to a hermeneutical analysis, evaluating its logical significance and the interplay of the results with some other points of Aristotle’s philosophy. Secondly, we point to a logical interpretation of the Aristotelian syllogisms from opposite premises in the antilogisms method proposed by Christine Ladd-Franklin in 1883, and we also present a logical treatment of the Aristotelian demonstration with inconsistent material in the paraconsistent logics Cn, 1 _ n _!, introduced by da Costa in 1963. These two issues seem having not yet been analysed in detail in the literature. (shrink)

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

Kant’s argument in § 38 of the *Critique of Judgment* is subject to a dilemma: if the subjective condition of cognition is the sufficient condition of the pleasure of taste, then every object of experience must produce that pleasure; if not, then the universal communicability of cognition does not entail the universal communicability of the pleasure. Kant’s use of an additional premise in § 21 may get him out of this difficulty, but the premises themselves hang in the air and (...) have no independent plausibility. What Kant offers as a proof of our right to make judgments of taste is more charitably construed as an indirect argument for the adequacy of a speculative explanation of a *presumed* right to make judgments of taste. (shrink)

I first criticize strawson's account of the transcendental deduction, And then argue that wittgenstein's considerations (in his later work) of the rule-Governed nature of judgment can be used to reconstruct a valid argument for a certain kind of objectivity, Which excludes solipsims. I suggest how kant's talk of synthesis can be reinterpreted in the light of this, As indeed can the doctrine of empirical realism and transcendental idealism.

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)

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.

In this paper we make some observations about Natural Deduction derivations [Prawitz, 1965, van Dalen, 1986, Bell and Machover, 1977]. We assume the reader is familiar with it and with proof-theory in general. Our development will be simple, even simple-minded, and concrete. However, it will also be evident that general ideas motivate our examples, and we think both our specific examples and the ideas behind them are interesting and may be useful to some readers. In a sentence, the bare (...) technical content of this paper is: Extending natural deduction with global well-formedness conditions can neatly and cheaply capture classical and intermediate logics. The interest here is in the ‘neatly’ and ‘cheaply’. By ‘neatly’ we mean ‘preserving proof-normalisation’,1 and ‘maintaining the subformula property’, and by ‘cheaply’ we mean ‘preserving the formal structure of deductions’ (so that a deduction in the original system is still, formally, a deduction in the extended system, and in particular it requires no extra effort to write just because it is in the extended system). To illustrate what we have in mind consider intuitionistic first-order logic (FOL) [van Dalen, 1986] as a paradigmatic example of a formal notion of deduction. A natural deduction derivation (or deduction) is an inductively defined tree structure where each node contains an instance of a formula. A deduction is valid when each successive node follows from its predecessors in accordance with some predetermined inference rules. A particular attraction of Natural Deduction is its clean and economical presentation. Here for example are deduction (fragments) proving A ∧ B from A and B, and ∀x. (P (x) ∧ Q(x)) from ∀x. P (x) and ∀x. Q(x): ∀x. P (x) (∀E). (shrink)

Accounting for the epistemic contribution of deduction has been a pervasive problem for logicians interested in deduction, such as, among others, Jakko Hintikka. The problem arises because the conclusion validly deduced from a set of premises is said to be “contained” in that set; because of this containment relation, the conclusion would be known from the moment the premises are known. Assuming this, it is problematic to explain how we can gain knowledge by deducing a logical consequence implied (...) by a set of known premises. To address this problem, we offer an alternative account of the epistemic contribution of deduction as the process required to deduce a conclusion or a theorem, understanding such a process not only in terms of the number of steps in the derivation but also, more importantly, in terms of the reason for or justification for every step. That is, we do not know a proposition unless we have a justification or proof of that proposition. With this goal in mind, we develop a justification logic system which exhibits the epistemic contribution of a deductive derivation as the resulting justified formula. (shrink)

This introduction to formal logic is one of the few paperbacks available that provides a broad survey of the field. In addition to a clear presentation of sentential and first order quantificational logic, there is a discussion of the philosophical significance of recent work by Church, Gödel, and Tarski. The proof technique employed throughout is the indirect argument. Since proofs of this sort can be converted into mechanical tests of validity, it is easier than most for a beginning student (...) to grasp. Furthermore, this style of argument has long been recognized as the most obvious method of showing arguments in ordinary language to be invalid. Interesting treatments of the construction of decision machines, the nature of formal systems, and the application of logic to mathematics are included in the last section. Numerous exercises are included for which some answers are provided.--R. P. M. (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)