George Boole published the pamphlet The Mathematical Analysis of Logic in 1847. He believed that logic should belong to a universal mathematics that would cover both quantitative and nonquantitative research. With his pamphlet, Boole signalled an important change in symbolic logic: in contrast with his predecessors, his thinking was exclusively extensional. Notwithstanding the innovations introduced he accepted all traditional Aristotelean syllogisms. Nevertheless, some criticisms have been raised concerning Boole’s view of Aristotelean logic as the solution of algebraic equations. In order (...) to circumvent such criticisms, we show here how Boole’s conclusions may be deduced from the axioms and inference rules proposed in his pamphlet, from which Aristotle’s deductions in Prior Analytics (including those that are completed through an impossibility) can be stated as theorems. We clarify his method for dealing with both universal and particular premises by means of his symbol v and demystify some criticisms concerning the way he expressed particular premises. Boole’s conclusions for some of those premises for which according to Aristotle there is no deduction are also discussed. The deductive system presented here has the potential to provide a new perspective on Boole’s approach to logic in The Mathematical Analysis of Logic. (shrink)

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

The paper is a continuation of A deductive-reductive form oflogic: general theory and intuitionistic case and considers the problem of definability of modal operators on the intuitionistic base. Contrary tothe classical case, it seems that the fact whether the connective is Heyting’sor Brouwerian is essential for the intuitionistic logic. The connective of possibility has the classical interpretation, i.e. w |= ✸α iff ∃t, if it is defined on the base of the logicwith Brouwerian connective of coimplication.

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)

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

Our article identifies and describes the metaphoric fallacy to a deductive inference (MFDI) that is an example of incorrect reasoning along the lines of the false analogy fallacy. The MFDI proceeds from informal semantical (metaphorical) claims to a supposedly formally deductive and necessary inference. We charge that such an inference is invalid. We provide three examples of the MFDI to demonstrate the structure of this invalid form of reasoning. Our goal is to contribute to the set of known informal fallacies.

The paper deals with reconstruction of the unique reductivecounterpart of the deductive logic. The procedure results in the deductivereductive form of logic. This extension is illustrated on the base of intuitionistic logics: Heyting’s, Brouwerian and Heyting-Brouwer’s ones.

The traditional No-Miracles Argument (TNMA) asserts that the novel predictive success of science would be a miracle, and thus too implausible to believe, if successful theories were not at least approximately true. The TNMA has come under fire in multiple ways, challenging each of its premises and its general argumentative structure. While the TNMA relies on explaining novel predictive success via the truth of the theories, we put forth a deductive version of the No-Miracles argument (DNMA) that avoids inference to (...) the best explanation entirely. Instead, a relatively simple empirical framework and a probabilistic analysis can accomplish the ambitious goals of the TNMA while entirely sidestepping its problems. This close-but-distinct argument has many independent strengths and comparatively few weaknesses. Indeed, objections tailored specifically to the DNMA reveal surprising insights into how exactly NMAs are neither circular nor question-begging, as has been widely speculated. (shrink)

I propose a deductive-nomological model for mathematical scientific explanation. In this regard, I modify Hempel’s deductive-nomological model and test it against some of the following recent paradigmatic examples of the mathematical explanation of empirical facts: the seven bridges of Königsberg, the North American synchronized cicadas, and Hénon-Heiles Hamiltonian systems. I argue that mathematical scientific explanations that invoke laws of nature are qualitative explanations, and ordinary scientific explanations that employ mathematics are quantitative explanations. I analyse the repercussions of this deductivenomological model (...) on causal explanations. (shrink)

This is a monumental study of the transcendental deduction—that argument of legendary obscurity lying at the heart of Immanuel Kant’s Critique of Pure Reason. Allison begins from the methodological precept that “Kant’s argument can best be understood in light of the internal development of his thought”, and his book thus provides a systematic historical account of the deduction and its emergence from earlier texts. It begins with two chapters on the major pre-critical writings, which trace the emergence of (...) Kant’s “methodological” concerns with the limits and possibilities of metaphysical theorizing. However, this concern... (shrink)

This paper attempts a deduction of Kant's concept of the highest good: that is, it attempts to prove, in accordance with Dieter Henrich.s interpretation of the notion of deduction, that the highest good is an end that is also a duty. It does this by appealing to features of practical reason that make up the legitimating facts that serve as the premises that any deduction must possess. According to Kant, the highest good consists of happiness, virtue, and (...) relations of proportionality and causation between happiness and virtue, such that happiness is proportional to and caused by virtue. I argue, by drawing on accepted Kantian notions, that Kant had compelling reasons for concluding that the highest good is in fact an end that is also a duty. If correct, then this argument provides the deduction promisedin my title. (shrink)

We propose a new schema for the deduction theorem and prove that the deductive system S of a prepositional logic L fulfills the proposed schema if and only if there exists a finite set A(p, q) of propositional formulae involving only prepositional letters p and q such that A(p, p) L and p, A(p, q) s q.

In this paper I explore the ways in which Alexander of Aphrodisias employs and develops so-called ‘common notions’ as reliable starting points of deductive arguments. He combines contemporary developments in the Stoic and Epicurean use of common notions with Aristotelian dialectic, and axioms. This more comprehensive concept of common notions can be extracted from Alexander’s commentary on Metaphysics A 1–2. Alexander puts Aristotle’s claim that ‘all human beings by nature desire to know’ in a larger deductive framework, and adds weight (...) to Aristotle’s use of the common understanding of the notion of ‘wisdom’. Finally I will indicate how these upgraded common notions are meant to play an important role in the general framework of metaphysics as a science. (shrink)

Our article identifies and describes the metaphoric fallacy to a deductive inference (MFDI) that is an example of incorrect reasoning along the lines of the false analogy fallacy. The MFDI proceeds from informal semantical (metaphorical) claims to a supposedly formally deductive and necessary inference. We charge that such an inference is invalid. We provide three examples of the MFDI to demonstrate the structure of this invalid form of reasoning. Our goal is to contribute to the set of known informal fallacies.

The aim of this article is to present and discuss the language of the «givens», a typical stylistic resource of Greek mathematics and one of the major features of the proof format of analysis and synthesis. I shall analyze its expressive function and its peculiarities, as well as its general role as a deductive tool, explaining at the same time its particular applications in subgenres of a geometrical proposition like the locus theorems and the so-called «porisms». The main interpretative theses (...) of this study are the following: the language of the «givens» (1) is the standard idiom in which “existence and uniqueness” of a mathematical object was proved, (2) was conceived as an unified framework reducing to a strictly deductive format disparate argumentative steps such as deductions, constructions, and calculations. (shrink)

Background As more people are living longer, they become frail and are affected by multi-morbidity, resulting in increased demands from the ambulance service. Being vulnerable, older patients may have reduced decision-making capacity, despite still wanting to be involved in decision-making about their care. Their needs may be complex and difficult to assess, and do not always correspond with ambulance assessment protocols. When needing an ambulance, older patients encounter ambulance clinicians who are under high workloads and primarily consider themselves as emergency (...) medical care providers. This situates them in the struggle between differing expectations, and ethical conflicts may arise. To resolve these, providing ethical care, focussing on interpersonal relationships and using ethical competence is needed. However, it is not known whether ambulance clinicians possess the ethical competence required to provide ethical care. Thus, the aim of this study was to deductively explore their ethical competence when caring for older patients with reduced decision-making ability. Methods A qualitative deductive and exploratory design was used to analyse dyadic interviews with ambulance clinicians. A literature review, defining ethical competence as comprising ethical sensitivity, ethical knowledge, ethical reflection, ethical decision-making, ethical action and ethical behaviour, was used as a structured categorization matrix for the analysis. Results Ambulance clinicians possess ethical competence in terms of their ethical knowledge, highlighting the need for establishing an interpersonal relationship with the older patients. To establish this, they use ethical sensitivity to interpret the patients’ needs. Doing this, they are aware of their ethical behaviour, signifying how they must act respectfully and provide the necessary time for listening and interacting. Conclusions Ambulance clinicians fail to see their gut feeling as a professional ethical competence, which might hinder them from reacting to unethical ways of working. Further, they lack ethical reflection regarding the benefits and disadvantages of paternalism, which reduces their ability to perform ethical decision-making. Moreover, their ethical knowledge is hampered by an ageist approach to older patients, which also has consequences for their ethical action. Finally, ambulance clinicians show deficiencies regarding their ethical reflections, as they reflect merely on their own actions, rather than on their values. (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)

A proof of ‘correctness’ for a mathematical algorithm cannot be relevant to executions of a program based on that algorithm because both the algorithm and the proof are based on assumptions that do not hold for computations carried out by real-world computers. Thus, proving the ‘correctness’ of an algorithm cannot establish the trustworthiness of programs based on that algorithm. Despite the (deceptive) sameness of the notations used to represent them, the transformation of an algorithm into an executable program is a (...) wrenching metamorphosis that changes a mathematical abstraction into a prescription for concrete actions to be taken by real computers. Therefore, it is verification of program executions (processes) that is needed, not of program texts that are merely the scripts for those processes. In this view, verification is the empirical investigation of: (a) the behavior that programs invoke in a computer system and (b) the larger context in which that behavior occurs. Here, deduction can play no more, and no less, a role than it does in the empirical sciences. (shrink)

I argue that the reasonable quest of looking for, and the desirable end of having better explanation of particular events are not allowed for by the deductive account of explanation except by total replacement of one theory by another. The article is also a trivialization of the deductive conception of complete explanation.

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.

The confusion/non-consequential thinking explanation proposed by Newstead, Girotto, and Legrenzi (1995) for poor performance on Wason's THOG problem (a hypothetico-deductive reasoning task) was examined in three experiments with 300 participants. In general, as the cognitive complexity of the problem and the possibility of non-consequential thinking were reduced, correct performance increased. Significant but weak facilitation (33-40% correct) was found in Experiment 1 for THOG classification instructions that did not include the indeterminate response option. Substantial facilitation (up to 75% correct) was obtained (...) in Experiment 2 with O'Brien et al.'s (1990) one-other-THOG classification instruction. In Experiment 3, a revised version of O'Brien et al.'s pre-test problem format also led to substantial facilitation, even with the use of the standard three-choice THOG classification instruction. These findings are discussed in terms of Newstead et al.'s theoretical proposal and possible attentional factors. (shrink)

A methodology is proposed as a tool for explanation of form in zoomorphology, in particular its design, diversity, and transformation. An alternate use of descriptive, inductive/comparative, and deductive methods is suggested. The basic concepts required are summarized. Following an extensive anatomical analysis a specific deductive methodology is developed, comprising three major parts: 1) Formal analysis of systems, using optimal design. 2) Transformation of an initial system's model by simulating modifications via maximizing the model for specific functional requirements. 3) Testing by (...) comparison of deduced modifications with real systems having the maximized functional capacity, trends in specific morphoclines, and empirical analogues from technology.The methodology produces a morpho-space of modifications that has the form of branching pathways. That pattern results from innovation, bifurcation, and radiation moments in the deduction. The morpho-space is converted in a mechano-space by including next to constructional characters those of operation and rolefulfilment. If the initial system is verified as the ancestor and the transformation-function is verified as the evolutionary theme of change, the explanation turns -in addition to a function anatomical one- also into an evolutionary one. The method is illustrated by reviewing some recent analyses about avian feeding: maximization of avian pecking for probing and filter feeding capacities. (shrink)

Juhani Yli-Vakkuri has argued that the Twin Earth thought experiments offered in favour of semantic externalism can be replaced by a straightforward deductive argument from premisses widely accepted by both internalists and externalists alike. The deductive argument depends, however, on premisses that, on standard formulations of internalism, cannot be satisfied by a single belief simultaneously. It does not therefore, constitute a proof of externalism. The aim of this article is to explain why.

The notion of normal proof theory, and yet it has been somewhat neglected by the systems of equational logic. The intention here is then to show the normalization procedure for the equational logic of the Labelled Natural Deduction system . With this we believe we are making a step towards filling a gap in the literature on equational logic. Besides presenting a normalization procedure for the LND equational fragment, we employ a new method to prove the normalization theorems for (...) equational logic: proof transformation by rewriting, based on a algebraic calculus on the 'rewrite reasons'. (shrink)