By considering the new notion of theinversesof syllogisms such asBarbaraandCelarent, we show how the rule ofIndirect Proof, in the form (no multiple or vacuous discharges) used by Aristotle, may be dispensed with, in a system comprising four basic rules of subalternation or conversion and six basic syllogisms.

While direct proof is widely considered the paradigm of the acquisition of knowledge by deductive means, indirect proof has traditionally been criticized as showing merely ‘that’ its conclusion is true and not ‘why’ it is true. This paper accounts for the traditional objection by emphasizing the argumentative role in indirect proof of logical principles such as excluded middle and non-contradiction.

Contends that Hegel's reconstruction of valid logic leads to a conception of indirect proof and syllogisms. Clarification of the concept of indirect proof; Reference to previous papers on the subject; Indirect proof as the natural form of deduction.

Frege's account of indirect proof has been thought to be problematic. This thought seems to rest on the supposition that some notion of logical consequence ? which Frege did not have ? is indispensable for a satisfactory account of indirect proof. It is not so. Frege's account is no less workable than the account predominant today. Indeed, Frege's account may be best understood as a restatement of the latter, although from a higher order point of view. I argue (...) that this ascent is motivated by Frege's conception of logic. (shrink)

It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of correct (...) independence arguments, deriving from the geometrical principle of duality and the associated idea of substitution invariance. The discussion also serves as a useful focal point for independently interesting details of Frege's mathematical environment. This feeds into a currently active scholarly debate because Frege's supposed attitude to independence arguments has been taken to support a widely accepted thesis (proposed by Ricketts among others) concerning Frege's attitude toward metatheory in general. I show that this thesis gains no support from Frege's puzzling remarks about independence arguments. (shrink)

In this paper it is shown that addition of certain reductions to the standard cut removing reductions of deductions in prepositional logic makes prepositional logic non-normalizable. From this follows that some provable propositions in prepositional logic has no direct proof.

In the last two decades, there has been increasing interest in a re-evaluation of Frege’s stance towards consistency- and independence proofs. Papers by several authors deal with Frege’s views on these topics. In this note, I want to discuss one particular problem, which seems to be a main reason for Frege’s reluctant attitude towards his own proposed method of proving the independence of axioms, namely his view that thoughts, that is, intensional entities are the objects of metatheoretical investigations. This (...) stands in contrast to more straightforward interpretations, which claim that Frege’s hesitancy is mainly due to worries concerning the logical constants or what counts as a logical inference. (shrink)

A proof of normalization for a classical system of Peano Arithmetic formulated in natural deduction is given. The classical rule of the system is the rule for indirect proof restricted to atomic formulas. This rule does not, due to the restriction, interfere with the standard detour conversions. The convertible detours, numerical inductions and instances of indirect proof concluding falsity are reduced in a way that decreases a vector assigned to the derivation. By interpreting the expressions of the vectors (...) as ordinals each derivation is assigned an ordinal less than ɛ0. The vector assignment, which proves termination of the procedure, originates in a normalization proof for Gödel’s T by Howard. (shrink)

In the first chapter, we discuss Dummett’s idea that the notion of truth arises from the one of the correctness of an assertion. We argue that, in a first-order language, the need of defining truth in terms of the notion of satisfaction, which is yielded by the presence of quantifiers, is structurally analogous to the need of a notion of truth as distinct from the one of correctness of an assertion. In the light of the analogy between predicates in Frege (...) and open formulas in Tarksi, we concentrate on the semantic status of predicates. We register a dual attitude of Dummett towards Frege’s ascription of reference to predicates. On the one it is needed to endow quantifiers with their appropriate meaning. On the other hand, the introduction of concepts, as semantic correlate of predicates, smuggles a realist flavor in the overall semantic picture. In concluding the excursus (and with it the chapter), we stress Dummett’s will of developing a semantic picture free from this realist trait. In the second chapter, we present the idea of a proof-theoretic semantics. As for Frege true sentences denotes the truth-value True, so here closed' (i.e. categorical) valid argumentations denotes proofs. If a language contains implication-like operators, in order to characterize the condition of validity of a closed argumentation one has to introduce a notion of validity applying to ‘open’ (i.e. hypothetical) argumentations. We argue that this problem is analogous to the one posed by quantifiers in Frege-Tarksi’s style semantics. That is, implication forces one to introduce notion of validity for argumentations, which is more substantial than the correctness of the assertion of their conclusions. As we saw, the corresponding claim in the truth-based approach—that quantifiers require one to introduce a notion of truth, which is more substantial than the one of an assertion being correct—was the source of realism. Hence, Dummett proposes to reduce the semantic contribution of open argumentations to the one of their ‘closed instances’. In the truth-based perspective, this would correspond to the denial of the need of introducing concepts as the semantic correlates of predicates. We argue that Dummett’s fear, that an irreducible notion of function (represented by the need of ascribing validity to open argumentations) would lead to realism, turns out to be ill-founded. In the third chapter, we discuss the role played by the notion of truth in the anti-realist account. The notion of truth is what the anti-realist needs to cope with the so-called paradox of deduction. The analysis of the paradox yields to distinguishing between the truth of a sentence and the truth of a sentence being recognized. In terms of these conceptual couple, we reconsider the relationship between truth and assertion in an anti-realist perspective. Grounds are provided for a thesis (which was already advanced in chapter two), according to which the notion of the assertion of a sentence being correct is primarily connected only with the canonical means of establishing a sentence. The possibility of establishing a sentence by indirect means is conceptually dependent on the practice of establishing logical relationship of dependence among sentences. That is, the notion of a closed valid non-canonical argumentation is of any theoretical relevance only in presence of a notion of validity applying to open argumentations. In the fourth chapter, we discuss the possibility of characterizing in the proof-theoretic-semantics a notion of refutation. We develop an original characterization of refutations starting from an informal inductive specification of the condition of refutations of logically complex sentences. A sub-structural logic, called dual-intuitionistic logic, stands to this notion in the same relationship in which intuitionistic logic stands to the notion of proof so far consdered. All notions developed in chapter 2 have their corresponding (dual) ones in the framework developed. In particular, the distinctions canonical/non-canonical and closed/open argumentations. In the refutation based perspective, elimination rules have priority over introductions and the (only) assumption over the (many) conclusions. In the conclusions, we indicate the ingredients that an anti-realist approach to meaning should incorporate, in order to avoid the difficulties we registered. The core of an alternative view is a different conception of the relationship between categorical and hypothetical notions, in which the validity of open argumentations is not reduced to that of their instances, but rather it is directly defined. As a limit case, one would get a notion of validity applying to closed argumentations. (shrink)

"Proof" has been and remains one of the concepts which characterises mathematics. Covering basic propositional and predicate logic as well as discussing axiom systems and formal proofs, the book seeks to explain what mathematicians understand by proofs and how they are communicated. The authors explore the principle techniques of direct and indirect proof including induction, existence and uniqueness proofs, proof by contradiction, constructive and non-constructive proofs, etc. Many examples from analysis and modern algebra are included. (...) The exceptionally clear style and presentation ensures that the book will be useful and enjoyable to those studying and interested in the notion of mathematical "proof.". (shrink)

Not focusing on the history of classical logic, this book provides discussions and quotes central passages on its origins and development, namely from a philosophical perspective. Not being a book in mathematical logic, it takes formal logic from an essentially mathematical perspective. Biased towards a computational approach, with SAT and VAL as its backbone, this is an introduction to logic that covers essential aspects of the three branches of logic, to wit, philosophical, mathematical, and computational.

Scholars who consider that the Transcendental Analytic contains the core of what Kant calls ‘transcendental idealism’ are mistaken. Indeed, Kant’s transcendental idealism of space, time and spatiotemporal objects is sufficiently proved in the Transcendental Aesthetic and does not depend on complementary claims made later on in the Critique. This does not mean, however, that we are allowed to subscribe to the so-called separability-thesis, which states that we can endorse Kant's views in the Transcendental Logic without endorsing the results of the (...) expositions and arguments laid out in the Aesthetic. Nor does it mean that the Aesthetic contains the only proof for transcendental idealism. Indeed, as Kant himself explicitly recognizes, the antinomies provide the premise for an indirect or mediate proof of his position with respect to the nature of space and time. First, I analyze Kant’s argument for the claim that the production of transcendental illusion is inherent to the function of human reasoning, and therefore inevitable. I then follow the presuppositions and the argumentation of the mediate proof of transcendental idealism and argue that it has a more general validity than could be assumed on the basis of Kant’s text, because it is in its essence independent from the particular determinations of the cosmological idea in the form of the four antinomies. Indeed, as I argue, the nature of reason itself and the content of the general cosmological idea suffice as elements for a mediate proof. Thus, I claim that transcendental idealism can be mediately established on the basis of the general antinomical relation between transcendental realism and material idealism, without any requirement to refer to the particular realisations of this antagonism as they are presented in the four antinomies. If the argument is correct, it shows that the mediate proof of transcendental idealism does not depend on a sympathetic account of the antinomies and is not automatically invalidated if the proofs of thesis and antithesis are considered unconvincing. (shrink)

This paper examines the role of reason in Shepherd's account of acquiring knowledge of the external world via first principles. Reason is important, but does not have a foundational role. Certain principles enable us to draw the required inferences for acquiring knowledge of the external world. These principles are basic, foundational and, more importantly, self‐evident and thus justified in other ways than by demonstration. Justificatory demonstrations of these principles are neither required, nor possible. By drawing on textual and contextual evidence, (...) I will show that Shepherd should have said that we know the first principles of any science, in general, and that “everything which begins to exist must have a cause”, in particular, via intuition, not via reason. Reasoning about such principles can help their self‐evidence shine through in certain cases; their justification, and our being justified in believing them, does not come from this reasoning, however. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Hume Studies Volume 29, Number 2, November 2003, pp. 205-221 The Origin of the Indirect Passions in the Treatise: An Analogy between Books 1 and 2 HARUKOINOUE 1. The Analogy Between Book 1 and Book 2 If the central design of the Treatise is to demonstrate that "the subjects of the Understanding and Passions make a complete chain of reasoning by themselves" (T 2; SBN xii), as Hume (...) advertises, it seems unquestionable that his intention lies in the illustration of the human mind as an integrated system dependent upon the cooperation of the understanding and the passions, which are discussed separately in Book 1 and Book 2, respectively.1 What is yet to be determined is whether any internal or systematic connection between these two books was originally intended by the author. Is it too fanciful to agree with John Passmore, and to suggest that some dynamic system of the human mind would emerge when we open the door which stands between the first two books?2 The key for this door seems to be found in Hume's account of personal identity. We may well remember how he distinguished two aspects of our identity, one regarding the understanding and another regarding the passions, and claimed that the latter "serves to corroborate" the former "by the making our distant perceptions influence each other, and by giving us a present concern for our past or future pains or pleasures" (T 1.4.6.19; SBN Haruko Inoue is Professor of Philosophy, Sapporo University 3-7-3-1 Toyohira-ku, Sapporo 062-8520, Japan. e-mail: [email protected] 206 Haruko Inoue 261). It seems reasonable to agree with Jane Mclntyre, who suggests that "Hume recognized questions about personal identity not addressed in Book 1, and that Book 2 makes an important contribution to our understanding of Hume's account of the self and its identity."3 And once we get a good understanding of this integrated mechanism of the human mind operating with both wheels, as it were, of the imagination and the passions, all we have to do is to accept his invitation and enjoy the efficiency of this two-wheeled car, driving through the fields of "Morals, Politics, and Criticism." For, if we take his words in his Advertisement seriously,4 it seems quite likely that the Treatise was published first as a set of Book 1 and Book 2, as Passmore suggests,5 and that Book 3 was written as the demonstration of the consistency of his system through the application of the hypothesis established in the first two books. This paper is a modest attempt to illustrate the intimate connection between Book 1 and Book 2: the latter, planned as a proof and reinforcement of the system Hume had established in Book 1, depends on the former. His basic strategy in the Treatise is to explain both systems of the mind by means of the easy transition of the imagination, which connects different perceptions with each other according to two kinds of principles or "properties of human nature " (T 2.1.4.2; SBN 283), viz. "the association both of impressions and ideas, and the mutual assistance they lend each other" (T 2.1.5.5; SBN 284). In Book 1 Hume establishes the system of the understanding in terms of the first association, viz. the association of ideas, and in Book 2 he proceeds to illustrate the system of passions by involving not only the second association but also the concurrence of both kinds of association which he calls "a double relation of impressions and ideas." His basic strategy is thus to apply the same method of reasoning to the illustration of both systems, assuming that "there is an attraction or association among impressions, as well as among ideas" (T 2.1.4.3; SBN 283). The intimate connection between the two systems of the understanding and the passions is illustrated through the demonstration of the "influence of the imagination upon the passions" (T 2.3.6.1; SBN 424), or in terms of the dependence of the latter on the former. It is owing to the dependence of the association of... (shrink)

Peter van Inwagen has developed two highly influential strategies for establishing incompatibilism about causal determinism and moral responsibility. These have come to be known as ‘the Direct Argument’ and ‘the Indirect Argument,’ respectively. In recent years, the two arguments have attracted closely related criticisms. In each case, it is claimed, the argument does not provide a fully general defense of the incompatibilist’s conclusion. While the critics are right to notice these arguments’ limitations, they have not made it clear what (...) the problem with the arguments is supposed to be. I suggest three possibilities, arguing that none proves to be well founded. I conclude that the scope of these arguments is fully adequate for their defenders’ purposes. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Hume Studies Volume 29, Number 2, November 2003, pp. 205-221 The Origin of the Indirect Passions in the Treatise: An Analogy between Books 1 and 2 HARUKOINOUE 1. The Analogy Between Book 1 and Book 2 If the central design of the Treatise is to demonstrate that "the subjects of the Understanding and Passions make a complete chain of reasoning by themselves" (T 2; SBN xii), as Hume (...) advertises, it seems unquestionable that his intention lies in the illustration of the human mind as an integrated system dependent upon the cooperation of the understanding and the passions, which are discussed separately in Book 1 and Book 2, respectively.1 What is yet to be determined is whether any internal or systematic connection between these two books was originally intended by the author. Is it too fanciful to agree with John Passmore, and to suggest that some dynamic system of the human mind would emerge when we open the door which stands between the first two books?2 The key for this door seems to be found in Hume's account of personal identity. We may well remember how he distinguished two aspects of our identity, one regarding the understanding and another regarding the passions, and claimed that the latter "serves to corroborate" the former "by the making our distant perceptions influence each other, and by giving us a present concern for our past or future pains or pleasures" (T 1.4.6.19; SBN Haruko Inoue is Professor of Philosophy, Sapporo University 3-7-3-1 Toyohira-ku, Sapporo 062-8520, Japan. e-mail: [email protected] 206 Haruko Inoue 261). It seems reasonable to agree with Jane Mclntyre, who suggests that "Hume recognized questions about personal identity not addressed in Book 1, and that Book 2 makes an important contribution to our understanding of Hume's account of the self and its identity."3 And once we get a good understanding of this integrated mechanism of the human mind operating with both wheels, as it were, of the imagination and the passions, all we have to do is to accept his invitation and enjoy the efficiency of this two-wheeled car, driving through the fields of "Morals, Politics, and Criticism." For, if we take his words in his Advertisement seriously,4 it seems quite likely that the Treatise was published first as a set of Book 1 and Book 2, as Passmore suggests,5 and that Book 3 was written as the demonstration of the consistency of his system through the application of the hypothesis established in the first two books. This paper is a modest attempt to illustrate the intimate connection between Book 1 and Book 2: the latter, planned as a proof and reinforcement of the system Hume had established in Book 1, depends on the former. His basic strategy in the Treatise is to explain both systems of the mind by means of the easy transition of the imagination, which connects different perceptions with each other according to two kinds of principles or "properties of human nature " (T 2.1.4.2; SBN 283), viz. "the association both of impressions and ideas, and the mutual assistance they lend each other" (T 2.1.5.5; SBN 284). In Book 1 Hume establishes the system of the understanding in terms of the first association, viz. the association of ideas, and in Book 2 he proceeds to illustrate the system of passions by involving not only the second association but also the concurrence of both kinds of association which he calls "a double relation of impressions and ideas." His basic strategy is thus to apply the same method of reasoning to the illustration of both systems, assuming that "there is an attraction or association among impressions, as well as among ideas" (T 2.1.4.3; SBN 283). The intimate connection between the two systems of the understanding and the passions is illustrated through the demonstration of the "influence of the imagination upon the passions" (T 2.3.6.1; SBN 424), or in terms of the dependence of the latter on the former. It is owing to the dependence of the association of... (shrink)

In order to explicate Gentzen’s famous remark that the introduction-rules for logical constants give their meaning, the elimination-rules being simply consequences of the meaning so given, we develop natural deduction rules for Sheffer’s stroke, alternative denial. The first system turns out to lack Double Negation. Strengthening the introduction-rules by allowing the introduction of Sheffer’s stroke into a disjunctive context produces a complete system of classical logic, one which preserves the harmony between the rules which Gentzen wanted: all indirect proof (...) reduces to direct proof. (shrink)

By rephrasing quantifier-free axioms as rules of derivation in sequent calculus, we show that the generalized Steiner–Lehmus theorem admits a direct proof in classical logic. This provides a partial answer to a question raised by Sylvester in 1852. We also present some comments on possible intuitionistic approaches.

What determines whether the procedures for proving the affirmative statement of God's existence may be called a proof? Certainly, it is necessary that all premises be true and that a reliable inference schemata be applied. One premise appears to be the most critical in the theistic argument. This premise is the principle of sufficient reason. I hold the view that the principle of sufficient reason cannot be found among the premises of any metaphysical explanation of reality, so I suggest that (...) the terms “proof” and “argument” not be used. Instead, we could speak of ways of acquiring discursive knowledge of God and ways of indirect substantiation of God's existence. (shrink)

What determines whether the procedures for proving the affirmative statement of God's existence may be called a proof? Certainly, it is necessary that all premises be true and that a reliable inference schemata be applied. One premise appears to be the most critical in the theistic argument. This premise is the principle of sufficient reason. I hold the view that the principle of sufficient reason cannot be found among the premises of any metaphysical explanation of reality, so I suggest that (...) the terms “proof” and “argument” not be used. Instead, we could speak of ways of acquiring discursive knowledge of God and ways of indirect substantiation of God's existence. (shrink)

A resurgence of scholarly work on proof of God?s existence is noticeable over the past decade, with considerable emphasis on attempts to provide?analytic proof? based on the meanings and logic of various identity statements which constitute premises of the syllogisms of the?proof?. Most recently perhaps, Emmanuel Rutten?s?modal-epistemic proof? has drawn serious academic attention. Like other?analytic? and strictly logical proofs of God?s existence, Rutten?s proof has been found flawed. In this paper I discuss the possibility of an?ethics-based? identity proof of (...) God?s existence. Such a proof, the first version of which, I believe, has been offered, indirectly, by Nikolai Lossky, utilizes the form and structure of the analytic proof, but fundamentally rests on the perception of moral values we associate with God and Godliness. The nature of the proof shifts the focus of the very attempt to?prove? God?s existence from what I believe is an unreasonable standard, unattainable even in?proving? the existence of the more mundane world, towards a more functional, practical and attainable standard. The proof proposed initially by Lossky, and in a more systematic form here, I believe, shows the indubitable existence of God in the sense of his moral presence in the lives of the faithful, at least with the same degree of certainty as the presence or?existence? of anything else that can be epistemically proven in principle. (shrink)

We prove an exponential lower bound on the size of proofs in the proof system operating with ordered binary decision diagrams introduced by Atserias, Kolaitis and Vardi [2]. In fact, the lower bound applies to semantic derivations operating with sets defined by OBDDs. We do not assume any particular format of proofs or ordering of variables, the hard formulas are in CNF. We utilize (somewhat indirectly) feasible interpolation. We define a proof system combining resolution and the OBDD proof (...) system. (shrink)

Immanuel Kant states that indirect arguments are not suitable for the purposes of transcendental philosophy. If he is correct, this affects contemporary versions of transcendental arguments which are often used as an indirect refutation of scepticism. I discuss two reasons for Kant’s rejection of indirect arguments. Firstly, Kant argues that we are prone to misapply the law of excluded middle in philosophical contexts. Secondly, Kant points out that indirect arguments lack some explanatory power. They can show (...) that something is true but they do not provide insight into why something is true. Using mathematical proofs as examples, I show that this is because indirect arguments are non-constructive. From a Kantian point of view, transcendental arguments need to be constructive in some way. In the last part of the paper, I briefly examine a comment made by P. F. Strawson. In my view, this comment also points toward a connection between transcendental and constructive reasoning. (shrink)

Ātman (soul) and Nairātmya (no soul) are, for the Brahmanical schools and the Buddhists respectively, equally fundamental tenets which neither side can concede to the other. Among the 16 formulations presented by Uddyotakara, the fifteenth, which is a proof of Ātman and is originally an indirect proof ( avīta/āvīta ), is presented in a prasaṅga -style, and contains double negation ( na nairātmyam ) in the thesis. However, it is perhaps Dharmakīrti who first transformed it into a normal style (...) ( sātmakam ). He is well aware of the law of excluded middle, and insisits that the negation is paryudāsa . On the Nyāya side, Uddyotakara at least seems to be unaware of the law of the logical equivalence of contraposition concerning pervasion ( vyāpti ). After Uddyotakara, however, Vyoman (Vyomaśiva), Bhāsarvajña and Vācaspatimiśra, all seem to be well aware of it. Dharmakīrti, in his conter-argument against the proof of ātman , discusses the negative expressions ‘‘ nairātmya ” and ‘‘ a-nairātmya ” Dharmakīrti here uses two logical arguments skillfully and tactically. As a critic of both the authenticity of the Veda and the existence of ātman , he insists on the theory of dichotomy and the equivalence of anvaya and vyatireka , whereas as an apologist he denies the application of these theories to the relation between the existence of ātman and the concept of nairātmya , because for him as a Buddhist the latter is not a negative but essentially positive state of affairs. (shrink)

What has become known as the ‘faith/history’ problem for historical religions like Christianity centres on the attempt to combine the ontological decisiveness, for faith, of an historical event characterized as an actual Incarnation of God with the epistemological indifference, or irrelevance, of historical information about that event which is decisive for faith. Without the former there is nothing to be related to or personally appropriated; without the latter faith is rendered vulnerable to the vagaries of historical research. Soren Kierkegaard's Climacus (...) writings – the Philosophical Fragments and the Concluding Unscientific Postscript – provide a classic formulation of the inherent tension. Twentieth-century variations on the same theme are found in the writings of Paul Tillich and Rudolf Bultmann, among others. In what follows I want to highlight an element in Kierkegaard's Climacus account which crucially distinguishes his attempt to deal with the faith/history relation from other attempts to deal with it, and thus offers what I suggest is a novel approach to overcoming the tension. This element – effectively a kind of a priori or conceptual proof – has been neglected, and whatever one's final judgement on its contribution to the problem as a whole, a consideration of it and its implications is, at the very least, integral to a full understanding of the thought-experiment in the Fragments . Moreover, in the process, useful light can be shed indirectly on twentieth-century attempts to deal with the faith/history problem. (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)

In this paper we revisit Pavel Tichý’s novel distinction between one-dimensional and two-dimensional conception of inference, which he presented in his book Foundations of Frege’s Logic (1988), and later in On Inference (1999), which was prepared from his manuscript by his co-author Jindra Tichý. We shall focus our inquiry not only on the motivation behind the introduction of this non-classical concept of inference, but also on further inspection of selected Tichý’s arguments, which we see as the most compelling or simply (...) most effective in providing support for his two-dimensional account of inference. Main attention will be given to exposing the failure of one-dimensional theory of inference in its explanation of indirect (reductio ad absurdum) proofs. Lastly, we discuss shortly the link between two-dimensional inference and deduction apparatus of Tichý’s Transparent Intensional Logic. (shrink)

Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" are devoted. (...) At the time of the first edition, 1920, the editor was apparently not acquainted with the secondary literature on Logica demonstrativa which continued to grow in the period preceding the second edition \ref[see D. J. Struik, in Dictionary of scientific biography, Vol. 12, 55--57, Scribner's, New York, 1975]. Of special interest in this connection is a series of three articles by A. F. Emch [Scripta Math. 3 (1935), 51--60; Zbl 10, 386; ibid. 3 (1935), 143--152; Zbl 11, 193; ibid. 3 (1935), 221--333; Zbl 12, 98]. (2) It seems curious that modern writers believe that demonstration of the "nondeducibility" of the parallel postulate vindicates Euclid whereas at first Saccheri seems to have thought that demonstration of its "deducibility" is what would vindicate Euclid. Saccheri is perfectly clear in his commitment to the ancient (and now discredited) view that it is wrong to take as an "axiom" a proposition which is not a "primal verity", which is not "known through itself". So it would seem that Saccheri should think that he was convicting Euclid of error by deducing the parallel postulate. The resolution of this confusion is that Saccheri thought that he had proved, not merely that the parallel postulate was true, but that it was a "primal verity" and, thus, that Euclid was correct in taking it as an "axiom". As implausible as this claim about Saccheri may seem, the passage on p. 237, lines 3--15, seems to admit of no other interpretation. Indeed, Emch takes it this way. (3) As has been noted by many others, Saccheri was fascinated, if not obsessed, by what may be called "reflexive indirect deductions", indirect deductions which show that a conclusion follows from given premises by a chain of reasoning beginning with the given premises augmented by the denial of the desired conclusion and ending with the conclusion itself. It is obvious, of course, that this is simply a species of ordinary indirect deduction; a conclusion follows from given premises if a contradiction is deducible from those given premises augmented by the denial of the conclusion---and it is immaterial whether the contradiction involves one of the premises, the denial of the conclusion, or even, as often happens, intermediate propositions distinct from the given premises and the denial of the conclusion. Saccheri seemed to think that a proposition proved in this way was deduced from its own denial and, thus, that its denial was self-contradictory (p. 207). Inference from this mistake to the idea that propositions proved in this way are "primal verities" would involve yet another confusion. The reviewer gratefully acknowledges extensive communication with his former doctoral students J. Gasser and M. Scanlan. ADDED 14 March 14, 2015: (1) Wikipedia reports that many of Saccheri's ideas have a precedent in the 11th Century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid, a fact ignored in most Western sources until recently. It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. (2) This book is another exemplification of the huge difference between indirect deduction and indirect reduction. Indirect deduction requires making an assumption that is inconsistent with the premises previously adopted. This means that the reasoner must perform a certain mental act of assuming a certain proposition. It case the premises are all known truths, indirect deduction—which would then be indirect proof—requires the reasoner to assume a falsehood. This fact has been noted by several prominent mathematicians including Hardy, Hilbert, and Tarski. Indirect reduction requires no new assumption. Indirect reduction is simply a transformation of an argument in one form into another argument in a different form. In an indirect reduction one proposition in the old premise set is replaced by the contradictory opposite of the old conclusion and the new conclusion becomes the contradictory opposite of the replaced premise. Roughly and schematically, P,Q/R becomes P,~R/~Q or ~R, Q/~P. Saccheri’s work involved indirect deduction not indirect reduction. (3) The distinction between indirect deduction and indirect reduction has largely slipped through the cracks, the cracks between medieval-oriented logic and modern-oriented logic. The medievalists have a heavy investment in reduction and, though they have heard of deduction, they think that deduction is a form of reduction, or vice versa, or in some cases they think that the word ‘deduction’ is the modern way of referring to reduction. The modernists have no interest in reduction, i.e. in the process of transforming one argument into another having exactly the same number of premises. Modern logicians, like Aristotle, are concerned with deducing a single proposition from a set of propositions. Some focus on deducing a single proposition from the null set—something difficult to relate to reduction. (shrink)

InMetaphysics Γ, Ch. 4, Aristotle speaks of both infinite regress and question-begging, but does not explicitly relate them. We get the impression that he thinks that to use one of these arguments to avoid the other is to jump from the frying-pan into the fire. This relationship is illustrated in terms of the ignorant belief that everything can be proved, and of attempts to prove the Law of Noncontradiction.

Kant conceived of 'critique' as a kind of winnowing exercise, with the aim of separating the wheat of good metaphysics from the chaff of bad. He used a less familiar metaphor to make this point, namely, that of 'the fiery test of critique'-not a medieval ordeal of trial by fire, but rather a metallurgical assay, or cupellation, a procedure in which ore samples are tested for their precious-metal content. When seen in this light, critique has a positive, investigatory side: it (...) seeks not merely to eliminate bad, 'dogmatic' metaphysics but also to uncover what of philosophical value might be contained in traditional speculative metaphysics. In this comprehensive study of the Transcendental Dialectic in Kant's Critique of Pure Reason, Proops argues that Kant uncovered two nuggets of value: the indirect proof of Transcendental Idealism afforded by the resolution of the Antinomies, and a defence of theoretically grounded 'doctrinal beliefs' in a wise and great originator, on the one hand, and in an afterlife, on the other. This examination of critique engages with Kant's views on a number of central problems in philosophy and meta-philosophy: the explanation of the enduring human impulse towards metaphysics, the correct philosophical method, the limits of self-knowledge, the possibility of human freedom, the resolution of metaphysical paradox, the justification of faith, the nature of scepticism, and the role of 'as if' reasoning in natural science. (shrink)

This paper offers a reconstruction of Wittgenstein's discussion on inductive proofs. A "algebraic version" of these indirect proofs is offered and contrasted with the usual ones in which an infinite sequence of modus pones is projected.

Cantor's diagonal argument provides an indirect proof that there is no one-one function from the power set of a set A into A. This paper provides a somewhat more constructive proof of Cantor's theorem, showing how, given a function f from the power set of A into A, one can explicitly define a counterexample to the thesis that f is one-one.

This paper analyzes and evaluates Bolzano's remarks on the apagogic method of proof with reference to his juvenile booklet "Contributions to a better founded presentation of mathematics" of 1810 and to his ?Theory of science? (1837). I shall try to defend the following contentions: (1) Bolzanos vain attempt to transform all indirect proofs into direct proofs becomes comprehensible as soon as one recognizes the following facts: (1.1) his attitude towards indirect proofs with an affirmative conclusion (...) differs from his stance to indirect proofs with a negative conclusion; (1.2) by Bolzano's lights arguments via consequentia mirabilis only seem to be indirect. (2) Bolzano does not deny that indirect proofs can be perfect certifications (Gewissmachungen) of their conclusion; what he denies is rather that they can provide grounds for their conclusions. (2.1) They cannot do the latter, since they start from false premises and (2.2) since they make an unnecessary detour. (3) The far-reaching agreement between his early and late assessment of apagogical proofs (in the Beyträge of 1810 and the Wissenschaftslehre of 1837) is partly due to the fact that he develops his own position always against the background of Wolff's and Lambert's views. (shrink)

A comprehensive introduction to formal logic, Logic and Philosophy: A Modern Introduction is a rigorous yet accessible text, appropriate for students encountering the subject for the first time. Abundant, carefully crafted exercise sets accompanied by a clear, engaging exposition build to an exploration of sentential logic, first-order predicate logic, the theory of descriptions, identity, relations, set theory, modal logic, and Aristotelian logic. And as its title suggests, Logic and Philosophy is devoted not only to logic but also to the philosophical (...) debates that led to the development of the field. Much new material has been added for the 13th edition. An introduction to set theory and its relationship to logic and mathematics, including philosophical issues, is now part of Chapter 13. Chapter 15 is an introduction to modal logic and Kripke semantics, concluding with a discussion of philosophical problems with any logical accommodation of modalities. Instructors who do not wish to present proof methods will find chapters on truth trees for both sentential and first-order logic, and a presentation of trees for modal logic. Special features of this text include presentations of the history of logic, alternatives to traditional methods of conditional and indirect proof, and a discussion of semantic problems with universal and existential instantiations. Throughout, the authors are sensitive to philosophical issues that arise from the relationship between ordinary language, symbolic logic, and justifications for the syntax and semantics of the various symbolic languages. Discussions range from the justification of the truth table for the sentential rendering of if... then statements to semantic and syntactic paradoxes, including some troubling paradoxes that arise in ordinary language (e.g., the so-called hangman or surprise quiz paradox). Logic and Philosophy includes ample material for a one-semester or two-semester course and provides a thorough preparation for more advanced logic courses. (shrink)

1. Despite the plain fact that there is nothing in this world that can be proved without reliance on some assumption or another, there is an inalienable difference between an argument that begins by assuming what it is designed to establish and one that begins by assuming the contradictory of what it is designed to establish. Arguments of the first kind are uncontroversially acknowledged to be circular, or question-begging; though valid they achieve nothing. Those of the second kind conform to (...) the classical pattern of reductio ad absurdum or indirect proof. A typical example is the proof of the irrationality of √2, which begins by assuming that for some integral m, n the identity m2/n2 = 2, and arrives at a contradiction. Nothing whatever is established or justified by arguments of the first kind, even granted the truth of all the other assumptions, and the validity of the rules of inference; while the second may establish that, if those assumptions are true, then the additional assumption is false. (shrink)

DEFINING OUR TERMS A “paradox" is an argumentation that appears to deduce a conclusion believed to be false from premises believed to be true. An “inconsistency proof for a theory" is an argumentation that actually deduces a negation of a theorem of the theory from premises that are all theorems of the theory. An “indirect proof of the negation of a hypothesis" is an argumentation that actually deduces a conclusion known to be false from the hypothesis alone or, more (...) commonly, from the hypothesis augmented by a set of premises known to be true. A “direct proof of a hypothesis" is an argumentation that actually deduces the hypothesis itself from premises known to be true. Since `appears', `believes' and `knows' all make elliptical reference to a participant, it is clear that `paradox', `indirect proof' and `direct proof' are all participant-relative. PARTICIPANT RELATIVITY In normal mathematical writing the participant is presumed to be “the community of mathematicians" or some more or less well-defined subcommunity and, therefore, omission of explicit reference to the participant is often warranted. However, in historical, critical, or philosophical writing focused on emerging branches of mathematics such omission often invites confusion. One and the same argumentation has been a paradox for one mathematician, an inconsistency proof for another, and an indirect proof to a third. One and the same argumentation-text can appear to one mathematician to express an indirect proof while appearing to another mathematician to express a direct proof. WHAT IS A PARADOX’S SOLUTION? Of the above four sorts of argumentation only the paradox invites “solution" or “resolution", and ordinarily this is to be accomplished either by discovering a logical fallacy in the “reasoning" of the argumentation or by discovering that the conclusion is not really false or by discovering that one of the premises is not really true. Resolution of a paradox by a participant amounts to reclassifying a formerly paradoxical argumentation either as a “fallacy", as a direct proof of its conclusion, as an indirect proof of the negation of one of its premises, as an inconsistency proof, or as something else depending on the participant's state of knowledge or belief. This illustrates why an argumentation which is a paradox to a given mathematician at a given time may well not be a paradox to the same mathematician at a later time. -/- The present article considers several set-theoretic argumentations that appeared in the period 1903-1908. The year 1903 saw the publication of B. Russell's Principles of mathematics, [Cambridge Univ. Press, Cambridge, 1903; Jbuch 34, 62]. The year 1908 saw the publication of Russell's article on type theory as well as Ernst Zermelo's two watershed articles on the axiom of choice and the foundations of set theory. The argumentations discussed concern “the largest cardinal", “the largest ordinal", the well-ordering principle, “the well-ordering of the continuum", denumerability of ordinals and denumerability of reals. The article shows that these argumentations were variously classified by various mathematicians and that the surrounding atmosphere was one of confusion and misunderstanding, partly as a result of failure to make or to heed distinctions similar to those made above. The article implies that historians have made the situation worse by not observing or not analysing the nature of the confusion. -/- RECOMMENDATION This well-written and well-documented article exemplifies the fact that clarification of history can be achieved through articulation of distinctions that had not been articulated (or were not being heeded) at the time. The article presupposes extensive knowledge of the history of mathematics, of mathematics itself (especially set theory) and of philosophy. It is therefore not to be recommended for casual reading. AFTERWORD: This review was written at the same time Corcoran was writing his signature “Argumentations and logic”[249] that covers much of the same ground in much more detail. https://www.academia.edu/14089432/Argumentations_and_Logic . (shrink)

Human Rights Ethics makes an important contribution to contemporary philosophical and political debates concerning the advancement of global justice and human rights. Butler's book also lays claim to a significant place in both normative ethics and human rights studies in as much as it seeks to vindicate a universalistic, rational approach to human rights ethics. Butler's innovative approach is not based on murky claims to "natural rights" that supposedly hold wherever human beings exist; nor does it succumb to the traditional (...) problems of justification associated with utilitarianism, Kantianism, and other procedural approaches to human rights studies. Instead, Butler proposes "a dialectical justification of human rights by indirect proof" that claims not to be question begging. Very much in the spirit of Hegel and Habermas, Butler proposes to vindicate a "totally rational account of human rights," but one that depends concretely and historically on a dialectically constructed "right to freedom of thought in its universal modes.". (shrink)

SummaryThis paper attempts to develop an interpretation of Kant's transcendental idealism which is based upon his critique of transcendental realism . It is argued that given Kant's transcendental distinction, all non‐ or pre‐critical philosophies, even Berkeleian phenomenalism are transcendentally realistic. This paradoxical result is used as the basis for an analysis of Kant's resolution of the mathematical antinomies, wherein this resolution is seen both as an “indirect proof” of transcendental idealism and as a refutation of transcendental realism. Finally, it (...) is claimed that Kant's idealism, at least insofar as it is established by means of a refutation of transcendental realism, is methodological rather than metaphysical, viz. it involves a claim about how the sensible world is to be considered in transcendental reflection, not a claim about the “real nature” of this world.RésuméCet article essaye de développer une interprétation de l'idéalisme transcendental de Kant, interprétation basée sur sa critique du réalisme transcendental . L'auteur montre que, si l'on accepte la distinction transcendentale de Kant, toutes les philosophies non critiques et précritiques sont transcendentalement réalistes, y compris le phénoménalisme de Berkeley. A partir de ce résultat paradoxal, il analyse la manière dont Kant résoud les antinomies mathématiques: cette résolution apparaît à la fois comme preuve indirecte de l'idéalisme transcendental et comme une réfutation du réalisme transcendental. L'auteur est finalement d'avis que l'idéalisme de Kant, du moins lorsqu'il est justifié par une réfutation du réalisme transcendental, est plus méthodologique que métaphysique, c'est‐à‐dire qu'il déclare comment le monde sensible doit être considéré dans la réflexion transcendentale et non quelle est la véritable nature de ce monde.ZusammenfassungEs wird versucht, eine Interpretation von Kants transzendentalem Idealismus zu entwickeln, die auf seiner Kritik am transzendentalen Realismus — verstanden als derjenige Standpunkt, der systematisch Erscheinung und Ding an sich verwechselt — beruht. Es wird argumentiert, dass — einmal Kants transzendentale Unterscheidung angenommen — aile früheren, vorkritischen Systeme der Philosophie einschliesslich Berkeleys Phänomenalismus transzendental realistisch sind. Dieses paradoxale Ergebnis wird als Basis für eine Analyse von Kants Auflösung der mathematischen Antinomien verwendet, wobei dièse Auflösung sowohl als ein indirekter Beweis für den transzendentalen Idealismus als auch als eine Widerlegung des transzendentalen Realismus gedeutet wird. Es wird schliesslich nahegelegt, dass Kants Idealismus, sofern wenigstens als er sich auf Widerlegung des transzendentalen Realismus abstützt, mehr ein rnethodologischer als ein metaphysischer Idealismus ist. Er statuiert eher über die Art, wie die Sinneswelt in einer transzendentalen Reflexion betrachtet werden soll, als über die Frage, wie die Welt wirklich beschaffen sei. (shrink)

A normalization procedure is given for classical natural deduction with the standard rule of indirect proof applied to arbitrary formulas. For normal derivability and the subformula property, it is sufficient to permute down instances of indirect proof whenever they have been used for concluding a major premiss of an elimination rule. The result applies even to natural deduction for classical modal logic.

This paper analyzes and evaluates Bolzano's remarks on the apagogic method of proof with reference to his juvenile booklet ‘Contributions to a better founded presentation of mathematics’ of 1810 and to his ‘Theory of science’ (1837). I shall try to defend the following contentions: (1) Bolzanos’ vain attempt to transform all indirect proofs into direct proofs becomes comprehensible as soon as one recognizes the following facts: (1.1) his attitude towards indirect proofs with an affirmative conclusion (...) differs from his stance to indirect proofs with a negative conclusion; (1.2) by Bolzano's lights arguments via consequentia mirabilis only seem to be indirect. (2) Bolzano does not deny that indirect proofs can be perfect certifications (Gewissmachungen) of their conclusion; what he denies is rather that they can provide grounds for their conclusions. (2.1) They cannot do the latter, since they start from false premises and (2.2) since they make an unnecessary detour. Equation(3) The far-reaching agreement between his early and late assessment of apagogical proofs (in the Beyträge of 1810 and the Wissenschaftslehre of 1837) is partly due to the fact that he develops his own position always against the background of Wolff's and Lambert's views. (shrink)

This thesis presents a proof theoretical investigation of some constructive set theories with restricted set induction. The set theories considered are various systems of Constructive Zermelo Fraenkel set theory, CZF ([1]), in which the schema of $\in$ - Induction is either removed or weakened. We shall examine the theories $CZF^\Sigma_\omega$ and $CZF_\omega$, in which the $\in$ - Induction scheme is replaced by a scheme of induction on the natural numbers (only for  formulas in the case of the first theory, (...) for arbitrary formulas in the second theory). Of particular interest is the system CZF− + INAC, which is obtained by removing $\in$ - Induction and by adding an axiom, INAC, for inaccessible sets. The main outcome is that CZF− + INAC is a predicative set theory with inaccessible sets, its proof theoretic ordinal being $\Gamma_0$. The proof theoretic strength of these theories, has been determined by means of proof theoretical reductions. The upper bounds have been obtained by interpreting the set theories in classical theories of (iterated) inductive definitions with ordinals. A significant feature of these interpretations is that they are realizability interpretations, which make an indirect use of Aczel’s interpretation of CZF in Martin Löf type theory, and, in the case of CZF− + INAC, also employ a maximum bisimulation relation to represent equality between sets. The lower bounds are obtained by interpreting appropriate subsystems of intuitionistic second order arithmetic in the set theories. At the conclusion of the thesis we present work unrelated to the previous chapters. It is a preliminary study of the constructible hierarchy in CZF. We begin by formalizing the notion of definability in constructive set theory and then introduce Gödel’s L in CZF. We prove that intuitionistic L is a model of a subsystem of CZF strong enough to include intuitionistic Kripke-Platek set theory, and also that L is a model of the axiom of constructibility, provably in CZF. (shrink)

PARC is an "appended numeral" system of natural deduction that I learned as an undergraduate and have taught for many years. Despite its considerable pedagogical strengths, PARC appears to have never been published. The system features explicit "tracking" of premises and assumptions throughout a derivation, the collapsing of indirect proofs into conditional proofs, and a very simple set of quantificational rules without the long list of exceptions that bedevil students learning existential instantiation and universal generalization. The system (...) can be used with any Copi-style set of inference rules, so it is quite adaptable to many mainstream symbolic logic textbooks. Consequently, PARC may be especially attractive to logic teachers who find Jaskowski/Gentzen-style introduction/elimination rules to be far less "natural" than Copi-style rules. The PARC system is also keyboard-friendly in comparison to the widely adopted Jaskowski-style graphical subproof system of natural deduction, viz., Fitch diagrams and Copi "bent arrow" diagrams. (shrink)