A central part of Kripke's influential interpretation of Wittgenstein's sceptical argument about meaning is the rejection of dispositional analyses of what it is for a word to mean what it does. In this paper I show that Kripke's arguments prove too much: if they were right, they would preclude not only the idea that dispositional properties can make statements about the meanings of words true, but also the idea that dispositional properties can make true statements about paradigmatic dispositional properties such (...) as a cup's fragility or a person's bravery. However, since dispositional properties can make such statements true, Kripke-Wittgenstein's arguments against dispositionalism about meaning are mistaken. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 38.2 (2000) 221-238 [Access article in PDF] Hume's Iterative Probability Argument: A Pernicious Reductio Kevin Meeker University of South Alabama In this essay I want to look afresh at David Hume's iterative probability argument, found in the section entitled "Of Scepticism with regard to Reason" in his A Treatise of Human Nature.1 Interestingly enough, after years of comparative neglect,2 this argument has (...) been basking in the glow of scholarly sunshine lately.3 But to what can we attribute its extended hiatus? The convergence of three major factors seems responsible. First, philosophers have oftentimes flung this argument into the ocean of obscurity because of its perceived inadequacies.4 Second, and on a related note, many assume that Hume himself permitted this argument to drown because it never resurfaces again in his writings. Most suspiciously, Hume's Enquiry Concerning Human Understanding, which is his popular (and preferred) presentation of the epistemological sections [End Page 221] of the Treatise, does not contain this argument.5 Last, but certainly not least, for a long time this argument sat uncomfortably with the naturalist interpretation of Hume6 that has gained ascendancy in the twentieth century. For prior to this century, Hume was almost unanimously regarded as a sceptic. Despite a few waves from those who interpreted Hume as a logical positivist earlier in this century, Norman Kemp Smith charted a new course for Humean interpretations with his emphasis on Hume's naturalism.7 And this is the current according to which most Hume interpreters set their sails today.Because the naturalist reading of Hume relies so heavily on the system he attempts to set forth in the Treatise, though, naturalists have come to see the importance of reading this argument in a naturalistically acceptable way. William E. Morris best expresses the significance of this argument from a naturalistic perspective in the following manner: If we ever are to understand Hume's view of the role of reason, it stands to reason that we should first figure out how to integrate "Of scepticism with regard to reason" into the picture. Only then will we be ready to move to a consideration of Hume's positive views about reason. These views are the least understood of any of Hume's doctrines. Gaining an adequate understanding of them is both the most pressing and the most difficult problem facing Hume studies today.8While I agree with Morris about the importance of getting a clear understanding of the reasoning of I, iv, 1, I will argue that a correct reading of this section reveals the implausibility of major naturalistic interpretations of Hume.This paper has four major sections. In the first important section, I will highlight the crucial aspects of the iterative argument that will be the focus of this paper and deal with a preliminary but important interpretive issue. In the second section, I will briefly look at Annette Baier's naturalistic reading of Hume's philosophy to show how this particular passage bears on broader [End Page 222] interpretive issues. The third main section will focus on explicating Morris' similar but more detailed interpretation. Finally in the last part I will show that while it is legitimate to characterize the argument here as a reductio (as Baier and Morris both do), the argument still yields, contrary to what naturalists have recently suggested, a very sceptical conclusion. 1. The Arguments of Section I, IV, 1 of the TreatiseHume's "Of Scepticism with regard to Reason" contains at least two important arguments. While Hume agrees that knowledge involves certainty (whereas beliefs based on probability are by their very nature uncertain9 ), the conclusion of his first argument is that "all knowledge resolves itself into probability" (Treatise, 181). His point here is simply that all of our beliefs are fallible. That is, nothing is absolutely certain; thus none of our beliefs attain the exalted status of certain knowledge. But once this consequence of our fallibility is recognized, he then argues roughly that the probability that any particular belief... (shrink)

Norman Finkelstein, a prominent political scientist specializing in the Palestine-Israel conundrum, on which he has authored five highly praised books, was denied tenure at DePaul University by the President, Rev. Dennis H. Holtschneider, on June 8, 2007. After examining the particulars of the case, it strikes me as so obviously wrong to deny him tenure that the tenure procedure at DePaul constitutes a reductio ad absurdum of a university system which allows such a thing to happen.

According to Carruthers ants and bees have minds. This claim is to be understood realistically. We do not interpret the overt behaviour of ants and bees by ascribing to them beliefs and desires in an instrumental manner. They rather possess minds in the relevant cognitive sense. In this paper, I propose to pave the way for a reductio against such a polemic view. In particular, I shall argue that if ants and bees have minds, by the same token, plants (...) do have minds too. In my view, the problem has to do with Carruthers' underlying technical concept of cognitive architecture; a concept which, as I shall argue, can be called into question both on empirical and conceptual grounds. Según Carruthers, las hormigas y las abejas tienen mente. Esta afirmación debe entenderse de modo realista. No es que interpretemos la conducta abierta de hormigas y abejas en términos de creencias y deseos atribuidos instrumentalmente. Se trata más bien de que tienen tales estados mentales en el sentido cognitivo relevante. En este trabajo, me propongo llevar esta polémica concepción a una reductio. En particular, argumentaré que si las hormigas y las abejas tienen mente, por la misma razón las plantas también la tendrían. A mi modo de ver, el problema tiene que ver con el concepto técnico de arquitectura cognitiva de Carruthers ; un concepto que puede ser cuestionado sobre bases empíricas y conceptuales, según argüiré. (shrink)

Free Semantics is based on normalized natural deduction for the weak relevant logic DW and its near neighbours. This is motivated by the fact that in the determination of validity in truth-functional semantics, natural deduction is normally used. Due to normalization, the logic is decidable and hence the semantics can also be used to construct counter-models for invalid formulae. The logic DW is motivated as an entailment logic just weaker than the logic MC of meaning containment. DW is the logic (...) focussed upon, but the results extend to MC. The semantics is called 'free semantics' since it is disjunctively and existentially free in that no disjunctive or existential witnesses are produced, unlike in truth-functional semantics. Such 'witnesses' are only assumed in generality and are not necessarily actual. The paper sets up the free semantics in a truth-functional style and gives a natural deduction interpetation of the meta-logical connectives. We then set out a familiar tableau-style system, but based on natural deduction proof rather than truth-functional semantics. A proof of soundness and completeness is given for a reductio system, which is a transform of the tableau system. The reductio system has positive and negative rules in place of the elimination and introduction rules of Brady's normalized natural deduction system for DW. The elimination-introduction turning points become closures of threads of proof, which are at the points of contradiction for the reductio system. (shrink)

This paper presents the main aspects of the views of the Tibetan logician Phya pa Chos kyi seng ge (1109–1169) on argumentation “by consequence” (thal ʼgyur, Skt. prasaṅga) based on his exposition of the topic in the fifth chapter of his Tshad ma yid kyi mun sel and on a parallel excursus in his commentary on Dharmakīrti’s Pramānaviniścaya. It aims at circumscribing primarily the nature and function of consequences (thal ʼgyur/thal ba) for this author—in particular the distinction between “proving consequences” (...) and “refuting consequences”—and the form prescribed for their enunciation in the context of debate. In addition to pointing out differences with the systems adopted by his predecessors, contemporaries and successors, the paper also discusses some of the similarities and differences between Phya pa’s understanding of argumentation by consequence and the notion of reductio ad absurdum in Western logic. (shrink)

Arguments by Sorkin (Impossible measurements on quantum fields. In: Directions in general relativity: proceedings of the 1993 International Symposium, Maryland, vol 2, pp 293–305, 1993) and Borsten et al. (Phys Rev D 104(2), 2021. https://doi.org/10.1103/PhysRevD.104.025012 ) establish that a natural extension of quantum measurement theory from non-relativistic quantum mechanics to relativistic quantum theory leads to the unacceptable consequence that expectation values in one region depend on which unitary operation is performed in a spacelike separated region. Sorkin [ 1 ] labels (...) such scenarios ‘impossible measurements’. We explicitly present these arguments as a no-go result with the logical form of a reductio argument and investigate the consequences for measurement in quantum field theory (QFT). Sorkin-type impossible measurement scenarios clearly illustrate the moral that Microcausality is not by itself sufficient to rule out superluminal signalling in relativistic quantum theories that use Lüders’ rule. We review three different approaches to formulating an account of measurement for QFT and analyze their responses to the ‘impossible measurements’ problem. Two of the approaches are: a measurement theory based on detector models proposed in Polo-Gómez et al. (Phys Rev D, 2022. https://doi.org/10.1103/physrevd.105.065003 ) and a measurement framework for algebraic QFT proposed in Fewster and Verch (Commun Math Phys 378(2):851–889, 2020). Of particular interest for foundations of QFT is that they share common features that may hold general morals about how to represent measurement in QFT. These morals are about the role that dynamics plays in eliminating ‘impossible measurements’, the abandonment of the operational interpretation of local algebras \({\mathcal {A}}(O)\) as representing possible operations carried out in region _O_, and the interpretation of state update rules. Finally, we examine the form that the ‘impossible measurements’ problem takes in histories-based approaches and we discuss the remaining challenges. (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)

The Aristotelian syllogistic cannot account for the validity of certain inferences involving relational facts. In this paper, we investigate the prospects for providing a relational syllogistic. We identify several fragments based on (a) whether negation is permitted on all nouns, including those in the subject of a sentence; and (b) whether the subject noun phrase may contain a relative clause. The logics we present are extensions of the classical syllogistic, and we pay special attention to the question of whether (...) class='Hi'>reductio ad absurdum is needed. Thus our main goal is to derive results on the existence (or nonexistence) of syllogistic proof systems for relational fragments. We also determine the computational complexity of all our fragments. (shrink)

Bilateralists hold that the meanings of the connectives are determined by rules of inference for their use in deductive reasoning with asserted and denied formulas. This paper presents two bilateral connectives comparable to Prior's tonk, for which, unlike for tonk, there are reduction steps for the removal of maximal formulas arising from introducing and eliminating formulas with those connectives as main operators. Adding either of them to bilateral classical logic results in an incoherent system. One way around this problem (...) is to count formulas as maximal that are the conclusion of reductio and major premise of an elimination rule and to require their removability from deductions. The main part of the paper consists in a proof of a normalisation theorem for bilateral logic. The closing sections address philosophical concerns whether the proof provides a satisfactory solution to the problem at hand and confronts bilateralists with the dilemma that a bilateral notion of stability sits uneasily with the core bilateral thesis. (shrink)

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the (...) given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations. (shrink)

Putnam and Searle famously argue against computational theories of mind on the skeptical ground that there is no fact of the matter as to what mathematical function a physical system is computing: both conclude (albeit for somewhat different reasons) that virtually any physical object computes every computable function, implements every program or automaton. There has been considerable discussion of Putnam's and Searle's arguments, though as yet there is little consensus as to what, if anything, is wrong with these arguments. (...) In the present paper we show that an analogous line of reasoning can be raised against the numerical measurement (i.e., numerical representation) of physical magnitudes, and we argue that this result is a reductio ad absurdum of the challenge to computational skepticism. We then use this reductio to get clearer about both (i) what's wrong with Putnam's and Searle's arguments against computationalism, and (ii) what can be learned about both computational implementation and numerical measurement from the shortcomings of both sorts of skeptical argument. (shrink)

‘Moral bioenhancement’ refers to the use of pharmaceuticals and other direct brain interventions to enhance ‘moral’ traits such as ‘empathy,’ and alter any ‘morally problematic’ dispositions, such as ‘aggression.’ This is believed to result in improved moral responses. In a recent paper, Tom Douglas considers whether medical interventions of this sort could be “provided as part of the criminal justice system’s response to the commission of crime, and for the purposes of facilitating rehabilitation : 101–122, 2014).” He suggests that (...) they could “at least in some cases, permissibly be provided without valid consent : 101–122, 2014)” as a form of rehabilitative punishment. He argues for this conclusion by ‘parity of reasoning,’ starting from the currently accepted practice of non-consensual incarceration. His argument appears to be dependent on the successful defence of the following two claims: that non-consensual incarceration is a morally justifiable practice, and that there is no meaningful distinction between the forcible imposition of this practice, and the forcible imposition of medical interventions on prisoners. From both claims, he deduces : if non-consensual incarceration is morally justifiable, so is the non-consensual imposition of medical correctives, in some cases. In this paper, I begin by suggesting that the basic argument behind the Parity Claim results in a reductio ad absurdum, whereby any practice that is sufficiently similar to incarceration in the ways Douglas presents, would also be considered permissible. This appears to be an unpalatable conclusion, casting doubt on the soundness of a key premise. Douglas appears to offer no means of deciding which practices are sufficiently similar to incarceration in terms of harm and threat to agency, and of the practices that I will present, which do seem to be, it does not seem that he could rule them out in any principled way. Next I turn to dispute claim relating to the purported justifiability of incarceration on rehabilitative grounds. If successful, this attack causes a break in reasoning from the justifiability of incarceration, to the justifiability of medical interventions. Medical interventions would then require an alternative, independent justification, through outlining the ways in which they are conducive to a particular aim of punishment, without relying on the justifiability of incarceration. This argument has not been provided, and I suggest that attempts to do so may run into difficulty. I untangle and make explicit the various assumptions made in Douglas’ contention that medical interventions “might be thought conducive to rehabilitation in some offenders,” locating my critique in the wider debate on the causes of crime. Finally, I seek to challenge the normative weight of the Parity Claim, arguing that to show that two practices are comparable in some sense is not sufficient to show that both are equally as permissible. Other social purposes must be considered, leading me to suggest that the forcible imposition medical correctives falls beyond the appropriate remit of the criminal justice system. (shrink)

In his 1910 book On the principle of contradiction in Aristotle, Jan Łukasiewicz claims that syllogistic is independent of the principle of contradiction . He also argues that Aristotle would have defended such a thesis in the Posterior Analytics. In this paper, we first show that Łukasiewicz's arguments for these two claims have to be rejected. Then, we show that the thesis of the independence of assertoric syllogistic vis-à-vis PC is nevertheless true. For that purpose, we first establish that there (...) is no possibility to account for Aristotle's original views on syllogistic without using per impossibile deductions. At last, following an idea due to Smith 1983, we prove that there is a complete reconstruction of assertoric syllogistic using ecthesis instead of reductio ad absurdum. Contrary to what is claimed in Smith 1983, we show that Smith's system SE has to be improved for such a reconstruction. This results in an ecthetic system which is complete with respect to syllogistic arguments, not explosi.. (shrink)

Data involving epistemic modals suggest that some classically valid argument forms, such as _reductio_, are invalid in natural language reasoning as they lead to modal collapses. We adduce further data showing that the classical argument forms governing the existential quantifier are similarly defective, as they lead to a _de re–de dicto_ collapse. We observe a similar problem for disjunction. But if the classical argument forms for negation, disjunction and existential quantification are invalid, what are the correct forms that govern the (...) use of these items? Our diagnosis is that epistemic modals interfere with hypothetical reasoning. We present a modal first-order logic and model theory that characterizes hypothetical reasoning with epistemic modals in a principled manner. One upshot is a sound and complete natural deduction system for reasoning with epistemic modals in first-order logic. (shrink)

The Lottery Paradox has been thought to provide a reductio argument against probabilistic accounts of inductive inference. As a result, much work in artificial intelligence has concentrated on qualitative methods of inference, including default logics, which are intended to model some varieties of inductive inference. It has recently been shown that the paradox can be generated within qualitative default logics. However, John Pollock's qualitative system of defeasible inference, does avoid the Lottery Paradox by incorporating a rule designed specifically (...) for that purpose. I shall argue that Pollock's system instead succumbs to a worse disease: it fails to allow for induction at all. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 38.2 (2000) 221-238 [Access article in PDF] Hume's Iterative Probability Argument: A Pernicious Reductio Kevin Meeker University of South Alabama In this essay I want to look afresh at David Hume's iterative probability argument, found in the section entitled "Of Scepticism with regard to Reason" in his A Treatise of Human Nature.1 Interestingly enough, after years of comparative neglect,2 this argument has (...) been basking in the glow of scholarly sunshine lately.3 But to what can we attribute its extended hiatus? The convergence of three major factors seems responsible. First, philosophers have oftentimes flung this argument into the ocean of obscurity because of its perceived inadequacies.4 Second, and on a related note, many assume that Hume himself permitted this argument to drown because it never resurfaces again in his writings. Most suspiciously, Hume's Enquiry Concerning Human Understanding, which is his popular (and preferred) presentation of the epistemological sections [End Page 221] of the Treatise, does not contain this argument.5 Last, but certainly not least, for a long time this argument sat uncomfortably with the naturalist interpretation of Hume6 that has gained ascendancy in the twentieth century. For prior to this century, Hume was almost unanimously regarded as a sceptic. Despite a few waves from those who interpreted Hume as a logical positivist earlier in this century, Norman Kemp Smith charted a new course for Humean interpretations with his emphasis on Hume's naturalism.7 And this is the current according to which most Hume interpreters set their sails today.Because the naturalist reading of Hume relies so heavily on the system he attempts to set forth in the Treatise, though, naturalists have come to see the importance of reading this argument in a naturalistically acceptable way. William E. Morris best expresses the significance of this argument from a naturalistic perspective in the following manner: If we ever are to understand Hume's view of the role of reason, it stands to reason that we should first figure out how to integrate "Of scepticism with regard to reason" into the picture. Only then will we be ready to move to a consideration of Hume's positive views about reason. These views are the least understood of any of Hume's doctrines. Gaining an adequate understanding of them is both the most pressing and the most difficult problem facing Hume studies today.8While I agree with Morris about the importance of getting a clear understanding of the reasoning of I, iv, 1, I will argue that a correct reading of this section reveals the implausibility of major naturalistic interpretations of Hume.This paper has four major sections. In the first important section, I will highlight the crucial aspects of the iterative argument that will be the focus of this paper and deal with a preliminary but important interpretive issue. In the second section, I will briefly look at Annette Baier's naturalistic reading of Hume's philosophy to show how this particular passage bears on broader [End Page 222] interpretive issues. The third main section will focus on explicating Morris' similar but more detailed interpretation. Finally in the last part I will show that while it is legitimate to characterize the argument here as a reductio (as Baier and Morris both do), the argument still yields, contrary to what naturalists have recently suggested, a very sceptical conclusion. 1. The Arguments of Section I, IV, 1 of the TreatiseHume's "Of Scepticism with regard to Reason" contains at least two important arguments. While Hume agrees that knowledge involves certainty (whereas beliefs based on probability are by their very nature uncertain9 ), the conclusion of his first argument is that "all knowledge resolves itself into probability" (Treatise, 181). His point here is simply that all of our beliefs are fallible. That is, nothing is absolutely certain; thus none of our beliefs attain the exalted status of certain knowledge. But once this consequence of our fallibility is recognized, he then argues roughly that the probability that any particular belief... (shrink)

Studies of Aristotle’s syllogistic system, since Corcoran’s deductionist interpretation supplanted Łukasiewicz’ axiomaticist interpretation, misrepresent Aristotle’s logic in two important respects. Following Corcoran, they take indirect deduction to occur only once in a deduction discourse; they then obviate the system having a reductio rule. Second, they represent reduction as a deductive process for deriving ‘imperfect’ syllogisms from ‘perfect’ syllogisms to impose an axiomatic interpretation on the logic. Denying that Aristotle's logic admits of a reductio rule results from (...) this misrepresentation of reduction. This discussion shows that the defects imputed to Aristotle's logic, and systems devised to resolve them, result from misunderstanding reduction, which itself results from misapprehending Prior Analytics expressly to identify in a metadiscourse deduction rules and not deductions per se. (shrink)

Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such asreductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th (...) century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titledSpecimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule ofreductio ad absurdumin a purely categorical calculus in which every proposition is of the formA is B. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule ofreductio ad absurdum, but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic known as RMI$_{{}_ \to ^\neg }$. (shrink)

This is a collection of papers, all but one of which were previously published, by one of England's leading logicians. Goodstein has described his position in the philosophy of mathematics as that of a "constructive formalist": leaning toward the Hilbert school, but emphasizing the constructive nature of mathematical entities. The papers are more or less technical and symbolic; those most difficult are "The Nature of Mathematics," "The Decision Problem," and "The Definition of Number." Other titles are "Proof by Reductio (...) ad Absurdum," "Logic Paradoxes", "Language and Experience," "The Meaning of Counting," "Mathematical Systems," "The Axiomatic Method," "Pure and Applied Mathematics," and "The Significance of Incompleteness Theorems"; the last-mentioned paper is also quite difficult, but it constitutes not only one of the best semi-formal presentations of Gödel's theorems, but also helps make quite clear what their relevance is to considerations of the expressive power of formal languages in general. This collection constitutes a fine introduction to the philosophy of mathematics.—P. J. M. (shrink)

In his logical foundation of arithmetic, Frege faced the problem that the semantic interpretation of his system does not determine the reference of the abstract terms completely. The contextual definition of number, for instance, does not decide whether the number 5 is identical to Julius Caesar. In a late writing, Quine claimed that the indeterminacy of reference established by Frege’s Caesar problem is a special case of the indeterminacy established by his proxy-function argument. The present paper aims to show (...) that Frege’s Caesar problem does not really support the conclusions that Quine draws from the proxy-function argument. On the contrary, it reveals that Quine’s argument is a non sequitur: it does not establish that there are alternative interpretations of our terms that are equally correct, but only that these terms are ambiguous. The latter kind of referential indeterminacy implies that almost all sentences of our overall theory of the world are either false or neither true nor false, because they contain definite descriptions whose uniqueness presupposition is not fulfilled. The proxy-function argument must therefore be regarded as a reductio ad absurdum of Quine’s behaviorist premise that the reference of terms is determined only by our linguistic behavior. (shrink)

This study investigated the kinds of knowledge necessary to learn an important troubleshooting strategy, elimination. A total of 50 college-level students searched for the source of failures in simple digital networks. Production system modelling suggested that students using a common but simpler backtracking strategy would learn the more advanced elimination strategy if they applied certain domain-specific knowledge and the general-purpose problem-solving strategy of reductio ad absurdum. In an experiment, students solved network troubleshooting problems after being trained with either (...) the domain-specific knowledge, the reductio ad absurdum strategy, both types of knowledge, or neither. Students needed both the domain-specific and general knowledge identified by the models in order to significantly increase their elimination use. (shrink)

This article concerns the treatment of propositional quantification in a framework of labelled natural deduction for modal logic developed by Basin, Matthews and Viganò. We provide a detailed analysis of a basic calculus that can be used for a proof-theoretic rendering of minimal normal multimodal systems with quantification over stable domains of propositions. Furthermore, we consider variations of the basic calculus obtained via relational theories and domain theories allowing for quantification over possibly unstable domains of propositions. The main result of (...) the article is that fragments of the labelled calculi not exploiting reductio ad absurdum enjoy the Church–Rosser property and the strong normalization property; such result is obtained by combining Girard’s method of reducibility candidates and labelled languages of lambda calculus codifying the structure of modal proofs. (shrink)

This paper offers an analysis of central features of modern world history which suggest a confirmation, and extension, of something resembling Fukuyama's Kojeve-Hegel *end of history' thesis. As is well known, Kojeve interpreted Hegel as having argued that in a meaningful sense history, as struggle and endeavour to achieve workable stasis in the mutual relations of selves and state-society collectivities, literally came to an end with Napoleon's 1806 victory at the battle of Jena. That victory led to the establishment or (...) consolidation of a European system which significantly embodied the conceptually ideal roles and mutual relations of individual, state, law, and culture (including religious culture), in the aggregated states ruled or presided over by Napoleon. For Hegel the universal structures which constitute the progression of the Absolute are importantly independent of the actual concrete historical individuals and doings which embody and implement them. Once realized upon the earth, the idea of a civil society living, under law, with a sustainable religious and national normative ideology is inexpungible. Even if it has for a time dimmed, it will resurface and re-present itself, and, for Kojeve, has done so, in the gradual articulation of European union and the formations of the League of Nations and its successor the United Nations, in the world that is still our present world. It is much of this model that Fukuyama adopted, and conceived, more explicitly than perhaps either Hegel or Kojeve had done, as a realized triangulation of democracy, liberal individual rights ideology, and capitalism. The realization came to the fore, in Fukuyama's view, in the matrix of the events set in motion by the fall of communism in the European world in 1989. Contrary to Fukayama, of course, there has been rather a lot of 'history' very dramatically in the years in and since 1989, and of course especially explosively in 2001. This recent history notwithstanding, the end of history thesis seems plausible and defensible. Four large geopolitical struggles may be identified, as constituting sequential clusters of argument aimed at determining the human telos, or end-state. They constitute also a sequence of reductios of blueprints that are rivals to the liberalism-democracy-capitalism complex. The four are World War I and the geopolitical struggles between Liberal modernism and fascism, communism, and Islamism. Analyses of these four struggles are offered and defended. (shrink)

This paper offers an analysis of central features of modern world history which suggest a confirmation, and extension, of something resembling Fukuyama's Kojeve-Hegel *end of history' thesis. As is well known, Kojeve interpreted Hegel as having argued that in a meaningful sense history, as struggle and endeavour to achieve workable stasis in the mutual relations of selves and state-society collectivities, literally came to an end with Napoleon's 1806 victory at the battle of Jena. That victory led to the establishment or (...) consolidation of a European system which significantly embodied the conceptually ideal roles and mutual relations of individual, state, law, and culture, in the aggregated states ruled or presided over by Napoleon. For Hegel the universal structures which constitute the progression of the Absolute are importantly independent of the actual concrete historical individuals and doings which embody and implement them. Once realized upon the earth, the idea of a civil society living, under law, with a sustainable religious and national normative ideology is inexpungible. Even if it has for a time dimmed, it will resurface and re-present itself, and, for Kojeve, has done so, in the gradual articulation of European union and the formations of the League of Nations and its successor the United Nations, in the world that is still our present world. It is much of this model that Fukuyama adopted, and conceived, more explicitly than perhaps either Hegel or Kojeve had done, as a realized triangulation of democracy, liberal individual rights ideology, and capitalism. The realization came to the fore, in Fukuyama's view, in the matrix of the events set in motion by the fall of communism in the European world in 1989. Contrary to Fukayama, of course, there has been rather a lot of 'history' very dramatically in the years in and since 1989, and of course especially explosively in 2001. This recent history notwithstanding, the end of history thesis seems plausible and defensible. Four large geopolitical struggles may be identified, as constituting sequential clusters of argument aimed at determining the human telos, or end-state. They constitute also a sequence of reductios of blueprints that are rivals to the liberalism-democracy-capitalism complex. The four are World War I and the geopolitical struggles between Liberal modernism and fascism, communism, and Islamism. Analyses of these four struggles are offered and defended. (shrink)

The relational syllogistic extends the classical syllogistic by allowing predicate phrases of the forms “rs every q”, “rs some q” and their negations, where q is a common noun and r a transitive verb. It is known that both the classical and relational syllogistic admit a finite set of syllogism-like rules whose associated derivation relation is sound and complete. In this article, we extend the classical and relational syllogistic by allowing ‘guarded’ predicate phrases of the form “rs onlyqs”, and their (...) negations. We show that, in both cases, the resulting logic is pspace-complete. It follows, on the assumption that NPTIME≠PSPACE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsc {nptime}\ne \textsc {pspace}$$\end{document}, that neither extension admits a finite set of syllogism-like rules whose associated derivation relation is sound and complete, even when reductio ad absurdum is allowed. We also show that further extending these systems with noun-complementation in sentence-subjects results in logics which are exptime-complete. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Idealism and Objectivity: Understanding Fichte’s Jena Project by Wayne M. MartinJanet RoccanovaWayne M. Martin. Idealism and Objectivity: Understanding Fichte’s Jena Project. Stanford: Stanford University Press, 1997. Pp. xvii + 177. Cloth, $45.00.This text claims to present a new interpretation of Fichte’s Jena system. Toward this end the author presents a primary thesis and several secondary theses as alternatives to standard or recent Fichte interpretations. Martin’s main thesis (...) is that the central and guiding project of Fichte’s Jena Wissenschaftslehre is to develop an idealist theory of objectivity. Most controversial of the subordinate theses will certainly be Martin’s claim that Fichte is a realist.The text is symmetrically divided into two parts, each composed of three chapters. Part I addresses the “Parameters of the Jena Project,” while Part II focuses on the “Elements of a Theory of Objectivity.”In section one of the first chapter Martin sides against the long-standing interpretation of Fichte’s Wissenschaftslehre as speculative metaphysics and with the more recent epistemological (and anthropological) reading of the Wissenschaftslehre, but warns against taking Fichte as a “classic epistemologist,” whose anti-skeptical project is to “provide a rational justification of our claims to empirical knowledge” (15). Rather, Martin contends, Fichte’s main goal in the Wissenschaftslehre is to develop a theory of the “objectivity” or “referential character” of consciousness, which is a concern more with the structure, rather than with the justification, of knowledge.Section two is where Martin lays out his particular notion of objectivity, according to which a conscious state is objective if it is “ ‘of’ or ‘about’ something that I take to exist independently of that conscious state” (18). The author emphasizes that objectivity, as [End Page 634] he understands it, is “less a function of [the] immediate content [of a conscious state] than of how I interpret or ‘take up’ that content” (19).The arguments of Chapters 2 and 3 are intended to lend support to Martin’s thesis that Fichte could, without inconsistency, be counted as a realist. In Chapter 2 Martin argues that Fichte’s rejection of dogmatism should not necessarily be interpreted as a rejection of realism. According to the author, Fichte indeed rejects any synthesis of idealism and dogmatism, although there is textual support for the claim that Fichte might support a synthesis of idealism and realism.In Chapter 3 Martin argues that the standard interpretation that Fichte utterly rejects things-in-themselves is incorrect, and proposes that the concept of things-in-themselves must necessarily be used by Fichte “to state the epistemological implications of [his] idealist theory of objectivity” (75). Martin’s ultimate goal in these chapters is to counter-balance the standard caricature of Fichte’s idealism by arguing that Fichte does not deny the possibility of a reality independent of the I.Part II is where Martin actually begins his interpretation of the Grundlage, especially sections four and five of the same, which, according to Martin, is where “Fichte tries to carry out the task of providing a theory of the objectivity of consciousness” (81). In Chapter 4 Martin examines the Fichtean concepts of (self-) positing and op-positing and determines that these are pre-representational cognitive acts that establish the schema of subject-object opposition, which is a condition for the possibility of conscious representation.Chapter 5 is a study of Fichte’s methodology, which Martin claims is an unsuccessful, but nonetheless “strikingly original,” “dialectical category theory,” or method of contradiction (5). Martin compares Fichte’s method to the ancient mathematical method of “regula falsi,” and concludes that the positive impact of Fichte’s method is to force a conceptual expansion. Fichte, in other words, uses contradiction to retain, rather than to reject (as in a reductio) the source of contradiction.Finally, in what is the most intriguing chapter of the text (Chapter 6) Martin examines Fichte’s striving doctrine in light of his attempt to develop an idealist theory of objectivity. Using ontogenetic analogies based on how an infant comes to experience an independent world, Martin argues here for the primacy of practice, and more specifically that there is a kind of striving that “is independent... (shrink)

In the movie, Memento, the hero, Leonard, suffers from a form of anterograde amnesia that results in an inability to lay down new memories. Nonetheless, he sets out on a quest to find his wife’s killer, aided by the use of notes, annotated polaroids, and body tattoos. Using these resources he attempts to build up a stock of new beliefs and to thus piece together the puzzle of his wife’s death. At one point in the movie, a character exasperated by (...) Leonard’s lack of biological recall, shouts: “YOU know? What do YOU know. YOU don’t know anything. In 10 minutes time YOU won’t even know you had this conversation” Leonard, however, believes that he does, day by day, come to know new things. But only courtesy of those photos, tattoos, tricks and ploys. Who is right?These are the kinds of question addressed at length in the paper ‘The Extended Mind’. Is the mind contained in the head? Or does the notion of thought allow mental processes to inhere in extended systems of body, brain and aspects of the local environment? The answer, we claimed, was that mental states, including states of believing, could be grounded in physical traces that remained firmly outside the head. As long as a few simple conditions were met, Leonard’s notes and tattoos could indeed count as new additions to his store of long-term knowledge and dispositional belief. In the present treatment I revisit this argument, defending our strong conclusion against a variety of subsequent observations and objections. In particular, I look at objections that rely on a contrast between the intrinsic content of neural symbols and the merely derived content of external inscriptions, at objections concerning the demarcation of scientific domains via natural kinds, and at objections concerning the ultimate locus of agentive control and the nature of perception versus introspection. I also mention a possible alternative interpretation of the argument as a reductio of the very idea of the mind as an object of scientific study. This is an interesting proposal. (shrink)

Aristotle was the first thinker to devise a logical system. He drew upon the emphasis on universal definition found in Socrates, the use of reductio ad absurdum in Zeno of Elea, claims about propositional structure and negation in Parmenides and Plato, and the body of argumentative techniques found in legal reasoning and geometrical proof. Yet the theory presented in Aristotle’s five treatises known as the Organon—the Categories, the De interpretatione, the Prior Analytics, the Posterior Analytics, and the Sophistical (...) Refutations—goes far beyond any of these. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 61.1 (2000) 19-37 [Access article in PDF] Eudaimonism and Theology in Stoic Accounts of Virtue Michael Gass The Stoics were unique among the major schools in the ancient world for maintaining that both virtue and happiness consist solely of "living in agreement with nature" (homologoumenos tei phusei zen). We know from a variety of texts that both Cleanthes and Chrysippus, if not also (...) Zeno, characterized such conduct both in essentialist terms, as a matter of living purposively in a manner natural for the human species, and in cosmological terms, as a matter of living in agreement with the purposes of nature as a whole, this being regarded as identical to Zeus. This is not to suggest, however, that cosmo-theological speculation figured prominently in the ethical systems of the early Stoics. Judging from extant summaries and discussions of Stoic ethical doctrines, those doctrines were justified in a dialectical fashion, not by appeal to facts about the purposive order of nature but simply from an analysis of ethical concepts patterned after similar analyses by earlier philosophers (most notably, Aristotle). It is this kind of learning that Chrysippus apparently had in mind when he maintained that ethics should be studied prior to undertaking a systematic investigation of nature: [T]here are three kinds of philosopher's theorems, logical, ethical and physical.... [W]hat should be ranked first of these are the logical, next the ethical, third the physical; and what should come last in the physical theorems is theology. 1On the other hand, as we learn from the testimony of Plutarch (and also from other indirect textual evidence examined recently by Jacques Brunschwig), 2 [End Page 19] Chrysippus also believed that physics in general and theology in particular yield a quite distinct, demonstrative knowledge of the matters first studied in ethics. A. A. Long and others have suggested that such knowledge was regarded by the Stoics as encompassing not just information about the ways in which virtuous living is encouraged by, and fits into, the wider aims of nature at large but also a deeper understanding of the very essence of virtue. 3 Specifically, virtue is revealed to consist in a conscious and deliberate harmonization of one's actions with the purposes of the divine architect. Thus, under this interpretation of their ethical intentions, the Stoics thought of the initial, analytical approach to ethics as necessarily incomplete even in its treatment of strictly ethical topics; the systematic study of nature was thought to impart additional wisdom regarding ethical matters.In her recent study of ancient ethical theories, Julia Annas seeks to counter this view of the place of cosmo-theological speculation in early Stoicism by arguing that such a religious characterization of virtue is incompatible with the eudaimonism to which all schools of ancient philosophy, including Stoicism, were committed. 4 She concedes that first Cleanthes and then much later the Roman Stoics did regard virtue in religious terms, but in her assessment this represents a radical departure from mainstream Stoic thinking on this subject (which is identified mostly with the views of Chrysippus and his disciples, all of whom were clearly and consistently committed to the eudaimonistic model of ethical inquiry). Annas argues that whereas the Stoic sage was traditionally depicted as a person whose commitment to morally appropriate conduct stems from his own rational reflection on his own needs and interests, a person who identified completely with the aims of the divine architect in his purposive activity would be someone who surrendered his right to decide for himself what his needs and interests are, and who therefore could even be required to act in violation of what he might judge by himself to be crucial to his happiness.In what follows I demonstrate that this argument, often advanced in the literature as a reductio ad absurdum of divine command theories of morality, falls wide of the mark in the present case due to the fact that those Stoics who characterized virtue in religious terms did so apparently because they regarded a complete identification with... (shrink)

Previous research into the formalisation of statute law identified a number of uses of language which posed problems for formalisation. A previous paper argued that these uses establish the requirement that a formalisation be isomorphic, but noted that this has odd consequences. This paper expands on what these consequences are and argues that they undermine the very idea of formalisation. Therefore, the whole argument constitutes a reductio ad absurdum of the idea of formalising statute law. The paper provides reasons (...) why this activity is impossible and places the conclusion in the context of the development of modem logic, in which context it appears uncontroversial. (shrink)

Both Putnam and Searle have argued that that every abstract automaton is realized by every physical system, a claim that leads to a reductio argument against Cognitivism or Strong AI: if it is possible for a computer to be conscious by virtue of realizing some abstract automaton, then by Putnam’s theorem every physical system also realizes that automaton, and so every physical system is conscious—a conclusion few supporters of Strong AI would be willing to accept. Dennett (...) has suggested a criterion of reverse engineering for identifying “real patterns,” and I argue that this approach is also very effective at identifying “real realizations.” I focus on examples of real-world implementations of complex automata because previous attempts at answering Putnam’s challenge have been overly restrictive, ruling out some realizations that are in fact paradigmatic examples of practical automaton realization. I also argue that some previous approaches have at the same time been overly lenient in accepting counter-intuitive realizations of trivial automata. I argue that the reverse engineering approach avoids both of these flaws. Moreover, Dennett’s approach allows us to recognize that some realizations are better than others, and the line between real realizations and non-realizations is not sharp. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Svātantrika-Prāsaṅgika Distinction: What Difference Does a Difference Make?William EdelglassThe Svātantrika-Prāsaṅgika Distinction: What Difference Does a Difference Make? Edited by Georges B. J. Dreyfus and Sara L. McClintock. Boston: Wisdom Publications, 2003. Pp. viii + 398.As early as Bhāvaviveka (sixth century), Indian Buddhist doxographers situated important philosophers in schools and sub-schools characterized by adherence to distinct views, thereby providing a coherent, hierarchical presentation of the Buddha's teaching. In (...) Tibet this practice continued and gained considerable importance: the pedagogical and hermeneutical significance of ordering texts occasioned widespread doxographical literary production.1 The doxographical projects, in which the systematic understanding of unchanging 'tenets' was privileged over the singular views of individual authors, were highly contested. Much of the disagreement was due to the multiplicity of interpretations bound to arise during a millennium of intellectual exchange. Doxographical hierarchy set the parameters of intellectual discourse, and thus the classification of tenets was not simply a matter of historical interest. Traditions were constructed in such a way that one's own thought would conform to an acceptable view, to say nothing of the complex relationship between political power and religious doctrine in Tibet. Despite the contested status of Tibetan doxographical categories, the classifications and distinctions that order much contemporary Western scholarship on Mahāyāna Buddhism in India were inherited from Tibet. Until recently, even the best research often uncritically employed dominant Tibetan classifications as basic interpretive categories and was thus able to bring order to an enormous body of literature.The most philosophically significant and vigorously contested classification in Tibetan doxography is the distinction between the Rang rgyud pa and the Thal 'gyur ba (sanskritized by modern scholars as 'Svātantrika' and 'Prāsaṅgika', respectively). The original locus of the discussion is three Indian commentaries on Nāgārjuna's (second century) Mūlamadhyamakakārikā. Buddhapālita (fifth-sixth centuries), in what is the oldest extant commentary on the Mūlamadhyamakakārikā by a known author, followed Nāgārjuna's reductio ad absurdam form of argumentation, leading from the opponent's premise to an untenable consequence (prasaṅga, thal 'gyur). Shortly thereafter Bhāvaviveka claimed that for every prasan.ga argument in Nāgārjuna's text there is an unstated positive argument, and he criticized Buddhapālita for not utilizing the new methods of Buddhist logic to demonstrate the validity of Madhyamaka thought. Bhāvaviveka argued that Mādhyamikas ought to present formal probative (prayoga; sbyor ba) and autonomous (svatantraprayoga; rang rgyud kyi sbyor ba) arguments and argued that inferential reasoning was necessary to establish the Madhyamaka view of emptiness. Candrakīrti (seventh century) then defended Buddhapālita, insisting that prasaṅga arguments were the only fitting method for a Mādhyamika because they do not require the affirmation of inherent nature. Roughly five centuries later, although Madhyamaka had already been divided according to a variety of classifications in India and Tibet, some Tibetan scholars retroactively characterized Bhāvaviveka as the founder of the Svātantrikas and Candrakīrti as the [End Page 415] founder of the Prāsaṅgikas. Tsong kha pa (1357-1419), the most famous proponent of the Svātantrika-Prāsaṅgika distinction, believed this debate was about more than methodological or epistemological questions: because autonomous argumentation requires inherently existing objects, even if only conventionally, he claimed that Svātantrikas were crypto-realists who deviated from Nāgārjuna's radical understanding of emptiness.Tsong kha pa's analysis of the Svātantrika-Prāsaṅgika distinction, and therefore Madhyamaka philosophy in general, came to dominate Western scholarly research. However, in the last few decades there has been considerable exploration of the forces and influences that contributed to the emergence of conflicting Tibetan doxographical systems.2 This research program has been significantly furthered with the publication of The Svātantrika-Prāsaṅgika Distinction: What Difference Does a Difference Make? Because the Svātantrika-Prāsaṅgika distinction is a Tibetan conceptual construction based on Indian commentaries, an understanding of the issues involved requires a careful analysis of the Indian sources, an investigation of the growth of the various doxographies in Tibet... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Idealism and Objectivity: Understanding Fichte’s Jena Project by Wayne M. MartinJanet RoccanovaWayne M. Martin. Idealism and Objectivity: Understanding Fichte’s Jena Project. Stanford: Stanford University Press, 1997. Pp. xvii + 177. Cloth, $45.00.This text claims to present a new interpretation of Fichte’s Jena system. Toward this end the author presents a primary thesis and several secondary theses as alternatives to standard or recent Fichte interpretations. Martin’s main thesis (...) is that the central and guiding project of Fichte’s Jena Wissenschaftslehre is to develop an idealist theory of objectivity. Most controversial of the subordinate theses will certainly be Martin’s claim that Fichte is a realist.The text is symmetrically divided into two parts, each composed of three chapters. Part I addresses the “Parameters of the Jena Project,” while Part II focuses on the “Elements of a Theory of Objectivity.”In section one of the first chapter Martin sides against the long-standing interpretation of Fichte’s Wissenschaftslehre as speculative metaphysics and with the more recent epistemological (and anthropological) reading of the Wissenschaftslehre, but warns against taking Fichte as a “classic epistemologist,” whose anti-skeptical project is to “provide a rational justification of our claims to empirical knowledge” (15). Rather, Martin contends, Fichte’s main goal in the Wissenschaftslehre is to develop a theory of the “objectivity” or “referential character” of consciousness, which is a concern more with the structure, rather than with the justification, of knowledge.Section two is where Martin lays out his particular notion of objectivity, according to which a conscious state is objective if it is “ ‘of’ or ‘about’ something that I take to exist independently of that conscious state” (18). The author emphasizes that objectivity, as [End Page 634] he understands it, is “less a function of [the] immediate content [of a conscious state] than of how I interpret or ‘take up’ that content” (19).The arguments of Chapters 2 and 3 are intended to lend support to Martin’s thesis that Fichte could, without inconsistency, be counted as a realist. In Chapter 2 Martin argues that Fichte’s rejection of dogmatism should not necessarily be interpreted as a rejection of realism. According to the author, Fichte indeed rejects any synthesis of idealism and dogmatism, although there is textual support for the claim that Fichte might support a synthesis of idealism and realism.In Chapter 3 Martin argues that the standard interpretation that Fichte utterly rejects things-in-themselves is incorrect, and proposes that the concept of things-in-themselves must necessarily be used by Fichte “to state the epistemological implications of [his] idealist theory of objectivity” (75). Martin’s ultimate goal in these chapters is to counter-balance the standard caricature of Fichte’s idealism by arguing that Fichte does not deny the possibility of a reality independent of the I.Part II is where Martin actually begins his interpretation of the Grundlage, especially sections four and five of the same, which, according to Martin, is where “Fichte tries to carry out the task of providing a theory of the objectivity of consciousness” (81). In Chapter 4 Martin examines the Fichtean concepts of (self-) positing and op-positing and determines that these are pre-representational cognitive acts that establish the schema of subject-object opposition, which is a condition for the possibility of conscious representation.Chapter 5 is a study of Fichte’s methodology, which Martin claims is an unsuccessful, but nonetheless “strikingly original,” “dialectical category theory,” or method of contradiction (5). Martin compares Fichte’s method to the ancient mathematical method of “regula falsi,” and concludes that the positive impact of Fichte’s method is to force a conceptual expansion. Fichte, in other words, uses contradiction to retain, rather than to reject (as in a reductio) the source of contradiction.Finally, in what is the most intriguing chapter of the text (Chapter 6) Martin examines Fichte’s striving doctrine in light of his attempt to develop an idealist theory of objectivity. Using ontogenetic analogies based on how an infant comes to experience an independent world, Martin argues here for the primacy of practice, and more specifically that there is a kind of striving that “is independent... (shrink)

It is well known that reductio ad absurdum arguments raise a number of interesting philosophical questions. What does it mean to assert something with the precise goal of then showing it to be false, i.e. because it leads to absurd conclusions? What kind of absurdity do we obtain? Moreover, in the mathematics education literature number of studies have shown that students find it difficult to truly comprehend the idea of reductio proofs, which indicates the cognitive complexity of these (...) constructions. In this paper, I start by discussing four philosophical issues pertaining to reductio arguments. I then briefly present a dialogical conceptualization of deductive arguments, according to which such arguments are best understood as a dialogue between two participants—Prover and Skeptic. Finally, I argue that many of the philosophical and cognitive difficulties surrounding reductio arguments are dispelled or at least further clarified once one adopts a dialogical perspective. (shrink)

Thomas Polger and Laurence Shapiro argue that Carl Gillett's much publicized dimensioned theory of realization is incoherent, being subject to a reductio. Their argument turns on the fact that Gillett's definition of realization makes property instances the exclusive relata of the realization relation, while his belief in multiple realization implies its denial, namely, that properties are the relata of the realization relation on occasions of multiple realization. Others like Sydney Shoemaker have also expressed their view of realization in terms (...) of property instances, yet they too have accepted the multiple realizability of properties. Thus I am interested in the more general issue raised by Polger and Shapiro's argument. Specifically, I show how to supplement a theory of realization with a category-inclusive auxiliary assumption, which avoids the stated reductio. I then offer a few reasons to justify the proposed category-inclusive view of realization, making some comparisons to supervenience and causation along the way. (shrink)

Emergence seems necessary for any naturalistic account of the world — none of our familiar world existed at the time of the Big Bang, and it does now — and normative emergence is necessary for any naturalistic account of biology and mind — mental phenomena, such as representation, learning, rationality, and so on, are normative. But Jaegwon Kim’s argument appears to render causally efficacious emergence impossible, and Hume’s argument appears to render normative emergence impossible, and, in its general form, it (...) precludes any emergence at all. I argue that both of these barriers can be overcome, and, in fact, that they each constitute reductios of their respective underlying presuppositions. In particular, causally efficacious ontological emergence can be modeled, but only within a process metaphysics, thus avoiding Kim’s argument, and by making use of non-abbreviatory forms of definition, thus avoiding Hume’s argument. I illustrate these points with models of the emergent nature of normative function and of representation. (shrink)

Ibn Sīnā proposed an analysis of arguments by reductio ad absurdum. His analysis contains, perhaps for the first time, a workable method for handling the making and discharging of assumptions in a formal proof. We translate the relevant text of Ibn Sīnā and put his analysis into the context of his general approach to logic.

Epistemic infinitism, advanced in different forms by Peter Klein, Scott Aikin, and David Atkinson and Jeanne Peijnenburg, is the theory that justification of a proposition for a person requires the availability to that person of an infinite, non-repeating chain of propositions, each providing a justifying reason for its successor in the chain. The reductio argument is the argument to the effect that infinitism has the consequence that no one is justified in any proposition, because there will be an infinite (...) chain of reasons supporting any proposition (and similarly, a chain supporting its negation). Four ways of defending infinitism against the reductio argument are considered and found wanting: Peijnenburg and Atkinson’s use of probabilistic chains of reasons; Klein’s concept of emergent justification; Aikin’s insistence that there be non-propositional input in the justification of any proposition; and Klein’s use of the distinction between reasons that are and are not available to a person. I contend that, in the absence of some further defence, the reductio argument makes infinitism untenable. (shrink)

Recent developments in Continental philosophy have included the emergence of a school of “speculative realism,” which rejects the human-centered orientation that has long dominated Continental thought. Proponents of speculative realism differ on several issues, but many agree on the need for an object-oriented ontology. Some speculative realists identify realism with materialism, while others accord equal reality to objects that are non-material, even fictional. Several thinkers retain a focus on difference, a well-established theme in Continental thought. This paper looks at speculative (...) realism from the perspective of systems theory. Many of the tenets of speculative realism have long been features of systems metaphysics and are expressed clearly in a systems framework. However, some views of some speculative realists differ substantially from systems theoretic ideas. (shrink)

We introduce a novel expansion of the four-valued Belnap–Dunn logic by a unary operator representing reductio ad contradictionem and study its algebraic semantics. This expansion thus contains both the direct, non-inferential negation of the Belnap–Dunn logic and an inferential negation akin to the negation of Johansson’s minimal logic. We formulate a sequent calculus for this logic and introduce the variety of reductio algebras as an algebraic semantics for this calculus. We then investigate some basic algebraic properties of this (...) variety, in particular we show that it is locally finite and has EDPC. We identify the subdirectly irreducible algebras in this variety and describe the lattice of varieties of reductio algebras. In particular, we prove that this lattice contains an interval isomorphic to the lattice of classes of finite non-empty graphs with loops closed under surjective graph homomorphisms. (shrink)