Harman [8] would concede that (1)–(3) are inconsistent, and (as a result) that something is wrong with premises (1)–(3). But, he would reject the relevantists’ diagnosis that (1) must be rejected. I take it he’d say it’s (2) that is to blame here. (2) is a bridge principle [12] linking entailment and inference. (2) is correct only for consistent B’s. [Even if B is consistent, the correct response may rather be to reject some Bi’s in B.].

In his paper, “On paradox without self-reference”, Neil Tennant proposed the conjecture for self-referential paradoxes that any derivation formalizing self-referential paradoxes only generates a looping reduction sequence. According to him, the derivation of the Liar paradox in natural deduction initiates a looping reduction sequence and the derivation of the Yablo's paradox generates a spiral reduction. The present paper proposes the counterexample to Tennant's conjecture for self-referential paradoxes. We shall show that there is a derivation of the Liar paradox which generates (...) a spiraling reduction procedure. Since the Liar paradox is a self-referential paradox, the result is a counterexample to his conjecture. Tennant has believed that classical reductio has no essential role to formalize paradoxes. As our counterexample applies the rule of classical reductio, he may reject the counterexample. In this sense, it will be briefly argued that classical reductio and his rules for the liar sentence share some inferential role. If classical reductio should not be used in paradoxical reasoning, neither should be his rules for the liar sentence. (shrink)

In his paper, “On paradox without self-reference”, Neil Tennant proposed the conjecture for self-referential paradoxes that any derivation formalizing self-referential paradoxes only generates a looping reduction sequence. According to him, the derivation of the Liar paradox in natural deduction initiates a looping reduction sequence and the derivation of the Yablo's paradox generates a spiral reduction. The present paper proposes the counterexample to Tennant's conjecture for self-referential paradoxes. We shall show that there is a derivation of the Liar paradox which generates (...) a spiraling reduction procedure. Since the Liar paradox is a self-referential paradox, the result is a counterexample to his conjecture. Tennant has believed that classical reductio has no essential role to formalize paradoxes. As our counterexample applies the rule of classical reductio, he may reject the counterexample. In this sense, it will be briefly argued that classical reductio and his rules for the liar sentence share some inferential role. If classical reductio should not be used in paradoxical reasoning, neither should be his rules for the liar sentence. (shrink)

Although many aspects of Bonaventure’s little classic De Reductione Artium ad Theologiam have been addressed in recent literature,1 the translation of the title remains problematic, not only from Latin into English but also from a Greek precedent into Latin. Calling it “On the Reduction of the Arts to Theology” always requires an explanation of the word “reduction.”2 How all the arts, indeed all of human learning, relate to theology and thus to God can hardly be considered a reduction in the (...) usual sense. “Leading back” is better, because it suggests the overall theological structure, but is still an incomplete translation, as revealed by the Greek background. The Latin word family reducere in this case reflects a .. (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)

Examines interrelated topics in Medieval and Renaissance Latin literature: the status of women as writers, the status of women as rhetorical figures, and the status of women in society from the fifth to the early seventeenth century.

This book offers an ideal first reader in ancient Greek. It presents a selection of extracts from a comprehensive range of Greek authors, from Homer to Plutarch, together with generous help with vocabulary and grammar. The passages have been chosen for their intrinsic interest and variety, and brief introductions set them in context. All but the commonest Greek words are glossed as they occur and a general vocabulary is included at the back. Although the book is designed to be used (...) by students who have only completed a beginner's course in Greek, students of all ages will find that it helps them to improve their ability and confidence in reading a wide range of prose and verse texts, while at the same time providing a fascinating and copiously illustrated introduction to the riches and variety of ancient Greek literature over a period of almost a thousand years. (shrink)

: Nagarjuna seems willing to embrace contradictions while at the same time making use of classic reductio arguments. He asserts that he rejects all philosophical views including his own-that he asserts nothing-and appears to mean it. It is argued here that he, like many philosophers in the West and, indeed, like many of his Buddhist colleagues, discovers and explores true contradictions arising at the limits of thought. For those who share a dialetheist's comfort with the possibility of true contradictions (...) commanding rational assent, for Nagarjuna to endorse such contradictions would not undermine but instead confirm the impression that he is indeed a highly rational thinker. It is argued that the contradictions he discovers are structurally analogous to many discovered by Western philosophers and mathematicians. (shrink)

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

In natural deduction classical logic is commonly formulated by adding a rule such as Double Negation Elimination (DNE) or Classical Reductio ad Absurdum (CRA) to a set of introduction and elimination rules sufficient for intuitionist first-order logic with conjunction, disjunction, implication, negation and the universal and existential quantifiers all taken as primitive. The natural deduction formulation of intuitionist logic, coming from Gentzen, has nice properties:— (i) the separation property: an intuitionistically valid inference is derivable using only the (...) introduction and elimination rules governing the connectives and/or quantifiers that occur in the premises (if any) and conclusion; (ii) the (strict) subformula property: more narrowly, there is a derivation of any intuitionistically valid inference that employs only subformulas of the formulas occurring in premises (if any) and conclusion. (Every formula is, of course, a subformula of itself.). (shrink)

According to a classical correspondence theory of truth, a proposition is true iff it corresponds to a fact. The approach has its competitors. One of them, the identity theory of truth, pushes for a surprising simplification. It says that true propositions do not correspond to facts, they are facts. Some find this view too bizarre to be taken seriously. Some are attracted to it because they worry that the correspondence theory opens a gap between our thoughts and reality--a gap (...) that, once opened, will turn out to be unbridgeable, thus making it impossible for our thoughts to come into contact with reality and for us to attain knowledge. They think the identity theory will avoid these nasty consequences because it does not open the gap to begin with. The no-gap theme will play a role in the background of the present paper. It will surface at times. But the paper is more concerned with a different theme, the collapse-charge. Opponents of the correspondence theory sometimes charge that the theory is unstable, that it must collapse into the identity theory, because there is not enough play between true propositions and facts to leave room for a genuine relation to hold between them. Those who regard the identity theory as absurd might see this a reductio of the correspondence theory. Others might see it as an argument for the identity theory. After some exploration of the identity theory, I will present one form of the collapse-charge, then I will discuss what a correspondence theorist has to offer by way of a response. (shrink)

Despite its short historical moment in the sun, behaviorism has become something akin to a theoria non grata, a position that dare not be explicitly endorsed. The reasons for this are complex, of course, and they include sociological factors which we cannot consider here, but to put it briefly: many have doubted the ambition to establish law-like relationships between mental states and behavior that dispense with any sort of mentalistic or intentional idiom, judging that explanations of intelligent behavior require reference (...) to qualia and/or mental events. Today, when behaviorism is discussed at all, it is usually in a negative manner, either as an attempt to discredit an opponent’s view via a reductio, or by enabling a position to distinguish its identity and positive claims by reference to what it is (allegedly) not. In this paper, however, we argue that the ghost of behaviorism is present in influential, contemporary work in the field of embodied and enactive cognition, and even in aspects of the phenomenological tradition that these theorists draw on. Rather than take this to be a problem for these views as some have, we argue that once the behaviorist dimensions are clarified and distinguished from the straw-man version of the view, it is in fact an asset, one which will help with task of setting forth a scientifically reputable version of enactivism and/or philosophical behaviorism that is nonetheless not brain-centric but behavior-centric. While this is a bit like “the enemy of my enemy is my friend” strategy, as Shaun Gallagher notes (2019), with the shared enemy of behaviorism and enactivism being classical Cartesian views and/or orthodox cognitivism in its various guises, the task of this paper is to render this alliance philosophically plausible. Doi: 10.1007/s11229-019-02432-1. (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 (...) 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)

In his paper ‘The possibility of vagueness’ (Fine in Synthese 194(10):3699–3725, 2017), Kit Fine proposes a new logic of vagueness, CL, that promises to provide both a solution to the sorites paradox and a way to avoid the impossibility result from Fine (Philos Perspect 22(1):111–136, 2008). The present paper presents a challenge to his new theory of vagueness. I argue that the possibility theorem stated in Fine (Synthese 194(10):3699–3725, 2017), as well as his solution to the sorites paradox, fail in (...) certain reasonable extensions of the language of CL. More specifically, I show that if we extend the language with any negation operator that obeys reductio ad absurdum, we can prove a new impossibility result that makes the kind of indeterminacy that Fine takes to be a hallmark of vagueness impossible. I show that such negation operators can be conservatively added to CL and examine some of the philosophical consequences of this result. Moreover, I demonstrate that we can define a particular negation operator that behaves exactly like intuitionistic negation in a natural and unobjectionable propositionally quantified extension of CL. Since intuitionistic negation obeys reductio, the new impossibility result holds in this propositionally quantified extension of CL. In addition, the sorites paradox resurfaces for the new negation. (shrink)

Though my title speaks of Kant’s mathematical realism, I want in this essay to explore Kant’s relation to a famous mathematical anti-realist. Specifically, I want to discuss Kant’s influence on L. E. J. Brouwer, the 20th-century Dutch mathematician who built a contemporary philosophy of mathematics on constructivist themes which were quite explicitly Kantian. Brouwer’s theory is perhaps most notable for its belief that constructivism requires us to abandon the traditional logic of mathematical reasoning in favor of different canon of reasoning, (...) called intuitionistic logic. Brouwer thought that classical logic is intrinsically bound up with a nonconstructive view of mathematics. This means that, according to Brouwer, when we do mathematics we must give up bivalence, we must no longer use such familiar logical laws as excluded middle, and we must sometimes forebear from the classic method of reductio ad absurdum. All of these are intuitionistically invalid classical principles. (shrink)

We extend the language of the classical syllogisms with the sentence-forms “At most 1 p is a q” and “More than 1 p is a q”. We show that the resulting logic does not admit a finite set of syllogism-like rules whose associated derivation relation is sound and complete, even when reductio ad absurdum is allowed.

There is an argument which seems to show that if all truths are knowable then all truths are known. It may be viewed as a "reductio ad absurdum" of certain forms of antirealism. However, The claim has been made elsewhere that the argument fails against antirealists who employ constructivist rather than classical logic. The paper defends and amplifies this claim against criticisms by crispin wright and others. Relations between knowability and time are discussed. Suggestions are also made about (...) the proper form of the constructivist semantics for the conditional, Involving a non-Platonistic version of the distinction between types and tokens of proofs. (shrink)

ABSTRACT I suggest how a broadly Kantian critique of classical logic might spring from reflections on constructibility conditions. According to Kant, mathematics is concerned with objects that are given through ‘arbitrary synthesis,’ in the form of ‘constructions of concepts’ in the medium of ‘pure intuition.’ Logic, by contrast, is narrowly constrained – it has no objects of its own and is fixed by the very forms of thought. That is why there is not much room for developments within logic, (...) as compared to the progress in mathematics. Kant’s view of logic remains critical, though – through considerations that are effectively on the scope and limits of classical logic and which play a part in his transcendental idealism. The most important ones are to be found in his critique of the use of reductio ad absurdum proofs in metaphysics and his solutions to the ‘antinomies of pure reason.’ Arguably, these considerations carry over to mathematics as well – by way of ‘analogues’ to the antinomies – in particular in resolving infinity paradoxes. (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)

This paper studies the relationship between Argumentation Logic, a recently defined logic based on the study of argumentation in AI, and classical Propositional Logic. In particular, it shows that AL and PL are logically equivalent in that they have the same entailment relation from any given classically consistent theory. This equivalence follows from a correspondence between the non-acceptability of sentences in AL and Natural Deduction proofs of the complement of these sentences. The proof of this equivalence uses a restricted (...) form of ND proofs, where hypotheses in the application of the Reductio of Absurdum inference rule are required to be “relevant” to the absurdity derived in the rule. The paper also discusses how the argumentative re-interpretation of PL could help control the application of ex-falso quodlibet in the presence of inconsistencies. (shrink)

In the Lysis, Socrates argues that friendship is driven by a desire to use others for one’s own gain. Some commentators take Socrates to be speaking for Plato on this point. By contrast, I shall argue that the Lysis is a reductio ad absurdum of this instrumental account of friendship. First, three arguments in the Lysis reach counterintuitive conclusions which may be avoided by abandoning the common premise that friendship is instrumental. Second, the dramatic context includes counterexamples to the (...) instrumental account of friendship. Third, Socrates distinguishes between people who are desirable because they are useful, and “true friends” who are desirable for their own sake. This is an explicit rejection of the instrumental account of friendship. (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)

While committed to the view that Buddhist Reductionism offers the best account of the Abhidharma distinction of the two truths, Siderits (2009) argues that Buddhist Reductionism has the surprising consequence of making itself inexpressible. This inexpressibility follows from the semantic insulation between conventional and ultimate discourses that Siderits argues is required in order to preserve classical logic for both types of discourse, avoiding contradiction and bivalence failure. I argue that inexpressibility is a problematic consequence that threatens to constitute a (...) reductio of Buddhist Reductionism. (shrink)

We study how to postpone the application of the reductio ad absurdum rule (RAA) in classical natural deduction. This technique is connected with two normalization strategies for classical logic, due to Prawitz and Seldin, respectively. We introduce a variant of Seldin’s strategy for the postponement of RAA, which induces a negative translation from classical to intuitionistic and minimal logic. Through this translation, Glivenko’s theorem from classical to intuitionistic and minimal logic is proven.

We study how to postpone the application of the reductio ad absurdum rule ) in classical natural deduction. This technique is connected with two normalization strategies for classical logic, due to Prawitz and Seldin, respectively. We introduce a variant of Seldin’s strategy for the postponement of \, which induces a negative translation from classical to intuitionistic and minimal logic. Through this translation, Glivenko’s theorem from classical to intuitionistic and minimal logic is proven.

Many interpreters have noted that §§138–242 of Wittgenstein's Philosophical Investigations is dominated by an attack on a platonist or classical realist conception of rules and meaning. In this paper, I address the lack of clarity that still exists concerning the nature and strength of the arguments in those sections. I argue that Wittgenstein's attack is genuinely compelling if viewed as an intricate reductio ad absurdum argument that runs all the way through §§138–201. On my reading, the well-known regress-of-interpretations (...) argument is merely one stage in the overall reductio and is not sufficient on its own to generate the rule-following paradox. (shrink)

Many interpreters have noted that §§138–242 of Wittgenstein's Philosophical Investigations is dominated by an attack on a platonist or classical realist conception of rules and meaning. In this paper, I address the lack of clarity that still exists concerning the nature and strength of the arguments in those sections. I argue that Wittgenstein's attack is genuinely compelling if viewed as an intricate reductio ad absurdum argument that runs all the way through §§138–201. On my reading, the well-known regress-of-interpretations (...) argument is merely one stage in the overall reductio and is not sufficient on its own to generate the rule-following paradox. (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)

The lack of specific arithmetical axioms in Book VII has puzzled historians of mathematics. It is hardly possible in our view to ascribe to the Greeks a conscious undertaking to axiomatize arithmetic. The view that associates the beginnings of the axiomatization of arithmetic with the works of Grassman [1861], Dedekind [1888] and Peano [1889] seems to be more plausible. In this connection a number of interesting historical problems have been raised, for instance, why arithmetic was axiomatized so late. This question (...) was first posed and quite conclusively answered by Yanovskaja [1956]. Her major and quite conclusive argument was that “algorithms in arithmetic have absolute character, while in geometry we have to do with reducibility algorithms”. In this paper we are going to draw attention on certain peculiarities of the construction of the arithmetical Books of Euclid’s "Elements" and show that Euclidean arithmetic is constructed by effective procedures. In spite of the use of inference by reductio ad absurdum and methods equivalent to mathematical induction, Euclidean arithmetic retains its finitary character. It is not full arithmetic, but a finitary fragment of classical arithmetic, and thereby there was not any internal reason for its axiomatization. (shrink)

Crispin Wright tried to refute classical 'Cartesian' skepticism contending that its core argument is extendible to a reductio ad absurdum (_Mind<D>, 100, 87-116, 1991). We show both that Wright is mistaken and that his mistakes are philosophically illuminating. Wright's 'best version' of skepticism turns on a concept of warranted belief. By his definition, many of our well-founded beliefs about the external world and mathematics would not be warranted. Wright's position worsens if we take 'warranted belief' to be implicitly (...) defined by the general principles governing it. Those principles are inconsistent, as shown by a variant of Godel's argument. Thus the inconsistency Wright found has nothing to do with the special premises of Cartesian skepticism, but is embedded in his own conceptual apparatus. Lastly, we show how a Cartesian skeptic could avoid Wright's critique by reconstructing a skeptical argument that does not use the claims Wright ultimately finds objectionable. (shrink)

Thus one must justify any category which he proposes as ultimate, and a classic justification is to raise imaginative thinking to a reductio ad absurdam, in which being, imagined without the category, is shown to be incompatible with being as we best know it, i.e., as including some other category which the philosophical community to whom he addresses his remarks will not deny.

This paper raises the question under what circumstances a plurality forms a set. My main point is that not always all things form sets. A provocative way of presenting my position is that, as a result of my approach, there are more pluralities than sets. Another way of presenting the same thesis claims that there are ways of talking about objects that do not always collapse into sets. My argument is related to expressive powers of formal languages. Assuming classical (...) logic, I show that if all plurality form a set and the quantifiers are absolutely general, then one gets a trivial theory. So, by reductio, one has to abandon one of the premiss. Then, I argue against the collapse of the pluralities into sets. What I am advocating is that the thesis of collapse limits important applications of the plural logic in model theory, when it is assumed that the quantifiers are absolutely general. (shrink)

Frege famously claimed that logic is the science of truth: “To discover truths is the task of all science; it falls to logic to discern the laws of truth” (Frege, 1956, p. 289). But just like the other foundational concept of set, truth at that time was intimately associated with paradox; in the case of truth, the Liar paradox. The set-theoretical paradoxes had their teeth drawn by being recognised as reductio proofs of assumptions that had seemed too obvious to (...) warrant stating explicitly, but were now seen to be substantive, and more importantly inconsistent. Tarski includes the Liar paradox in his classic discussion of the concept of truth (Tarski, 1956), and developed it, in the form of his famous theorem on the undefinability of truth, as a reductio of the assumption that a language could be semantically closed, in the sense of being able to contain its own truth-predicate.Frege famously claimed that logic is the science of truth: “To discover truths is the task of all science; it falls to logic to discern the laws of truth” (Frege, 1956, p. 289). But just like the other foundational concept of set, truth at that time was intimately associated with paradox; in the case of truth, the Liar paradox. The set-theoretical paradoxes had their teeth drawn by being recognised as reductio proofs of assumptions that had seemed too obvious to warrant stating explicitly, but were now seen to be substantive, and more importantly inconsistent. Tarski includes the Liar paradox in his classic discussion of the concept of truth (Tarski, 1956), and developed it, in the form of his famous theorem on the undefinability of truth, as a reductio of the assumption that a language could be semantically closed, in the sense of being able to contain its own truth-predicate. (shrink)

In 'Chrysippus' Puzzle about Identity', John Bowin (thereafter JB) cogently strengthens David Sedley's reading of the puzzle of Chrysippus as a reductio ad absurdum of the Growing Argument. For Sedley, Chrysippus reduces to absurdity the assumption that matter is the sole principle of identity by refuting its presupposition that the two protagonists of the puzzle, namely Theon and Dion, are related as part to the whole. According to Plutarch's Comm. not. 1083 a8-c1, however, the Growing Argument concludes by posing (...) that growth is actually 'generation' and 'destruction'. In order to avoid the contradiction, Theon should have perished rather than become a part of Dion. JB attempts to answer the questions of whether within the Growing Argument there are elements against Theon being a living part of Dion. He shows that in both Epicharmus' fragment 2 and Plutarch's Comm.not. 1083b 308 "there is nothing to block the inference from matter being the sole principle of identity to the possibility that Theon could be a part of Dion" (246). Again, in exploring whether the above contradiction can be solved, he convincingly argues against Epicharmus' and Plutarch's reading of growth as generation and destruction. In the last part of his article, JB stresses that the reductio ad absurdum of the Growing Argument can be tackled without introducing the concept of 'peculiarly qualified individuals'. (Bryn Mawr Classical Review 2004.04.12). (shrink)

Kierkegaard in Fear and Trembling presents a reductio ad absurdum regarding the time-spans subject to moral evaluation. The text's classic dilemma depends on assuming that we only evaluate discrete, contextless instants. The pseudonymous author constantly seeks the single instant or moral “photograph” that indicates Abraham's status. Doing so, however, extracts scripture's moral legislation out from narrative, resulting in theological paralysis and thereby requiring an alternative temporal vocabulary for evaluating Abraham. Fear and Trembling contains an under-explored alternative that sets Abraham (...) within the covenantal narrative's temporality. The paper explores the consequences of shifting from evaluating instants to evaluating narrative durations, showing that while Genesis 22 remains a challenging episode, it also offers a model of imaginative faith, fidelity to promises, and hope in the face of trauma that focusing on moral instants overlooks. (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)

In lieu of an abstract, here is a brief excerpt of the content:Illusions of KnowingMatthew T. Kapstein (bio)Knowing Illusion: Bringing a Tibetan Debate into Contemporary Discourse, Volume I: A Philosophical History of the Debate, and Volume II: Translations. By The Yakherds ( José Cabezón, Ryan Conlon, Thomas Doctor, Douglas Duckworth, Jed Forman, Jay Garfield, John Powers, Sonam Thakchöe, Tashi Tsering, and Geshé Yeshes Thabkhas). New York: Oxford University Press, 2020.Metaphysics is a subject much more curious than useful, the knowledge of (...) which, like that of a sunken reef, serves chiefly to enable us to keep clear of it.Charles S. Peirce, "How to Make Our Ideas Clear"I. The Elusive Way to the MiddleKnowing Illusion: Bringing a Tibetan Debate into Contemporary Discourse is the fruit of collaborative efforts by the Yakherds, a group of scholars stemming from the earlier Cowherds, in which two of the present herdsmen—Sonam Thakchöe and Jay Garfield—were active as well.1 In moving from the broad issues of Buddhist philosophy that occupied the Cowherds to a tightly focused study of a particular debate in Tibet, the group has parted from the gentle cattle of the Indian plains to tend more rugged Himalayan stock. The present work, in two volumes, offers an exceptionally lucid guide to some of the main lines of Madhyamaka thought in Tibet and the disputes that these engendered. It constitutes an essential contribution to our knowledge of the philosophical traditions of Tibetan Buddhism, but, beyond this, merits the attention of those focusing on their Indian background and on Buddhist philosophies more generally. Indeed, Knowing Illusion is among the rare works in this area that may be profitably taken up by philosophers who are not specifically concerned with Buddhist or other Asian ways of thought, but who may wish to explore approaches beyond those of the Euro-American tradition to some of the central topics of philosophy overall: truth and falsehood, knowledge and ignorance, existence and illusion, belief and argument. Of the many interesting [End Page 1023] questions raised in Knowing Illusion, whether from the perspective of specialized Tibetan Buddhist Studies or of philosophy more broadly, only a sample can be addressed in the space of this review.The unifying thread in Knowing Illusion is a series of written disputes that were provoked by a fifteenth-century teacher belonging to the Sakyapa order of Tibetan Buddhism, Taktsang Sherab Rinchen (1405–1477), in his treatise titled Freedom from Extremes Accomplished through Comprehensive Knowledge of Philosophy (hereinafter Comprehensive Knowledge of Philosophy).2 Central to these debates was the challenge of squaring the Madhyamaka philosophy of Nāgārjuna (second century c.e.), whose teaching of ultimate emptiness (śūnyatā) seemed to undermine most—perhaps all—claims to knowledge, with the positive theory of knowledge (pramāṇa) advanced by Dignāga (fifth century c.e.) and Dharmakīrti (sixth century c.e.). The problem had already preoccupied Indian Buddhist thinkers, including philosophers Bhāviveka (sixth century c.e.) and Śāntarakṣita (eighth century c.e.), who sought to harmonize the insights of Madhyamaka with those of the pramāṇa school, and Dharmakīrti's contemporary Candrakīrti, whose trenchant interpretations of Nāgārjuna appeared to undermine the entire project of a 'theory of knowledge'.3 In Tibet, thinkers of the former camp came to be known as proponents of Svātantrika-Mādhyamika (so-called because they affirmed that emptiness could be established through positive proof, svatantra), while those allied with Candrakīrti were said to follow Prāsaṅgika-Mādhyamika (who allowed only indirect proof, reductio ad absurdum, or prasaṅga).4The debate reemerged forcefully in Tibet, and not merely as a historical exercise, in tandem with the expansion of monastic colleges there during the twelfth century, as Tibetan teachers increasingly elaborated original perspectives on the Indian classics. Many aspects of the Svātantrika/Prāsaṅgika split were co. (shrink)

In the author's view, the proper verdict on the reconcilability of the content of Christian revelation with the full‐blown natural theological concept of God found in the works of classical theologians is much less clear than many contemporary theologians would have it. The author argues that one can reasonably accept the philosophical concept of God as necessary being while rejecting the more problematic notions of immutability and simplicity. This chapter briefly discusses the strands of thought offered by natural theology. (...) It describes Luis de Molina's problematic notion of middle knowledge to assess the timeless eternity theorist's conception of God's knowledge of and actions in response to the world. It argues that the eternity thesis is committed to the doctrine of middle knowledge (or, as with Leftow's alternative, something very much like it), and tries to persuade the reader that this constitutes something of a reductio of the eternity thesis. (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)