In this paper I critically engage with Pauline Kleingeld’s ‘volitional self-contradiction’ interpretation of Kant’s formula of universal law. I make three remarks: first, I seek to clarify what it means for a contradiction to be volitional as opposed to logical; second, I suggest that her interpretation might need to be closer to Korsgaard’s ‘practical contradiction’ interpretation than she thinks; and third, I suggest that more work needs to be done to explain how a (...) volitional self-contradiction generates both a ‘contradiction in conception’ and a ‘contradiction in will.’. (shrink)

Kant’s Formula of Universal Law (FUL) is generally believed to require you to act only on the basis of maxims that you can will without contradiction to become universal laws. In “Contradiction and Kant’s Formula of Universal Law” (2017), I have proposed to read the FUL instead as requiring that, for any maxim on which you act, you can will two things simultaneously, without volitional self-contradiction: (1) willing the maxim as your own action principle and (2) willing (...) that it become a universal law. In the present essay, I reply to comments by Mark Timmons, Michael Walschots, Paola Romero, and Stefano Lo Re. In response to their comments concerning the application of the FUL, I expand the interpretive framework of the Volitional Self-Contradiction Interpretation. I argue that Kant also constructs the diagnostic volitional self-contradiction as a contradiction between (1) willing a tested maxim to become a universal law and (2) willing what humans, qua finite rational beings, necessarily will, namely the means to their actual and possible future ends. I also clarify how the two ways in which Kant specifies the test are related. Furthermore, I clarify the relation between the Volitional Self-Contradiction interpretation and other interpretations of the FUL, in particular the ‘Practical Contradiction’ and ‘Logical Contradiction’ interpretations, as well as its difference from the Golden Rule. I also address the objection that the volitional self-contradiction is superfluous, clarify the relation between the Formula of Universal Law and the Formula of the Law of Nature, and explain why the will of an immoral agent does not contain a volitional self-contradiction of the type at issue. (shrink)

Derrida on Formal Logic: An Interpretive Essay develops the dominant themes in Jacques Derrida's texts on the principles of formal logic, especially identity and contradiction, as these themes emerged from Derrida's discussion of Joyce's Ulysses. For students of Derrida and his place as a critic of western metaphysics, the work provides a clear account and critical evaluation of implications drawn from Derrida's conclusions concerning the strength of formal logic.

This article examines Pauline Kleingeld’s “volitional self-contradiction” (VSC) interpretation of Kant’s formula of universal law. It begins in §1 with an outline of Kleingeld’s interpretation and then proceeds in §2 to raise some worries about how the interpretation handles Kant’s egoism example. §3 considers VSC’s handling of the false promise example comparing it in §4 with the Logical/Causal Law (LCL) interpretation, which arguably does better than its VSC competitor in handling this example. §5 deploys (...) the LCL interpretation to consider the related objections that the VSC interpretation is “superfluous” or at least “misguided” as a (formal) criterion of the permissibility of action on a maxim. These objections, it is argued, help reveal two distinct roles for contradictions in Kant’s applications of the formula of universal law – one for delivering moral judgments and one reflective of a formal coherence constraint on all deliberation having psychological significance for agents who contemplate violating duty. Disentangling these helps reveal some of the complexity in the use to which Kant puts the formula of universal law in his sample applications revealing the true significance of what Kleingeld refers to as volitional self-contradiction. In §6 is summary. (shrink)

According to Kant’s Universal Law Formula, maxims that cannot be conceived as universal laws denote duties of perfect obligation. In the recent literature, two versions of the Contradiction in Conception test have received the most attention. When acting on a maxim would violate a perfect duty, according to the Logical Contradiction Interpretation (LCI), universalizing the maxim would make it literally impossible to perform the action as described in the original maxim. According to the Practical Contradiction (...) Interpretation (PCI), the locus of the contradiction is as follows: the agent acts on the maxim in order to achieve some purpose, but were the agent’s maxim universalized, the purpose would be unattainable. Having examined the most widely accepted versions of both interpretations, I argue that i) PCI cannot generate contradictions for the maxims of any actions that violate perfect duties beyond those generated by LCI, ii) despite claims of its proponents, PCI cannot solve the vexing problem of relevant descriptions that Kant’s account faces, and iii) other arguments in favor of PCI are at best inconclusive. I therefore conclude that since LCI better coheres with the text, sympathetic interpreters should focus their efforts on LCI, not PCI. (shrink)

This book offers a clear, analytic, and innovative interpretation of Heidegger's late work. This period of Heidegger's philosophy remains largely unexplored by analytic philosophers, who consider it filled with inconsistencies and paradoxical ideas, particularly concerning the notions of Being and nothingness. This book takes seriously the claim that the late Heidegger endorses dialetheism--namely the position according to which some contradictions are true--and shows that the idea that Being is both an entity and not an entity is neither incoherent nor (...) logically trivial. The author achieves this by presenting and defending the idea that reality has an inconsistent structure. In doing so, he takes one of the most discussed topics in current analytic metaphysics, grounding theory, into a completely unexplored area. Additionally, in order to make sense of Heidegger's concept of nothingness, the author introduces an original axiomatic mereological system that, having a paraconsistent logic as a base logic, can tolerate inconsistencies without falling into logical triviality. This is the first book to set forth a complete and detailed discussion of the late Heidegger in the framework of analytic metaphysics. It will be of interest to Heidegger scholars and analytic philosophers working on theories of grounding, mereology, dialetheism, and paraconsistent logic. (shrink)

The view that contradictions cannot be true has been part of accepted philosophical theory since at least the time of Aristotle. In this regard, it is almost unique in the history of philosophy. Only in the last forty years has the view been systematically challenged with the advent of dialetheism. Since Graham Priest introduced dialetheism as a solution to certain self-referential paradoxes, the possibility of true contradictions has been a live issue in the philosophy of logic. Yet, despite the arguments (...) advanced by dialetheists, many logicians and philosophers still hold the opinion that contradictions cannot be true. -/- Rather than advocating the truth of certain contradictions, this thesis offers a different challenge to the classical logician. By showing that it can be philosophically coherent to propose that true contradictions are metaphysically possible, the thesis suggests that the classical logician must do more than she currently has to justify her confidence in the impossibility of true contradictions. Simply fighting off the dialetheist’s putative examples of true contradictions at the actual world isn’t enough to justify the classical logician’s conclusion that true contradictions are impossible. -/- To aid the thesis dialectically, we introduce a new position, absolutism, which hypothesises that it’s metaphysically possible for at least one contradiction to be true, contrasting with the dialetheic hypothesis that some contradictions are true in the actual world. We demonstrate that absolutism can be given a philosophically coherent interpretation, an appropriate logic, and that certain criticisms are completely toothless against absolutism. The challenge put to the classical logician is then: On what logical or philosophical grounds can we rule out the metaphysical possibility of true contradictions? (shrink)

Chapter VI discusses a few assumptions which underlie the proposed reconstruction of Hegel's procedures. It is shown that certain equivalents of such assumptions are either explicitly accepted by Hegel, or they are consequences of theses he subscribed to. Finally, it is suggested that some of these assumptions envisage a conception of language and philosophy which has an interesting parallel in Wittgenstein's later work. Such a conception sets philosophy sharply apart from the sciences, and deemphasizes the formation of contradictions. The general (...) relationship between vagueness and contradiction is briefly explored to show that some familiar contradiction-generating procedures can be seen to be based on vagueness of the relevant theoretical expressions. ;Chapter V deals with the Aufhebung. After an examination of Hegel's own theory of sublation, and an analysis of some textual examples, it is shown that some Aufhebung-procedures can be modelled by certain patterns of argument concerning simple algebraic structures. A few Hegelian concepts, such as "opposition," "unity," "passing over," "reflection," are partially explicated in the process. ;Chapter IV plays a pivotal role in the dissertation. It explains the generation of dialectical contradictions as stemming from syntactic and intensional indeterminacy of the theoretical terms. It is shown that Hegel's conceptual terms have no well-determined sense, so that their senses can be identified in different and possibly inconsistent ways. It is also shown that they can be made to play different syntactic roles, for no rigid determination of their syntactic function is assumed. This, again, may create contradictions. Both kinds of indeterminacy are related to Hegel's borrowing of his conceptual terms from a "natural" philosophical koine. ;Chapter III is an analysis of Hegel's language in the Logic. It tries to determine the grammatical and semantical status of Hegel's conceptual terms and to explicate the Hegelian use of the copula as occurring in certain typical sentential forms. ;The presence of the so-called "dialectical contradictions" is the most evident logico-linguistic peculiarity of Hegel's text. After a discussion of Hegel's theory of contradiction, and of some of its interpretations, it is argued that any satisfactory account of the dialectical method must explain the formation of contradictions on the basis of other logico-linguistic features of Hegel's text. ;Chapter II deals with a few such explanations which have been offered by modern interpreters of Hegel. Use of contemporary analytic concepts in the logical and linguistic analysis of Hegel's text is justified with reference to the Carnapian categories of "explication" and "rational reconstruction." ;The literature on Hegel's dialectic has made important contributions to an understanding of its nature both as a metaphysical theory of reality and as a philosophical attitude towards the human world. However, many recent interpreters agree that no satisfactory progress has been made in understanding dialectic as a method. This dissertation identifies the dialectical method as the specific form of Hegel's philosophical discourse, and tries to bring out its basic logico-linguistic features. The Science of Logic is chosen as the field of inquiry, because of its crucial position among Hegel's mature works. (shrink)

Classical logic, of first or higher order, is extended with sentential operators and quantifiers, interpreted substitutionally over unrestricted substitution class. Operators mark a single layered, consistent metalanguage. Self-reference, arising from substitutional quantification over sentences, allows to express paradoxes which, unlike contradictions, do not lead to explosion. Semantics of the resulting language, using semi-kernels of digraphs, is non-explosive yet two-valued and has classical semantics as a special case for clasically consistent theories. A complete reasoning is obtained by extending LK with two (...) rules for sentential quantifiers. Adding (cut) yields a complete system for the explosive semantics. (shrink)

In this paper I defend the traditional interpretations of Kant’s Formula of a Law of Nature from recent attacks leveled by Faviola Rivera-Castro, James Furner, Ido Geiger, Pauline Kleingeld and Sven Nyholm. After a short introduction, the paper is divided into four main sections. In the first, I set out the basics of the three traditional interpretations, the Logical Contradiction Interpretation, the Practical Contradiction Interpretation and the Teleological Contradiction Interpretation. In the second, I (...) examine the work of Geiger, Kleingeld and Nyholm: these three commentators reject the traditional interpretations entirely, but I argue that this rejection is ill-founded. In the third and fourth, I take a detailed look at Furner’s work, work in which he seeks to revise (rather than reject) the traditional interpretations. I argue that, despite his more modest aims, Furner’s revision is also ill-founded. (shrink)

This article challenges the prevailing interpretations of Hegel's account of the concept "contradiction" in the Science of Logic by arguing that it is concerned with the principle of Hegel's method rather than with the classical law of non-contradiction. I first consider Hegel's Doctrine of Essence in view of Kant's discussion of the concepts of reflection in the first Critique. On this basis, I examine Hegel's account of the logical principles based on the concepts "identity," "opposition," and " (...) class='Hi'>contradiction." Finally, I point out how the principle Hegel derives from the concept of contradiction actually informs his own method. (shrink)

In 1971, Wolfgang Müller-Lauter introduced his study of Nietzsche as an investigation into the history of modern nihilism in which “contradiction” forms the central thread of the argument. For Müller-Lauter, the interpretive task is not to demonstrate the overall coherence or incoherence of Nietzsche’s philosophy, but to examine Nietzsche’s “philosophy of contradiction.” Against those such as Karl Jaspers, Karl Löwith and Martin Heidegger, Müller-Lauter argued that contradiction is the foundation of Nietzsche’s thought, and not a problem to (...) be corrected or cast aside for exegetical or political purposes. For Müller-Lauter, contradiction qua incompatibility (not just mere opposition) holds a key to Nietzsche’s affective vision of philosophy. Beginning with the relationship between will to power and eternal recurrence, in this paper I examine aspects of Müller-Lauter’s account of Nietzsche’s philosophy of contradiction specifically in relation to the counter-interpretations offered by two other German commentators of Nietzsche, Leo Strauss and Karl Löwith, in order to confirm Müller-Lauter’s suggestion that contradiction is indeed an operative engine of Nietzsche’s thought. Indeed contradiction is a key Nietzschean theme and an important dynamic of becoming which enables the subject to be revealed as a “multiplicity” ( BGE §12) and as a “fiction” (KSA 12:9[91]). Following Müller-Lauter’s assertion that for Nietzsche the problem of nihilism is fundamentally synonymous with the struggle of contradiction experienced by will to power, this paper interprets Nietzsche’s philosophy of contradiction in terms of subjective, bodily life (rather than in terms of logical incoherences or ontological inconsistencies). Against the backdrop of nihilism, the “self” (and its related place holder the “subject”), I will argue, becomes the psycho-physiological battlespace for the struggle and articulation of “contradiction” in Nietzsche’s thought. (shrink)

We present probabilistic approaches to check the validity of selected connexive principles within the setting of coherence. Connexive logics emerged from the intuition that conditionals of the form If ∼A, then A, should not hold, since the conditional’s antecedent ∼A contradicts its consequent A. Our approach covers this intuition by observing that for an event A the only coherent probability assessment on the conditional event A|~A is p(A|~A)=0 . Moreover, connexive logics aim to capture the intuition that conditionals should express (...) some “connection” between the antecedent and the consequent or, in terms of inferences, validity should require some connection between the premise set and the conclusion. This intuition is covered by a number of principles, a selection of which we analyze in our contribution. We present two approaches to connexivity within coherence-based probability logic. Specifically, we analyze connections between antecedents and consequents firstly, in terms of probabilistic constraints on conditional events (in the sense of defaults, or negated defaults) and secondly, in terms of constraints on compounds of conditionals and iterated conditionals. After developing different notions of negations and notions of validity, we analyze the following connexive principles within both approaches: Aristotle’s Theses, Aristotle’s Second Thesis, Abelard’s First Principle and selected versions of Boethius’ Theses. We conclude by remarking that coherence-based probability logic offers a rich language to investigate the validity of various connexive principles. (shrink)

The starting point of this paper concerns the apparent difference between what we might call absolute truth and truth in a model, following Donald Davidson. The notion of absolute truth is the one familiar from Tarski’s T-schema: ‘Snow is white’ is true if and only if snow is white. Instead of being a property of sentences as absolute truth appears to be, truth in a model, that is relative truth, is evaluated in terms of the relation between sentences and models. (...) I wish to examine the apparent dual nature of logical truth (without dwelling on Davidson), and suggest that we are dealing with a distinction between a metaphysical and a linguistic interpretation of truth. I take my cue from John Etchemendy, who suggests that absolute truth could be considered as being equivalent to truth in the ‘right model’, i.e., the model that corresponds with the world. However, the notion of ‘model’ is not entirely appropriate here as it is closely associated with relative truth. Instead, I propose that the metaphysical interpretation of truth may be illustrated in modal terms, by metaphysical modality in particular. One of the tasks that I will undertake in this paper is to develop this modal interpretation, partly building on my previous work on the metaphysical interpretation of the law of non-contradiction (Tahko 2009). After an explication of the metaphysical interpretation of logical truth, a brief study of how this interpretation connects with some recent important themes in philosophical logic follows. In particular, I discuss logical pluralism and propose an understanding of pluralism from the point of view of the metaphysical interpretation. (shrink)

One of the most interesting debates in Hegel scholarship today comes from the question of how to interpret Hegel’s treatment of contradiction in the Science of Logic.1 Some interpreters claim that Hegel defiantly disregards the basic law of noncontradiction, which states that something cannot both be and not be in the same time, manner, or place, proposing instead that for Hegel true contradictions really do exist, and not only in rational conception but equally in the very fabric of reality. (...) However, other interpreters propose less direct readings of Hegel’s thesis from productive contradiction, either by limiting Hegel’s conception of contradiction to only certain forms, such as only in... (shrink)

In 2004, the Department for Education and Skills in England published its Five Year Strategy for Children and Learners (DfES, 2004). It was preceded by Excellence and Enjoyment: a strategy for primary schools (DfES, 2003). 'Excellence and enjoyment' seems to constitute an ambiguity, even a contradiction. The government's view is otherwise. It states that enjoyment (for pupils) is a consequence of excellent teaching. In turn, excellent teaching is said to be more assured if it is personalised and creative. This (...) official logic is questionable. A different interpretation is offered. Global capitalism is placing fiscal pressures on the public expenditure of the nation-state, and new accommodations and justifications have to be made by government as it continues to re-focus the education system towards an economic purpose. In this endeavour it must gain public and professional assent. Informed by critical theorists, it is argued here that 'excellence' associates with a producer ethic; 'enjoyment' with a consumer ethic. The former enables the capital accumulation process within a social-democratic welfare state; the latter justifies the policies of accumulation. (shrink)

A common reaction to Hegel’s suggestion that we collapse Kant’s distinction between form and content is that, since such a move would also deprive us of any way of distinguishing the merely logical from the real possibility of our concepts, it is incoherent and ought to be rejected. It is true that these two distinctions are intimately related in Kant, such that if one goes, the other does as well. But it is less obvious that giving them up as (...) Kant conceives them is as incoherent a proposal as many of Hegel’s critics think. It has been the point of a recent account of Hegel’s idealism to demonstrate that his critique of Kant’s dichotomy between form and content or concept and intuition does not commit him to the view that human cognition is materially creative of its content in the manner of a God-like or intuitive intellect—does not, in other words, signify a return to a pre-Critical metaphysics. This is the interpretative stance I adopt here as well, in hope of giving defenders of Kant reasons for taking Hegel more seriously. (shrink)

In a recent work, Walter Carnielli and Abilio Rodrigues present an epistemically motivated interpretation of paraconsistent logic. In their view, when there is conflicting evidence with regard to a proposition A (i.e. when there is both evidence in favor of A and evidence in favor of ￢A) both A and ￢A should be accepted without thereby accepting any proposition B whatsoever. Hence, reasoning within their system intends to mirror, and thus, should be constrained by, the way in which we (...) reason about evidence. In this article we will thoroughly discuss their position and suggest some ways in which this project can be further developed. The aim of the paper is twofold. On the one hand, we will present some philosophical critiques to the specific epistemic interpretation of paraconsistent logic proposed by Carnielli & Rodrigues. First, we will contend that Carnielli & Rodrigues’s interpretation implies a thesis about what evidence rationally justifies to accept or believe, called Extreme Permissivism, which is controversial among epistemologists. Second, we will argue that what agents should do, from an epistemic point of view, when faced with conflicting evidence, is to suspend judgment. On the other hand, despite these criticisms we do not believe that the epistemological motivation put forward by Carnielli & Rodrigues is entirely wrong. In the last section, we offer an alternative way in which one might account for the epistemic rationality of accepting contradictions and, thus, for an epistemic understanding of paraconsistency, which leads us to discuss the notion of diachronic epistemic rationality. (shrink)

Naive speakers find some logical contradictions acceptable, specifically borderline contradictions involving vague predicates such as Joe is and isn’t tall. In a recent paper, Cobreros et al. (J Philos Logic, 2012) suggest a pragmatic account of the acceptability of borderline contradictions. We show, however, that the pragmatic account predicts the wrong truth conditions for some examples with disjunction. As a remedy, we propose a semantic analysis instead. The analysis is close to a variant of fuzzy logic, but conjunction and (...) disjunction are interpreted as intensional operators. (shrink)

Using logic-laden terms to translate and interpret what the ancient Indian Buddhist thinkers said when we are not sure what they spoke about when they spoke about ‘contradictions’, etc. in natural languages can sometimes make things frustrating. Keeping in mind Wittgenstein’s exhortation, “don’t think, but look!”, I approach the issues of contradiction, negation, and the catuṣkoṭi via case-by-case study on several pertinent passages in Dharmapāla’s Dasheng Guangbailun Shilun. The following are some interrelated observations which should not be overgeneralized, especially (...) considering the limited scope of this study and its methodological considerations. First, there is an implicit rule of non-opposition and there could be no real oppositions for apparent oppositions because of implicit qualifications. Moreover, these are not new since or after Dignāga. Second, Dharmapāla and his contemporaries are familiar with the two negative usages, and prasajya-pratiṣedha is used for negating the opponents’ theses and is related to the no-thesis view. It’s not a good idea to assign truth values to sentences using prasajya-pratiṣedha or to a thesis in which the terms have no real objects. And Dharmapāla’s theses are just therapeutic tools. Third, in Dharmapāla’s discussion of the negative catuṣkoṭi, one can see the rule of non-opposition, the strategy of qualifications, and the two negative usages. Although the four positions in the catuṣkoṭi are regarded as mutually exclusive and jointly exhaustive, they are jointly exhaustive in pragmatic context, and are mutually exclusive under some interpretations, but not all. My tentative suggestion is that the catuṣkoṭi is a loosely term-based way of categorization. Last but not least, in the text discussed, I do not see Buddhist thinkers endorsing any contradictions. (shrink)

We present two approaches to investigate the validity of connexive principles and related formulas and properties within coherence-based probability logic. Connexive logic emerged from the intuition that conditionals of the form if not-A, thenA, should not hold, since the conditional’s antecedent not-A contradicts its consequent A. Our approaches cover this intuition by observing that the only coherent probability assessment on the conditional event $${A| \overline{A}}$$ A | A ¯ is $${p(A| \overline{A})=0}$$ p ( A | A ¯ ) = 0. (...) In the first approach we investigate connexive principles within coherence-based probabilistic default reasoning, by interpreting defaults and negated defaults in terms of suitable probabilistic constraints on conditional events. In the second approach we study connexivity within the coherence framework of compound conditionals, by interpreting connexive principles in terms of suitable conditional random quantities. After developing notions of validity in each approach, we analyze the following connexive principles: Aristotle’s theses, Aristotle’s Second Thesis, Abelard’s First Principle, and Boethius’ theses. We also deepen and generalize some principles and investigate further properties related to connexive logic (like non-symmetry). Both approaches satisfy minimal requirements for a connexive logic. Finally, we compare both approaches conceptually. (shrink)

The contradiction in conception test is one of two tests posed by Kant’s Formula of the Law of Nature. This article proposes a new interpretation of this test: a causal-teleological version of the Logical Contradiction Interpretation. Its distinctive feature is that it identifies causal and teleological implications in the thought of a universal law of nature. A causal-teleological version of LCI has two advantages. While the established view of the Groundwork’s applications of the CC test (...) is a hybrid view that treats the Groundwork’s arguments as different in kind, a causal-teleological version of LCI unifies the Groundwork’s applications of the CC test. Relatedly, a causal-teleological version of LCI provides a solution to the problem of how the CC test can confirm the impermissibility of a self-directed maxim. (shrink)

According to Kant’s Universal Law Formula, maxims that cannot be conceived as universal laws denote duties of perfect obligation. In the recent literature, two versions of the Contradiction in Conception test have received the most attention. When acting on a maxim would violate a perfect duty, according to the Logical Contradiction Interpretation (LCI), universalizing the maxim would make it literally impossible to perform the action as described in the original maxim. According to the Practical Contradiction (...) Interpretation (PCI), the locus of the contradiction is as follows: the agent acts on the maxim in order to achieve some purpose, but were the agent’s maxim universalized, the purpose would be unattainable. Having examined the most widely accepted versions of both interpretations, I argue that i) PCI cannot generate contradictions for the maxims of any actions that violate perfect duties beyond those generated by LCI, ii) despite claims of its proponents, PCI cannot solve the vexing problem of relevant descriptions that Kant’s account faces, and iii) other arguments in favor of PCI are at best inconclusive. I therefore conclude that since LCI better coheres with the text, sympathetic interpreters should focus their efforts on LCI, not PCI. (shrink)

This is a short story about vision, thought, and contradiction and the role they play in the first half of Hegel's Science of Logic. The Logic begins with a descent, in this case, the fall from Being into Nothingness. Later, at nearly the exact middle of each text, there is a certain paradox in which everything is at stake, the category of contradiction. At this exact moment, thinking both fails and is birthed anew in a speculative guise. In (...) this section, we engage some of Analytic philosophy's influential interpretations of Hegel's strange use of contradiction. In order to get there, we turn to a curious art work, James Turrell's Pleiades, as an aesthetic example of that first fall. We will then progress through the text, with thought and vision as our dual guide, at quite a quick pace, not slowing down until we enter Hegel's story of contradiction, where I will show the explosive nature of contradiction. This will allow us to see how Hegel harnesses the power of contradiction in order to generate the second half of the story of the Science of Logic. I begin with the descent of being into nothingness, the moment when darkness. (shrink)

In this paper we present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language in such a way that consistency may be logically independent of non- contradiction. We defend the view according to which logics of formal inconsistency may be interpreted as theories of logical consequence of an epistemological character. We also argue that in order (...) to philosophically justify paraconsistency there is no need to endorse dialetheism, the thesis that there are true contradictions. Furthermore, we argue that an intuitive reading of the bivalued semantics for the logic mbC, a logic of formal inconsistency based on classical logic, fits in well with the basic ideas of an intuitive interpretation of contradictions. On this interpretation, the acceptance of a pair of propositions A and ¬A does not mean that A is simultaneously true and false, but rather that there is conflicting evidence about the truth value of A. (shrink)

Proper interpretation of the problem of contradiction in motion has acquired pressing importance in recent years in conjunction with the development of dialectical materialist concepts of motion, space and time, continuity and discontinuity, infinity, and the like, and also in relations to problems of dialectical logic. In a number of writings , we have attempted to demonstrate that internal contradiction is inherent in motion that inseparably connected with it are such contradictory factors as variability and stability, movement (...) and rest, with variability playing the leading role. Moreover, generally speaking, the concept of preservation, stability or rest means the preservation or maintenance of the very state of motion. (shrink)

In Book Γ of the Metaphysics Aristotle states and attempts to prove what he calls the basic principle of the science of being as being: the law of non‐contradiction. In this paper I defend an interpretation of his proof, inspired by Elizabeth Anscombe's 1961 essay in ‘Three Philosophers’, though some of its features were remarked on by Lukasiewicz in 1910, according to which Aristotle is proving this principle only for substance predicates, and that it is to be understood (...) as the basic principle of metaphysics rather than of logic. It is the principle that every substance has an essential nature, and if it is of the essential nature of x to be F then it cannot be of the essential nature of x to be not‐F since then x's being F and x's being not‐F would be the same. He puts it by saying that in such a case F and not‐F would ‘signify the same’ for x. Since they do not, x cannot be both. (shrink)

A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems (...) to change this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more 'big picture' ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics. (shrink)

Valentin Asmus made a significant contribution to the formation of key interpretations, analyses and evaluations of Immanuel Kant’s work in the Russian-Soviet tradition of studying the “history of foreign philosophy”. This article shows precisely which principles and developmental models Asmus laid down in his interpretation of the transcendental dialectic section of Kant’s philosophical system. The article attempts to show that in his reading of Kant, Asmus actively relies on Hegel’s philosophical legacy, namely, on his theory of dialectics, the ontological (...) status of contradiction and the highly significant role of “error” in the formation and advancement of knowledge. Asmus reads Kant through the lens of Hegel’s philosophy and adheres to Hegelian philosophy as a benchmark of fidelity and heuristics, as a canon and organon, through which Kant’s thought should be evaluated. Pursuing this path, he notes the significant progress of Kant’s thought compared to the metaphysical philosophy of previous centuries, but points out the insufficiency of Kant’s determination to fully think through contradictions ontologically and dialectically. Kant confines himself to pointing out the natural origin of the principle of “appearance” in the limits of reason, but is not ready to revise the foundations of classical logic, where any contradictions should be avoided. As a consequence, he settles on the epistemological interpretation of contradiction seeing it as an error of reason, albeit a natural one, leaving Asmus deeply disenchanted with his Hegelian expectations. (shrink)

The present book discusses all aspects of paraconsistent logic, including the latest findings, and its various systems. It includes papers by leading international researchers, which address the subject in many different ways: development of abstract paraconsistent systems and new theorems about them; studies of the connections between these systems and other non-classical logics, such as non-monotonic, many-valued, relevant, paracomplete and fuzzy logics; philosophical interpretations of these constructions; and applications to other sciences, in particular quantum physics and mathematics. Reasoning with contradictions (...) is the challenge of paraconsistent logic. The book will be of interest to graduate students and researchers working in mathematical logic, computer science, philosophical logic, linguistics and physics. (shrink)

In this paper we propose to take seriously the claim that at least some kinds of paraconsistent negations are subcontrariety forming operators. We shall argue that from an intuitive point of view, by considering paraconsistent negations as formalizing that particular kind of opposition, one needs not worry with issues about the meaning of true contradictions and the like, given that “true contradictions” are not involved in these paraconsistent logics. Our strategy will consist in showing that, on the one hand, the (...) natural translation for subcontrariety in formal languages is not a contradiction in natural language, and on the other, translating alleged cases of contradiction in natural language to paraconsistent formal systems works only provided we transform them into a subcontrariety. Transforming contradictions into subcontrariety shall provide for an intuitive interpretation for paraconsistent negation, which we also discuss here. By putting all those pieces together, we hope a clearer sense of paraconsistency can be made, one which may liberate us from the need to tame contradictions. (shrink)

Recent experiments have shown that naive speakers find borderline contradictions involving vague predicates acceptable. In Cobreros et al. we proposed a pragmatic explanation of the acceptability of borderline contradictions, building on a three-valued semantics. In a reply, Alxatib et al. show, however, that the pragmatic account predicts the wrong interpretations for some examples involving disjunction, and propose as a remedy a semantic analysis instead, based on fuzzy logic. In this paper we provide an explicit global pragmatic interpretation rule, based (...) on a somewhat richer semantics, and show that with its help the problem can be overcome in pragmatics after all. Furthermore, we use this pragmatic interpretation rule to define a new consequence-relation and discuss some of its properties. (shrink)

In the *Science of Logic*, Hegel states unequivocally that the category of “life” is a strictly logical, or pure, form of thinking. His treatment of actual life – i.e., that which empirically constitutes nature – arises first in his *Philosophy of Nature* when the logic is applied under the conditions of space and time. Nevertheless, many commentators find Hegel’s development of this category as a purely logical one especially difficult to accept. Indeed, they find this development only comprehensible (...) as long as one simultaneously assumes that Hegel breaks his promise to let the logic do the leading. However, if Hegel were to in fact allow the logical development to be led by biological analogies at this point, problems would ensue. Not only would it contradict his own speculative method, which should secure the necessity of the categories, but it would also endanger the ontological generality of the category of life itself. Beyond undermining his method and the logical integrity of the category, however, I will argue that such a reading makes the transition to the next category of “cognition” unintelligible and problematic. My aim in the first part of this paper is to argue how logical life can be read as a pure category. I then argue in the second part how my reconstruction makes the transition to cognition intelligible without resorting to profane or supernatural interpretations. (shrink)

In De Interpretatione 6-9, Aristotle considers three logical principles: the principle of bivalence, the law of excluded middle, and the rule of contradictory pairs (according to which of any contradictory pair of statements, exactly one is true and the other false). Surprisingly, Aristotle accepts none of these without qualification. I offer a coherent interpretation of these chapters as a whole, while focusing special attention on two sorts of statements that are of particular interest to Aristotle: universal statements not (...) made universally and future particular statements. With respect to the former, I argue that Aristotle takes them to be indeterminate and so to violate the rule of contradictory pairs. With respect to the latter, the subject of the much discussed ninth chapter, I argue that the rule of contradictory pairs, and not the principle of bivalence, is the focus of Aristotle's refutation. Nevertheless, Aristotle rejects bivalence for future particular statements. (shrink)

This work proposes a logical analysis for the kind of knowledge or insight provided by Buddhist enlightenment, which is often presented only in the form of contradictions and riddles. The comprehension of contradictions is perplexing to most western logic, and yet developed here is a theory demonstrating how a non truth-functional interpretation can be attached to certain naturalistic contradictions. In this way, the logical and psychological status of Enlightenment can be analysed in a manner consistent with the (...) claims of much Eastern thought. For example, it is argued that by breaking the logical principle of identity, it is possible to understand the contradiction "Everybody's friend is nobody's friend" in such a way as to learn something new about friendship, thereby causing us to modify our concept of friendship. In this way our grasp of contradictions can lead to conceptual change. -/- It is further noted that conflicts in all areas of life can be expressed by contradictions, so that our capacity in general to grasp naturalistic contradictions is also a basis for conflict-resolution -thus justifying the term 'Enlightenment'. This pragmatic comprehension of contradictions is thus shown to formalise, not only the Middle Way of Buddhism, but also Socratic dialectic. Finally, the analysis explains why gnomic utterances in every religion and Zen in particular, are sometimes expected to trigger spiritual enlightenment. (shrink)

Hume’s account of contrariety in Book I of the Treatise poses several interpretive puzzles. I consider each in turn and offer a novel interpretation of contrariety based on Hume’s discussion of the passions. That Book II and Book I form a complete chain of reasoning suggests that the way in which passions are related is analogous to the way in which ideas are related in the understanding. I argue that Hume identifies three species of empirical contrariety in Book II: (...) contraries, oppositions, and contradictions. All three species help clarify Hume’s remarks concerning contrariety in Book I. I further identify a second genus I call logical contrariety by drawing on Hume’s discussion of fictitious duration and distance. (shrink)

The author provides another metaphysical argument to bolster his thesis of the logical harmony of determinism and volition. He shows how the typical mainstream and Objectivist doctrine of “libertarian” free will commits the same logical error as the Sophist attacks on the Law of Non-Contradiction—namely, an out-of-context interpretation of, respectively, the Law of Causality and the Law of Identity as being unconditional absolutes, which they are not and cannot be.

We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others (Among whom we may count or not E. W. Beth, depending on how we interpret some of his statements), our answer is a clear "no". Philosophically, our argument is based on combining a formal semantic approach, in the spirit of E. W. Beth's proposal of applying Tarski's semantical methods to the analysis of physical theories, (...) with an empirical-experimental approach to Logic, as advocated by both Beth and Putnam, but understood by us in the view of the operationalrealistic tradition of Jauch and Piron, i. e. as an investigation of "the logic of yes-no experiments" (or "questions"). Technically, we use the recently-developed setting of Quantum Dynamic Logic (Baltag and Smets 2005, 2008) to make explicit the operational meaning of quantum-mechanical concepts in our formal semantics. Based on our recent results (Baltag and Smets 2005), we show that the correct interpretation of quantum-logical connectives is dynamical, rather than purely propositional. We conclude that there is no contradiction between classical logic and (our dynamic reinterpretation of) quantum logic. Moreover, we argue that the Dynamic-Logical perspective leads to a better and deeper understanding of the "non-classicality" of quantum behavior than any perspective based on static Propositional Logic. (shrink)

In update logic as studied by Baltag, Moss, Solecki and van Benthem, little attention is paid to the interpretation of an action by an agent, which is just assumed to depend on the situation. This is actually a complex issue that nevertheless complies to some logical dynamics. In this paper, we tackle this topic. We also deal with actions that change propositional facts of the situation. In parallel, we propose a formalism to accurately represent an agent's epistemic state (...) based on hyperreal numbers. In that respect, we use infinitesimals to express what would surprise the agents (and by how much) by contradicting their beliefs. We also use a subjective probability to model the notion of belief. It turns out that our probabilistic update mechanism satisfies the AGM postulates of belief revision. (shrink)

This article focuses on the applications of philosophical logic in the discipline of philosophy of religion of both ‘Eastern’ and ‘Western’ traditions, in which the problem of apparent ontological contradictions can be found. A number of philosophers have proposed using the work of those non-classical logicians who countenance the violation of the law of non-contradiction (LNC) to address this problem. I discuss (1) whether classical or non-classical account of logic is universal in applying to all true theories, and (2) (...) whether there might be extra-logical considerations which affect what is the correct account of logic for the doctrines in question. With regard to Jc Beall’s application of non-classical (FDE) logic to the doctrine of the Incarnation, I argue using the evidence from the writings of church fathers that the meaning of negation found in the core claims of the doctrine of the Incarnation should not be interpreted in accordance with Beall’s FDE account, and that this extra-logical consideration refutes Beall’s project. Moreover, the FDE’s acceptance of the possibility of statements that are both true and false is contrary to what is allowed by the definition of negation in classical logic; therefore (contrary to Beall), Beall is in fact using a different definition of negation compared with the definition used by the classical account. I develop this point in interaction with contemporary philosophy of religion literature and explain its implications and significance for this discipline. (shrink)

This article is an inquiry into how the relationship between the principle of non-contradiction and the limits of thought has been understood by thinkers as diverse as Hegel, Heidegger, Levinas, and Graham Priest. While Heidegger and Levinas focus on the question of temporality and Priest takes a formal approach, all these philosophers effectively maintain that the principle of non-contradiction imposes a restriction on thought that disables it from adequately accounting for its own limits and thus what lies beyond (...) those limits, the implication being that the violation of the principle is necessary for such an accounting to take place. However, the ultimate argument here is that, contrary to Priest’s interpretation, Hegel’s philosophy can be convincingly read as supporting the idea that the mind’s ability to go beyond any particular limit of thought can actually be said to involve an adherence to a normative demand to locate and dispel the contradictions that emerge through the very setting of determinative limits. This is a non-formal consistency that evinces a “logic” that is unknowingly followed by the Heideggerian and Levinasian phenomenological philosophies of transcendence. (shrink)

In a forthcoming paper, Walter Carnielli and Abilio Rodrigues propose a Basic Logic of Evidence whose natural deduction rules are thought of as preserving evidence instead of truth. BLE turns out to be equivalent to Nelson’s paraconsistent logic N4, resulting from adding strong negation to Intuitionistic logic without Intuitionistic negation. The Carnielli/Rodrigues understanding of evidence is informal. Here we provide a formal alternative, using justification logic. First we introduce a modal logic, KX4, in which \ can be read as asserting (...) there is implicit evidence for X, where we understand evidence to permit contradictions. We show BLE embeds into KX4 in the same way that Intuitionistic logic embeds into S4. Then we formulate a new justification logic, JX4, in which the implicit evidence motivating KX4 is made explicit. KX4 embeds into JX4 via a realization theorem. Thus BLE has both implicit and explicit possibly contradictory evidence interpretations in a formal sense. (shrink)

Table of Contents Volume I Preface to Volumes I and II: A Guide to the Primer Chapter 1, Basic Ideas and Tools Chapter 2, Transcription between English and Sentence Logic Chapter 3, Logical Equivalence, Logical Truths, and Contradictions Chapter 4, Validity and Conditionals Chapter 5, Natural Deduction for Sentence Logic: Fundamentals Chapter 6, Natural Deduction for Sentence Logic: Strategies Chapter 7, Natural Deduction for Sentence Logic: Derived Rules and Derivations without Premises Chapter 8, Truth Tree for Sentence Logic: (...) Fundamentals Chapter 9, Truth Trees for Sentence Logic: Applications Index for Volume 1 Solutions Manual for Volume 1 Table of Contents to Volume II Introduction to Predicate Logic Notes Chapter 1, Predicate Logic: Syntax Chapter 2, Predicate Logic: Semantics and Validity Chapter 3, More about Quantifiers Chapter 4, Transcription Chapter 5, Natural Deduction for Predicate Logic: Fundamentals Chapter 6, More on Natural Deduction for Predicate Logic Chapter 7, Truth Tress for Predicate Logic: Fundamentals Chapter 8, More on Truth Tress for Predicate Logic Chapter 9, Identity, Functions, and Definite Descriptions Chapter 10, Metatheory: The Basic Concepts Chapter 11, Mathematical Induction Chapter 12, Soundness and Completeness for Sentence Logic Trees Chapter 13, Soundness and Completeness for Sentence Logic Derivations Chapter 14, Koenig’s Lemma, Compactness, and Generalization to Infinite Sets of Premises Chapter 15, Interpretations, Soundness, and Completeness for Predicate Logic Index for Volume II Solutions Manual for Volume II Diagrammatic Summary of Rules Corrections to the Text. (shrink)

The author argues against Christine Korsgaard's influential interpretation of Kant's contradiction in conception test of the categorical imperative. Korsgaard's rejection of the ‘teleological' interpretation is shown to be based on a misunderstanding of the role that teleology plays for Kant in ruling out immoral maxims, and her defence of the ‘practical' interpretation is shown to be less faithful to the text than the competing ‘logical' interpretation. The works of Barbara Herman and Allen Wood are (...) also discussed and evaluated. (shrink)

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, (...) such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)

Logics L F (M) are considered, in which M ("most") is a new first-order quantifier whose interpretation depends on a given filter F of subsets of ω. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that F is not of the form $\{(\bigcap F) \cup X:|\omega - X| with $|\omega - \bigcap F| = \omega$ . Moreover the set of validities of L F (M) and even of L F ω 1 ω (M) (...) depends only on a few basic properties of F. Similar characterizations are given of the class of filters F for which L F (M) has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models (U, q), where the collection q is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindstrom's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter F). However, Lindstrom's argument is applied to characterize the invariant sentences as just those of first-order logic. (shrink)

Logics $L^F(M)$ are considered, in which $M$ ("most") is a new first-order quantifier whose interpretation depends on a given filter $F$ of subsets of $\omega$. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that $F$ is not of the form $\{(\bigcap F) \cup X: |\omega - X| < \omega\}$ with $|\omega - \bigcap F| = \omega$. Moreover the set of validities of $L^F(M)$ and even of $L^F_{\omega_1\omega}(M)$ depends only on a few basic properties (...) of F. Similar characterizations are given of the class of filters $F$ for which $L^F(M)$ has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models $(\mathfrak{U}, q)$, where the collection $q$ is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindstrom's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter $F$). However, Lindstrom's argument is applied to characterize the invariant sentences as just those of first-order logic. (shrink)

In this paper I present four interpretations of so-called negative theology and provide a number of attempts to model this theory within a formal system. Unfortunately, all of them fail in some manner. Most of them are simply inconsistent, some contradict the usual religious praxis and discourse, and some do not correspond to the key theses of negative theology. I believe that this paper shows how challenging this theory is from a logical perspective.