This paper aims to argue that Jean-Luc Marion’s philosophical theology is an axio-meonto-Theo-logy which proposes a new way of approaching God. The traditional way of approaching God in theo-logy attained God by the predication and the predicate in the categories of being. However, Marion’s theology attempts to bring out the freedom of God from all categories of being. It provides a critique of the traditional way of approaching God and two arguments for Marion’s alternative approach. On the grounds of the (...) axiological argument and the meontological argument, I defend Marion’s theology from some recent criticisms. (shrink)

The characterization of thought as a subject reflection about a given object is expressed by an enunciation of predicative order. The introduction of the possibility of a type of thought that it is not constituted in virtue of this presupposition brings a lot of difficulties, which is responsible for why Lloyd treats this theme as an enigma of Greek philosophy, i.e, non-discursive thinking. Plotinus seems to make a distinction between rational and intellectual thought, taking as a starting point the (...) _sui generis_ definition of a thinking that thinks itself. The intriguing nature of this description, i.e, the self-thinking as non-discursive thought, has raised studies with the primary aim to understand how it works this type of thinking. Then to clarify what does it mean to adopt the non-discursive approach. This paper intends to analyze the cited problem in the light of the “non-predicative” of Santos. The hypothesis is that by applying this concept to the discussion it will contribute to increase clarification on the distinction between reason and Intellect. That exercise will be conducted in majority through the reading of V.3 [49] of Plotinus _Enneads_. The nature of the intellectual thought would be non-predicative and the rational would be predicative. (shrink)

Predicate modal formulas with non-modalized quantifiers (call them Q-formulas) are considered as schemata of arithmetical formulas, where is interpreted as the provability predicate of some fixed correct extension T of arithmetic. A method of constructing 1) non-provable in T and 2) false arithmetical examples for Q-formulas by Kripke-like countermodels of certain type is given. Assuming the means of T to be strong enough to solve the (undecidable) problem of derivability in QGL, the Q-fragment of the predicate version of the logic (...) GL, we prove the recursive enumerability of the sets of Q-formulas all arithmetical examples of which are: 1) T-provable, 2) true. In. particular, the first one is shown to be exactly QGL and the second one to be exactly the Q-fragment of the predicate version of Solovay's logic S. (shrink)

. Three logical squares of predication or quantification, which one can even extend to logical hexagons, will be presented and analyzed. All three squares are based on ideas of the non-traditional theory of predication developed by Sinowjew and Wessel. The authors also designed a non-traditional theory of quantification. It will be shown that this theory is superfluous, since it is based on an obscure difference between two kinds of quantification and one pays a high price for differentiating in this way: (...) losing the definability between the existence- and all-quantifier. Therefore, a combination of non-traditional predication and classical quantification is preferred here. (shrink)

Where can I find a man who has forgotten words, so I can have a word with him?Why another article on logos and dao 道? Is it not the case that enough scholars have looked into the similarities between the term logos and the notion of dao? Although it may seem so, I will argue that when another perspective is employed, logos and dao might fruitfully be compared on a different level from the one used by most (...) of these comparisons. In this essay I will argue first that in many instances the approach of some of the scholars who have compared logos and dao has been one-sided and has mostly consisted in a comparison of these two key notions that has sought to portray both notions as denoting some kind of metaphysical... (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)

Two connected themes have been at the core of the old perplexity regarding thinking and speaking about non-existent objects. One involves a question of reference. Can we refer to non-existent objects without, thereby, recognizing, in some sense, non-existent entities as objects of reference? The other involves a question about existence. Is existence a property representable by a predicate in a logically adequate symbohsm? It is argued (1) that existence is not to be construed as an attribute represented by a predicate, (...) (2) that nonnaming names introduce problems, not solutions to problems, (3) that purported properties such as self-identical are specious, and (4) that the Russell property is also seen to be specious by our consideration of predication. (shrink)

Two connected themes have been at the core of the old perplexity regarding thinking and speaking about non-existent objects. One involves a question of reference. Can we refer to non-existent objects without, thereby, recognizing, in some sense, non-existent entities as objects of reference? The other involves a question about existence. Is existence a property representable by a predicate in a logically adequate symbohsm? It is argued (1) that existence is not to be construed as an attribute represented by a predicate, (...) (2) that nonnaming names introduce problems, not solutions to problems, (3) that purported properties such as self-identical are specious, and (4) that the Russell property is also seen to be specious by our consideration of predication. (shrink)

Two connected themes have been at the core of the old perplexity regarding thinking and speaking about non-existent objects. One involves a question of reference. Can we refer to non-existent objects without, thereby, recognizing, in some sense, non-existent entities as objects of reference? The other involves a question about existence. Is existence a property representable by a predicate in a logically adequate symbohsm? It is argued (1) that existence is not to be construed as an attribute represented by a predicate, (...) (2) that nonnaming names introduce problems, not solutions to problems, (3) that purported properties such as self-identical are specious, and (4) that the Russell property is also seen to be specious by our consideration of predication. (shrink)

At Sophist 260e3-261a2, the Eleatic Stranger claims that in order to demonstrate that falsehood is, he and Theaetetus must first track down what speech (logos), opinion (doxa), and appearance (phantasia) are, and then observe the communion (koinōnia) that speech, opinion, and appearance have with non-being. The Stranger, however, never explicitly discusses the communion of speech, opinion, and appearance with non-being. Yet presumably their communion is implicit in his account of falsehood, given his claim that observing that communion is needed (...) in order to demonstrate that falsehood is (260e5-a2). This essay seeks to make the communion that speech has with non-being explicit. I argue that speech has communion with non-being in that the things and actions speech combines together by means of nouns and verbs need not be combined in a way that reveals how the being a given speech is about combines ontologically with other beings. (shrink)

Plural definite descriptions across many languages display two well-known properties. First, they can give rise to so-called non-maximal readings, in the sense that they ‘allow for exceptions’. Second, while they tend to have a quasi-universal quantificational force in affirmative sentences, they tend to be interpreted existentially in the scope of negation. Building on previous works, we offer a theory in which sentences containing plural definite expressions trigger a family of possible interpretations, and where general principles of language use account for (...) their interpretation in various contexts and syntactic environments. Our theory solves a number of problems that these previous works encounter, and has broader empirical coverage in that it offers a precise analysis for sentences that display complex interactions between plural definites, quantifiers and bound variables, as well as for cases involving non-distributive predicates. The resulting proposal is briefly compared with an alternative proposal by Križ, which has similar coverage but is based on a very different architecture and sometimes makes subtly different predictions. (shrink)

In this essay, I deal with the problem of the attribution of psychological predicates to non-human animals. The first section illustrates three research topics where it has become scientifically legitimate to explain the conduct of non-human animals by means of the attribution of psychological predicates. The second section discusses several philosophical objections to the legitimacy of such attributions provided by central thinkers from the last decades. I try to show that these objections —which are related among other questions to the (...) holism of the mental, the indeterminacy of the attributions, and the strangeness of animal concepts— can be alleviated. In the third section, I propose to adopt a literalist view of the attributions in the sense articulated by Figdor. At the same time, I argue that one must draw limits to the conceptual change forwarded by her literalist view, taking into account holistic considerations and the fact that the psychological concepts must retain their core notes. (shrink)

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also (...) model other axioms of Church’s intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin’s intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions. (shrink)

Plural predication is a pervasive part of ordinary language. We can say that some people are fifty in number, are surrounding a building, come from many countries, and are classmates. These predicates can be true of some people without being true of any one of them; they are non-distributive predications. However, the apparatus of modern logic does not allow a place for them. Thomas McKay here explores the enrichment of logic with non-distributive plural predication and quantification. His book will be (...) of great interest to philosophers of language, linguists, metaphysicians, and logicians. (shrink)

This paper is devoted to the formulation and investigation of a dynamic semantic interpretation of the language of first-order predicate logic. The resulting system, which will be referred to as ‘dynamic predicate logic’, is intended as a first step towards a compositional, non-representational theory of discourse semantics. In the last decade, various theories of discourse semantics have emerged within the paradigm of model-theoretic semantics. A common feature of these theories is a tendency to do away with the principle of compositionality, (...) a principle which, implicitly or explicitly, has dominated semantics since the days of Frege. Therefore the question naturally arises whether non-compositionality is in any way a necessary feature of discourse semantics. Since we subscribe to the interpretation of compositionality as constituting primarily a methodological principle, we consider this to be a methodological rather than an empirical question. As a consequence, the emphasis in the present paper lies on developing an alternative compositional semantics of discourse, which is empirically equivalent to its non-compositional brethren, but which differs from them in a principled methodological way. Hence, no attempts are made to improve on existing theories empirically. Nevertheless, as we indicate in section 5, the development of a compositional alternative may in the end have empirical consequences, too. First of all, it can be argued that the dynamic view on interpretation developed in this paper suggests natural and relatively easy to formulate extensions which enable one to deal with a wider range of phenomena than can be dealt with in existing theories. Moreover, the various approaches to the model-theoretic semantics of discourse that have been developed during the last decade, have constituted a ‘fresh start’ in the sense that much of what had been accomplished before was ignored, at least for a start. Of course, this is a justified strategy if one feels one is trying to develop a radically different approach to recalcitrant problems. However, there comes a time when such new approaches have to be compared with the older one, and when an assessment of the pros and cons of each has to be made. One of the main problems in semantics today, we feel, is that a semantic theory such as Montague grammar, and an approach like Kamp’s discourse representation theory, are hard to compare, let alone that it is possible to unify their insights and results.. (shrink)

In Book 3 of 'On Abstinence from Animal Food', Porphyry is traditionally taken to be arguing in favour of the belief that animals are rational. However, elsewhere in his corpus, he endorses the opposite view, declaring that man differs from other mortal animals because he is rational and they are irrational. Jonathan Barnes offers a way of understanding Porphyry’s logical theory which is intended to make it consistent with the traditional interpretation of 'On Abstinence'. He suggests that the same predicate (...) can be related to different subjects in different ways. Porphyry’s claim that man differs from other mortal animals with respect to the predicate ‘rational’, is best understood to mean that whereas ‘rational’ is specifically predicated of man, it is non-specifically predicated of animals. This explains how they differ from man with respect to this predicate, without failing to be rational. This paper argues that there are reasons for rejecting Barnes’ interpretation. Barnes must maintain, for example, that animals possess the apparently incompatible properties of being specifically irrational and non-specifically rational simultaneously. Since Barnes fails to resolve the conflict, Porphyry’s claims that animals are irrational in his logical works pose a significant problem for the traditional interpretation of 'On Abstinence'. (shrink)

Antisthenes (c. 445- c. 365 BC), was a prominent follower of Socrates and bitter rival of Plato. In this revisionary account of his philosophy in all its aspects, P. A. Meijer claims that Plato and Aristotle have corrupted our perspective on this witty and ingenious thinker. The first part of the book reexamines afresh Antisthenes' ideas about definition and predication and concludes from these that Antisthenes never held the (in)famous theory that contradiction is impossible. The second part of the book (...) argues that Antisthenes' logical theories bear directly on his activities as an exegete of Homer and hence as a theological thinker. Part three, finally, offers innovative readings of Antisthenes' ethical fragments. (shrink)

The present work takes the decease of Horst Wessel as an opportunity to present and honour his work (and that of his group), which has not received the attention it deserves. The focus will be on works which might not be sufficiently well-known. Wessel was, as we aim to show, familiar with the international debate concerning logical and philosophical issues and strived to solve them by considering theories of logical consequence, a non-traditional theory of predication and the theory of logical (...) terms, all of which he developed in joint work with Alexander Sinowjew. Wessel had to significantly refine these theories in order to formulate his criticism towards alternative logics and to apply the theories to a treatment of intensional contexts. He was vehemently opposed to approaches which revised fundamental laws of classical logic. Questions concerning the history of Philosophy are addressed in Wessel’s criticism of the Kantian antinomies of pure reason, in Wessel’s contributions to the problem of universals and his analyses of fatalism, tychism and antifatalism. (shrink)

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying second-order logic—and (...) investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension. (shrink)

According to the well–known argument, the traditional conception of existence and predication leads to the infamous paradox of non-existence. For instance, the sentence ‘Pegasus does not exist’ commits us to accept that there is something that does not exist. The easy way out is to analyze existence as a second–order concept expressing that there is at least one instance of some first–order concept. In this article, I argue that the traditional conception of existence and predication does not lead to the (...) paradox of non-existence. Negative existentials are not subject-predicate sentences that select existing beings only to deny them the property of existence. Aristotle’s insistence that being is not a genus, i.e. that being is not a predicate of some subject, is pivotal. Furthermore, Aristotle does not have a uniform conception of existence and predication. According to Aristotle, to be is to be an item in the categorical scheme, and consequently, every predication of some being should be analysable as one of the most general types of predication according to the categorical scheme. However, in the conceptual framework of Aristotle’s philosophy, a different problem of non-existence arises. For instance, ‘Pegasus is thought-about’ might commits us to the claim that Pegasus is a being of some kind. I reconstruct Aristotle’s solution as being satisfactory in the light of his understanding of predication and existence. Moreover, I suggest that a similar puzzle leads many contemporary philosophers to an abandonment of the Quinean concept of existence and predication in favour of a more traditional approach. (shrink)

We give an extension of the Jónsson‐Tarski representation theorem for both normal and non‐normal modal algebras so that it preserves countably many infinite meets and joins. In order to extend the Jónsson‐Tarski representation to non‐normal modal algebras we consider neighborhood frames instead of Kripke frames just as Došen's duality theorem for modal algebras, and to deal with infinite meets and joins, we make use of Q‐filters, which were introduced by Rasiowa and Sikorski, instead of prime filters. By means of the (...) extended representation theorem, we show that every predicate modal logic, whether it is normal or non‐normal, has a model defined on a neighborhood frame with constant domains, and we give a completeness theorem for some predicate modal logics with respect to classes of neighborhood frames with constant domains. Similarly, we show a model existence theorem and a completeness theorem for infinitary modal logics which allow conjunctions of countably many formulas. (shrink)

Kant said that existence is not a predicate and Russell agreed, arguing that a sentence such as ‘The king of France exists’, which seems to attribute existence to the king of France, really has a logical form that is not reflected in the surface structure of the sentence at all. While the surface form of the sentence consists of a subject and a predicate, the underlying logical form, according to Russell, is the formula given in. This formula obviously has no (...) subjectpredicate form and in fact has no single constituent that corresponds to the verb phrase ‘exists’ in the surface sentence. ∃x∀y The importance of Russell’s analysis becomes clear when we consider ‘The king of France does not exist’. If this sentence would attribute non-existence to the king it should entail that there is someone who does not exist, just as ‘Mary doesn’t like bananas’ entails that there is someone who doesn’t like bananas. Thus the idea that all sentences have subject-predicate form has led some philosophers to the view that there are objects that lack existence. This embarrassing position can be avoided once Russell’s analysis is accepted: if ‘The king of France does not exist’ is formalised as the negation of formula, no unwanted consequences follow. (shrink)

In the framework of set theory we cannot distinguish between natural and non-natural predicates. To avoid this shortcoming one can use mathematical structures as conceptual spaces such that natural predicates are characterized as structurally nice subsets. In this paper topological and related structures are used for this purpose. We shall discuss several examples taken from conceptual spaces of quantum mechanics (orthoframes), and the geometric logic of refutative and affirmable assertions. In particular we deal with the problem of structurally distinguishing between (...) natural colour predicates and Goodmanian predicates like grue and bleen. Moreover the problem of characterizing natural predicates is reformulated in such a way that its connection with the classical problem of geometric conventionalism becomes manifest. This can be used to shed some new light on Goodman's remarks on the relative entrenchment of predicates as a criterion of projectibility. (shrink)

In so-called Kripke-type models, each sentence is assigned either to true or to false at each possible world. In this setting, every possible world has the two-valued Boolean algebra as the set of truth values. Instead, we take a collection of algebras each of which is attached to a world as the set of truth values at the world, and obtain an extended semantics based on the traditional Kripke-type semantics, which we call here the algebraic Kripke semantics. We introduce algebraic (...) Kripke sheaf semantics for super-intuitionistic and modal predicate logics, and discuss some basic properties. We can state the Gödel-McKinsey-Tarski translation theorem within this semantics. Further, we show new results on super-intuitionistic predicate logics. We prove that there exists a continuum of super-intuitionistic predicate logics each of which has both of the disjunction and existence properties and moreover the same propositional fragment as the intuitionistic logic. (shrink)

Despite the moral gravity and far-reaching consequences of ethical judgment, evidence shows that such judgment is surprisingly malleable, prone to bias, informed by intuition and implicit associations, and swayed by mere circumstance. In this vein, this research examines how mere colors featured in logos can bias consumers’ ethical judgments about a retailer. Exposure to a logo featuring an eco-friendly color makes an ethically ambiguous practice seem more ethical; however, exposure to a logo featuring a non-eco-friendly color makes the same (...) practice seem less ethical. This effect is due to the embodied meaning of color, not referential meanings associated with the names of colors, and it is mediated by perceptions of a retailer’s eco-friendliness. Furthermore, although the word “green” appears to influence ethical ratings of retail practices more than the word “blue,” visual exposure to either color evokes similar perceptions of eco-friendliness and influences ethical judgments. Study 2c assesses and rules out alternative explanations for this effect. Critically, an eco-friendly color can skew judgments even when the practices judged are not ethically ambiguous. Individual differences in ethical sensitivity moderate the observed effect, such that individuals who are less ethically sensitive are less influenced by color. The article concludes with a discussion on how logo colors shape consumers’ perceptions of retailer ethicality. (shrink)

The knowledge of truth, in Plato, is an experience that calls for conversion of the soul (μεταστροφή, περιστροφή). The basic feature of this experience consists in some sort of connection, which is constantly at work, between rational arguments and their non-rational conditions, briefly, lógos and páthos. How does this connection show up in Plato? Its crucial importance emerges many times at both narrative (récit) and theoretical level. In the three parts of my contribution, I show how logos-pathos intertwines with (...) Plato's notion of conversion considered as a double-sided phenomenon, as both "process" of moral education and "noetic event". (shrink)

The primary purpose of this note is to demonstrate that predicative Frege arithmetic naturally interprets certain weak but non-trivial arithmetical theories. It will take almost as long to explain what this means and why it matters as it will to prove the results.

Philosophers have had difficulty in explaining the difference between disjunctive and non-disjunctive predicates. Purely syntactical criteria are ineffective, and mention of resemblance begs the question. I draw the distinction by reference to relations between borderline cases. The crucial point about the disjoint predicate 'red or green', for example, is that no borderline case of 'red' is a borderline case of 'green'. Other varieties of disjunctive predicates are: inclusively disjunctive (such as 'red or hard'), disconnected (such as 'grue' on the usual (...) definitions), and skew (such as 'grue' on an emended definition). 'Green' is not a disjunctive predicate. Nelson Goodman's new riddle of induction elicits yet another response. (shrink)

Descartes’ physics is dependent on his metaphysics, which is to say, on knowledge of the nature of God and of the human soul. This is clear throughout Descartes’ work, but it is especially so in th...

A new formalism for predicate logic is introduced, with a non-standard method of binding variables, which allows a compositional formalization of certain anaphoric constructions, including donkey sentences and cross-sentential anaphora. A proof system in natural deduction format is provided, and the formalism is compared with other accounts of this type of anaphora, in particular Dynamic Predicate Logic.

We formulate Suppes predicates for various kinds of space-time: classical Euclidean, Minkowski's, and that of General Relativity. Starting with topological properties, these continua are mathematically constructed with the help of a basic algebra of events; this algebra constitutes a kind of mereology, in the sense of Lesniewski. There are several alternative, possible constructions, depending, for instance, on the use of the common field of reals or of a non-Archimedian field (with infinitesimals). Our approach was inspired by the work of Whitehead (...) (1919), though our philosophical stance is completely different from his. The structures obtained are idealized constructs underlying extant, physical space-time. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Logos and Alogon: Thinkable and Unthinkable in Mathematics, from the Pythagoreans to the Moderns by Arkady PlotnitskyNoam CohenPLOTNITSKY, Arkady. Logos and Alogon: Thinkable and Unthinkable in Mathematics, from the Pythagoreans to the Moderns. Cham: Springer, 2023. xvi + 294 pp. Cloth, $109.99The limits of thought in its relations to reality have defined Western philosophical inquiry from its very beginnings. The shocking discovery of the incommensurables in (...) Greek mathematics not only reintroduced these limits with greater force but also instigated two different forms of coping with the unthinkable in Western philosophy, mathematics, and science. As Arkady Plotnitsky sets forth, in response to the incommensurability crisis in the ancient Greek intellectual world, there appeared two different attitudes toward the alogon, that is, the unthinkable or irrational. Plato and his followers sought to overcome the unthinkable, of which the incommensurables in mathematics are only an example, through philosophy. In contrast, the Pythagoreans continued to hold on to mathematics as the primary means of knowing the world, but put a larger emphasis on geometry. Plotnitsky [End Page 359] claims that by not putting aside the primacy of mathematics, the Pythagorean shift from arithmetic to geometry established in Western thought a certain tolerance toward the unthinkable within the realm of the thinkable, “a possibility of the alogon in the logos.” Until modern times this remained no more than a possibility. However, in modern mathematics and mathematical physics Plotnitsky identifies a certain fulfillment of this potential, a “radical Pythagorean mathematics.” In his comprehensive and thought-provoking study, Plotnitsky recounts the development of such radical modern Pythagoreanism, not only to tell its history but also to suggest that the unthinkable is a condition of possibility for thought, and that a proper recognition of this role it plays is crucial for the advancement of mathematical thinking and thinking in general.Plotnitsky begins by portraying in broad strokes the two defining characteristics of radical Pythagorean mathematics: the presence of the unthinkable within mathematical thought itself, and the interplay of algebra and geometry. Relying on an extensive knowledge of the history of modern mathematics and physics, in the first two chapters of the book Plotnitsky gradually demonstrates the key role that the alogon plays in some theories of modern mathematical thought, such as Gödel’s incompleteness theorems and quantum mechanics. He also presents and clarifies his own conception of the history of Western thought as a creative process of concept-forming. By creating rather than discovering concepts, great mathematicians break new ground for thinking. The book as a whole argues in length for these two main theses.In chapter 3, the author addresses the thought of Bernhard Riemann to make the case that Riemann’s mathematical thought puts an emphasis on creative conceptual thinking and is not defined merely by calculations or logic. Such thinking is problem-posing rather than axiomatic. Riemann’s central innovation in this respect is the concept of manifold (Mannigfaltigkeit), which enabled him to think anew space and geometry in a radical non-Euclidean fashion, implying an infinite number of possible geometries. Plotnitsky argues—borrowing Deleuze and Guattari’s terminology—that this opened up a “plane of immanence,” which is to say that Riemann created a concept-generating movement of thought. However, we cannot grasp the generation of concepts in modern mathematics without giving proper attention to the interplay of algebra and geometry that defines it. Chapter 4 demonstrates this interweaving of the two fields by recounting the history of the idea of the curve in modernity, from the analytic geometry and calculus of Fermat and Descartes to its modification in the concept of a Riemann surface, up to its more recent developments in the algebraic geometry of Weil and Grothendieck. It gives a historical account of the algebraization of the curve that at the same time brings into relief the irreducible geometrical aspect, continuity, of all modern mathematical conceptualizations of the curve.The next two chapters provide further case studies. Chapter 5 expands and solidifies the argument of chapter 2. It discusses three examples of mathematical practice, each displayed by a pair of important [End Page 360] mathematicians: Abel and Galois, Lobachevsky and Riemann, Weil and Grothendieck. Each case... (shrink)

Presupposing that most predicates do not correspond directly to genuine relations, I argue that all genuine relations are symmetric. My main argument depends on the premise that there are no brute necessities, interpreted so as to require logical and metaphysical necessity to coincide for sentences composed entirely of logical vocabulary and primitive predicates. Given this premise, any set of purportedly primitive predicates by which one might hope to express the facts about non-symmetric relations order their relata will generate an objectionable (...) multiplication of possibilities. In the final section I give a different argument, based on the weaker premise that brute necessities should not be multiplied without necessity. (shrink)

A free logic is one in which a singular term can fail to refer to an existent object, for example, `Vulcan' or `5/0'. This essay demonstrates the fruitfulness of a version of this non-classical logic of terms (negative free logic) by showing (1) how it can be used not only to repair a looming inconsistency in Quine's theory of predication, the most influential semantical theory in contemporary philosophical logic, but also (2) how Beeson, Farmer and Feferman, among others, use it (...) to provide a natural foundation for partial functions in programming languages. Vis à vis (2), the question is raised whether the Beeson-Farmer-Feferman approach is adequate to the treatment of partial functions in all programming languages. Gumb and the author say No, and suggest a way of handling the refractory cases by means of positive free logic. Finally, Antonelli's solution of a problem associated with the Gumb-Lambert proposal is mentioned. (shrink)

Graham Priest takes the relation of identity to be non-transitive. In this paper, we are going to discuss several consequences of identity as a non-transitive relation. We will consider the Henkin-style completeness proof for classical first-order logic with a non-transitive “identity” predicate, Leibniz-identity in Priest’s second-order minimal logic of paradox, and the question whether or not identity of individuals should be defined as Leibniz-identity.