The state of affairs of some things falling under a predicate is supposedly a single entity that collects these things as its constituents. But whether we think of a state of affairs as a fact, a proposition or a possibility, problems will arise if we adopt a plural logic. For plural logic says that any plurality include themselves, so whenever there are some things, the state of affairs of their plural self-inclusion should be a single thing that collects them all. (...) This leads to paradoxes analogous to those that afflict naïve set theory. Here I suggest that they are the very same paradoxes, because sets can be reduced to states of affairs. However, to obtain a consistent theoretical reduction we must restrict the usual axiom scheme of Comprehension for plural logic to ‘stratified’ formulas, to avoid viciously circular definitions. I prove that with this modification to the background plural logic, the theory of states of affairs is consistent; moreover, it yields the axioms of the familiar set theory NFU. (shrink)

In “Function and Concept” and “On Concept and Object”, Frege argued that certain differences between dependent and independent meanings were inviolable and “founded deep in the nature of things” but, in those articles, he was not explicit about the actual consequences of violating such differences. However, since by creating a law that permitted one to pass from a concept to its extension, he himself mixed dependent and independent meanings, we are in a position to study some of the actual consequences (...) of his having done so. To make certain of Frege’s ideas about the inviolability of logical form more tangible, I describe a string of very interrelated consequences that his attempt to transform dependent meanings into independent meanings actually brought in its wake. (shrink)

Pour projeter de la lumière dans de nombreux coins et recoins obscurs de la logique pure de Husserl et dans les rapports entre sa logique formelle et sa logique transcendantale, et combler des lacunes empêchant qu’on arrive à une appréciation juste de sa Mannigfaltigkeitslehre, ou théorie de multiplicités, on examine comment, en prônant une théorie des systèmes déductifs, ou systèmes d’axiomes, comme tâche suprême de la logique pure, Husserl cherchait à résoudre certains problèmes épineux auxquels il s’était heurté en écrivant (...) Philosophie de l’arithmétique. Ces problèmes sont décrits. Ensuite, on rassemble les éléments nécessaires pour caractériser ce que Husserl, à travers les textes présentement disponibles, voulait dire des Mannigfaltigkeiten. Pour conclure, il est indiqué comment Husserl pouvait considérer que sa théorie représentait une solution aux problèmes qui avaient conduit à son élaboration.To shed light on the numerous dark areas surrounding Husserl’s ideas about pure logic and the relationship between his formal logic and his transcendental logic, and so provide an accurate characterisation of his Mannigfaltigkeitslehre, a close look is taken at how, by advocating a theory of deductive systems, or axiomatic systems as the highest task of pure logic, Husserl sought to resolve certain thorny problems he encountered when writing the Philosophy of Arithmetic. Those problems are described. Then, his ideas about axiomatization and manifolds are drawn together from the various sources now available to provide a more complete picture of his Mannigfaltigkeitslehre. In conclusion, it is shown how Husserl considered his theory to be a solution to the problems leading to its development. (shrink)

On the received view, the Representational Theory of Measurement reduces measurement to the numerical representation of empirical relations. This account of measurement has been widely criticized. In this article, I provide a new interpretation of the Representational Theory of Measurement that sidesteps these debates. I propose to view the Representational Theory of Measurement as a library of theorems that investigate the numerical representability of qualitative relations. Such theorems are useful tools for concept formation that, in turn, is one crucial aspect (...) of measurement for a broad range of cases in linguistics, rational choice, metaphysics, and the social sciences. (shrink)

Why should one think Frege's definition of the ancestral correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that seems to undermine Frege's claim to have justified induction in purely logical terms. I discuss such circularity objections and then offer a new definition of the ancestral intended to be intensionally correct; its extensional correctness then follows without proof. This new definition can be proven equivalent to Frege's without any use of arithmetical induction. This (...) proves, without any use of arithmetical induction, that Frege's definition is extensionally correct and so answers the circularity objections. (shrink)

There has been considerable discussion in the literature of one kind of identity problem that mathematical structuralism faces: the automorphism problem, in which the structure is unable to individuate the mathematical entities in its domain. Shapiro (Philos Math 16(3):285–309, 2008) has partly responded to these concerns. But I argue here that the theory faces an even more serious kind of identity problem, which the theory can’t overcome staying within its remit. I give two examples to make the point.

This paper is about the meaning and function of identity statements involving proper names. There are two prominent views on this topic, according to which identity statements ascribe a relation: the object-view, on which identity statements ascribe a relation borne by all objects to themselves, and the name-view, on which an identity statement 'a is b' says that the names 'a' and 'b' codesignate. The object- and name-views may seem to exhaust the field. I make a case for treating identity (...) statements as sui generis instead of attempting to explain them by means of the idea that they ascribe a relation. My contention is that once we do this, no analysis is required. -/- I do not wish to insist that we stop saying that identity statements ascribe a relation. The point is that there is a fundamental disanalogy between identity statements and other two-termed statements which we overlook to our peril. This will be seen to parallel the more recognized disanalogy between existence statements and other one-termed statements. One way of registering the fundamental disanalogy is to say that identity statements are not relational, but this is not essential. Following my negative arguments in section 2, I employ some simple diagrammatical models in section 3 to exhibit the fundamental disanalogy. In a final section I respond to some possible objections which may be raised against this kind of approach. (shrink)

If Sophroniscus is the father of Socrates, then Socrates is the son of Sophroniscus. If Socrates is similar to Plato, then Plato is similar to Socrates. But how many relations does Sophroniscus and Socrates being so related involve? How many does Plato and Socrates being thus related? Is there a difference between the two cases? These are questions that have featured prominently in discussions of relations in recent years, but they are by no means new. Focusing on a text by (...) the later Archbishop of Canterbury, Robert Kilwardby, this paper explores some of the replies and main arguments advanced by a number of philosophers working in the Latin west in the mid-to-late thirteenth century. (shrink)

Can institutional objects be identified with physical objects that have been ascribed status functions, as advocated by John Searle in The Construction of Social Reality (1995)? The paper argues that the prospects of this identification hinge on how objects persist – i.e., whether they endure, perdure or exdure through time. This important connection between reductive identification and mode of persistence has been largely ignored in the literature on social ontology thus far.

In 1913 Wittgenstein raised an objection to Russell’s multiple relation theory of judgment that eventually led Russell to abandon his theory. As he put it in the Tractatus, the objection was that “the correct explanation of the form of the proposition, ‘A makes the judgement p’, must show that it is impossible for a judgement to be a piece of nonsense. (Russell’s theory does not satisfy this requirement,” (5.5422). This objection has been widely interpreted to concern type restrictions on the (...) constituents of judgment. I argue that this interpretation is mistaken and that Wittgenstein’s objection is in fact a form of the problem of the unity of the proposition. (shrink)

How are philosophical questions about what kinds of things there are to be understood and how are they to be answered? This paper defends broadly Fregean answers to these questions. Ontological categories—such as object , property , and relation —are explained in terms of a prior logical categorization of expressions, as singular terms, predicates of varying degree and level, etc. Questions about what kinds of object, property, etc., there are are, on this approach, reduce to questions about truth and logical (...) form: for example, the question whether there are numbers is the question whether there are true atomic statements in which expressions function as singular terms which, if they have reference at all, stand for numbers, and the question whether there are properties of a given type is a question about whether there are meaningful predicates of an appropriate degree and level. This approach is defended against the objection that it must be wrong because makes what there depend on us or our language. Some problems confronting the Fregean approach—including Frege’s notorious paradox of the concept horse—are addressed. It is argued that the approach results in a modest and sober deflationary understanding of ontological commitments. (shrink)

The paper describes the evolution of russell's theory of judgment between 1910 and 1913, With especial reference to his recently published "theory of knowledge" (1913). Russell abandoned the book and with it the theory of judgment as a result of wittgenstein's criticisms. These criticisms are examined in detail and found to constitute a refutation of russell's theory. Underlying differences between wittgenstein's and russell's views on logic are broached more sketchily.

On investigating a theorem that Russell used in discussing paradoxes of classes, Graham Priest distills a schema and then extends it to form an Inclosure Schema, which he argues is the common structure underlying both class-theoretical paradoxes (such as that of Russell, Cantor, Burali-Forti) and the paradoxes of ?definability? (offered by Richard, König-Dixon and Berry). This article shows that Russell's theorem is not Priest's schema and questions the application of Priest's Inclosure Schema to the paradoxes of ?definability?.1 1?Special thanks to (...) Francesco Orilia for criticisms of an early draft of this article. (shrink)

This paper examines Russell's substitutional theory of classes and relations, and its influence on the development of the theory of logical types between the years 1906 and the publication of Principia Mathematica (volume I) in 1910. The substitutional theory proves to have been much more influential on Russell's writings than has been hitherto thought. After a brief introduction, the paper traces Russell's published works on type-theory up to Principia. Each is interpreted as presenting a version or modification of the substitutional (...) theory. New motivations for Russell's 1908 axiom of infinity and axiom of reducibility are revealed. (shrink)

Frege seems to hold two incompatible theses:(i) that sentences differing in structure can yet express the same sense; and (ii) that the senses of the meaningful parts of a complex term are determinate parts of the sense of the term. Dummett offered a solution, distinguishing analysis from decomposition. The present paper offers an embellishment of Dummett?s distinction by providing a way of depicting the internal structures of complex senses?determinate structures that yield distinct decompositions. Decomposition is then shown to be adequate (...) as a foundation for the informativity and analyticity of logic. (shrink)

This paper offers an interpretation of Russell's multiple-relation theory of judgment which characterizes it as direct application of the 1905 theory of definite descriptions. The paper maintains that it was by regarding propositional symbols (when occurring as subordinate clauses) as disguised descriptions of complexes, that Russell generated the philosophical explanation of the hierarchy of orders and the ramified theory of types of _Principia mathematica (1910). The interpretation provides a new understanding of Russell's abandoned book _Theory of Knowledge (1913), the 'direction (...) problems' and Wittgenstein's criticisms. (shrink)

SummaryIn June 1913 the 18-year-old Norbert Wiener presented to Harvard University a doctoral thesis comparing the logical systems of Schröder and Russell, with special reference to their treatment of relations. Shortly afterwards he visited Russell in Cambridge (England) and showed him a copy of the thesis. Russell wrote out some comments, to which Wiener replied.None of these documents has been published. In this paper I summarise the contents of Wiener's thesis, and describe and quote from the subsequent discussion with Russell. (...) I preface the account with some remarks on the personal relationship of Wiener and Russell, and conclude it with the description of the location of the various documents cited. (shrink)

It is widely assumed that Russell's problems with the unity of the proposition were recurring and insoluble within the framework of the logical theory of his Principles of Mathematics. By contrast, Frege's functional analysis of thoughts (grounded in a type-theoretic distinction between concepts and objects) is commonly assumed to provide a solution to the problem or, at least, a means of avoiding the difficulty altogether. The Fregean solution is unavailable to Russell because of his commitment to the thesis that there (...) is only one ultimate ontological category. This, combined with Russell's reification of propositions, ensures that he must hold concepts and objects to be of the same logical and ontological type. In this paper I argue that, while Frege's treatment of the unity of the proposition has immediate advantages over Russell's, a deeper consideration of the philosophical underpinnings and metaphysical consequences of the two approaches reveals that Frege's supposed solution is, in fact, far from satisfactory. Russell's repudiation of the Fregean position in the Principles is, I contend, convincing and Russell's own position, despite its problems, conforms to a greater extent than Frege's with common sense and, furthermore, with certain ideas which are central to our understanding of the origins of the analytical tradition. (shrink)

This paper describes the materials in the Russell Archives relevant to Russell's work on logic and the foundations of mathematics, and suggests the kinds of information that may and may not be drawn about the historical development of his ideas. By way of illustration, a couple of episodes are described. The first concerns a logical system closely related to his theory of denoting, which preceeds the system used in Principia mathematics, while the second describes a delay in publishing the second (...) volume of that work due to the discovery by Whitehead of a conceptual error. (shrink)

This paper discusses some of the ways in which circular definitions and circular explanations entail or fail to entail infinite regresses. And since not all infinite regresses are vicious, a few criteria of viciousness are examined in order to determine when the entailment of a regress refutes a circular definition or a circular explanation.

We seek means of distinguishing logical knowledge from other kinds of knowledge, especially mathematics. The attempt is restricted to classical two-valued logic and assumes that the basic notion in logic is the proposition. First, we explain the distinction between the parts and the moments of a whole, and theories of ?sortal terms?, two theories that will feature prominently. Second, we propose that logic comprises four ?momental sectors?: the propositional and the functional calculi, the calculus of asserted propositions, and rules for (...) (in)valid deduction, inference or substitution. Third, we elaborate on two neglected features of logic: the various modes of negating some part(s) of a proposition R, not only its ?external? negation not-R; and the assertion of R in the pair of propositions ?it is (un)true that R? belonging to the neglected logic of asserted propositions, which is usually left unstated. We also address the overlooked task of testing the asserted truth-value of R. Fourth, we locate logic among other foundational studies: set theory and other theories of collections, metamathematics, axiomatisation, definitions, model theory, and abstract and operator algebras. Fifth, we test this characterisation in two important contexts: the formulation of some logical paradoxes, especially the propositional ones; and indirect proof-methods, especially that by contradiction. The outcomes differ for asserted propositions from those for unasserted ones. Finally, we reflect upon self-referring self-reference, and on the relationships between logical and mathematical knowledge. A subject index is appended. (shrink)

(2000). CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM: THE PHILOSOPHY OF THE EXACT SCIENCES IN THE BACKGROUND TO EARLY LOGICAL EMPIRICISM. British Journal for the History of Philosophy: Vol. 8, No. 1, pp. 71-106.

This paper addresses a number of closely related questions concerning Kant's model of intentionality, and his conceptions of unity and of magnitude [Gröβe]. These questions are important because they shed light on three issues which are central to the Critical system, and which connect directly to the recent analytic literature on perception: the issues are conceptualism, the status of the imagination, and perceptual atomism. In Section 1, I provide a sketch of the exegetical and philosophical problems raised by Kant's views (...) on these issues. I then develop, in Section 2, a detailed analysis of Kant's theory of perception as elaborated in both the Critique of Pure Reason and the Critique of Judgment; I show how this analysis provides a preliminary framework for resolving the difficulties raised in Section 1. In Section 3, I extend my analysis of Kant's position by considering a specific test case: the Axioms of Intuition. I contend that one way to make sense of Kant's argument is by juxtaposing it with Russell's response to Bradley's regress; I focus in particular on the concept of ‘unity’. Finally, I offer, in Section 4, a philosophical assessment of the position attributed to Kant in Sections 2 and 3. I argue that, while Kant's account has significant strengths, a number of key areas remain underdeveloped; I suggest that the phenomenological tradition may be read as attempting to fill precisely those gaps. (shrink)

One of the most striking features of twentieth-century philosophy has been its obsession with language. For the most part, this phenomenon is greeted with hostile incredulity by external observers. Surely, they say, if philosophy is the profound and fundamental discipline which it has purported to be for more than two millennia, it must deal with something more serious than mere words, namely the things they stand for, and ultimately the essence of reality or of the human mind.

English mass noun phrases & count noun phrases differ only minimally grammatically. The basis for the difference is ascribed to a difference in the features +/-CT. These features serve the morphosyntactic function of determining the available options for the assigment of grammatical number, itself determined by the features +/-PL: +CT places no restriction on the available options, while -CT, in the unmarked case, restricts the available options to -PL. They also serve the semantic function of determining the sort of denotation (...) associated with demonstrative & quantified noun phrases. The feature -CT requires that the associated denotation be the set whose sole member is the greatest aggregate of which the noun phrase or noun is true; the feature +CT requires that the associated denotation be the set whose members are all & only those minimal aggregates of which the noun phrase or noun is true. At the same time, neither mass NPs nor count NPs that are arguments of a predicate have their predicate evaluated with respect to their denotations. Rather, the predicate is evaluated with respect to an aggregation, a set of aggregates constructed from the denotation of the noun phrase that is an argument of the predicate. 3 Tables, 4 Figures, 74 References. AA. (shrink)

La théorie russellienne des relations est ordinairement conçue comme le résultat d'une réflexion logique et ontologique sur l'ordre et l'asymétrie. Le présent article vise à présenter une autre généalogie, centrée sur les concepts de grandeur et de vecteur. Nous montrons en premier lieu que la thèse de l'irréductibilité des relations est avancée pour la première fois en 1897, à l'occasion d'une reformulation de la dialectique hégélienne de la quantité. Nous soulignons, en second lieu, que la notion de grandeur fait, autour (...) des années 1898-1899, l'objet d'une formalisation mathématique, complètement indépendante de la théorie des relations et fondée exclusivement sur les algèbres vectorielles. Nous montrons enfin que Russell, en possession de sa nouvelle doctrine logique, développe dès 1901 une théorie relationnelle des grandeurs vectorielles. Tout se passe donc comme si un des intérêts de la nouvelle théorie était, pour le philosophe, de permettre la substitution des relations aux grandeurs. (shrink)

In Appendix B of Russell's The Principles of Mathematics occurs a paradox, the paradox of propositions, which a simple theory of types is unable to resolve. This fact is frequently taken to be one of the principal reasons for calling ramification onto the Russellian stage. The paper presents a detaiFled exposition of the paradox and its discussion in the correspondence between Frege and Russell. It is argued that Russell finally adopted a very simple solution to the paradox. This solution had (...) nothing to do with ramified types but marked an important shift in his theory of propositions. (shrink)

According to the received view, Russell rediscovered about 1900 the logical definition of cardinal number given by Frege in 1884. In the same way, we are told, he stated and developed independently the idea of logicism, using the principle of abstraction as the philosophical ground. Furthermore, the role commonly ascribed in this to Peano was only to invent an appropriate notation to be used as mere instrument. In this paper I hold that the study of Russell's unpublished manuscripts and Peano's (...) (and disciples) writings (as part of a larger investigation only pointed out here) shows, on the contrary, that Russell obtained the method to transform definitions by abstraction into nominal definitions and the general logicist idea from Peano (and his school). The only original insight from Russell, the principle of abstraction, partially derived from Moore's early philosophy, was finally abandoned because it was not of practical use. (shrink)

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new (...) analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom). (shrink)

A report of a person's desire can be true even if its embedded clause underspecifies the content of the desire that makes the report true. It is true that Fiona wants to catch a fish even if she has no desire that is satisfied if she catches a poisoned minnow. Her desire is satisfied only if she catches an edible, meal-sized fish. The content of her desire is more specific than the propositional content of the embedded clause in our true (...) report of her desires. Standard semantic accounts of belief reports require, however, that the embedded clause of a true belief report specify precisely the content of the belief that makes it true. Such accounts of belief reports therefore face what I call "the problem of underspecification" if they are extended to desire reports. Such standard accounts are sometimes refined by requiring that a belief report can be true not only if its subject has a belief with precisely the propositional content specified by its embedded clause, but also only if its subject grasps that content in a particular way. Such refinements do not, however, help to address the problem of underspecification for desire reports. (shrink)

There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms that parses an adjunction into two separate parts. Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory and (...) is focused on the interpretation and application of the mathematical concepts. (shrink)

Soames (Philos Top 15:44–87, 1987 , J Philos Logic 37:267–276, 2008 ) has argued that propositions cannot be sets of truth-supporting circumstances. This argument is criticized for assuming that various singular terms are directly referential when in fact there are good grounds to doubt this.

One of the many ways that ‘deflationary’ and ‘inflationary’ theories of truth are said to differ is in their attitude towards truth qua property. This difference used to be very easy to delineate, with deflationists denying, and inflationists asserting, that truth is a property, but more recently the debate has become a lot more complicated, owing primarily to the fact that many contemporary deflationists often do allow for truth to be considered a property. Anxious to avoid inflation, however, these deflationists (...) are at pains to point out that the truth property, on their view, is not a property of any significant interest. Correspondingly, inflationists have seen this as an opportunity to refine what kind of property they think truth is, which—according to them—moves their views beyond deflationism. The upshot of this is that there are number of different accounts in the literature of what distinguishes an inflationary truth property from a deflationary one, or—as it is sometimes put—what distinguishes a ‘substantive’ property from an ‘insubstantive’ one. This has made it hard to pin down exactly what is at issue at the metaphysical level between deflationists and inflationists, which makes it increasingly hard to see how debates between them are properly phrased. Given that these positions represent the two central attitudes towards truth in contemporary debates, this makes for a serious obstacle for the project of discerning the correct theory of truth. The aim of this paper is to discern the best way to distinguish between substantive and insubstantive properties, and thus to restore some focus to these debates. I argue that the three central distinctions in the literature fail, and offer what I take to be a more promising distinction in terms of a graded distinction between abundant and sparse properties. (shrink)

Frege held that logical objects are objective but not wirklich, and that psychologism follows from the mistake of believing whatever is not wirklich to be subjective. It has been suggested that Frege's use of the terms ?objective? and ?wirklich? is in line with that found in Lotze's Logic; from this it has been inferred that Frege's doctrines have been misinterpreted as being ontological in character, but that they really belong to epistemology. In fact, Lotze held that something may be the (...) same for all thinkers, and yet may exist only in thought, not independently of it. For Frege, by contrast, there is nothing intermediate between the content of a single consciousness and what exists independently of being thought at all. This crucial disagreement underlies the divergence between Frege's realism and Lotze's idealism. (shrink)

Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicist's conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no principles of (...) inference specific to a given local topic. Poincaré, a Kantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical reasoning (proof). In this essay, I try to isolate and clarify this idea and to describe the mathematical epistemology which underlies it. Central to this epistemology (which is basically Kantian in orientation, and closely similar to that advocated by Brouwer) is a principle of epistemic conservation which says that knowledge of a given type cannot be extended by means of an inference unless that inference itself constitutes knowledge belonging to the given type. (shrink)

A feature of Frege's philosophy of arithmetic that has elicited a great deal of attention in the recent secondary literature is his contention that numbers are ‘self‐subsistent’ objects. The considerable interest in this thesis among the contemporary philosophy of mathematics community stands in marked contrast to Kreisel's folk‐lore observation that the central problem in the philosophy of mathematics is not the existence of mathematical objects, but the objectivity of mathematics. Although Frege was undoubtedly concerned with both questions, a goal of (...) the present paper is to argue that his success in securing the objectivity of arithmetic depends on a less contentious commitment to numbers as objects than either he or his critics have supposed. As such, this paper is an articulation and defense of both Frege's analysis of arithmetic and Kreisel's observation. (shrink)

What is logical relevance? Anderson and Belnap say that the “modern classical tradition [,] stemming from Frege and Whitehead-Russell, gave no consideration whatsoever to the classical notion of relevance.” But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson–Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and (...) Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein’s Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains. (shrink)

We discuss the concept of a cardinal number and its history, focussing on Cantor's work and its reception. J'ay fait icy peu pres comme Euclide, qui ne pouvant pas bien >faire< entendre absolument ce que c'est que raison prise dans le sens des Geometres, definit bien ce que c'est que memes raisons. (Leibniz) 1.

The article concerns the treatment of the so-called denoting phrases, of the forms ?every A?, ?any A?, ?an A? and ?some A?, in Russell's Principles of Mathematics. An initially attractive interpretation of what Russell's theory was has been proposed by P.T. Geach, in his Reference and Generality (1962). A different interpretation has been proposed by P. Dau (Notre Dame Journal, 1986). The article argues that neither of these is correct, because both credit Russell with a more thought-out theory than he (...) actually had. The conclusion is mainly negative: at this date Russell has no coherent theory of these phrases. An appendix notes that his understanding of the quantifiers in predicate logic is also, at this date, not entirely secure. (shrink)

Causal slingshots are formal arguments advanced by proponents of an event ontology of token-level causation which, in the end, are intended to show two things: (i) The logical form of statements expressing causal dependencies on token level features a binary predicate ‘‘... causes ...’’ and (ii) that predicate takes events as arguments. Even though formalisms are only revealing with respect to the logical form of natural language statements, if the latter are shown to be adequately captured within a corresponding formalism, (...) proponents of slingshots usually take the adequacy of their formalizations for granted without justifying it. The first part of this paper argues that the most discussed version of a causal slingshot, viz. the one e.g. presented by Davidson (Essays on actions and events. Oxford, Clarendon Press, 1980), can indeed be refuted for relying on an inadequate formal apparatus. In contrast, the formal means of Gödel’s (The philosophy of Betrand Russell. New York, Tudor, 1944) often neglected slingshot are shown to stand on solid ground in the second part of the paper. Nonetheless, I contend that Gödel’s slingshot does only half the work friends of event causation would like it to do. It provides good reasons for (i) but not for (ii). (shrink)