The simplest quantified modal logic combines classical quantification theory with the propositional modal logic K. The models of simple QML relativize predication to possible worlds and treat the quantifier as ranging over a single fixed domain of objects. But this simple QML has features that are objectionable to actualists. By contrast, Kripke-models, with their varying domains and restricted quantifiers, seem to eliminate these features. But in fact, Kripke-models also have features to which actualists object. Though these philosophers have introduced variations (...) on Kripke-models to eliminate their objectionable features, the most well-known variations all have difficulties of their own. The present authors reexamine simple QML and discover that, in addition to having a possibilist interpretation, it has an actualist interpretation as well. By introducing a new sort of existing abstract entity, the contingently nonconcrete, they show that the seeming drawbacks of the simplest QML are not drawbacks at all. Thus, simple QML is independent of certain metaphysical questions. (shrink)

In "Actualism or Possibilism?" (Philosophical Studies, 84 (2-3), December 1996), James Tomberlin develops two challenges for actualism. The challenges are to account for the truth of certain sentences without appealing to merely possible objects. After canvassing the main actualist attempts to account for these phenomena, he then criticizes the new conception of actualism that we described in our paper "In Defense of the Simplest Quantified Modal Logic" (Philosophical Perspectives 8: Philosophy of Logic and Language, Atascadero, CA: Ridgeview, 1994). We respond (...) to Tomberlin's criticism by showing that we wouldn't analyze the problematic claim (e.g., "Ponce de Leon searched for the fountain of youth") in the way he suggests. (shrink)

In this paper, we develop an alternative strategy, Platonized Naturalism, for reconciling naturalism and Platonism and to account for our knowledge of mathematical objects and properties. A systematic (Principled) Platonism based on a comprehension principle that asserts the existence of a plenitude of abstract objects is not just consistent with, but required (on transcendental grounds) for naturalism. Such a comprehension principle is synthetic, and it is known a priori. Its synthetic a priori character is grounded in the fact that it (...) is an essential part of the logic in which any scientific theory will be formulated and so underlies (our understanding of) the meaningfulness of any such theory (this is why it is required for naturalism). Moreover, the comprehension principle satisfies naturalist standards of reference, knowledge, and ontological parsimony! As part of our argument, we identify mathematical objects as abstract individuals in the domain governed by the comprehension principle, and we show that our knowledge of mathematical truths is linked to our knowledge of that principle. (shrink)

This study offers a novel integration of distinct aspects of Russell's thought.

In this paper, we investigate (1) what can be salvaged from the original project of "logicism" and (2) what is the best that can be done if we lower our sights a bit. Logicism is the view that "mathematics is reducible to logic alone", and there are a variety of reasons why it was a non-starter. We consider the various ways of weakening this claim so as to produce a "neologicism". Three ways are discussed: (1) expand the conception of logic (...) used in the reduction, (2) allow the addition of analytic-sounding principles to logic so that the reduction is not to "logic alone" but to logic and truths knowable a priori, and (3) revise the conception of "reducible". We show how the current versions of neologicism fit into this classification scheme, and then focus on a kind of neologicism which we take to have the most potential for achieving the epistemological goals of the original logicist project. We argue that that the "weaker" the form of neologicism, the more likely it is to be a new form of logicism, and show how our preferred system, though mathematically weak, is metaphysically and epistemogically strong, and can "reduce" arbitrary mathematical theories to logic and analytic truths, if given a legitimate new sense of "reduction". (shrink)

According to Scott Soames’ Beyond Rigidity, there are two important pieces of unfinished business left over from Saul Kripke’s influential Naming and Necessity. Soames reads Kripke’s arguments about names as primarily negative, that is, as proving that names don’t have a meaning expressible by definite descriptions or clusters of them. The famous Kripkean doctrine that names are rigid designators is really only part of the negative case. The thesis that names refer to the same object with respect to every possible (...) world is a byproduct of their meaning, not a positive account of what they mean. As well, the hints about causal chains and dubbings are no more than a picture, as Kripke says, and not a positive theory of meaning. Thus one piece of unfinished business, to which Soames devotes the most attention, is to give a positive account of the meanings of names. To do this Soames proposes that the meaning of a singular term is the contribution it makes to the semantic content of the sentences in which it occurs. The semantic content of a sentence is ordinarily a proposition, that proposition expressed by the most commonly intended assertion using the sentence. Soames’ proposal for a positive account is that the meaning of a proper name is its contribution to those propositions, simply the object to which it refers. Arguing for this positive account occupies the bulk of the book but I will not discuss it in my contribution to this symposium. (shrink)

The views of David Lewis and the Meinongians are both often met with an incredulous stare. This is not by accident. The stunned disbelief that usually accompanies the stare is a natural first reaction to a large ontology. Indeed, Lewis has been explicitly linked with Meinong, a charge that he has taken great pains to deny. However, the issue is not a simple one. "Meinongianism" is a complex set of distinctions and doctrines about existence and predication, in addition to the (...) famously large ontology. While there are clearly non-Meinongian features of Lewis' views, it is our thesis that many of the characteristic elements of Meinongian metaphysics appear in Lewis' theory. Moreover, though Lewis rejects incomplete and inconsistent Meinongian objects, his ontology may exceed that of a Meinongian who doesn't accept his possibilia. Thus, Lewis explains the truth of "there might have been talking donkeys" by appealing to possibilia which are talking donkeys. But the Meinongian need not accept that there exist things which are talking donkeys. Indeed, we show that a Meinongian even need not accept that there are nonexistent things which are talking donkeys! (shrink)

This paper investigates the strange case of an argument that was directed against a positivist verification principle. We find an early occurrence of the argument in a talk by the phenomenologist Roman Ingarden at the 1934 International Congress of Philosophy in Prague, where Carnap and Neurath were present and contributed short rejoinders. We discuss the underlying presuppositons of the argument, and we evaluate whether the attempts by Carnap actually succeed in answering this argument. We think they don’t, and offer instead (...) a few sociological thoughts about why the argument seems to have disappeared from the profession’s evaluaton of the positivist criterion of verifiability. (shrink)

In this paper, the authors briefly summarize how object theory uses definite descriptions to identify the denotations of the individual terms of theoretical mathematics and then further develop their object-theoretic philosophy of mathematics by showing how it has the resources to address some objections recently raised against the theory. Certain ‘canonical’ descriptions of object theory, which are guaranteed to denote, correctly identify mathematical objects for each mathematical theory T, independently of how well someone understands the descriptive condition. And to have (...) a false belief about some particular mathematical object is not to have a true belief about some different mathematical object. (shrink)

Originally published in 1910, Principia Mathematica led to the development of mathematical logic and computers and thus to information sciences. It became a model for modern analytic philosophy and remains an important work. In the late 1960s the Bertrand Russell Archives at McMaster University in Canada obtained Russell's papers, letters and library. These archives contained the manuscripts for the new Introduction and three Appendices that Russell added to the second edition in 1925. Also included was another manuscript, 'The Hierarchy of (...) Propositions and Functions', which was divided up and re-used to create the final changes for the second edition. These documents provide fascinating insight, including Russell's attempts to work out the theorems in the flawed Appendix B, 'On Induction'. An extensive introduction describes the stages of the manuscript material on the way to print and analyzes the proposed changes in the context of the development of symbolic logic after 1910. (shrink)

To mark the centenary of the 1910 to 1913 publication of the monumental Principia Mathematica by Alfred N. Whitehead and Bertrand Russell, this collection of fifteen new essays by distinguished scholars considers the influence and history of PM over the last hundred years.

In the case of an actual proper name such as ‘Aristotle’ opinions as to the Sinn may differ. It might, for instance, be taken to be the following: the pupil of Plato and teacher of Alexander the Great. Anybody who does this will attach another Sinn to the sentence ‘Aristotle was born in Stagira’ than will a man who takes as the Sinn of the name: the teacher of Alexander the Great who was born in Stagira. So long as the (...) Bedeutung remains the same, such variations of Sinn may be tolerated, although they are to be avoided.. (shrink)

This is a companion article to the translation of ‘Zasada sprzeczności a logika symboliczna’, the appendix on symbolic logic of Jan Łukasiewicz's 1910 book O zasadzie sprzeczności u Arytotelesa (On the Principle of Contradiction in Aristotle). While the appendix closely follows Couturat's 1905 book L'algebra de la logique (The Algebra of Logic), footnotes show that Łukasiewicz was aware of the work of Peirce, Huntington and Russell (before Principia Mathematica). This appendix was influential in the development of the Polish school of (...) logic, directly inspiring Stanisław Leśniewski and Leon Chwistek and more widely by serving as a text of the new symbolic logic. This appendix was an important source of the dominant algebraic logic in Poland, but also indicates that Łukasiewicz appreciated Russell's axiomatic approach to logic. (shrink)

In ‘On Denoting’ and to some extent in ‘Review of Meinong and Others, Untersuchungen zur Gegenstandstheorie und Psychologie’, published in the same issue of Mind (Russell, 1905a,b), Russell presents not only his famous elimination (or contextual defi nition) of defi nite descriptions, but also a series of considerations against understanding defi nite descriptions as singular terms. At the end of ‘On Denoting’, Russell believes he has shown that all the theories that do treat defi nite descriptions as singular terms fall (...) logically short: Meinong’s, Mally’s, his own earlier (1903) theory, and Frege’s. (He also believes that at least some of them fall short on other grounds—epistemological and metaphysical—but we do not discuss these criticisms except in passing). Our aim in the present paper is to discuss whether his criticisms actually refute Frege’s theory. We fi rst attempt to specify just what Frege’s theory is and present the evidence that has moved scholars to attribute one of three different theories to Frege in this area. We think that each of these theories has some claim to be Fregean, even though they are logically quite different from each other. This raises the issue of determining Frege’s attitude towards these three theories. We consider whether he changed his mind and came to replace one theory with another, or whether he perhaps thought that the different theories applied to different realms, for example, to natural language versus a language for formal logic and arithmetic. We do not come to any hard and fast conclusion here, but instead just note that all these theories treat defi nite descriptions as singular terms, and that Russell proceeds as if he has refuted them all. After taking a brief look at the formal properties of the Fregean theories (particularly the logical status of various sentences containing nonproper defi - nite descriptions) and comparing them to Russell’s theory in this regard, we turn to Russell’s actual criticisms in the above-mentioned articles to examine the extent to which the criticisms hold.. (shrink)

This chapter summarizes Russell’s The Problems of Philosophy, presents new biographical details about how and why Russell wrote it, and highlights its continued significance for contemporary philosophy. It also surveys Russell’s famous distinction between “knowledge by acquaintance” and “knowledge by description,” his developing views about our knowledge of physical reality, and his views about our knowledge of logic, mathematics, and other abstract objects.

This article presents notes that Russell made while reading the works of Gottlob Frege in 1902. These works include Frege's books as well as the packet of off-prints Frege sent at Russell's request in June of that year. Russell relied on these notes while composing "Appendix A: The Logical and Arithmetical Doctrines of Frege" to add to The Principles of Mathematics, which was then in press. A transcription of the marginal comments in those works of Frege appeared in the previous (...) issue of this journal. (shrink)

This critical notice of Stephen Neale's "Descriptions", (MIT Press, 1990) summarizes the content of the book and presents several objections to its arguments, as well as praising Neale for showing just how close the linguistic notion of L F is to the analytic philosopher's notion of "logical form". It is claimed that Neale's use of generalized quantifiers to represent definite descriptions from Russell's account by which descriptions are "incomplete symbols". I also argue that his assessment of the Quine/Smullyan exchange about (...) "Necessarily the number of the planets is greater than seven" is incorrect. (shrink)

Leon Chwistek's 1924 paper ?The Theory of Constructive Types? is cited in the list of recent ?contributions to mathematical logic? in the second edition of Principia Mathematica, yet its prefatory criticisms of the no-classes theory have been seldom noticed. This paper presents a transcription of the relevant section of Chwistek's paper, comments on the significance of his arguments, and traces the reception of the paper. It is suggested that while Russell was aware of Chwistek's points, they were not important in (...) leading him to the adoption of extensionality that marks the second edition of PM. Rudolf Carnap seems to have independently rediscovered Chwistek's issue about the scope of class expressions in identity contexts in his Meaning and Necessity in 1947. (shrink)

Acquaintance, Knowledge, and Logic (awarded the 2016 Bertrand Russell Society Book Prize) brings together ten new essays on Bertrand Russell's best-known work, The Problems of Philosophy. These essays, by some of the foremost scholars of his life and works, reexamine Russell's famous distinction between “knowledge by acquaintance” and “knowledge by description,” his developing views about our knowledge of physical reality, and his views about our knowledge of logic, mathematics, and other abstract objects. In addition, this volume includes an editors' introduction, (...) which summarizes Russell's influential book, presents new biographical details about how and why Russell wrote it, and highlights its continued significance for contemporary philosophy. (shrink)

This note presents a transcription of Russell's letter to Hawtrey of 22 January 1907 accompanied by some proposed emendations. In that letter Russell describes the paradox that he says "pilled" the "substitutional theory" developed just before he turned to the theory of types. A close paraphrase of the derivation of the paradox in a contemporary Lemmon-style natural deduction system shows which axioms the theory must assume to govern its characteristic notion of substituting individuals and propositions for each other in other (...) propositions. Other discussions of this paradox in the literature are mentioned. I conclude with remarks about the significance of the paradox for Russell. (shrink)

A correspondence theory of truth involves at least three constituents; the truth bearer, propositions, which stand in a relation of correspondence to the third element, the truth maker, some objects or fact with which the truth maker must correspond. Correspondence theories differ about the nature of truth makers, over whether one needs to include properties, and in particular over whether facts must be assumed in addition in order to give a correct account not merely of the conditions under which propositions (...) are true, but also what makes them true. Simple modal propositions state that objects necessarily or possibly have certain properties. What makes such propositions true? In this paper I want to consider various candidates for an account of the truth makers of modal propositions. After rejecting several alternatives that rely only on objects and properties, i.e., truth conditions, I will present several refinements of a fact based theory culminating in the view that truth makers for modal propositions are actual or non-actual facts which have facts and possible worlds as constituents. (shrink)

Bertrand Russell presented three systems of propositional logic, one first in Principles of Mathematics, University Press, Cambridge, 1903 then in “The Theory of Implication”, Routledge, New York, London, pp. 14–61, 1906) and culminating with Principia Mathematica, Cambridge University Press, Cambridge, 1910. They are each based on different primitive connectives and axioms. This paper follows “Peirce’s Law” through those systems with the aim of understanding some of the notorious peculiarities of the 1910 system and so revealing some of the early history (...) of classical propositional logic. “Peirce’s Law” is a valid formula of elementary propositional logic: [ ⊃ p] ⊃ p This sentence is not even a theorem in the 1910 system although it is one of the axioms in 1903 and is proved as a theorem in 1906. Although it is not proved in 1910, the two lemmas from the proof in 1906 occur as theorems, and Peirce’s Law could have been derived from them in a two step proof. The history of Peirce’s Law in Russell’s systems helps to reconstruct some of the history of axiomatic systems of classical propositional logic. (shrink)

As Alasdair Urquhart has noted, Bertrand Russell asserted that developing the theory of definite descriptions from 1905 was the first step towards solving the paradoxes that were finally resolved after 1908 in Principia Mathematica with the theory of types. I extend Urquhart’s suggestion that Russell was referring to the use of the notion of incomplete symbol in his solution to the paradoxes in his doomed theory “substitutional theory” of “Russellian propositions” in 1906. The Introduction to PM states that expressions for (...) propositions are incomplete symbols. This paper assesses the status of propositions in PM and connects the theory of types with the theory of descriptions. (shrink)

A transcription of Russell's marginal comments in his copies of Frege's works, from his readings of Frege in 1902. The greatest number are in the early sections of Grundgesetze der Arithmetik, Vol. I, but there are also marginal comments in Begriffsschrift, Grundlagen der Arithmetik, "Ueber Formale Theorien der Arithmetik", "Ueber Begriff und Gegenstand", "Function und Begriff", "Kritische Beleuchtung einiger Punkte in E. Schroeders ..." and two corrections of typographical errors in "Ueber Sinn und Bedeutung".

Three half-leaves of the final manuscript of Principia Mathematica have come to light in the Bertrand Russell Archives. They were originally tucked in Russell's own copy but avoided archival notice because their versos had been employed for an index of propositions used in theorem *350·62. The leaves form the whole of a folio 152 and the top half of 153 and include *336·51 through part of *336·52, on pages 400–1 of Volume III. Markings by the Cambridge University Press add to (...) our knowledge of the typesetting and proofreading of PM and give some indication of the fate of the remainder of the approximately 5–6,000 manuscript leaves, of which only one had been known to have survived. (shrink)

A Companion to Analytic Philosophy is a comprehensive guide to many significant analytic philosophers and concepts of the last hundred years. Provides a comprehensive guide to many of the most significant analytic philosophers of the last one hundred years. Offers clear and extensive analysis of profound concepts such as truth, goodness, knowledge, and beauty. Written by some of the most distinguished philosophers alive, some of whom have entries in the book devoted to them.

This note presents a transcription of Russell's letter to Hawtrey of 22 January 1907 accompanied by some proposed emendations. In that letter Russell describes the paradox that he says "pilled" the "substitutional theory" developed just before he turned to the theory of types. A close paraphrase of the derivation of the paradox in a contemporary Lemmon-style natural deduction system shows which axioms the theory must assume to govern its characteristic notion of substituting individuals and propositions for each other in other (...) propositions. Other discussions of this paradox in the literature are mentioned. I conclude with remarks about the significance of the paradox for Russell. (shrink)

A discussion of views first presented by this author and Edward Zalta in 1995 in the paper “Naturalized Platonism vs. Platonized Naturalism”. That paper presents an application of Zalta’s “object theory” to the ontology of mathematics, and claims that there is a plenitude of abstract objects, all the creatures of distinct mathematical theories. After a summary of the position, two questions concerning the view are singled out for discussion: just how many mathematical objects there are by our account, and the (...) nature of the properties we use to characterize abstract objects. The difference between the authors in more recent developments of the view are also discussed. (shrink)

Ernst Mally’s Gegenstandstheoretische Grundlagen der Logik und Logistik proposes that the abstract object “the circle” does not satisfy the properties of circles, but instead “determines” the class of circles. In this he anticipates the notion of “encoding” that Edward Zalta proposes for his theory ofObjects. It is argued that Mally did anticipate the notion of “encoding”, but sees it as a way of taking the concept as the subject of a proposition, rather than as a primitive notion in the theory (...) of a new ontological category of abstract objects, as Zalta does. (shrink)

According to Richard Gaskin, The Problem of the Unity of the Proposition is to explain 'what distinguishes propositions from mere aggregates, and enables them to be true or false' (18).1 This problem arises from the simpler problem of distinguishing a sentence from a 'mere list' of words (1). The unity of a sentence is due to its syntax, a level of structure which is not apparent in the string of words which are uttered or written, and which distinguishes a sentence (...) from a list. However, if one holds that sentences express propositions that are composed of objects which serve as semantic referents of its words, then the problem of unity becomes one about the metaphysics of propositions. What constitutes a .. (shrink)

In the Grundlagen Frege says that "line a is parallel to line b" differs from "the direction of a = the direction of b" in that "we carve up the content in a way different from the original way". It seems that such recarving is crucial to Frege's logicist program of defining numbers, but it also seems incompatible with his later theory of sense and reference. I formulate a restriction on recarving, in particular, that no names may be introduced that (...) introduce new possibilities of reference failure, which is observed by Frege's examples. This restriction discriminates between various relatives of the "slingshot" argument which rely on a step of recarving. I offer an argument for the restriction based on Fregean principles, which I formalize in Church's "Logic of Sense and Denotation", and briefly discuss various axioms of his "Alternative (0)" which are incompatible with recarving. (shrink)

This is the first English translation directly based on the original Polish ‘Zasada sprzeczności a logika symboliczna’, the appendix on symbolic logic of Jan Łukasiewicz's 1910 book O zasadzie sprzeczności u Arytotelesa (On the Principle of Contradiction in Aristotle).

I compare Russell’s theory of mathematical functions, the “descriptive functions” from Principia Mathematica ∗30, with Frege’s well known account of functions as “unsaturated” entities. Russell analyses functional terms with propositional functions and the theory of definite descriptions. This is the primary technical role of the theory of descriptions in P M . In Principles of Mathematics and some unpublished writings from before 1905, Russell offered explicit criticisms of Frege’s account of functions. Consequenly, the theory of descriptions in “On Denoting” can (...) be seen as a crucial part of Russell’s larger logicist reduction of mathematics,aswellasanexcursionintothetheoryof reference.
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