This paper considers two reasons that might support Russell’s choice of a ramified-type theory over a simple-type theory. The first reason is the existence of purported paradoxes that can be formulated in any simple-type language, including an argument that Russell considered in 1903. These arguments depend on certain converse-compositional principles. When we take account of Russell’s doctrine that a propositional function is not a constituent of its values, these principles turn out to be too implausible to make these arguments troubling. (...) The second reason is conditional on a substitutional interpretation of quantification over types other than that of individuals. This reason stands up to investigation: a simple-type language will not sustain such an interpretation, but a ramified-type language will. And there is evidence that Russell was tacitly inclined towards such an interpretation. A strong construal of that interpretation opens a way to make sense of Russell’s simultaneous repudiation of propositions and his willingness to quantify over them. But that way runs into trouble with Russell’s commitment to the finitude of human understanding. (shrink)

Certain commentators on Russell's “no class” theory, in which apparent reference to classes or sets is eliminated using higher-order quantification, including W. V. Quine and (recently) Scott Soames, have doubted its success, noting the obscurity of Russell’s understanding of so-called “propositional functions”. These critics allege that realist readings of propositional functions fail to avoid commitment to classes or sets (or something equally problematic), and that nominalist readings fail to meet the demands placed on classes by mathematics. I show that Russell (...) did thoroughly explore these issues, and had good reasons for rejecting accounts of propositional functions as extra-linguistic entities. I argue in favor of a reading taking propositional functions to be nothing over and above open formulas which addresses many such worries, and in particular, does not interpret Russell as reducing classes to language. (shrink)

In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell’s Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his philosophy (...) due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege’s extant logical system, if Frege’s system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege’s logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therein. (shrink)

The paper critically scrutinizes the widespread idea that Russell subscribes to a "Universalist Conception of Logic." Various glosses on this somewhat under-explained slogan are considered, and their fit with Russell's texts and logical practice examined. The results of this investigation are, for the most part, unfavorable to the Universalist interpretation.

It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church's “lambda calculus” for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903–1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the lambda calculus. Russell also anticipated Schönfinkel's combinatory logic approach of treating multiargument functions as functions having other functions as value. (...) Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and “value-ranges”. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Russell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the lambda calculus. (shrink)

I investigate why Frege rejected the theory of types, as Russell presented it to him in their correspondence. Frege claims that it commits one to violations of the law of excluded middle, but this complaint seems to rest on a dogmatic refusal to take Russell’s proposal seriously on its own terms. What is at stake is not so much the truth of a law of logic, but the structure of the hierarchy of the logical categories, something Frege seems to neglect. (...) To come to a better understanding of Frege’s response, I proceed to investigate his conception of the nature of the logical categories, and how it differs from Russell’s. I argue that, for Frege, our grasp of the logical categories cannot be severed from our grasp of the Begriffsschrift notation itself. Russell, on the other hand, attaches no such importance to notation. From Frege’s point of view, Russell has not succeeded in presenting an alternative conception of the logical hierarchy, since such a conception must go in tandem with the development of a notation. Moreover, Frege has good reasons to think that Russell’s proposal does not admit of a suitable notation. (shrink)

Russell's version of the multiple-relation theory from the "Theory of Knowledge" manuscript is presented and defended against some objections. A new problem, related to defining truth via correspondence, is reconstructed from Russell's remarks and what we know of Wittgenstein's objection to Russell's theory. In the end, understanding this objection in terms of correspondence helps to link Russell's multiple-relation theory to his later views on propositions.

It is fairly well known that Wittgenstein's criticisms of Russell's multiple‐relation theory of judgement had a devastating effect on the latter's philosophical enterprise. The exact nature of those criticisms however, and the explanation for the severity of their consequences, has been a source of confusion and disagreement amongst both Russell and Wittgenstein scholars. In this paper, I offer an interpretation of those criticisms which shows them to be consonant with Wittgenstein's general critique of Russell's conception of logic and which serves (...) to elucidate some of the notoriously enigmatic passages of the Tractatus. In particular, I seek to show the continuity of Wittgenstein's criticisms of the theory of judgement with his remarks on Russell's paradox and the theory of types. In addition, I place these issues in the context of Russell's own philosophical ambitions in order to reveal the deep divisions between the two over the nature of logical form and the analysis of propositional content. (shrink)

Since its publication in 1967, van Heijenoort’s paper, “Logic as Calculus and Logic as Language” has become a classic in the historiography of modern logic. According to van Heijenoort, the contrast between the two conceptions of logic provides the key to many philosophical issues underlying the entire classical period of modern logic, the period from Frege’s Begriffsschrift (1879) to the work of Herbrand, Gödel and Tarski in the late 1920s and early 1930s. The present paper is a critical reflection on (...) some aspects of van Heijenoort’s thesis. I concentrate on the case of Frege and Russell and the claim that their philosophies of logic are marked through and through by acceptance of the universalist conception of logic, which is an integral part of the view of logic as language. Using the so-called “Logocentric Predicament” (Henry M. Sheffer) as an illustration, I shall argue that the universalist conception does not have the consequences drawn from it by the van Heijenoort tradition. The crucial element here is that we draw a distinction between logic as a universal science and logic as a theory. According to both Frege and Russell, logic is first and foremost a universal science, which is concerned with the principles governing inferential transitions between propositions; but this in no way excludes the possibility of studying logic also as a theory, i.e., as an explicit formulation of (some) of these principles. Some aspects of this distinction will be discussed. (shrink)

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)

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 article aims first at showing that Russell's general doctrine according to which all mathematics is deducible ‘by logical principles from logical principles’ does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV–V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: Russell's logicism (...) does not only contain the claim that mathematics is no more than logic, it also contains the claim that the differences between the various mathematical sciences can be logically justified—and thus, that, contrary to the arithmetization stance, analysis, geometry and mechanics are not merely outgrowths of arithmetic. The second aim of this article is to set out the neglected Russellian theory of quantity. The topic is obviously linked with the first, since the mere existence of a doctrine of magnitude, in a work dated from 1903, is a sign of a distrust vis-à-vis the arithmetization programme. After having shown that, despite the works of Cantor, Dedekind and Weierstrass, many mathematicians at the end of the 19th Century elaborated various axiomatic theories of the magnitude, I will try to define the peculiarity of the Russellian approach. I will lay stress on the continuity of the logicist's thought on this point: Whitehead, in the Principia, deepens and generalizes the first Russellian 1903 theory. (shrink)

There are many wonderful puzzles concerning Principia Mathematica, but none are more striking than those arising from the crisis that befell Whitehead in November of 1910. Volume 1 appeared in December of 1910. Volume 2 on cardinal numbers and Russell's relation arithmetic might have appeared in 1911 but for Whitehead's having halted the printing. He discovered that inferences involving the typically ambiguous notation ‘Nc‘α’ for the cardinal number of α might generate fallacies. When the volume appeared in 1912, it was (...) extensively emended by Whitehead and accompanied by a Prefatory Statement of Symbolic Conventions. This paper endeavors to recover from Whitehead's bad emendations—including his bewildering thesis that since ‘‘α’ is ‘true whenever significant,’ ‘α is to be accepted. It is supposedly a fallacy to apply Modus Ponens and infer Nc‘α from ‘α and‘‘α. (shrink)

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)

Bernard Linsky, The Evolution of Principia Mathematica; Bertrand Russell's Manuscripts and Notes for the Second Edition. Cambridge: Cambridge University Press. 2011. 407 pp. + two plates. $150.00/£...

Hume's Principle, dear to neo-Logicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations that Cantor's power-class theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's ‘no-classes’ theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theory of cardinal numbers as objects set out in Frege's Grundgesetze. (...) Though Frege did not realize it, Cantor's power-theorem entails that Frege's cardinals as objects do not always obey Hume's Principle. (shrink)

Reply to Beaney: the closing of the historical mindIn his comments, Michael Beaney sets himself up as the arbiter of what is genuine history and what isn’t. While celebrating the outpouring of specialized scholarship on Frege, he has no patience with the enterprise outlined in the Précis, which attempts to construct a large-scale picture of the richness of the analytic tradition. That enterprise is one in which great figures of our recent past are challenged by aspects of contemporary thought, and (...) our current struggles are enriched by insights of theirs that haven’t been fully absorbed. Although some work of this sort can be done piecemeal, one historical figure at a time, there is much to be learned from a more encompassing perspective. While no picture constructed by a single author can be authoritative about everything, the attempt to give informative, connected assessments of major milestones in the tradition is our best hope of understanding who we are and where we have come from. (shrink)

The present text discusses whether there is a tension between aphorisms 6.1-6.13 of the Tractatus and the Church-Turing theorem about the decidability of predicate logic. We attempt to establish the following points: (i) Aphorisms 6.1-6.13 are not consistent with the Church-Turing theorem. (ii) The logical symbolism of the Tractatus, built from the N-operator, can (and should) be interpreted as expressively complete with respect to first-order formulas. (iii) Wittgenstein’s reasons for believing that Logic is decidable were purely philosophical and the undecidability (...) result shows that there are aspects of his criticisms of Frege and Russell that become unjustified in light of these results. (shrink)

Fitch's basic logic is an untyped illative combinatory logic with unrestricted principles of abstraction effecting a type collapse between properties (or concepts) and individual elements of an abstract syntax. Fitch does not work axiomatically and the abstraction operation is not a primitive feature of the inductive clauses defining the logic. Fitch's proof that basic logic has unlimited abstraction is not clear and his proof contains a number of errors that have so far gone undetected. This paper corrects these errors and (...) presents a reasonably intuitive proof that Fitch's system K supports an implicit abstraction operation. Some general remarks on the philosophical significance of basic logic, especially with respect to neo-logicism, are offered, and the paper concludes that basic logic models a highly intensional form of logicism. (shrink)

This paper is an attempt to explain why Russell abandoned the ontology of propositions, mind-independent complex entities that are possible objects of judgements. I argue that he did so not because of any decisive argument but because he found it better to endorse the existential account of truth, according to which a judgement is true if and only if there exists (or in his view subsists) a corresponding fact. In order to endorse this account, he had examined various theories of (...) judgement before he adopted the multiple-relation theory of judgement, the most feasible way he then had of espousing it. I also attempt to explain why he preferred the existential account of truth. (shrink)

L’article vise à établir un lien entre les travaux de Russell sur les fondements de la géométrie et ses célèbres textes des années 1911-1914 sur la perception, la matière et les sense-data. Nous insistons d’abord sur le fait que la notion russellienne de donnée sensorielle n’est pas phénoménaliste : les sense-data des Problèmes de Philosophie sont des objets extérieurs aussi peu mentaux que les corps matériels. L’accent est ensuite mis sur l’article On Matter : Russell y introduit pour la première (...) fois l’idée que les sense-data n’ont pas une, mais deux positions — une position où ils sont vus et une position d’où ils sont vus. Cette manœuvre permet à Russell de considérer la donnée sensorielle comme le point d’intersection neutre de deux ensembles fondamentalement différents : une série « physique », matérielle, et une série de « perspectives », mentales. L’idée qu’une même chose peut être classée sous différentes catégories, donc que le mode de classification aristotélicien en espèce et en genre n’est pas le seul concevable, est à la fois la clé de voûte de la théorie des relations entre « sense-data » et « physical things » de 1914 et le nerf du ralliement ultérieur au « monisme neutre ». Nous défendons, dans une troisième partie, que ce canevas n’est lui-même qu’un lointain écho de la conception de la géométrie projective définie comme théorie générale des relations d’incidence développée par Russell dans The Principles of Mathematics (1903). (shrink)

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)

The second printing of Principia Mathematica in 1925 offered Russell an occasion to assess some criticisms of the Principia and make some suggestions for possible improvements. In Appendix A, Russell offered *8 as a new quantification theory to replace *9 of the original text. As Russell explained in the new introduction to the second edition, the system of *8 sets out quantification theory without free variables. Unfortunately, the system has not been well understood. This paper shows that Russell successfully antedates (...) Quine's system of quantification theory without free variables. It is shown as well, that as with Quine's system, a slight modification yields a quantification theory inclusive of the empty domain. (shrink)

Logical universalism, a label that has been pinned on to Frege, involves the conflation of two features commonly ascribed to logic: universality and radicality. Logical universality consists in logic being about absolutely everything. Logical radicality, on the other hand, corresponds to there being the one and the same logic that any reasoning must comply with. The first part of this paper quickly remarks that Frege’s conception of logic makes logical universality prevail and does not preclude the admission of different contexts (...) of discourse. The paper then aims to make it clear how Frege’s universalism can make sense of contextuality. Drawing on a suggestion made by Frege in his discussion of Hilbert, it shows that a properly Fregean notion of model can be devised. Taking up a suggestion from Wilfrid Hodges and William Demopoulos that the non-logical constants of a formal language can be compared to indexicals, this paper shows,paceHodges and Demopoulos, that such an understanding of non-logical constants is not beyond Frege’s horizon. A formal framework, based on the modern tool of fibrations, is set out to explain and justify this point. This framework allows one to compare Frege and Tarski, by formalizing Frege’s suggestion and by presenting Tarski’s semantics in a generalized setting. (shrink)

In this article, I examine the ramified-type theory set out in the first edition of Russell and Whitehead's Principia Mathematica. My starting point is the ‘no loss of generality’ problem: Russell, in the Introduction (Russell, B. and Whitehead, A. N. 1910. Principia Mathematica, Volume I, 1st ed., Cambridge: Cambridge University Press, pp. 53–54), says that one can account for all propositional functions using predicative variables only, that is, dismissing non-predicative variables. That claim is not self-evident at all, hence a problem. (...) The purpose of this article is to clarify Russell's claim and to solve the ‘no loss of generality’ problem. I first remark that the hierarchy of propositional functions calls for a fine-grained conception of ramified types as propositional forms (‘ramif-types’). Then, comparing different important interpretations of Principia’s theory of types, I consider the question as to whether Principia allows for non-predicative propositional functions and variables thereof. I explain how the distinction between the formal system of the theory, on the one hand, and its realizations in different epistemic universes, on the other hand, makes it possible to give us a more satisfactory answer to that question than those given by previous commentators, and, as a consequence, to solve the ‘no loss of generality’ problem. The solution consists in a substitutional semantics for non-predicative variables and non-predicative complex terms, based on an epistemic understanding of the order component of ramified types. The rest of the article then develops that epistemic understanding, adding an original epistemic model theory to the formal system of types. This shows that the universality sought by Russell for logic does not preclude semantical considerations, contrary to what van Heijenoort and Hintikka have claimed. (shrink)

In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead's Principia Mathematica ([71], 1910-1912) and Church's simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege's (...) Grundgesetze der Arithmetik for which Russell applied his famous paradox and this led him to introduce the first theory of types, the Ramified Type Theory (RTT). We present RTT formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from RTT leading to the simple theory of types STT. We present STT and Church's own simply typed λ-calculus (λ → C) and we finish by comparing RTT, STT and λ → C. (shrink)

The so-called “Slingshot” argument purports to show that an ontology of facts is untenable. In this paper, we address a minimal slingshot restricted to an ontology of physical facts as truth-makers for empirical physical statements. Accepting that logical matters have no bearing on the physical facts that are truth-makers for empirical physical statements and that objects are themselves constituents of such facts, our minimal slingshot argument purportedly shows that any two physical statements with empirical content are made true by one (...) and the same fact. It is well-known that Russell’s theory of descriptions may be employed to reveal a scope fallacy in the slingshot argument. This paper reveals that there is a quite independent Russellian criticism of the slingshot argument based on the thesis that facts are structured entities. (shrink)