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We classify two of Bertrand Russell's theories of events within the point-free ontology. The first of such approaches was presented informally by Russell in 'The World of Physics and the World of Sense' (Lecture IV in Our Knowledge of the External World of 1914). Based on this theory, Russell sketched ways to construct instants as collections of events. This paper formalizes Russell's approach from 1914. We will also show that in such a reconstructed theory, we obtain all axioms of Russell's second theory from 1936 and all axioms of Thomason's theory of events from 1989. Russell's work certainly influenced the works of Stanisław Leśniewski, his student Alfred Tarski, and Czesław Lejewski – prominent members of the Lvov-Warsaw School (LWS). We see our work in the tradition of the research of Leśniewski and Tarski. Building on the technical tools developed in this environment and in the spirit of the traditional research of the LWS, we engage here, in particular, with two classic works by Russell on fundamental ontology. [ABSTRACT FROM AUTHOR]

Reviews the essay, `Foundations of Logic 1903-1905, The Collected Papers of Bertrand Russell, Volume 4,' edited by A. Urquhart.

In this paper I argue that Hegel’s treatment of dialectical inferences, in particular of Plato’s dialectics in the Lectures on the History of Philosophy, belongs to the history of the logical rule that, from Gerolamo Cardano to Bertrand Russell, is known as consequentia mirabilis (CM). In 1906 Russell formalises it as follows: and its correspondent positive form as My paper has two parts. First, I show that dialectical inferences, for Hegel, involve sentences of the form and . Hegel, following Plato, stresses that these inferences are das Wunderbare, the marvellous element of dialectical reasoning. In this sense Hegel’s view belongs to the history of CM from a perspective that is both terminological and logical. Second, I stress the peculiarity of Hegel’s CM. In a classical setting, the conditional is admissible and from we can, by CM, derive , and from , by CM, infer . However, in Hegel’s analysis it is evident that we can neither infer from , nor from . Rather, in dialectical inferences what we can validly infer from and is only . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

In this paper, I will show to what extent we can use our modern understanding of the Square of Opposition in order to make sense of Kant's double standard solution to the cosmological antinomies. Notoriously, for Kant, both theses and antitheses of the mathematical antinomies are false, while both theses and antitheses of the dynamical antinomies are true. Kantian philosophers and interpreters (including Schopenhauer, for example) have criticized Kant's solution as artificial and prejudicial. In the paper, I do not dispute such claims, but I show that our modern understanding of the Square of Opposition enables us to more naturally deliver the result Kant was aiming at. Accordingly, the paper does not pretend to be exegetically accurate. It is an attempt to revise the antinomies with the help of standard classical logic. And although such a revision entails some re-interpretation, in the end, it will actually help to unveil some of Kant's thoughts. [ABSTRACT FROM AUTHOR]

Jin Yuelin (1895–1984), a Chinese logician and philosopher, is greatly influenced by Hume's and Russell's philosophies. How should we respond to Hume's problem of induction? This is an important clue to understand Jin's whole philosophical career. The first section of this paper gives a brief historical review of Russell and Jin. The second section outlines Hume's skeptical arguments against causality and induction. The third section expounds Russell's justification of induction by discussing his views on Hume's skepticism, causality, principle of induction, and empirical postulates. The fourth section clarifies Jin's justification of induction by discussing his critique of Hume's epistemology and his arguments for the reliability of causality and the eternal truth and apriority of the principle of induction. The final section compares Jin's justification of induction with Russell's and concludes that there are similarities and differences between their projects and that both their attempts fail. This paper takes the similar responses to the problem of induction by Jin and Russell to demonstrate the communication that there has been between Chinese philosophers and the Western ones. [ABSTRACT FROM AUTHOR]

This article presents three extracts from the introductory course in mathematical logic that Godel gave at the University of Notre Dame in 1939. The lectures include a few digressions, which give insight into Godel's views on logic prior to his philosophical papers of the 1940s. The first extract is Godel's first lecture. It gives the flavour of Godel's leisurely style in this course. It also includes a curious definition of logic and a discussion of implication in logic and natural language. The second extract is a discussion on undecidability and on Leibniz. The third extract concerns the paradoxes and Russell's theory of types. [ABSTRACT FROM AUTHOR]

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 (where non-homogeneous relations are allowed) 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. [ABSTRACT FROM PUBLISHER]

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'a' for the cardinal number of α (i.e. the class of all classes similar with 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 ∈ Nc'α' is 'true whenever significant,' ⊢ α ∈ Nc'α is to be accepted. It is supposedly a fallacy to apply Modus Ponens and infer ⊢ ⁆!Nc'α from ⊢ α ∈ Nc'α and ⊢ α ∈ Nc'α ⊃ ⁆!Nc'α. [ABSTRACT FROM AUTHOR]

The paper offers a historical survey of the emergence of logical formalization in twentieth-century analytically oriented philosophy of religion. This development is taken to have passed through three main ‘stages’: a pioneering stage in the late nineteenth and early twentieth centuries (led by Frege and Russell), a stage of crisis in the 1920s and early 1930s (occasioned by Wittgenstein, logical positivists such as Carnap, and neo-Thomists such as Maritain), and a stage of rehabilitation in the 1930s, 1940s, and 1950s (led by the Cracow Circle and Quine). [ABSTRACT FROM PUBLISHER]

Principia Mathematic goes to great lengths to hide its order/type indices and to make it appear as if its incomplete symbols behave as if they are singular terms. But well-hidden as they are, we cannot understand the proofs in Principia unless we bring them into focus. When we do, some rather surprising results emerge – which is the subject of this paper. [ABSTRACT FROM AUTHOR]

‘The Principles of the Pure Type Theory’ is a translation of Leon Chwistek's 1922 paper ‘Zasady czystej teorii typów’. It summarizes Chwistek's results from a series of studies of the logic of Whitehead and Russell's Principia Mathematica which were published between 1912 and 1924. Chwistek's main argument involves a criticism of the axiom of reducibility. Moreover, ‘The Principles of the Pure Type Theory’ is a source for Chwistek's views on an issue in Whitehead and Russell's ‘no-class theory of classes’ involving the notion of ‘scope’. [ABSTRACT FROM PUBLISHER]

Bertrand Russell was one of the most prominent figures in the formation and development of mathematical logic. It is widely acknowledged that his work in this field exerted tremendous influence in the West, especially in the first three decades of the twentieth century. The important role he played in inspiring Chinese interest in this subject, however, is virtually unknown. This paper describes Russell's contributions to the introduction of mathematical logic in China through a discussion of his lectures in Beijing in March of 1921, and the subsequent Chinese translations of his Introduction to Mathematical Philosophy and Principia Mathematica . [ABSTRACT FROM AUTHOR]

Focuses on Bertrand Russell's arguments for restricting modes of specification and domains of quantification. Achievements of Russell's paper entitled `On Denoting'; Definition of the substitutional theory; Arguments for the `no-class' view; Vicious principle and the theory of types of 1910; Incomplete symbols in Principia mathematica.

Bertrand Russell's 1906 article 'The Theory of Implication' contains an algebraic weak completeness proof for classical propositional logic. Russell did not present it as such. We give an exposition of the proof and investigate Russell's view of what he was about, whether he could have appreciated the proof for what it is, and why there is no parallel of the proof in Principia Mathematica. [ABSTRACT FROM AUTHOR]

Influenced by G. E. Moore, Russell broke with Idealism towards the end of 1898; but in later years he characterized his meeting Peano in August 1900 as 'the most important event' in 'the most important year in my intellectual life'. While Russell discovered his paradox during his post-Peano period, the question arises whether he was already committed, during his pre-Peano Moorean period, to assumptions from which his paradox may be derived. Peter Hylton has argued that the pre-Peano Russell was thus vulnerable to (at least one version of) Russell's paradox and hence that the paradox exposes a pre-existing difficulty in Russell's Moorean philosophy. Contrary to Hylton, I argue that the Moorean Russell adhered to views which insulated him against the paradox. Further, I argue that Russell became vulnerable to his paradox as a result of changes in his Moorean position occasioned, first, by his acceptance of Cantor's theory of the transfinite, and, second, by his correspondence with Frege. I conclude with some general comments regarding Russell's acceptance of naïve set theory. [ABSTRACT FROM AUTHOR]

We re-examine the problem of existential import by using classical predicate logic. Our problem is: How to distribute the existential import among the quantified propositions in order for all the relations of the logical square to be valid? After defining existential import and scrutinizing the available solutions, we distinguish between three possible cases: explicit import, implicit non-import, explicit negative import and formalize the propositions accordingly. Then, we examine the 16 combinations between the 8 propositions having the first two kinds of import, the third one being trivial and rule out the squares where at least one relation does not hold. This leads to the following results: (1) three squares are valid when the domain is non-empty; (2) one of them is valid even in the empty domain: the square can thus be saved in arbitrary domains and (3) the aforementioned eight propositions give rise to a cube, which contains two more (non-classical) valid squares and several hexagons. A classical solution to the problem of existential import is thus possible, without resorting to deviant systems and merely relying upon the symbolism of First-order Logic (FOL). Aristotle's system appears then as a fragment of a broader system which can be developed by using FOL. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

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, Konig-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 [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Russell alleged that the version of the cosmological argument he debated with Copleston involved type confusions, but the definitions of plural descriptive functions and the ancestral in Principia Mathematica can be used to reformulate the argument in a type-safe way via a notion of causally self-sufficient classes. Although the argument depends on the assumption that the class of contingent things is not causally self-sufficient, if that assumption is weakened to say only that it may not be so, then a new modalized version emerges which has similarities with recent ontological arguments but weaker and more intuitive assumptions. (Familiarity with Principia notation and conventions is presupposed throughout.) [ABSTRACT FROM AUTHOR]

Rejects as spurious the contrast between a theory that takes 'Every philosopher is smart' and 'Socrates is smart' to express propositions with the same structure and a theory that does not. Bertrand Russell's 'On denoting' (1905) and "The Principles of Mathematics" (1903) (PoM); Implications of the PoM for the notion of predication; Focus of argument on s-p propositions.

Focuses on attempts made by Richard Brinkley and B Russell to understand the unity proposition, while highlighting the basis of these attempts. Information on treaties made by Brinkley; Justification of theories made by Brinkley and Russell.