Russell's Theory Research Articles

A Re-Examination of the Russellian Theory of Descriptions

The theory of descriptions occupies a very prominent place in Russell's system of logic and indeed in his system of philosophy. Since the publication of the now classical paper “On Denoting” inMindfor 1905 the theory had been incorporated intoPrincipia Mathematica, the first volume of which appeared in 1910. In 1918 Russell discussed descriptions in his lectures on the Philosophy of Logical Atomism, which subsequently were published inThe Monistfor 1919. A very lucid exposition of the main tenets of the doctrine is to be found in theIntroduction to Mathematical Philosophydating from the same year. Epistemological aspects of the theory of descriptions are examined in “Knowledge by Acquaintance and Knowledge by Description“, in the Proceedings of the Aristotelian Society for 1910–11, and also in Chapter V of The Problems of Philosophy, first published in 1912.

The 'Meaning' of Russell's Theory of Descriptions

The ‘Meaning’ of Russell's Theory of Descriptions Alan R. White Alan R. White University of Hull Search for other works by this author on: Oxford Academic Google Scholar Analysis, Volume 20, Issue 1, October 1959, Pages 8–9, https://doi.org/10.1093/analys/20.1.8 Published: 01 October 1959

The 'Meaning' of Russell's Theory of Descriptions

The 'Meaning' of Russell's Theory of Descriptions

Margaret Macdonald. Introduction. Philosophy and analysis, A selection of articles published in Analysis between 1933–40 and 1947–53, edited by Margaret Macdonald; Basil Blackwell, Oxford1954, and Philosophical Library, New York 1954; pp. 1–14. - A. M. MacIver. Demonstratives and proper names. A reprint of XII 104(4). Philosophy and analysis, A selection of articles published in Analysis between 1933–40 and 1947–53, edited

Margaret Macdonald. Introduction. Philosophy and analysis, A selection of articles published in Analysis between 1933–40 and 1947–53, edited by Margaret Macdonald; Basil Blackwell, Oxford1954, and Philosophical Library, New York 1954; pp. 1–14. - A. M. MacIver. Demonstratives and proper names. A reprint of XII 104(4). Philosophy and analysis, A selection of articles published in Analysis between 1933–40 and 1947–53, edited

Axiomatic Syntax: The Construction and Evaluation of a Syntactic Calculus

This paper discusses methods for presenting syntactic information in the form of a calculus, and for measuring its goodness of fit to a language.' Given a morpheme list for a language, the aim of a syntactic system is to tell us how to put together the sequences of morphemes which are used as sentences in the language. Such directions we shall call the FORMATION RULES. Additionally, works on syntax usually give a certain amount of information about the equivalences between some sequences and others, e.g. that John discovered the path = The path was discovered by John. We shall call such statements TRANSFORMATION RULES. Only a small part of the transformation rules is covered in syntax; some others are discussed in mathematical logic, e.g. Russell's theory of descriptions, procedures of generalization and abstraction. So far in mathematical logic, most attention has been directed to developing formation and transformation rules of artificial languages, and no complete treatment has yet been made of the transformation rules of a natural language. The division between formation and transformation rules can to some extent be altered by altering the units of the system. We shall concentrate on syntax as formation rules, and give a procedure for putting this material into a precise form so that it may be evaluated.2 Let us consider the syntax of a language L, say that of the English statement-type sentences, which may be constructed out of some finite word list. (For a con-

The Scaffolding of Russell's Theory of Descriptions

The Scaffolding of Russell's Theory of Descriptions

Note on a Formula in my ≫Bertrand Russell's Theories of Causation>.

TheoriaVolume 19, Issue 3 p. 177-178 Note on a Formula in my ≫Bertrand Russell's Theories of Causation>. Erik Götlind, Erik Götlind UppsalaSearch for more papers by this author Erik Götlind, Erik Götlind UppsalaSearch for more papers by this author First published: December 1953 https://doi.org/10.1111/j.1755-2567.1953.tb01016.xAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume19, Issue3December 1953Pages 177-178 RelatedInformation

Bertrand Russell's Theories of Causation.

Bertrand Russell's Theories of Causation.

Correction: Whitehead and Russell's Theory of Types: A Reply

Correction: Whitehead and Russell's Theory of Types: A Reply

Whitehead and Russell's Theory of Types: A Reply

Whitehead and Russell's Theory of Types: A Reply

Whitehead and Russell's Theory of Types A Reply

Whitehead and Russell's Theory of Types A Reply

P. T. Geach. Russell's theory of descriptions. Analysis (Oxford), vol. 10 no. 4 (1950), pp. 84–88.

P. T. Geach. Russell's theory of descriptions. Analysis (Oxford), vol. 10 no. 4 (1950), pp. 84–88.

J. J. C. Smart. Whitehead and Russell's theory of types. Analysis (Oxford), vol. 10 no. 4 (1950), pp. 93–96.

J. J. C. Smart. Whitehead and Russell's theory of types. Analysis (Oxford), vol. 10 no. 4 (1950), pp. 93–96.

Whitehead and Russell's Theory of Types

Whitehead and Russell's Theory of Types

Whitehead and Russell's Theory of Types

Whitehead and Russell's Theory of Types

A formal system of logic

The main purpose of this paper is to present a formal systemPin which we enjoy a smooth-running technique and which countenances a universe of classes which is symmetrical as between large and small. More exactly,Pis a system which differs from the inconsistent system of [1] only in the introduction of a rather natural new restrictive condition on the defining formulas of the elements (sets, membership-eligible classes). It will be proved that if the weaker system of [2] is consistent, thenPis also consistent.After the discovery of paradoxes, it may be recalled, Russell and Zermelo in the same year proposed two different ways of safeguarding logic against contradictions (see [3], [4]). Since then various simplifications and refinements of these systems have been made. However, in the resulting systems of Zermelo set theory, generation of classes still tends to be laborious and uncertain; and in the systems of Russell's theory of types, complications in the matter of reduplication of classes and meaningfulness of formulas remain. In [2], Quine introduced a system which seems to be free from all these complications. But later it was found out that in it there appears to be an unavoidable difficulty connected with mathematical induction. Indeed, we encounter the curious situation that although we can prove in it the existence of a class V of all classes, and we can also prove particular existence theorems for each of infinitely many classes, nobody has so far contrived to prove in it that V is an infinite class or that there exists an infinite class at all.

Practical forms of type theory

Russell's theory of types, though probably not providing the soundest possible foundation for mathematics, follows closely the outlook of most mathematicians. The present paper is an attempt to present the theory of types in forms in which the types themselves only play a rather small part, as they do in ordinary mathematical argument. Two logical systems are described (called the “nested-type” and “concealed-type” systems). It is hoped that the ideas involved in these systems may help mathematicians to observe type theory in proofs as well as in doctrine. It will not be necessary to adopt a formal logical notation to do so.

On ordered pairs

Wiener, in 1914, reduced the theory of relations to that of classes by construing relations as classes of ordered pairs and defining the ordered pair in turn on the basis of class theory alone.1 The definition, as improved by Kuratowski,2 identifies the ordered pair x;y with uxyi(ix U iy).In terms of Russell's theory of types, x;y in the above sense is two types higher than x and y. Even when we abandon Russell's theory of fixed types of objects in favor of a theory of stratified formulae,3 there is still significance in saying that ‘x;y’ is of type 2 relative to ‘x’ and ‘y’—meaning that a test of the stratification of any context involves assigning a higher number by 2 to ‘x;y’ than to ‘x’ and ‘y’.

A homogeneous system for formal logic

Two more or less standard methods exist for the systematic, logical construction of classical mathematics, the so-called theory of types, due in the main to Russell, and the Zermelo axiomatic set theory. In systems based upon either of these, the connective of membership, “ε”, plays a fundamental role. Usually although not always it figures as a primitive or undefined symbol.Following the familiar simplification of Russell's theory, let us mean by a logical type in the strict sense any one of the following: (i) the totality consisting exclusively of individuals, (ii) the totality consisting exclusively of classes whose members are exclusively individuals, (iii) the totality consisting exclusively of classes whose members are exclusively classes whose members in turn are exclusively individuals, and so on. Any entity from (ii) is said to be one type higher than any entity from (i), any entity from (iii), one type higher than any entity from (ii), and so on. In systems based upon this simplified theory of types, the only significant atomic formulae involving “ε” are those asserting the membership of an entity in an entity one type higher. Thus any expression of the form “(x∈y)” is meaningless except where “y” denotes an entity of just one type higher than the type of the entity denoted by “x” It is by means of general type restrictions of this kind that the Russell and other paradoxes are avoided.

The consistency of the ramified Principia

When the axioms of infinity and choice are added to the system of logic of the second edition of Whitehead and Russell's Principia mathematica, and when, as in the second edition, the axiom of reducibility is omitted (so that the ramified nature of Russell's theory of types, especially his distinction between “orders,” is not in effect obliterated), the resulting system will hereafter be referred to as “the ramified Principia.” It will be shown that a certain system S, slightly stronger than the ramified Principia, is consistent, so that the ramified Principia itself is consistent. From Gödel's theorem it will then follow that the sort of mathematical induction employed in the consistency proof of S cannot be adequately handled within S or within the ramified Principia.The method of proving the consistency of S will be roughly as follows: A consistent non-constructive system S′ will be defined by means of induction with respect to a serial well-ordering of all the propositions of S. It will then be shown that every true proposition of S is a true proposition of S′, so that S must also be consistent.1. Preliminary conventions.1.1. Definition. By a “symbol” is to be understood any typographical expression which does not consist of a horizontal collinear sequence of expressions. Expressions formed by adding indices to either of the two upper or two lower corners of a symbol will also be considered symbols.