Synopsis of Bertrand Russell, ‘La Théorie des types logiques’
[‘The Theory of Logical Types’]
Bertrand Russell (1872-1970) wrote ‘The Theory of Logical Types’ (1910) for the Revue de Métaphysique et de Morale as a clarification of his earlier paper ‘Mathematical Logic as Based on the Theory of Types’ (1908)1, which described a formal method of blocking logical and semantic paradoxes, and in particular, the paradox that bears his name. Russell’s paradox infects naïve set theory via the assumption that every predicate determines a class – for example, that ‘red’ presupposes a class containing all and only red things. Along these lines however, ‘class’ is itself a predicate, and so ‘the class of all classes’ determines a class, as does ‘the class of all classes that are not members of themselves.’ This latter class is paradoxical: it is a member of itself if it is not – although if it is not, it is.
Russell’s solution blocks logical paradoxes by blocking self-membership. This is accomplished through a hierarchy of ‘types’, beginning with individuals (type 0), followed by classes whose members are individuals (type 1), classes of classes (type 2), and so on, such that a class of type n + 1 can only have members of type n. Since a class and its members must be of different types, self-membership is checked.
In ‘The Theory of Logical Types,’ Russell applies the process of restriction to the arguments of a propositional function, rather than regulate the membership relation itself. A propositional function is ‘something which contains a variable x, and expresses a proposition as soon as a value is assigned to x’ (CpA 10.4:54; English version 216); for example, ‘x is a man’ is a propositional function to which ‘Socrates’; is a possible argument. Expressed in terms of propositional functions, a hierarchy of types would begin with individuals of type 0 (for example, ‘;Socrates’), followed by functions that take an argument of type 0 (for example, ‘Socrates is a man’), followed by ‘functions to which such functions are possible arguments, and so on’ (66/231).
These schemes are versions of the ‘simple theory of types,’ which Frank P. Ramsey separated from the more complicated ‘ramified’ theory presented in ‘The Theory of Logical Types.’ While the simple theory is adequate to solve logical paradoxes (such as Russell’s paradox), the semantic paradoxes (such as the Cretan liar) require the ramified theory. Russell applies the latter to both kinds of paradox because he does not distinguish them, taking them both as instances of the same ‘vicious circle’ (53/215).
As an example of a vicious circle, Russell notes that the collection of all propositions contains the proposition ‘all propositions are either true or false’ (53/216). Russell argues that ‘such a statement could not be legitimate unless ‘all propositions’ referred to some already definite collection, which it cannot do if new propositions are created by statements about ‘all propositions’’ (53/216). That is to say, the collection of all propositions could not be ‘already definite’ if a statement about ‘all propositions’ adds another proposition to it. Statements about ‘all propositions’ are therefore not true or false, but ‘meaningless’ and a vicious circle principle emerges: ‘given any set of objects such that, if we suppose the set to have a total, it will contain members which presuppose this total, then such a set cannot have a total’ (54/216).
In order to block such illegitimate totalities, Russell proposes a hierarchy of orders in which statements about ‘all propositions’ are only statements about propositions of a lower order. The assertion that ‘all propositions asserted by Cretans are false’ is now a proposition of order n + 1 asserting that all propositions of a lower order ‘n’ are false if asserted by Cretans. The assertion ‘all propositions asserted by Cretans are false’ therefore no longer presupposes a totality of which it is part.
In this manner, the ‘ramified’ theory blocks semantic paradox, but at great cost: mathematical induction is among those parts of classical mathematics that no longer hold. So, Russell adds an axiom ‘in order to legitimate a great mass of reasoning, in which [...] we are concerned with such notions as ‘all properties of a’’ (73/241). Problems arise with such reasoning in the form of the proposition ‘Napoleon had all the qualities that make a great general’ (74/241). The quality of ‘having all the qualities that make a great general’ is of a different type than the qualities of ‘daring, strategic brilliance,’ etc. Yet, ‘x had all the qualities that make a great general’ is formally equivalent to ‘x was daring, strategically brilliant’ etc. (both are true). So, Russell stipulates that any higher order function with an object ‘a’ among its arguments is formally equivalent to a first-order function with ‘a’ among its arguments via the ‘axiom of reducibility’ (76/243). The axiom is extra-logical however, posing a problem for Russell’s ambition of making mathematics a branch of logic. In addition, both ramified and simple type theory exhibit a ‘systematic ambiguity of truth and falsehood’: ‘the words ‘true’ and ‘false’ have many different meanings, according to the kind of proposition to which they are applied’ (59/222). This ‘systematic ambiguity’ also affects the identity sign and other parts of logic.
The article ‘The Theory of Logical Types’ shows the extent to which self-reference insists as a structural feature of mathematical logic, both by analyzing the role of self-reference in paradoxes, and by demonstrating the effort and cost of removing it from mathematical reasoning.Russell and Frege also increasingly realized the role that the definite article (‘;the’) plays in generating self-referential paradoxes and pseudo names; for example, in the case of Frege’s paradox of ‘the concept horse.’ From Badiou’s perspective, Russell situates self-reference as a ‘real’ of language that is not just a failure to express something, but an indicator of a properly ontological ‘excess’: ‘Russell’s paradox could be interpreted as an excess of the multiple over the capacity of language to present it without falling apart. One could just as well say that it is language which is excessive in that it is able to pronounce properties such as ¬(a∈a) [i.e. it is false that a is an element of a].’2 For Badiou, language must therefore be restricted in its ‘presentative pretensions,’ through which every predicate determines a class.
Considered along these lines, the legislations against paradox (type theory and set theory’s axiom of separation) are directed against language itself – or at least a capacity of it, taken as self-reference (Russell.), or as what Badiou condemns as ‘idealinguistery’, or as what Frege denounces as ‘the fatal tendency of language to form apparent proper names’ (i.e. names that have no referent), a tendency that Frege locates in fiction and poetry.3 Russell’s approach might thus be usefully contrasted with the neutralization of paradox achieved through non-classical (intuitionist or paraconsistent) logics – or the algebra of Russell’s student, G. Spencer Brown (who tried to ‘use’ self-reference rather than foreclose it).
References to this text in other articles in the Cahiers pour l’Analyse:
- Russell, Bertrand. ‘The Theory of Logical Types’. In Russell, Essays in Analysis, ed. Douglas Lackey. London: George Allen & Unwin Ltd., 1973. 215-252. Though originally published in French as ‘La Théorie des types logiques’ in the Revue de métaphysique et de la morale (1910), the English draft was written by Russell himself, and published with minor revisions as the second chapter of his Principia Mathematica volume I (co-authored with Alfred North Whitehead) (Cambridge: Cambridge University Press, 1910), 37-65.
- Badiou, Alain. Being and Event, trans. Oliver Feltham. London: Continuum, 2005.
- Frege, Gottlob. The Frege Reader, ed. Michael Beaney. Oxford: Blackwell Publishing, 1997.
- Russell, Bertrand. Essays in Analysis, ed. Douglas Lackey. London: George Allen & Unwin Ltd., 1973.
Selected secondary sources:
- Stevens, Graham. The Russellian Origins of Analytical Philosophy: Bertrand Russell and the Unity of the Proposition. New York: Routledge, 2005.
- Van Heijenoort, Jean (ed). From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Cambridge: Harvard University Press, 1967.
1. Bertrand Russell, ‘Mathematical Logic as Based on the Theory of Types’, American Journal of Mathematics 30 (1908): 222-262. ↵
2. Alain Badiou, Being and Event, 47. For Badiou, set theory’s axiom of separation confirms that ‘being is anterior to language’, since we can only use language to isolate a set if we first presume the existence of that set (501). ↵
3. Badiou, Being and Event, 47; Gottlob Frege, The Frege Reader, 369. ↵