Universals

5. Frege exhumes universals

In the twentieth century, the problem of universals has re-emerged under its familiar name, accompanied by more or less the same guiding illustrations used by Plato and Aristotle. This rebirth has occurred in the tradition of analytic philosophy, notably in the work of Frege, Russell, Wittgenstein, Quine and Armstrong.

A new twist to the theory of universals can be traced to groundbreaking work by Frege on the nature of natural numbers in his Grundlagen der Arithmetik (The Foundations of Arithmetic) (1884). As for Plato, so too for Frege, Russell and others in recent times, advances in mathematics have been the source of a philosophical focus on the problem of universals. Frege’s analysis of natural numbers (1, 2, 3,…) proceeded in three very different stages (see Frege, G. §9).

In the first stage of his analysis of numbers, Frege introduced the idea that numbering individuals essentially involves not the attribution of properties to individuals but, rather, the attribution of properties to properties. To illustrate: when asked ‘How many are on the table?’, Frege notes that there will be many different possible answers, as for instance (1) ‘Two packs of playing cards’ or (2) ‘104 playing cards’. The metaphysical truth-makers identified by Frege for these two sample answers are (1) that the property of being a pack of playing cards on the table is a property which has the property of having two instances, and (2) that the property of being a playing card on the table is a property which has the property of having one hundred and four instances. In general, natural numbers number individuals only via the intermediary of contributing to second-order properties, or properties of properties, namely properties of the form ‘having n instances’. Like Kant, Frege speaks of concepts (Begriffe) rather than of ‘universals’. Yet Frege’s concepts are definitively not private mental episodes, but are thoroughly mind-independent, more like Plato’s Forms than Aristotelian universals.

In the second stage of his analysis of numbers, Frege gives a very new twist to the theory of universals. He argues that the nature of universals, or concepts, is such as to make it impossible in principle ever to refer to a universal by any name or description. Thus for instance, in saying ‘Socrates is wise’, the universal which is instantiated by Socrates is something which is expressed by the whole arrangement of symbols into which the name ‘Socrates’ is embedded to yield the sentence ‘Socrates is wise’. Suppose you were to try to name this universal by the name ‘wisdom’. Then, compare ‘Socrates is wise’ with the concatenation of names – ‘Socrates wisdom’. The mere name ‘wisdom’ clearly leaves out something which was present in the attribution of wisdom to Socrates. Hence a universal cannot be referred to by a name.

Thus, a property can only be expressed by a predicate, never by a name or by any logical device which refers to individuals. Indeed, if we wish to attribute existence to universals, we cannot do so by the use of the same sort of device (the first-order quantifier) which is used to attribute existence to individuals. Thus, for instance, from ‘Socrates is wise’ we may infer ‘There exists something which is wise’, and ‘There exists something which is Socrates’:

Yet we may not infer that ‘There exists something which Socrates possesses’, or that ‘There exists something which is wisdom’:

Frege does, however, allow us to attribute existence to universals, using logical devices called higher-order quantifiers, which he introduced in his Begriffsschrift (1879). That is, we can infer from ‘Socrates is wise’ to ‘There is somehow such that: Socrates is that-how’:

But although there is somehow that Socrates is, this does not entail that there is anything which is the somehow that Socrates is: universals (concepts) can only have second-order existence, not first-order existence.

For Frege, numbering things essentially involved attribution of properties to properties. So the sorts of things being attributed are not the sorts of things which can be named. Yet, Frege argued, numbers can be named – numbers are abstract individuals, he says, objects not concepts. Hence the third stage of Frege’s analysis of numbers consists in the attempt to find individuals – objects – which could be identified with the numbers. It was this stage of the analysis which resulted in the emergence of modern set theory. For every property, Frege argued, there is a corresponding individual: the extension of that universal, the set of all the things (or all the actual and possible things) which instantiate that universal. Thus, for instance, corresponding to the property of being a property which has two instances, there will be a set of sets which have two members. Modern mathematics has selected different candidates for identification with the natural numbers, but it has followed Frege hook, line and sinker with respect to the broad strategy of identifying numbers, and functions and relations, with sets.

Frege’s legacy has significantly changed the agenda for any theory of universals which, like Plato’s, aspires to do justice to mathematics. It leaves three courses open for exploration. One course is that charted by Quine (1953, 1960), of allowing the existence of sets but not of any other nameable things which might be called universals, nor of any of Frege’s higher-order, unnameable universals. Another course is that of allowing the existence of nameable things other than sets: this is a course charted, for example, by Armstrong (1978). A third course allows also the irreducible significance of higher-order quantification (Boolos 1975; Bigelow and Pargetter 1990).