# 1. Hans Herzberger and the inexpressible

Hans Herzberger as a philosopher and logician has shown deep interest both in the philosophy of Gottlob Frege, and in the topic of the inexpressible and the ineffable. In the fall of 1982, he taught at the University of Toronto, together with André Gombay, a course on Frege's metaphysics, philosophy of language, and foundations of arithmetic. Again, in the fall of 1986, he taught a seminar on the philosophy of language that dealt with 'the limits of discursive symbolism in several domains of human experience.' The course description continues by saying: 'Special attention will be given to the paradoxes underlying various doctrines of the inexpressible and the tensions inherent in those paradoxes. Some doctrines of semantic, ethical and religious mysticism will be critically examined.'

It seems appropriate in an essay written to honour my old colleague and friend that I discuss a doctrine of the semantically inexpressible, namely Frege's theory of incomplete expressions. I propose a simple, minimalistic interpretation of what Frege might have meant by 'unsaturated' entities, but then go on to argue that Frege's associated doctrine that these entities are unnameable cannot be sustained.

# 2. Frege's doctrine of incomplete entities

Frege's logical universe is made up of two fundamentally different types of entity, functions and objects; this division reflects on the ontological level a syntactical division. The essence of a function, Frege tells us in his essay *Function and Concept*, is a syntactical expression with gaps in it, gaps that are to be filled with names. Thus the **[End Page 119]** arithmetical expressions '2 · 1^{3} + 1', '2 · 2^{3} + 2', '2 · 4^{3} + 4' represent values of the function given by the variable expression '2 · *x ^{3}* +

*x'*, and the 'essential peculiarity' of a function is given by the common element of these expressions, a common element indicated by Frege by the gappy expression '2 · ( )

^{3}+ ( )' (Frege 1984, 140).

This syntactical doctrine is both familiar and innocuous. The puzzles in interpreting Frege arise from the fact that he insists that the syntactical division between complete expressions like proper names (that do not need any syntactical supplementation to have a well-determined reference) and 'gappy' expressions containing free variables has an ontological correlate. The functions themselves, that are the referents (*Bedeutungen*) of gappy functional expressions, are themselves gappy. Frege expresses this idea over and over again in his writings. He says in *Function and Concept*: '… a function by itself must be called incomplete, in need of supplementation, or "unsaturated." And in this respect functions differ fundamentally from numbers' (Frege 1984, 140). In the first section of the *Grundgesetze der Arithmetik*, vol. 1, entitled 'The function is unsaturated,' Frege writes:

The essence of the function manifests itself rather in the connection it establishes between the numbers whose signs we put for '

x' and the numbers that then appear as denotations of our expression — a connection intuitively represented in the course of the curve whose equation in rectangular coordinates is 'y= (2 + 3x^{2})x.' Accordingly the essence of thefunctionlies in that part of the expression which is there over and above the 'x.' The expression fora functionisin need of completion, unsaturated. Thus the argument is not to be counted a part ofthe function, but serves to complete the function, which in itself isunsaturated

Again, in *What is a Function?*, Frege writes:

The peculiarity of functional signs, which we here called 'unsaturatedness,' naturally has something answering to it in the functions themselves. They too may be called 'unsaturated,' and in this way we mark them out as fundamentally different from numbers. Of course this is no definition; but likewise none is here possible. I must confine myself to hinting at what I have in mind by means of a metaphorical

[End Page 20]expression, and here I rely on my reader's agreeing to meet me halfway.

There is no real difficulty in insisting, as Frege does, that functional expressions, just like proper names, have both sense and reference (*Sinn und Bedeutung*). The hard puzzles, both for Frege and...