Ian Rumfitt's theory of predication

Frege put forward a revolutionary theory of predication which he used to solve the traditional problem of multiple generality. Before Frege, logicians were unable to account for the validity of inferences which involved sentences containing more than one expression of generality. Frege's theory makes use of what he called incomplete or unsaturated expressions. These can be understood in two main ways. They can be thought of as being linguistic functions or as patterns. Geach favours the former interpretation, whereas Dummett favours the latter interpretation. Ian Rumfitt favours the function-interpretation and so do I. Elsewhere on this website I present my version of the function-interpretation. Unfortunately, Ian Rumfitt's account of this has a number of flaws to which I draw attention in my paper "Rumfitt's Theory of Predication". Here, I summarise my main objections to his interpretation of Frege.

Ian Rumfitt's theory cannot account for the validity of inferences involving more than one expression of generality

An inference involving multiple generality that Ian Rumfitt cannot account for — Figure 1. Multiple generality.

Consider the three sentences shown in Fig. 1. Ian Rumfitt's theory cannot account for the validity either of the inference of (B) from (A) or of that of (C) from (B). This is because his formation rules do not allow us to form either the incomplete expression "ξ loves everybody" or the proposition "Somebody loves everybody" containing two expressions of generality. It should be noted that Ian Rumfitt uses the term "proposition" as Geach does and Geach explains how he uses it on p. 139 of his paper "Names and Identity":

As in other works of mine, I use "proposition" in the medieval sense: for a sentence serving, as grammarians say, to express a complete thought, to say what is or is not so, rather than for the thought so expressed.

That Ian Rumfitt's formation rules do not allow us to form the incomplete expression "ξ loves everybody" would not be a disaster if his theory contained something like the extraction-procedure for forming incomplete expressions. This procedure produces an unsaturated expression by removing one or more occurrences of an expression from another expression. For example, the extraction-procedure allows us to remove the proper name "Jack" from the unasserted proposition "Jack loves everybody" to form the incomplete expression "ξ loves everybody". Unfortunately, the way Ian Rumfitt sets up his theory of predication the extraction-procedure cannot be coherently added to it. What I call "the extraction-procedure" is a generalisation of Frege's "second procedure for forming names of first-level functions" (Grundgesetze, p. 47).

For Rumfitt predicables are linguistic functions returning asserted propositions

Ian Rumfitt borrows the term "predicable" from Geach and Geach explains how he uses it on p. 52 of his book Reference and Generality:

A predicable is an expression that gives us a proposition about something if we attach it to another expression that stands for what we are forming the proposition about; the predicable then becomes a predicate, and the other expression becomes its subject; I call such a proposition a predication.

Frege explicitly prohibits the assertion sign being used to form incomplete expressions in the footnote on p. 22 of Funktion und Begriff:

The assertion sign [Urtheilsstrich] cannot be used to construct a functional sign; for it does not serve, in conjunction with other signs, to designate an object. "⊢ 2 + 3 = 5" does not designate anything; it asserts something.

The value of a predicable has to be an unasserted proposition. For Frege, the expression "⊢ 2 + ξ = 5" is not an incomplete expression. I say more about such pathological linguistic functions elsewhere.

An example of an incomplete expression and an example of one that isn't, but which Ian Rumfitt thinks is — Figure 2. Incomplete expressions?

Rumfitt completely ignores combination problems

A simple example of a combination problem concerns conjunction. This is usually defined as an operator that makes a proposition out of two propositions. So defined it cannot be used to make a one-place predicable out of two one-place predicables. However, we sometimes want to use it in that way. That's a combination problem. Potts has studied combination problems extensively in many publications, including his article "Fregean grammar" and his book Structures and Categories for the Representation of Meaning, which Rumfitt reviewed.

The function-interpretation can be rehabilitated in order to overcome these failings as I show in "Rumfitt's Theory of Predication". The version of the function-interpretation that I favour also has other uses. It allows us to provide a coherent account of Fregean incomplete expressions on the linguistic level (as I show in "On the Interpretation of Incomplete Expressions"), in the realm of sense (as I show in "On the Sense of Unsaturated Expressions") and also in the realm of reference (as I show in "Is the Concept Horse an Object?"). What Rumfitt has done is valuable, however, because it makes clear that the full potential of Frege's theory of predication has still not been realised and that it is likely to give rise to further developments in the philosophy of language. Unfortunately, Rumfitt's inaccurate presentation of this theory may delay this potential being realised.

References

Gottlob Frege, Funktion und Begriff, Jena, Pohle, 1891.
Gottlob Frege, Grundgesetze der Arithmetik, volume 1, Jena, Pohle, 1893.
Antoni Diller, Frege's Theory of Functions in Application to Linguistic Structures, PhD thesis, Department of Philosophy, University of Leeds, 1987; a PDF version of my thesis is available on this website.
Antoni Diller, "On the Interpretation of Incomplete Expressions", Logique et Analyse, vol. 36 (1993), pp. 75–104;a PDF version of this paper is available on this website.
Antoni Diller, "On the Sense of Unsaturated Expressions", Philosophical Papers, vol. XXII (1993), pp. 71–79; the full text of this paper is freely available.
Antoni Diller, "Is the Concept Horse an Object?", Modern Logic, vol. 4 (1993), pp. 345–366; the full text of this paper is freely available.
Antoni Diller, "Rumfitt's Theory of Predication". This paper was presented at the Fifth European Congress for Analytic Philosophy (ECAP5) which was held in Lisbon from the 27th to the 31st of August 2005
Peter Thomas Geach, "Names and Identity", in Samuel Guttenplan (ed.), Mind and Language, Oxford, Oxford University Press, 1975, pp. 139–158.
Peter Thomas Geach, Reference and Generality, third edition, Ithaca and London, Cornell University Press, 1980.
Timothy C. Potts, "Fregean grammar", Studia Logica, vol. XXXVII (1978), pp. 7–26.
Timothy C. Potts, Structures and Categories for the Representation of Meaning, Cambridge, Cambridge University Press, 1994.
Ian Rumfitt, "Frege’s theory of predication", The Philosophical Review, vol. 103 (1994), pp. 599–637.
Ian Rumfitt, Review of Potts's Structures and Categories for the Representation of Meaning, The Philosophical Review, vol. 105.2 (April 1996), pp. 264–267.

I have provided a more detailed bibliography of Rumfitt's writings on another page.