Fregean incomplete expressions
My study of Frege has focused on his notion of an unsaturated or incomplete expression. I use Peter Geach's idea of a linguistic function to unpack the notion of an incomplete expression. There follow some of the theses I have argued for.
Incomplete expressions are linguistic functions
A linguistic function is a function whose arguments are either linguistic expressions or linguistic functions and whose values are also either linguistic expressions or linguistic functions. For example, "ξ snores" is that linguistic function which, for any proper name X taken as argument, yields the expression obtained by concatenating X and "snores" with the inclusion of a space between them. Thus, if we apply the function "ξ snores" to the argument "Jack", we get the unasserted proposition "Jack snores".
Not all linguistic functions are unsaturated expressions
It is relatively straightforward to construct pathological linguistic functions. These are genuine linguistic functions, but Frege would not count them as being incomplete expressions. An example of such a function is the one that maps a person's name onto his father's name. This function takes, for example, the arguments "Isaac", "Jacob" and "Judah" onto the values "Abraham", "Isaac" and "Jacob", respectively. The word "therefore" can also be used to construct a pathological function. This makes a sentence out of a list of propositions and a proposition. For example, the sentence "Jack snores, therefore someone snores" can be seen as having been produced from the singleton list containing "Jack snores" and the proposition "someone snores" by means of that linguistic function which takes a list and a proposition and yields the sentence obtained by concatenating the elements of the list, separated by commas if necessary, and terminated with a comma, followed by the word "therefore" followed by the proposition which is the function's second argument. (All the commas involved are followed by a single space.)
Only nonpathological linguistic functions are incomplete expressions
In the light of what Frege says about unsaturated expressions it is possible to formulate the following criterion of demarcation between incomplete expressions and pathological linguistic functions:
An unsaturated expression is a linguistic function that can be represented by means of Frege's ξnotation (or an extension of this in order to cope with functions of higher level or greater arity) and which is capable of having as a referent an entity which is of a type that occurs somewhere in the Fregean hierarchy of types and which is not a basic type.
To complete this account of what counts as a Fregean incomplete expression we just need to specify the Fregean hierarchy of types:
Let B be the set of basic types. It contains the type of objects, J, and that of truthvalues, H. (As is well known in his later writings Frege thought that truthvalues were objects. Here it is not important whether or not we follow him in this.) Then, the set T, consisting of all the types in the Fregean hierarchy, can be defined as follows:
 B ⊂ T.
 If α, β ∈ T, then α → β ∈ T.
An entity which has type α → β is a function whose arguments are drawn from α and whose values are drawn from β.
More information about these theses can be found in my paper "On the Interpretation of Incomplete Expressions" (1993).
Relevant publications
The papers "On the Interpretation of Incomplete Expressions", "On the Sense of Unsaturated Expressions" and "Is the Concept Horse an Object?" are based on my PhD thesis. In the more recent paper "Rumfitt's Theory of Predication" I show that Ian Rumfitt's interpretation of Frege's theory of predication has a number of defects and I argue that my interpretation of Frege's theory is better than Rumfitt's.
 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.
 Kevin Klement criticises my defence of the position that the sense of an incomplete expression is itself a function on pp. 70–72 of his book Frege and the Logic of Sense and Reference, [London, Routledge, 2002, ISBN 0415937906].
 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", in Teresa Marques (ed.), Book of Abstracts ECAP5, [Lisbon, Faculty of Letters, University of Lisbon, 2005], p. 37.
© Antoni Diller (3 July 2018)