Fregean incomplete expressions

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 non-pathological 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 ξ-notation (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 truth-values, H. (As is well known in his later writings Frege thought that truth-values 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:

1. B ⊂ T.
2. 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.