  The approach to language analysis developed by Frege rests essentially on the idea of a strict discrimination between two main kinds of expressions: proper names (singular terms) and functional expressions. Proper names designate (signify, denote, or refer to) singular objects, and functional expressions designate (signify, denote, or refer to) functions. [Note: In the literature, the expressions ‘designation’, ‘signification’, ‘denotation’, and ‘reference’ are usually taken to be synonymous. This practice is used throughout the present entry.] The name ‘Ukraine’, for example, refers to a certain country, and the expression ‘the capital of’ denotes a one-place function from countries to cities, in particular, a function that maps Ukraine to Kyiv (Kiev). Whereas names are “saturated” (complete) expressions, functional expressions are “unsaturated” (incomplete) and may be saturated by applying them to names, producing in this way new names. Similarly, the objects to which singular terms refer are saturated and the functions denoted by functional expression are unsaturated. Names to which a functional expression can be applied are called the arguments of this functional expression, and entities to which a function can be applied are called the arguments of this function. The object which serves as the reference for the name generated by an application of a functional expression to its arguments is called the value of the function for these arguments. Particularly, the above mentioned functional expression ‘the capital of’ remains incomplete until applied to some name. An application of the function denoted by ‘the capital of’ to Ukraine (as an argument) returns Kyiv as the object denoted by the compound expression ‘the capital of Ukraine’ which, according to Frege, is a proper name of Kyiv. Note that Frege distinguishes between an \(n\)-place function \(f\) as an unsaturated entity that can be completed by and applied to arguments \(a_1\),…, \(a_n\) and its course of values, which can be seen as the set-theoretic representation of this function: the set