Gottlob Frege, answered from the texts and cited to the page.
Concepts and objects are categorially distinct entities. An object is saturated (gesättigt) — complete, self-standing, capable of figuring as an argument to a function. A concept is unsaturated (ungesättigt) — incomplete, function-like, requiring an argument to yield a truth-value. The grammatical sign of the distinction is that an object-expression is a saturated singular term ("Socrates," "the morning star") while a concept-expression is an unsaturated predicate ("…is a man," "…is a horse").12
The function/argument analysis of propositions replaces the older subject/predicate structure: a proposition Fa is the value of the concept F (function) for the argument a (object). The famous Concept Horse paradox — that the saturated phrase "the concept horse" cannot itself refer to a concept, because referring to a concept by means of a saturated singular term has turned the concept into an object — is a feature of the analysis, not a refutation: it reflects the strict requirement that the categorial distinction be preserved at the level of the symbolic apparatus, even where ordinary language is willing to blur the boundary.34
In the first three examples, on the other hand, 'is' is used like the 'equals' sign in arithmetic, to express an equation.^ In the sentence 'The morning star is Venus', we have two proper names, 'morning star' and 'Venus', for the same object. In the sentence 'the morning star is a planet' we have a proper name, 'the morning star', and a concept-word, 'planet'.Collected Papers on Mathematics, Logic, and Philosophy, p. 190
Of course two complete wholes can stand in a relation to one another; but then this relation is a third element—and one that is doubly unsaturated! In the case of a concept we can also call the unsaturatedness its predicative nature.Posthumous Writings, pp. 191–192
A logical connection into a whole can come about only through this, that an unsaturated part is saturated or completed by one or more parts.Collected Papers on Mathematics, Logic, and Philosophy, p. 288
But nobody can require that my mode of expression shall agree with Kerry's. It must indeed by recognized that here we are confronted by an awkwardness of language, which I admit cannot be avoided, if we say 197 that the concept horse is not a concept,® whereas, e.g., the city (of Berlin is a city, and the volcano Vesuvius is a volcano. Language is here in a predicament that justifies the departure from custom.Collected Papers on Mathematics, Logic, and Philosophy, pp. 192–193
The distinction is foundational for Frege's mature logical and semantic apparatus and for the function/argument analysis of propositions that founded modern quantification theory. The Concept Horse paradox, first published in On Concept and Object (1892), has been continuously discussed in the secondary literature. Wittgenstein's Tractatus §4.126 grants the underlying point and resolves it through the showing/saying distinction (the unsaturatedness of a concept can be shown by the use of an unsaturated symbol but cannot be said in a saturated sentence). Russell discusses the paradox in Principles of Mathematics (1903) §49. Geach defends the distinction in Reference and Generality (1962/1980) and argues that the paradox reveals a deep truth about predication. Dummett's "Incompleteness of Concepts and Functions" (FPL Chapter 8) reads the paradox as a real puzzle that can be resolved by distinguishing levels — the predicate refers to a concept; the saturated phrase "the concept horse" refers to a related but distinct object (the extension or value-range). Strawson's Subject and Predicate (1974) develops an alternative analysis. Wright's "Why Frege Does Not Deserve His Grain of Salt" (1998) defends Frege against the suggestion that the paradox shows the distinction itself to be incoherent.