The problem was discussed under this name by Bertrand Russell, but can be traced back to Plato. In Plato's Sophist, the simplest kind of sentence consists of just a proper name and a universalterm. The name refers to or picks out some individualobject, and the predicate then says something about that individual. The difficulty is to explain how the predicate does this. If, as Plato thinks, the predicate is the name of some universal concept or , how do we explain how the sentence comes to be true or false? If, for example, "Socrates is wise" consists of just a name for Socrates, and a name for the universal concept of Wisdom, how could the sentence be true or false? In either case, the "Socrates" signifies Socrates, and the predicate signifies Wisdom. But the sentence asserts that Socrates is wise. The assertion of wisdom must consist in the assertion of some relation between Socrates and Wisdom. What is this relation? The problem was discussed much later by Francis Bradley. If we assume that a sentence consists of two objects and a relation that connects them, and we represent this by three names, say John, loving, Mary, how do we express the fact that John loves Mary? For "John", "loving" and "Mary" would name the objects they do, even if this were not a fact. This is known as Bradley's regress.
The problem became significant in the early development of set theory. Set membership is a formalrepresentation of the relation between the two parts of the proposition, and there are certain philosophical problems connected with this, as Frege realised when he investigated the distinction between concept and object. Assume that "Shergar is a horse" analyses into what "Shergar" names, and what "is a horse" names. Objects are fundamentally different from concepts, otherwise we get the problem of the unity of the proposition. A predicate cannot function as the subject of a sentence. But what are we doing when we talk about the concept is a horse? Aren't we using the expression "the concept is a horse", and isn't that a subject expression, which refers to an Object? Yes, says Frege, and on that account the concept is a horse is not a concept at all. This is a dogma that even Frege's most faithful followers found difficult to swallow. The difficulty was discussed in detail in The Principles of Mathematics by Russell, who saw no resolution. Consider e.g. "A differs from B". The constituents of this proposition are simply A, difference and B. The proposition relates A and B, using the words "is... from" in "A is different from B". But if we represent this contribution by words for relations, as e.g. "A difference B" we are back to a list of terms, we are essentially back at Bradley's regress. Ludwig Wittgenstein addresses the problem early on in the Tractatus Logico-Philosophicus. In section 2.01 he claims that "states of affairs" are combinations of objects. In section 2.03 he explains that nothing is needed to link the objects, since the objects hang together. The arrangement of words that in the sentence corresponds to the arrangement or structure of objects in the state of affairs expressed by the sentence. This is the so-called picture theory of the proposition.
