the simple statement that the ring is lying on the book : the structure which is represented in the proposition is just one feature of the fact. All facts (and there is an infinity of them) which have this feature in common would make the proposition true. The range of facts described by a certain proposition is larger or smaller according to the nature of the proposition. If I say, "the ring is lying in the middle of the top-cover of the book" the range is smaller; if I say "the ring is lying on the book or on the table" it becomes still larger. The proposition "the ring is lying on the book or somewhere else" would describe still a great many more possibilities, and, finally, the proposition "there is a ring and there is no ring" embraces all possible facts, it is always true. If it is always true whatever the facts may be, it must be a priori, and, as a matter of fact, if we look at it a little more closely, we recognize it as a tautology. We see: a tautology or an analytic proposition is a boundary case of a proposition when the range of facts with which it is compatible embraces all possibilities, or, we may say, the whole world (Wittgenstein). In this case the proposition ceases to express anything; it is true, not because the structure corresponds to a particular range of facts in the world, but because it does not point to any particular fact at all. It is true by virtue of its own structure, or, in the language of old-fashioned logic, it possesses "formal truth" only, where as a synthetic propositions has "material truth", it expresses an actual fact.
Tautologies (or analytic judgments) are the only propositions a priori, they have absolute validity, but they own it to their own form, not to a correspondence to facts, they tell us nothing about the world, they represent simply structures.
Kant had seen correctly, although rather vaguely, that if a proposition is valid a priori it must own its validity to the form of knowledge, not to its material, because our understanding cannot possibly know beforehand what material will present itself to the mind in experience, while it might very well impress its own form on any material. Thus he concluded that the synthetic judgments a priori, in which he believed, had the ground of their validity in the forms of our reason (the categories) and of our intuition (space and time). But alas! he failed to see that even the whole complicated apparatus of cognition which he invented could not explain the possibility of synthetic propositions a priori; and he was unable to realise this because his "forms"
