At first glance the claim of empiricism that knowledge must come from sense experience seems obvious: How else could one hope to make contact with the world around one? Consequently, rationalism has been sharply challenged—in the 19th century by the empiricism of John Stuart Mill (1806–73) and in the 20th by that of the logical positivists, among other movements. Mill argued that all a priori certainties are illusory: Why do people believe, for example, that two straight lines cannot enclose a space? Is it because they see it as logically necessary? No, it is because they have experienced so long and so unbroken a row of instances of it—a new one whenever they see the corner of a table or the bordering rays of a light beam—that they have formed the habit of thinking in this way and are now unable to break it. A priori propositions, Mill claimed, are merely empirical statements of very high generality.

This theory has now been abandoned by most empiricists themselves. Its implication that such statements as “2 + 2 = 4” are only probably true and may have exceptions has proved quite unconvincing. The rationalist’s rejoinder is that one cannot, no matter how hard one tries, conceive 2 + 2 as making 5, for its equaling 4 is necessary. But a priori knowledge is also universal. Neither of these two characteristics can be accounted for by sense experience. That a crow is black can be perceived, but not that it must be black or that crows will always be black; no run of perceptions, however long, could assure us of such truths. On the other hand, a priori truths can be seen with certainty—that if a figure, for instance, is a plane triangle within a Euclidean space, its angles must and always will equal two right angles.

One of the most formidable challenges to rationalism came in the 20th century from such logical positivists as the Oxford empiricist A.J. Ayer (1910–89) and Rudolf Carnap (1891–1970), who had been a central figure in the Vienna Circle, where this movement first arose. Unlike Mill, they accepted a priori knowledge as certain; but they laid down a new challenge—the denial of its philosophical importance. A priori propositions, they said, are (1) linguistic, (2) conventional, and (3) analytic: (1) They are statements primarily of how one proposes to use words; if one says that “a straight line is the shortest line between two points,” this merely reports one’s definition of “straight” and declares one’s purpose to use it only of the shortest. (2) Being a definition, such a statement expresses a convention to which there are alternatives; it may be defined in terms of the paths of light rays if one chooses. (3) The statement is analytic in that it merely repeats in its predicate a part or the whole of the subject term and hence tells nothing new; it is not a statement about nature but about meanings only. And since rationalistic systems depend throughout upon statements of this kind, their importance is illusory.

To this clear challenge some leading rationalists have replied as follows: (1) positivists have confused real with verbal definition. A verbal definition does indeed state what a word means; but a real definition states what an object is, and the thought of a straight line is the thought of an object, not of words. (2) The positivists have confused conventions in thought with conventions in language. One is free to vary the language in which a proposition is expressed but not the proposition itself. Start with the concept of a straight line, and there is no alternative to accepting it as the shortest. (3) Some a priori statements are admittedly analytic, but many are not. In “whatever is coloured is extended,” colour and extension are two different concepts of which the first entails the second but is not identical with it in whole or part. Contemporary rationalists therefore hold that the a priori has emerged victorious from the empiricists’ efforts to discredit such knowledge and the positivists’ attempts to trivialize it.