In Chapter IV, Ayer deals with a priori knowledge, and his views are still of some interest, not only for their own sake, but also because there is confusion about what logical positivists thought about a priori knowledge and analytic statements.
For Ayer, propositions with “factual content” are essentially synthetic propositions, and synthetic propositions never have necessary and universal truth, but only probable truth (Ayer 1971 : 64–65, 80).
So how, then, does the logical positivist account for the truths of mathematics and logic?
Ayer made it clear that he regarded Mill’s belief that the truths of mathematics and logic were merely very well confirmed inductive generalisations to be unconvincing (Ayer 1971 : 67).
Ayer’s answer is of course that the truths of mathematics and logic are analytic propositions (Ayer 1971 : 71).
Kant had famously defined analytic propositions as those where the sense of the predicate is already contained in the sense of the subject.
But Ayer used an extended definition of analyticity as the key to establishing the epistemological insight that maths and logical truths are analytic. For Ayer,
“a proposition is analytic when its validity depends solely on the definitions of the symbols it contains, and synthetic when its validity is determined by the facts of experience.” (Ayer 1971 : 73).An analytic proposition is thus “entirely devoid of factual content” (Ayer 1971 : 73) in the sense that it says nothing necessarily true about empirical reality, and so no experience is able to refute it.
Nevertheless, analytic proposition have real meaning and sense. Ayer’s crucial passage about analytic propositions is here:
“When we say that analytic propositions are devoid of factual, content, and consequently that they say nothing, we are not suggesting that they are senseless in the way that metaphysical utterances are senseless. For, although they give us no information about any empirical situation, they do enlighten us by illustrating the way in which we use certain symbols. Thus if I say, ‘Nothing can be coloured in different ways at the same time with respect to the same part of itself’, I am not saying anything about the properties of any actual thing; but I am not talking nonsense. I am expressing an analytic proposition, which records our determination to call a colour expanse which differs in quality from a neighbouring colour expanse a different part of a given thing. In other words, I am simply calling attention to the implications of a certain linguistic usage. Similarly, in saying that if all Bretons are Frenchmen, and all Frenchmen Europeans, then all Bretons are Europeans, I am not describing any matter of fact. But I am showing that in the statement that all Bretons are Frenchmen, and all Frenchmen Europeans, the further statement that all Bretons are Europeans is implicitly contained. And I am thereby indicating the convention which governs our usage of the words ‘if’ and ‘all’.So the logical positivist view is that analytic a priori propositions are
We see, then, that there is a sense in which analytic propositions do give us new knowledge. They call attention to linguistic usages, of which we might otherwise not be conscious, and they reveal unsuspected implications in our assertions and beliefs.” (Ayer 1971 : 73–74).
(1) not meaningless or nonsense as “metaphysical” propositions were presumed to be;In sense (3), Ayer was clear that a priori reasoning from analytic statements can yield new knowledge: what Ayer was denying here was that analytic a priori reasoning gives us necessarily true knowledge of the real, external world.
(2) do have real meaning and sense, and
(3) could and do provide human beings with “new knowledge,” such as revealing “unsuspected implications in our assertions and beliefs.”
For Ayer, all necessity is therefore logical de dicto necessity. This view was very influential in modern analytic philosophy until the revolution in analytic metaphysics from the 1970s onwards in the work of Saul Kripke, David Lewis, Hilary Putnam, and Keith Donnellan.
Ayer, A. J. 1971 . Language, Truth and Logic. Penguin Books, London.