What is the use of analytic propositions?

Question

[Note: I'm using Kant's terms.]

Analytic proposition, for example "all bodies are extended", or "a=a", seems like an unhelpful type of proposition (contra to the synthetic a priori and a postriori propositions). What do we get from analyzing the concept in the terms we already know of it? It seems that every bit of useful analysis we can get from what can appear to be analytical proposition would in fact be a synthetic proposition ("extendness" is a concept that we synthetically provided to the concept "body", "equalness" is a concept we synthetically provided to the concept "a"; Only after providing those subject concepts with those object concepts, we can provide an analytic proposition that'll simply attach the two together as though they where always one and the same, making the analytical proposition something of a fake synthetic a priori proposition at best).

Answer 1

Analyticity, the a priori and necessity

What does Kant say ? That if a judgment is analytic, its denial will involve a contradiction because the predicate is contained in the subject :

That a body is extended is a proposition that holds a priori and is not empirical. For, before appealing to experience, I have already in the concept of body all the conditions required for my judgment. I have only to extract from it, in accordance with the principle of contradiction, the required predicate, and in so doing can at the same time become conscious of the necessity of the judgment - and that is what experience could never have taught me (B 11-2).

The clearest expression is perhaps the following:

For, if the judgment is analytic, whether affirmative or negative, its truth can always be known in accordance with the principle of contradiction (B. 190, italics in original).

The principle of contradiction must therefore be recognised as being the universal and completely sufficient principle of all analytic knowledge; but beyond the sphere of analytic knowledge it has, as a sufficient criterion of truth, no authority and no field of application (B. 191).

These quotations need to be taken in conjunction with Kant's definition of the principle of contradiction: 'The proposition that no predicate contradictory of a thing can belong to it, is entitled the principle of contradiction, and is a universal, though merely negative, criterion of all truth' (B. 190). To take his own earlier example, to say 'This body is not extended' is contradictory because we are both affirming and denying extension of it. But we know we are doing this because the predicate 'extended' is contained in the subject 'body' as part of its definition; it is because 'All bodies are extended' is an analytic truth that the principle of contradiction manages to get a grasp on the situation. In ordinary cases we have to extract a contradiction by putting together an earlier statement of the speaker with a later one: 'Yesterday you said that it was square, but today you refer to its three sides'. In such cases one of the contradictory statements must be given up, though it is open to us which one to surrender. In the case of analytic judgments we have no option; by using that particular word as the subject we are com- mitted to the definition and this is why a 'merely negative' criterion of truth is applicable. (Anthony Manser, 'How Did Kant Define 'Analytic'?', Analysis, Vol. 28, No. 6 (Jun., 1968), pp. 197-199 : 198.)

The (or a) 'use' of analytic propositions appears to be then that in empirical matters they reveal a necessity which 'experience could never have taught me'. Kant does not deny that I need experience (e.g. of bodies and the meanings of words) in order to formulate an analytic judgement or proposition. When Kant says, 'before appealing to experience, I already have in the concept of body all the conditions required for my judgment', he does not mean 'before having had any experience'; he means 'merely using what experience I have' and without having to make further, extra or special investigative appeal to experience.

Answer 2

Mathematics consists of axiom systems and analytical propositions, nothing more. Every mathematical statement is fundamentally a tautology, saying that those axioms imply this theorem. Mathematics is useful. It enables us to transform systems of synthetic propositions into other non-obvious synthetic propositions.

In general, simple analytical propositions aren't going to be useful, but more complicated ones can be.

In the case of Kant's example of a body, a body is defined in a certain way, and from that we know that it is extended. That's not very useful.

If we had minds that could instantly grasp all the consequences of something, math wouldn't be useful. That isn't the case.

Suppose we are going from point A to point B in a vehicle that can maintain a certain speed. Now, the distance is a synthetic proposition, as is the speed. We use the analytic proposition that time taken to traverse a distance at a given speed is the distance divided by the speed to determine how long the trip will take. That's useful. (It's a very simple example, of course.)