[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).

2 Answers 2


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.

  • So basically, analytic propositions are used merely to affirm the state (the sum of its general properties) of the subject? Jul 2, 2018 at 10:22
  • In considering an analytic judgement I am 'conscious of the necessity of the judgment - and that is what experience could never have taught me '(B 11-2). The necessity of the judgment is objective; it is only the consciousness of that necessity which is a state of the subject. That is my view of what Kant is saying; I have kept the discussion within the bounds of Kant's Critical philosophy because you indicated at the start that you were using a Kantian framework - or so I understood.
    – Geoffrey Thomas
    Jul 2, 2018 at 12:26
  • yes, I do try to understand it under Kant's philosophy. So analytic judgment gives the objectivity to the properties we subjectively assigned the object? That's actually interesting, but I'm not sure if it's a good proposition, as according to that we can "give" objectivity to any properties we want from the object. So I'm probably mistaken in my understanding. Jul 2, 2018 at 15:19
  • Do you mean that e.g. we subjectively assign extension to body ? Extension is 'contained in' the concept of body : when we say 'body is extended' we learn nothing empirical but we do realise that the analytic statement is necessarily true. The statement acquaints us with necessity which neither the concept of extension nor the concept of body did on its own. Kant can hardly stop us from 'subjectively assigning' properties to objects, nor can anyone else. If analyticity has its proper uses, as in the body/ extension case, it is hardly discredited if 'subjective properties' can be fed into it.
    – Geoffrey Thomas
    Jul 2, 2018 at 17:26
  • 1
    @Yechiam Weiss. Thanks - a rich crop of answers !
    – Geoffrey Thomas
    Jul 3, 2018 at 17:31

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.)

  • There's no way we defined something in a way that was unknown to us, and according to that definition we learned that the definition is true. It's circular argumenting. Aug 1, 2018 at 4:48
  • Excuse me, I'm not following. Definitions, in the mathematical sense, are neither true nor false. In the case of the body, if the term "body" is defined as something that implies extension, then bodies are extended. The appropriate question is whether something is a body by the definition we're using. The original question was not whether analytical propositions were circular arguments, but whether they were useful. Aug 2, 2018 at 20:10
  • the problem is specifically shown in the last sentence in your answer - "body defined in a certain way, from that we know that it is extended": if we define something in a certain way that doesn't make it useful to us that we "know" it's defined that way, because we defined it. We have essentially added no new knowledge on the matter. It's like I can say that a is b, when I don't know what a nor b represents. So now I have a well-defined object, a, that is equal to b. I have gathered no useful knowledge from that statement, unless I know what a and b represent - Aug 2, 2018 at 20:51
  • - and if I do know, that it's not an analytical sentence anymore. Aug 2, 2018 at 20:52
  • Sure. Repeating definitions is not normally useful. Not all analytical sentences are useful. However, I claim that mathematics is useful, and it's nothing but analytical sentences. Aug 3, 2018 at 20:01

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