I just started to read about Kant's metaphysical distinction between analytic vs synthetic truths (necessary vs contingent) and his epistemological distinction between a priori vs a posteriori truths. I'm reading about this from a book that explores logic both from a mathematical and a philosophical point of view, although I personally don't have a lot of knowledge of philosophy, I mainly come from mathematics.

So, I understand that Kant claims that all mathematical judgments are both a priori and contingent (synthetic). But I thought I read somewhere that Kant believed also that all and only the a priori truths are necessary and all and only the a posteriori truths are contingent. If it's really true that he believed so, than doesn't that contradict his view about mathematical judgments?

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    Kant was extremely sensitive to the minutiae of modal categories and so it's not that mathematics is contingent full-stop but that it's logically but not metaphysically contingent. Jul 13 at 18:41
  • @KristianBerry I'm a bit confused. This will take me more than one comment to explain: (1) In my book it says that one sometimes speaks of logically necessary and logically contingent truths instead of analytical and synthetic truths to distinguish from the physically necessary and physically contingent truths. So, that means that the logically necessary and the physically necessary truths are two different subcategories of the necessary truths in general (also known as the analytic truths) and the similarly goes to the contingent and synthetic? Jul 13 at 21:12
  • @KristianBerry (2) Also, in the book they say that the necessary contingent distinction is metaphysical, but you say that the logically necessary and logically contingent distinction is not metaphysical, but aren't the logically necessary and logically contingent subcategories of the general necessary and contingent? Does necessary and contingent perhaps are abriviations for physically necessary and contingent? Because when we say that a distinction is metaphysical we mean that it is about the nature of physical reality, right? Jul 13 at 21:16
  • Necessary and contingent have various meanings: plato.stanford.edu/entries/modality-varieties Jul 13 at 21:31
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    @MichaelNovak one of Kant's relevant other distinctions is between general and transcendental logic. Analytical necessity goes with general logic, synthetic necessity goes with transcendental logic (although it is applied in empirical thought via schematism, not in itself). There's a section "The Postulates of Empirical Thought" that details Kant's theory of (meta)physical modality fairly well, although his analysis of the Fourth Antinomy illuminates his viewpoint even more. Jul 13 at 22:10

1 Answer 1


Kant does not claim that “all mathematical judgments are both a priori and contingent (synthetic).”

For Kant, all a priori truths are necessary truths, including a priori synthetic truths.

Prior to Kant the empiricists had believed that what Kant called synthetic judgements were not a priori judgements, that they were empirical judgements. Kant argued that if metaphysics is to be possible, then a priori synthetic knowledge must be possible and the example he gave was that of geometry. Thus, for Kant, geometry is a priori, necessary, and universal - i.e., in no way contingent.

It is important to keep in mind here that Kant is writing at a time when nobody had thought of the idea that there could be other geometries. For Kant there was one geometry - Euclidean geometry - and it was a priori and synthetic.

You may have some confusion about Kant’s use of the word synthetic. Kant thought of analytic judgements as those where the concept of the predicate is contained in the concept of the subject and synthetic judgements as those where this is not the case.

For example, if we define a triangle as a three-sided figure in the plane, then the statement “A triangle has three sides” is analytic since the concept of a triangle contains the concept of it having three sides. On the other hand, the statement “The interior angles of a triangle add up to 180 degrees” is synthetic since our concept of a triangle as a three-sided figure does not include the concept of it having interior angles that sum to 180 degrees.

Rightly or wrongly, I like to think of it like this: a priori synthetic judgements are the synthesis of our concepts and our reasoning, while a posteriori synthetic judgements are the synthesis of our empirical considerations, our concepts, and our reasoning.

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    Thank you very much. After doing dome additional reading, together with your answer, I now understand that my confusion was because this subject is summed rather shortly in only a few paragraphs in my book and it make it seem like analytic and necessary are the same thing and synthetic and contingent are the same thing. While they are two different distinctions: a metaphysical one and a linguistic one. Jul 16 at 0:41
  • The only thing that still I'm not entirely sure about is where does logically necessary and physically necessary fit in. So, as I understand it the physically necessary truths are the broadest of the 3; these are the truths that cannot be false according to the laws of physics. The necessary truths are a subcategory of the physically necessary: they defined simply as the truths cannot be false. And logically necessary are a subcategory of the necessary truths; they are the truths that cannot be false according to the laws of logic. Jul 16 at 0:41
  • Is that an accurate classification of these 3 terms? If so, then I honestly don't really understand, why does physically necessary is more general than necessary? What's the difference between them? If necessary is a metaphysical term, then how do we classify physically necessary? Doesn't metaphysical means about the physical? I don't understand the difference between them. And also, if logically necessary is a subcategory of necessary, then logically necessary is also metaphysical, right? Jul 16 at 0:41
  • @MichaelNovak It is an interesting analysis that you give, but not one which Kant would have entertained. For Kant, all empirical judgements are contingent and knowledge of the physical world as it "truly is" is impossible. All we know is how things appear to us - i.e., our perceptions. E.g., we experience things spatially and temporally because that is how our minds represent the information it receives via our senses. We don't know if the world is truly spatial and temporal. So one might say that for Kant, "physically necessary" is almost a contradiction in terms.
    – nwr
    Jul 16 at 15:33
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    @MichaelNovak Sorry for the confusion. Philosophy is not like mathematics. There is no consensus on where logic fits into philosophy. Some view logic as metaphysics, others view logic as epistemology, while philosophical considerations of "the physical" are rooted in ontology. Qualifiers like necessary and contingent are dependent on the logic being applied and the subject being discussed. The Stanford Encyclopedia of Philosophy offers a good introduction to these concepts as used in philosophy.
    – nwr
    Jul 17 at 1:35

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