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Consider the following statement:

Cats have four legs

This is an analytic statement, since its truth can be established by looking at the subject and predicate; one does not need to look further into the world to establish it.

Its reverse:

What has four legs is a cat

Is synthetic, since we need to look at the world to establish its truth value, which, in fact, is false.

Now consider the following statement:

What can be proved is true

Going by the usual understanding of proof and truth, one can say that this is an analytic proposition, and true.

Its reverse:

What is true can be proved

Seems more problematic; is it synthetic or analytic? And if so, why?

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  • Speaking of Kant, I would say that it is problematic to speak of synthetic and analytic propositions (non-epistemic) instead of synthetic and analytic judgements (epistemic) in the first place. What you present here seems to be posited rather in terms of Frege, isn't it?
    – Philip Klöcking ♦
    Aug 16, 2016 at 8:59
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    Post-Kantian versions of analyticity were already relativized to something like Carnap's "linguistic frameworks", so what is analytic depends on what use of terms is accepted within a framework. "Truth is provability" is essentially the definition of truth in intuitionism, so to them it is analytic. But realists of course reject it, and accept unprovable truths (like facts about what Aristotle did on his 17th birthday). Quine complained that he still coudn't tell if "green is extended" is analytic or synthetic, and Grice-Strawson responded that there are always grey areas.
    – Conifold
    Aug 16, 2016 at 22:12
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    @mobileink That was Quine's original contention, but only in the general sense of Analyticity that applies to arbitrary frameworks. Within a framework its users are free to designate whatever claims they wish as "analytic" by fiat, as long as they can more or less agree on it. The designation may be vague, but if people keep using the term a pragmatist must take note of a practice as it exists, analyticity for practical purposes is as fine as the fruit/vegetable distinction. Quine's pragmatic description of analytic sentences is that their truth is learned by learning to use words in them.
    – Conifold
    Aug 18, 2016 at 0:44
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    @mobileink: your critique doesn't work: you say that 'batchelor' is a standard example of an analytic concept, and when I put it to you that I expected people to read 'cat' as 'a healthy cat', you say this is another way of saying 'all four-legged cats have four legs'; exactly, so this is no different from saying 'batchelors' are another way of saying 'unmarried men'; this simply shows that you're disturbed merely by rephrasing a standard example, and hardly touches upon the nub of the question at all.
    – Mozibur Ullah
    Aug 20, 2016 at 7:36
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    @mobileink: Not really, you're muddying the issue with a lot of virtual ink.
    – Mozibur Ullah
    Aug 23, 2016 at 12:53

3 Answers 3

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I believe there are a few problems with the two statements you consider to be analytic:

1) Cats have four legs -- Cats with serious injuries and cats born with deformities are still cats. Even if we pretend that injuries and deformities are somehow irrelevant, or if we imagine that the only cats that exist are fully-formed and healthy, there is still no concept of "4" in "cat". Still, if we insist that a thoroughly robust concept of cat must contain the notion of 'having four legs', we'd be relying on a very quirky understanding of analyticity; one that allows analytic statements to be falsifiable (for example, if a previously-pristine feline undergoes an amputation).

We might as well claim that such an amputee STILL HAS FOUR LEGS (but perhaps in a weaker sense) in order to inject all the desired analyticity into unsuspecting felines, but why in the world would we want to do that?

2) What can be proved is true -- This seems very wrong, primarily because we can prove something false by establishing that a contradiction was obtained. "Proof" might commonly refer to formal or empirical demonstrations aimed to fortify alleged veridicality, but we can also construct proofs that successfully establish that something is not the case. Sure, one might suggest that a valid proof of contradiction (and therefore falsity) results in a judgment like: "it's TRUE that x is false" but this doesn't really get us anywhere closer to the original goal, for reasons articulated by Kant a couple years ago.

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This is a bit old. But not the only problematic aspect of Kant's philosophy that I encountered after reading several introductions to his philosophy (from Stephen Korner's On Kant to several Stanford Encyclopedia entries), was his distinction between analytic and synthetic judgments, as admitted by the commentators themselves.

Analytic according to Kant is where the predicate "inheres" in the subject rendering the judgment true by definition. A synthetic on other hand is when the predicate doesn't inhere in the subject and therefore requires proof.

But the above theory is problematic at least in two respects.

First if our definitions follow the Aristotle's theory where concepts are defined by their genus and differentia, then any concept defined as such would have its genus and differentia "inhered" in it! So if you define a cat as "a carnivorous with ability for ambush or stalking and short pursuit hunting" (as per WP:Felidae) then all concepts used in the definition would "inhere" in it, and hence the judgement corresponding to it would be analytic by Kant's definition.

But this does not seem to be what Kant seems to have suggested by his definition of analytic judgement since he confines all analytic judgments to a priori.

However even with a priori qualification another problem rises which I call the phase problem of Kant's theory. This consists in the fact that once any empirical judgement is made, we no longer require an empirical investigation for expressing the same judgement afterwards, because then the same judgement would be ever true by definition which then leads to the first problem!

So we can conclude that Kant couldn't properly frame and capture his concepts of analytic and synthetic judgments. And this I read is widely acknowledge by Kant scholars. Coming from a background of acquaintance with Muslim philosophy I've found their logical and epistemological theories far more nuanced and sophisticated than Kant. However the Modern Analytic philosophy must have introduced many improvements to Kant's tradition.

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"Cats have four legs" is obviously not analytic! You could amputate all the legs in all the cats in the world and they would still be cats. So your first premise/example is wrong. That made me stop reading because you have not thought your question thru.

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