It sounds like you have proved that, if we assume there are no a priori synthetic truths then it follows, by proof by contradiction, that there are no a priori synthetic truths.

I must be missing something, because that doesn't sound like progress.

Perhaps you can clarify what it is that you are saying. For a start it's not clear what statement 'the statement' refers to in 1, 2 and 3.

Perhaps you can clarify what it is that you are saying. For a start it's not clear what statement 'the statement' refers to in 1, 2 and 3. — andrewk

I apologize for my lack of clarity. I may be poor writer but it is never my intention to be unclear.

I'm analyzing the statement 'there are no a priori synthetic truths'. I'm asking how is the statement, 'there are no a priori synthetic truths' known (assuming that it is known)? I break down all possibilities given the analytic-synethic, and the a priori-a posteriori distinction. I comment on each of the break down. And in case you missed it, the statement that I keep referring to is 'there are no a priori synthetic truths'.

Is that clearer to you?

Thoughts? — Purple Pond

Statement: A necessitates B

1. A priori analytic? No, it's not deductive according to Hume.
2. A posterori synthetic? No, experience only gives us constant conjunction.
3. A posterori analytic. No, unlike water is H20, causation is not shown to be a necessary relation by experience.

However, doesn't that mean the truth of A necessitating B being a priori synthetic is itself a posterori analytic? After finding that experience doesn't not show it to the be case, we conclude that it must be apriori.

thanks for that.

Given the clarification, it seems to me that if we assume that statement S: "There are no a priori synthetic truths" is true and known, then either it is

- analytic and known a priori; or
- analytic and known a posteriori; or
- synthetic and known a posteriori.

which seems to line up with your 1, 2, 3 (perhaps in a different order).

Like you, I think only the first one is feasible. To say one knows S to be true by mode 1, one would have to have a proof of it. Such a proof would be just a matter of selecting suitable premises and proceeding from there. I see no obstacle in principle to doing that. Kantians may object to some of the premises, but it wouldn't be philosophy if there wasn't an argument over premises.

The key point of interest is your suggestion that if S is true by virtue of 1 then the negation of S must be self-contradictory. Can you explain why you think that? Bear in mind that 'analytic' (in so far as it is defined at all, which is only very loosely and vaguely) is only a property of truths, not of propositions generally, so it doesn't mean anything to say that a proposition is 'analytically false'..

The key point of interest is your suggestion that if S is true by virtue of 1 then the negation of S must be self-contradictory. — andrewk

It's just something that I hear a lot. A proposition's truth is analytic if and only if the negation of that proposition implies a contradiction. To quote the Encyclopedia Britannica:"Some philosophers prefer to define as analytic all statements whose denial would be self-contradictory, and to define the term synthetic as meaning “not analytic.” ". Here's a link.

The trouble with that Britannica version is that 'self-contradictory' is not a defined term. It is not at all clear what it means. I think we can agree that A & ~A is self-contradictory. It is not so immediately clear that A & (B v ~A) & (~B v ~A) is self-contradictory, yet the proposition A & ~A can be deduced from it in a few steps.

We might say that a proposition is self-contradictory if a contradiction (statement of the form P & ~P) can be deduced from it without using any other non-logical axioms. But with that definition, very few statements would be self contradictory and, in particular, the statement 'This bachelor is married' would not be self-contradictory, even though the statement 'No bachelor is married' is used as a canonical example of an analytic truth.

The trouble is that the whole Analytic/Synthetic distinction is a hot mess of poorly defined terms, and attempts to tighten up the definitions just end up dissolving the whole problem.

This bachelor is married' would not be self-contradictory, even though the statement 'No bachelor is married' is used as a canonical example of an analytic truth. — andrewk

I don't understand this. Saying the bachelor is married contradicts the definition. I took the point of contradiction to mean analytical statements have their truth contained in the definition or rules of the respective concept or statement. So we can derive all sorts of mathematical or logical truths that don't rely on facts about the world being added to the mix.

Saying the bachelor is married contradicts the definition. — Marchesk

A contradiction can be deduced from the statement together with additional axioms. But a contradiction cannot be deduced from the statement on its own, so it is not self-contradictory.

Take the definition of bachelor to be:

B is a bachelor at time t iff B has never married at any time <= t and B is adult at time t and B is human at time t and B is male at time t and B is alive at time t

We can deduce 'B is not married' from 'B has never married', but only by using an axiom that says that 'X has never been the case' entails 'X is not now the case'. Or in mathematical terms, that (x=y -> x<=y). That axiom is part of the Peano system of axioms for arithmetic, and is needed to complete the contradiction on the bachelor statement. So the statement is not self-contradictory because it requires additional axioms to reach a contradiction.

My goal in the OP was to point out a problem with knowing the proposition that, 'there are no a priori synthetic truths'. I tried, in vain, to find a problem with each possibility given all the distinctions mentioned. But as you point out the very philosophical distinctions are the problem. I have nothing more to say other than language has defeated me once again.

So the statement is not self-contradictory because it requires additional axioms to reach a contradiction. — andrewk

Okay, but those additional axioms aren't based on facts about the world. They're just further steps in logical reasoning based on the definition given.

The assertion (1) "There are no a priori synthetic truths" can be tested by showing a example of a priori and synthetic truth, as Kant showed. Mathematics theorems are a good example. Thus, (1) is a posteriori and synthetic, due to it refers to something that either exist or not.

Thus, (1) is a posteriori and synthetic, due to it refers to something that either exist or not. — Belter

How about the assertion that, 'there are no even prime numbers'? It refers to something that exists or not. Is that too a posteriori and synthetic?

I agree, and so I would say that the bachelor statement is analytic (ie I would not use the 'negation is self-contradictory' definition of analytic).
But by the same reasoning the statement 7+5=12 is analytic, and Kant denied that it was.

Perhaps it's you and me against Kant. (Only on that issue. I think Kant's fabulous on many other issues)

I have nothing more to say other than language has defeated me once again. — Purple Pond

Say not 'the language has defeated me' but rather 'the language and I failed to reach an agreement'. Personally, I blame the language. :wink:

The Goldbach conjecture, for example, is true a priori, and synthetic. If someone proves it, the conjecture will become into a theorem, and so it will be true a posteriori. A priori and a posteriori in my opinion are able to capture the distinction between predictive and retrospective knowledge.

In my view, Kant's examples of synthetic a priori are not theorems, as I said wrongly in the first commentary. The synthetic a priori fits for example to mathematical conjectures and physics predictions.

The synthetic a priori fits for example to mathematical conjectures and physics predictions. — Belter

Mathematical conjectures are not known a priori, nor a posterori, but they're are guessed. The same is true for some physical predictions. Furthermore, it's possible for mathematical conjectures and physics predictions to turn out to be false, and knowledge includes only true propositions.