Is "there are synthetic a priori truths" a synthetic a priori truth?

Question

Disregarding any modern objections to the division of synthetic/analytic and a priori/a posteriori, how would one argue for or against this claim, using Kant's definitions and assumptions?

Also, is the answer to this (SE) question and the answer to this question also synthetic a priori? e.g., are all of the following synthetic a priori?

"There are synthetic a priori truths" is synthetic a priori

""There are synthetic a priori truths" is synthetic a priori" is synthetic a priori

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Edit, following some answers: This might be a more precise description of the statement I'm curious about:

There exists at least one SAPT (in our world)

Answer 1

Although Kant does not have the clearest account of some of these propositions in his system, I think he could have formulated them (if the question of their "existence" is posed) in an analytic way, for example in this case:

"There are synthetic a priori truths" = "If there are any truths, these are either analytic or synthetic"/"Every truth is analytic or synthetic"

Now as for individual synthetic a priori truths, the demonstration of their truth requires synthetic means, so in a sense they represent themselves as, "I am a synthetic a priori truth," and arriving at this representation will involve synthesizing their truth.

On the other hand, perhaps we might be able to prove by analysis first that some proposition S is not analytically true if it is true at all, but this will eventually also analytically lead to, "Therefore, S is synthetically a priori true if true at all." Again, discharging the conditional renders a proposition that can be construed ontologically, hence synthetically; but if framed as an existence question, we run into Kant's claim elsewhere that existence claims as such are the purview of experience: a priori we can decide a few vague possibilities and necessities too (vs. the categories of modality), but not absolutely possible or necessary existence itself. So the specific existence of a synthetic a priori truth might seem intuitive from the outside, but perhaps instead it is analytically a posteriori true that some synthetic a priori truth is as it is at all. Kant does not officially allow for analytic aposteriority, and his own anticipation of a Kripkesque semantics for water terms does not frame the empirical relativity of water as analytic; on the other hand, when Kant discusses the historically "famous" reference to "truth, unity, and goodness," he says:

Thus the criterion of the possibility of a conception (not of its object) is the definition of it, in which the unity of the conception, the truth of all that may be immediately deduced from it, and finally, the completeness of what has been thus deduced, constitute the requisites for the reproduction of the whole conception. Thus also, the criterion or test of an hypothesis is the intelligibility of the received principle of explanation, or its unity (without help from any subsidiary hypothesis) – the truth of our deductions from it (consistency with each other and with experience) – and lastly, the completeness of the principle of the explanation of these deductions, which refer to neither more nor less than what was admitted in the hypothesis, restoring analytically and a posteriori, what was cogitated synthetically and a priori [emphasis added].

Answer 2

According to Kant a synthetic a priori truth (SAPT) is

• a true statement,

• obtainable without the corresponding experience

• and not obtainable by only analyzing the meaning of the words.

You ask question Q:

Is the statement ‚There exists at least one SAPT‘ a SAPT?

1.) On one hand, a person, who denies the existence of any SAPT, negates Q. Because he considers ‚There exists at least one SAPT‘ a false statement.

2.) On the other hand, a person, who assumes the existence of at least one SAPT, has first to determine what the statement in Q means, e.g.

It is a SAPT that there exists at least one SAPT.

Hence it means: Without referring to at least one particular SAPT one can prove that there must exist at least one SAPT. In particular:

In our world there must exist at least one SAPT.

I question this statement. Because all of Kant’s SAPT from geometry presuppose the development of mathematical science. And similarly for his SAPT from physics. But I consider the development of geometry and of the science of physics a contingent, not a necessary fact in our world.

Hence I have no argument to affirm Q, but I cannot negate Q. I have to leave open Q in case 2.)

I‘m curious to hear the arguments of those who affirm Q.

Edited: Replaced 'physics' by 'science of physics' to prevent misunderstanding.

Answer 3

The answer seems in the positive to me, assuming Kant's position that there are synthetic a priori truths (e.g. 7+5=12).

First, let's bring forth another answer's criteria for a synthetic a priori truth (SAPT):

• a true statement,

• obtainable without the corresponding experience

• and not obtainable by only analyzing the meaning of the words.

Now let's analyze this statement and check these conditions, one at a time:

In our world there must exist at least one SAPT.

• 1st condition: Kant repeatedly says, sometimes with apparent huge conviction, that some of his statements are synthetic a priori, e.g. 7+5=12. So, we can hold that according to Kant, "there are synthetic a priori truths" is a true statement;
• 2nd condition: it does not seem a posterori, or dependent on experience, because if this were the case we would have a bunch of synthetic a priori truths in Kant's system that would depend on a synthetic a posteriori fact, or depend indirectly on experience. Kant himself uses in the transcendental deductions arguments like this (if A is a necessary condition to B and B is synthetic a priori, A must be a priori);
• 3rd condition: it does not seem analytic, or obtainable through the meaning of words, because it is nowhere in the definition of the concept "synthetic a priori truth" that it exists

If all conditions are met, then we can assert the following:

It is a SAPT that there exists at least one SAPT.

A fourth possible argument is that Kant requires metaphysics to be synthetic a priori, and this judgment is, at least apparently, metaphysical.