What are the more complex/interesting examples of synthetic a priori statements?

Question

The usual examples of synthetic a priori statements are – it seems at least since Kant:

• "Nothing can be simultaneously red and green all over"

• 7 + 5 = 12 (or any other basic arithmetic statements).

Are there more complex examples than those two? Or at least a more diverse list of examples?

I'm just baffled by the extreme sparsity of examples here, especially since "synthetic a priori" is a difficult concept to grasp (ok, for analytic a priori the "bachelor" example gets recycled since hundreds of years, too. But at least the concept of analytic a priori is easy to understand).

Are there philosophers who study examples of synthetic a priori more seriously?

Answer 1

The notion of a priori changed a lot since Kant, see Did Kant consider Newtonian mechanics a priori? Today they are seen as potentially fallible, even if not empirical.

The Austrian school, including Brentano’s pupils Stumpf, Husserl and Reinach, and more recently "Manchester three" Mulligan, Simons, and Barry Smith, focused on more immediate and elementary a priori. The idea is that they are a priori because when we try to conceive of a counterexample not only can we not, but we "see" that it is impossible. Barry Smith wrote an interesting essay In Defense of Extreme (Fallibilistic) Apriorism arguing that attempts to do without such a priori invariably end up relying on them in a guise. Polish philosopher Wojciech Zelaniec catalogued a list of prototypical examples of Austrian a priori (see below).

As for more "serious" a priori, the conception was developed by some neo-Kantians (Cassirer, recently Friedman) and logical positivists (Reichenbach, Carnap). These are also fallible and revisable, but they are by no means obvious, and their discovery may require a lot of work and thought. Although they may be established in empirical sciences it does not make sense to call them empirical, because they need to be assumed to enable empirical measurements and their theoretical interpretation to begin with. Friedman classifies them into "coordination principles" connecting theoretical parts of scientific theories to observations (e.g. the law of inertia in classical mechanics, or the equivalence principle in general relativity), and "philosophical meta-principles", that act as extra-empirical selection rules (like locality, causality, gauge invariance, general covariance, etc., in physics). See his Einstein, Kant, and the Relativized A Priori and Dynamics of Reason.

Stjernfelt's Diagrammatology (2007) has a nicely written review chapter on synthetic a priori. Zelaniec's list is quoted from there, names in parantheses are philosophers that discussed the example.

Examples of Austrian a priori

1. every color is extended (Kant, Berkeley, Hume, Husserl, Stumpf)
2. for every two events, if one of them is later than the other, the other is not later than the first one (Pap)
3. if something is beautiful and real, then it is good (Roth)
4. everything red is colored (Chisholm)
5. every three tones are ordered linearly with regard to their pitch (Roth, Husserl, Stumpf)
6. the pleasant is preferable to the unpleasant (Scheler)
7. no surface, if it is red all over, is at the same time green all over (Schlick, Wittgenstein, Russell, Ayer, Pap, (Aristotle))
8. everything that is square has a shape (Chisholm)
9. only good actions can be the object of a duty (Scheler)
10. man acts (Hoppe, von Mises, (Aristotle))
11. if any tone-quality is eliminated, a tone-intensity will also be eliminated (Husserl)
12. every promise gives rise to – mutually correlated – claim and obligation (Reinach, Lipps)
13. pink is more like red than black (Austin, similar examples in Locke, Hume, Reinach, Hering)
14. every judgment comprises a presentation within itself (Stegmüller, (Brentano))

Answer 2

"Blue + Yellow = Green."

It's basically a reformulation of 7+5=12, but it helps get the point across for beginners. There's by definition nothing "green" about "blue," nor anything by definition "green" about "yellow," so there's no way to logically deduce this (i.e., it's not a tautology). However, when you add them together you get green, and you get it just as universally as 7+5=12.

And while this may seem rather a posteriori, it doesn't seem any more a posteriori than Austin's example "pink is more like red than black." Furthermore, it also seems to me just as a posteriori as 7+5=12, since one would indeed need to synthesize 7 things with 5 things to get 12, at least once, precisely because the property 12 isn't suggested in either 7 or 5. Were this not the case, 7+5=12 would seem to me just as analytic as "A triangle has three-sides."

Answer 3

This is a misunderstanding of Kant. Although he speaks of '5 + 7 = 12' and 'space has three dimensions' as synthetic a priori statements which might suggest this simplistic idea, Kant aims at something different than proving that there we can determine truth of certain non-trivial statements a priori (of course his examples are ridiculous, neither of these statements is either synthetic or analytic, a posteriori or a priori, because a single statement doesn't even have a meaning of its own - in contexts of applied mathematics 5 + 7 = 12 might be "synthetic a posteriori" if you want to... doesn't really matter).

An interesting feature of Kant's philosophy is that Kant, strictly speaking, doesn't need the notion of analycity in the modern sense - even though he was the one to invent it (Leibniz and Hume had similar ideas but nevertheless). Kant in the preface to the 1787 edition of the Critique of Pure Reason speaks of the intellect constraining itself only by what it itself posits and then in Postulates of Empirical thought in General his explanation of modality doesn't reference thinkability without contradictions like Leibniz's notion of a possible world (which was taken up by the early Wittgenstein in the Tractatus). If we deem all true statements analytic, then thinkability without contradictions in Leibniz's sense constraints what is possible to what is actual. If we follow Quine by claiming that there's no distinction, then only obvious contradictions like "p and ~p" would exclude a world from being possible in the virtue of non-contradictoriness. Leibnizian modality is dependent on analycity.

But Kant isn't satisfied by that because he knows, 200 years before Kripke and thanks to Hume's skepticism, that statements of physics, for example, are not analytic but are anyways necessary. Kant's 'synthetic a priori' is partly due to an apparent confusion of necessity with a priority - it is clear that Kant identifies the two - but for Kant a priority rather should be understood in terms of necessity (Kant's other achievement in this regard is noticing that notwendigkeit exists in the sphere of deontic and aletheic modality, as moral duty and laws of, i.e. physics, although Leibniz apparently noted that earlier in his logical writings, which weren't published until the end of the nineteenth century).

Kant's Copernican Revolution then follows due to the need to explain necessity in terms of the intellect positing laws as conditions of experience which is then understood as being equally an objective constraint on the objects of experience: "the conditions of the possibility of experience in general are likewise conditions of the possibility of the objects of experience" (A158/B197). Postulates of Empirical Thought in General is where Kant very clearly outlines this intent. This system of subjective but equally objective posits which is elaborated in the Transcendental Analytic can be equally understood as an answer to the question of How is determinate (contentful) use of logical notions (in judging) possible? which is nothing but How are synthetic a priori judgements possible?... or, in other words, How do apparent mere-thought-determinations of logic relate to the world?.

That's why Kant claims the synthetic a priori question unifies his whole critical project - because it does, the originality of the question is not due to the fact that the pre-Kantians didn't think substantial a priori knowledge was possible (Kant himself thought synthetic a priori is merely formal... its not some magical intuition) but rather because it comprises of numerous questions which weren't asked before Kant. Kant's own answer - given fully in the schematism chapter of the Analytic - relies, of course, on time, which Heidegger unsurprisingly greatly appreciated.

The American philosopher Wilfrid Sellars's account of modality is similar to Kant's. It makes no reference to possible worlds but understands necessity and possibility as metalinguistic constraints on our application of concepts - this is a Carnapian idea... which is interesting because Carnap was taught by neo-Kantians.