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."