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My understanding of Logical Positivists is that, following Wittgenstein, they accepted only 2 types of proposition as meaningful:

  1. Propositions based on formal logic (i.e. tautologies)
  2. Empirically verifiable statements

It seems to me that these are restatements of Hume's classification of knowledge into either analytic a priori (relations of ideas) or synthetic a posteriori (matters of fact).

(And Logical Positivism is just Hume on steroids)

My question: Did the Logical Positivists accept Kant's assertion that synthetic a priori knowledge is possible? If they didn't, what was their objection to it?

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The Logical Positivists did not accept synthetic a priori knowledge. They accepted only Hume's Fork, two kinds of knowledge, as you suggested in the question.

Logical Positivism was not a single shared opinion, but a variety of opinions and arguments under a shared general approach. We can take A.J.Ayer's Language, Truth And Logic (1936) as one representative. Ayer argues that all the cases that Kant took to be synthetic a priori are in fact either analytic or a posteriori. Ayer takes Kant to have been misled by the dominance of Euclidean Geometry and of Newtonian mechanics in his time. The discovery of Non Euclidean Geometry, the gradual refutation of Newtonian mechanics and its replacement, have taught us better. Every geometry is, within its own framework, simply a consequence of its axioms, and in that sense analytic. The fit of a geometry to the world, on the other hand, is a purely empirical, hence a posteriori matter. In any case, there is no place for the synthetic a priori any more. Ayer summarizes (p.80):

And thus we are able to dismiss Kant’s transcendental aesthetic without having to bring forward the epistemological difficulties which it is commonly said to involve. For the only argument which can be brought in favour of Kant’s theory is that it alone explains certain ‘facts’. And now we have found that the ‘facts’ which it purports to explain are not facts at all. For while it is true that we have a priori knowledge of necessary propositions, it is not true, as Kant supposed, that any of these necessary propositions are synthetic. They are without exception analytic propositions, or, in other words, tautologies.

  • The same holds for ethics, too. Kant's categorical imperative is a synthetic a priori judgement. No synthetic a priori knowledge, no categorical imperative. This is often left out of the story, but it is the reason for the non-cognitivist stance of most logical empiricists in ethics. (I don't have the time to write an answer myself, hence the comment.) – DBK Apr 30 '15 at 18:52
  • What about Descartes' cogito ergo sum? – user76284 Sep 27 '18 at 4:35
  • @user76284 Ayer, which I have taken to represent the logical positivists, rejected Cogito Ergo Sum, like Hume. An act of thinking does not, according to them, imply the existence of a substance that thinks. – Ram Tobolski Sep 27 '18 at 17:14
  • Thanks. But what about the existence of thoughts themselves? – user76284 Sep 28 '18 at 1:53
  • @user76284 Well the existence of a thought of mine is something that I am simply aware of, and therefore it is known to me, like any knowledge of experience, in synthetic a posteriori way. – Ram Tobolski Sep 28 '18 at 7:34

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