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In the Stanford Encyclopedia of Philosophy's entry on Karl Popper, several steps of Popper's deductive approach are enumerated. I understand all of these but the second:

The second step is semi-formal, the axiomatising of the theory to distinguish between its empirical and its logical elements. In performing this step the scientist makes the logical form of the theory explicit. Failure to do this can lead to category-mistakes—the scientist ends up asking the wrong questions, and searches for empirical data where none are available. Most scientific theories contain analytic (i.e., a priori) and synthetic elements, and it is necessary to axiomatise them in order to distinguish the two clearly.

What are some concrete examples of

  • "empirical elements" vs "logical elements", and of
  • "analytic elements" vs "synthetic elements"?

How do these differ? What kinds of "category-mistakes" occur when these are not properly distinguished?

  • @MauroALLEGRANZA: Can you provide some "concrete examples" of the elements listed (that's the question)? – orome Jan 15 '17 at 14:55
  • @MauroALLEGRANZA: Also, I think your last quote describes what the reference lists as the first step ("The first is formal, a testing of the internal consistency of the theoretical system to see if it involves any contradictions") and not the step in question here. – orome Jan 15 '17 at 14:57
  • @MauroALLEGRANZA: Sorry but none of this is remotely helpful. Focus on the question as stated: (1) stick with the reference provided and (2) aim at concrete examples. E.g., in the Kepler's theory of planetary orbital paths, what would be an "empirical element"; what would be an example of a "logical element", etc. – orome Jan 15 '17 at 15:09
  • @MauroALLEGRANZA: And a "logical element"? – orome Jan 22 '17 at 12:17
  • @MauroALLEGRANZA: I mean of Kepler (it's helpful to stick to one context). – orome Jan 22 '17 at 14:37
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By 'empirical', Popper means 'a posteriori'. By 'logical', Popper means 'a priori'. The distinction between a priori and a posteriori is a distinction of kinds of truth/falsity rather than kinds of propositions. On the other hand, the analytic-synthetic distinction is a Kantian distinction about the subject-predicate relations of propositions. Examples of all of these will depend on the particular philosophy.

First, I'll give examples distinguishing a posteriori (i.e. empirical) from a priori (i.e. "logical"):

  • If someone tells you that they're happy then it is true a posteriori that someone told you that they're happy.
  • If every natural number is the sum of its preceding whole number and one then it is true a priori that 5=4+1.

Next, is the analytic-synthetic distinction. Analytic statements are those in which the subject's concept strictly implies the predicate's concept, and synthetic statements are those in which the subject's concept does not imply the predicate's concept. I'll give examples distinguishing analytic from synthetic:

  • "The eukaryotic cells are animal" is a synthetic statement, because many non-animal species (e.g. the entire plant & fungi kingdoms) have eukaryotic cells such that the subject "eukaryotic cells" does not imply the predicate "are animal".
  • "Three is a number" is an analytic statement because the subject "three" strictly implies the predicate "number".

Popper is pointing out that, when these are not properly distinguished by scientists, category mistakes like "it was before the beginning of time" or "something came from nothing" can be made. "It was before the beginning of time" is false a priori because "before" has the extension "is at a time that is followed by" such that "before" is in the category "time" and, thus, it is a contradiction for there to be a time before time if the extension of time includes every referent for "before" and "after". "Something came from nothing" is defective in that identifying any referent for "nothing" falls into a paradox of self-reference and, moreover, such a statement usually is a vague way of saying, "Something presently known came from nothing presently known", which is a categorically different idea from "something came from nothing".

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