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In Proper and Improper concepts (1927) Carnap argued for the distinction between proper concepts (the ones that are explicitly defined)

”It is essential to a proper concept that for any object it is in principle decidable whether the object falls under that concept or not; and the decision can be made in practice given sufficient knowledge of the object.” (Carnap, 1927, p. 13)

and improper concepts (which are implicitly defined by axiomatic systems),

“for an improper concept, the question whether a particular object falls under it is not decidable, and thus has no sense, regardless of how much is known about the object." (Carnap, 1927, p. 13).

I wonder whether this might provide some light on the concepts of information, matter and energy which are quite commonly referred to as fundamental. Each of them seems to be implicitly defined by their corresponding theories (or structurally dependend on the implicit concepts of those theories). For instance, quantum information (von Neumann entropy) is structurally dependent on the axioms of abstract quantum theory (and on the implicit definition of what is a physical system - that which can be assigned state vector in Hilbert space, is measured according to the measurement postulate, can be composed according to the fourth postulate and evolves in time according to the second one). Is my understadning correct - are information, matter and energy improper concepts?

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  • You have to apply Carnap's distinction to theories: consider e.g. Information theory and check if there is a formal definition of information in it or the concept of information is defined only "implicitly" through the axioms of the theory that involves it. Feb 21, 2023 at 13:09
  • See for an example (and maybe the source): Hilbert's axiom for geometry (1899). Feb 21, 2023 at 13:22
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    If those are improper, you should be able to point to objects that can't be assigned as belonging to, or outside of either category. Is this object matter or not? Etc. It also depends on what concept of "matter" or "energy" you use of course. I bet Catnap argued that a concept is not inherently proper or improper, but can be refined until it becomes proper, or to the contrary, become improper after the finding of an unknown object that can't be classified (like "mammal" before discovering the platypus, or "planet" before the Kuiper belt objects were found).
    – armand
    Feb 21, 2023 at 23:25
  • @MauroALLEGRANZA I agree, but consider that quantum theory (QT) rests on implicit definitions of what is a physical system, measurement (improper concepts) ... If you then define von Neumann entropy from these concepts, you will get a "structural" improper concepts, i.e. to decide how much information sth has, that something must firstly satisfy the whole axiomatic system of QT. So, in fact, "the question whether a particular object falls under it is not decidable" for von Neumann entropy.
    – Eauriel
    Feb 26, 2023 at 15:19
  • I mean you can decide the question for mathematical objects, but not for the external ones. For external objects, firstly you need to show that these satisfy axioms (and so proper concept of the whole axiomatic system), and then there is a von Neumann entropy for that which is satisfied.
    – Eauriel
    Feb 26, 2023 at 16:04

2 Answers 2

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I have only skimmed Carnap's paper- I haven't studied it- but my very quick and superficial reading suggests that matter and energy, at least, are not improper concepts. I am inclined to assume that information is not an improper concept, in the sense intended by Carnap, but I have reservations because information is such a broad heading.

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What are the objects that we could decide (or not) would fall under the concepts of information, matter and energy? You seem to imply that information is derived from other concepts - but that doesn't seem to align with Carnap's definition: information would have to be taken as the concept at hand itself, and we would try to determine if information is a proper/improper concept and we would have to decide for particular objects if they fall under that concept of "information".

If we want to make information into an improper concept, we have to show another concept for which we could not decide if information "falls under it". But you show a bunch of "antecedent" concepts to information that seem very well defined, and for which you clearly say that information "falls under" those, so that would make information a proper concept - understanding "falls under" as "derived from" - which I don't think is quite correct. But again, what would be the concept that information would "fall under" (i.e. most likely, be a particular example of, not derived from)?

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