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Carnap on empirical significance

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Carnap’s search for a criterion of empirical significance is usually considered a failure. I argue that the results from two out of his three different approaches are at the very least problematic, but that one approach led to success. Carnap’s criterion of translatability into logical syntax is too vague to allow for definite results. His criteria for terms—introducibility by chains of reduction sentences and his criterion from “The Methodological Character of Theoretical Concepts”—are almost trivial and have no clear relation to the empirical significance of sentences. However, his criteria for sentences—translatability, verifiability, falsifiability, confirmability—are usable, and under the assumptions needed for the Carnap sentence approach, verifiability, falsifiability, and translatability become equivalent. As a result of the Carnap sentence approach, metaphysics is rendered analytic.

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Fig. 1


  1. 1.

    More or less in keeping with Carnap’s and common terminology, I will use ‘term’ synonymously with ‘non-logical symbol’.

  2. 2.

    Arthur Pap translates the title more loosely as “The Elimination of Metaphysics Through Logical Analysis of Language” (Carnap 1959b).

  3. 3.

    Rolf A. George’s translation of the Aufbau (Carnap 1928a) uses the more literal translations ‘object sphere’ for ‘type’ (Carnap 1967a, §29), and ‘confusion of spheres’ for ‘type confusion’ (§30).

  4. 4.

    Note that the translation into logical syntax is different from the translation into the formal mode of speech (Carnap 1934, §74): Logical syntax is used to phrase object-sentences, which express claims of the empirical sciences, while the formal mode of speech is used to phrase syntactical sentences, which express philosophical claims.

  5. 5.

    “What is to be investigated is being only and—nothing else; being alone and further—nothing; solely being, and beyond being—nothing. What about this Nothing?” (Carnap 1931b, p. 69; cf. Heidegger 1931, pp. 9–10).

  6. 6.

    Marhenke (1949/1950) comes to the opposite conclusion because he ignores the problems with the informal translatability condition and, contrary to Carnap, assumes that inference rules are only applicable to significant sentences.

  7. 7.

    Even in Carnap’s early works, however, his later position occurs as an undercurrent (Lutz 2012, §3.6.1).

  8. 8.

    He also claims that it is in general possible to translate all scientific sentences into a variety of different basic languages, for example the physical language (Carnap 1928a, §§54–60).

  9. 9.

    Carnap (1928a, §179) of course does not think that metaphysics in the sense of “basic knowledge”, i. e., “logical, experiential, constructional order” is non-significant. In this case, however, he suggests using the terms ‘basic science’ or ‘cosmology’.

  10. 10.

    “Aber die logische Analyse kommt zu dem Ergebnis [ ...], daß die sog. metaphysischen Sätze Scheinsätze sind, da sie in keinem Ableitungsverhältnis (weder einem positiven noch einem negativen) zu den Sätzen der Protokollsprache stehen.”

  11. 11.

    “Erlebnisse, Wahrnehmungen, aber auch Gefühle, Gedanken usw.”

  12. 12.

    This terminology follows that of Hempel (1950, pp. 45–48).

  13. 13.

    As Justus’s criticism (2011, pp. 429–430) of a criterion by Sober (2008, §2.14) shows, this dilemma is still under discussion.

  14. 14.

    Note that this puts additional restrictions on protocols.

  15. 15.

    For the ground of \(S\), the equivalence can be expressed only in languages that allow disjunctions with infinitely many disjuncts.

  16. 16.

    There is an additional wrinkle whose implications I will not discuss in the following: According to Carnap (1932), it was Neurath (1932) who first suggested treating protocol sentences and system-sentences as from the same language, which suggests that Carnap assumed in earlier discussions that the protocol sentences are in one language, the remaining (“system-”)sentences in another. But Carnap already assumed in the Aufbau that all scientific terms are definable in basic terms, and thus the basic sentences are a subset of the scientific language. Thus it is not clear in what way Neurath’s suggestion was novel.

  17. 17.

    He adds: “If by verification is meant a definitive and final establishment of truth, then no (synthetic) sentence is ever verifiable, as we shall see. We can only confirm a sentence more and more.” As far as the deducibility from \(\fancyscript{B}\)-sentences is concerned, Carnap’s argument for his claim is essentially that universally quantified sentences cannot be verified. Carnap’s more general point is that \(\fancyscript{B}\)-sentences themselves are never absolutely secure and thus also holds for falsifiability (Carnap 1936, §6).

  18. 18.

    This is what Carnap (1936, pp. 433–434) seems to assume.

  19. 19.

    I thank Pierre Wagner for very helpful discussions of this publication of Carnap’s.

  20. 20.

    Carnap (1950a, §87) discusses the conditions at length.

  21. 21.

    Note the difference to Carnap’s recursive condition in “Testability and Meaning”, where the background assumptions are fixed, but (essentially) the set \(\fancyscript{B}\) changes in each recursive step.

  22. 22.

    This is not true for the strongest criterion, translatability: Assume \(\sigma \) can be translated into \(\fancyscript{B}\)-sentence \(\beta \) by \(\vartheta \) and \(\vartheta \) can be translated into \(\fancyscript{B}\)-sentence \(\beta '\) without background assumptions. If \(\fancyscript{B}\) is only restricted by the terms it contains, \(\vartheta \) is then a \(\fancyscript{B}\)-sentence and therefore \(\sigma \) itself is a \(\fancyscript{B}\)-sentence. In this case, a definition of empirical significance as recursive translatability is safe, but also pointless.

  23. 23.

    Carnap (1928a, §38) also discusses the need for “definitions in use”. As far as terms (i. e., non-logical symbols) are concerned, these are equivalent to explicit definitions because of the eliminability theorems (cf. Gupta 2009, §2.3).

  24. 24.

    The definitions of the definability of constant or function symbols additionally contain uniqueness conditions for the constants and function values, respectively. The conditions are philosophically interesting because they introduce restrictions on the sets of sentences in which constant and function symbols can be defined (Essler 1982, §14, §15; Hodges 1993, p. 59), but also introduce technical subtleties that would lead the current discussion too far afield.

  25. 25.

    Carnap also defines single reduction sentences, but these will not be relevant in the following. As in the Aufbau, Carnap (1936, §16) assumes that there are different “sufficient bases” for the reduction of scientific terms.

  26. 26.

    Carnap uses ‘reducible’ for another property of predicates. See the end of this section.

  27. 27.

    In short, ‘being empirical significant’ is a categorical predicate, while ‘the meaning of’ is a function.

  28. 28.

    Incidentally, Claim 12 can be cheaply proved by choosing \(\vartheta =\{\beta \leftrightarrow \sigma \}\) for any \(\fancyscript{B}\)-sentence \(\beta \) and any contingent sentence \(\sigma \) containing only \(\fancyscript{A}\)-terms. However, since \(\vartheta \) is a claim of purely propositional logic, this may go against the spirit of Carnap’s suggestion.

  29. 29.

    Besides explicit definitions and reduction pairs, Przełęcki (1969, Chaps. 7–8) also discusses systematic weakenings of reduction pairs.

  30. 30.

    This is just a shorthand: \(C\) stays the same, but for the recursive step, it does not matter whether a term is in \(C\) or introducible through a chain of reduction pairs on the basis of \(C\).

  31. 31.

    There could, of course, again be modifications of Carnap’s recursive step that do not trivialize his criterion, but since even the recursion base is problematic, the search may not be worth the effort.

  32. 32.

    Presumably because according to Carnap (1936, p. 447, thm. 7), it ensures that the abstract terms are reducible to elementary terms.

  33. 33.

    Each abstract term is an inus condition for \(B\) (Mackie 1980, p. 62).

  34. 34.

    The assumption that \(\vartheta \) is a conjunction is not essential.

  35. 35.

    Since Carnap (1956, pp. 51, 62) assumes that logical implication is given by semantic entailment, I use ‘\(\vDash \)’ rather than ‘\(\vdash \)’.

  36. 36.

    Note that here, too, it is the set \(K\) of terms that is recursively defined, not \(\vartheta \) (here: \(T.C\)).

  37. 37.

    Formula (3) is closely related to the method of “de-Occamization” suggested by Van Cleve (reported by Creath 1976, 397). Using Claim 14, it is easy to see that in formula (3), none of the \(\fancyscript{A}\)-terms is Carnap-significant relative to \(\varnothing \).

  38. 38.

    However, the recursion base of Creath’s criterion seems to me more promising than the full definition, because it already allows for determining whether a whole set of terms is significant. It thus could arguably be a non-trivial substitute for Carnap’s recursive definition.

  39. 39.

    In Tarski’s fortunate case, the necessary condition is so strong that it provides a sufficient condition as well.

  40. 40.

    In “Testability and Meaning”, Carnap (1937, p. 34) states that in confirmable sentences (which are used to define the principle of empiricism), “[p]redicates which are confirmable [ ...] are admitted”, but this does not mean that only confirmable predicates are admitted. As far as I can tell, Carnap (1936, p. 457, thm. 10) also proves only that a sentence containing exclusively confirmable predicates is itself confirmable. Hempel (1950, p. 51) may have intended his claim that the approach defining significant sentences via significant terms “has its origin in Carnap’s essay, Testability and Meaning” as a historical rather than logical remark. The criterion he discusses (definition 3.2), which demands that a significant sentence must contain only observable terms or those definable in observable terms, is his own. It did not originate with “Testability and Meaning”, as Soames (2003, p. 292) claims.

  41. 41.

    Accordingly, I do not agree with the optimistic evaluation of Carnap’s criterion by Justus (forthcoming), although I am optimistic about some of Carnap’s other criteria.

  42. 42.

    I thank Richard Creath for this point.

  43. 43.

    Besides this tension, the passage quoted above is connected to another lacuna. Carnap follows it up by in effect spelling out the different conditions of his criterion, and concludes with: “On the basis of the preceding considerations, I shall now give definitions for the concept of significance[.]” (Carnap 1956, p. 51). But as a first step of an explication, the determination of the explicandum, it is remarkably poor. There is no investigation of actual usage or practically clear cases as Carnap (1950a, p. 4) in his elucidation of explication demands, but rather the use of pure intuition on comparably technical matters of deductive inferences, which one may share or not. Furthermore, Carnap also does not show the criterion’s fruitfulness for the formulation of many universal statements, which is arguably the central requirement of a good explication (p. 7). Finally, Carnap provides a justification for his criterion for terms, but gives no justification at all for his criterion for sentences. (I thank Christopher French for discussions and his spirited defense of Carnap in this matter).

  44. 44.

    Psillos (2000a, §4) argues that Carnap rediscovered the Ramsey sentence while trying to generalize Craig’s theorem to type theory.

  45. 45.

    For the conjunction of two reduction pairs, Carnap (1936, §10) already describes an equivalent method in “Testability and Meaning” (cf. Carnap 1963b, pp. 964–965).

  46. 46.

    That this does not render claims of empirical significance preliminary or knowledge dependent is shown in Sect. 6.1.

  47. 47.

    The criticism by Demopoulos (2007, vii) that, essentially, the Carnap sentence renders a theory analytically equivalent to its content in \(\fancyscript{B}\) is therefore but an expression of Carnap’s conception of empiricism.

  48. 48.

    Whether his interpretation of Thomas Aquinas is correct is irrelevant for my discussion.

  49. 49.

    The term ‘God’ thus must be defined as entailing a first cause.

  50. 50.

    The discussion here and in the following ignores problems stemming from the incompleteness of deduction in higher order logic for Tarski semantics. One could avoid any such problems by switching either to semantic entailment or to Henkin semantics.

  51. 51.

    This strategy is not that uncommon, for example when \(\ulcorner x\ne y\urcorner \) is defined in the metalanguage of a logic as an abbreviation for \(\ulcorner \lnot x=y\urcorner \), without introducing a corresponding axiom into the language.

  52. 52.

    Thus it seems that “pure optatives” (proposals, requests, demands, etc. that do not assert any matters of fact) are analytic, contrary to Carnap’s claims (cf. Carnap 1963b, p. 32.b).

  53. 53.

    The recursion base of the criterion by Creath (1976) may fit the bill.


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A previous version was presented in 2013 at the workshop Carnap on Logic at the Ludwig-Maximilians-Universität München, some aspects were presented in 2013 at the workshop Formal Epistemology and the Legacy of Logical Empiricism at the University of Texas at Austin and in 2012 at the Groningen/Munich Summer School Formal Methods in Philosophy under the title “The Criteria for the Empirical Significance of Terms”. I thank the audiences for helpful comments and discussions, and I especially thank two anonymous reviewers for a multitude of excellent comments, corrections, and suggestions. Research for this article was supported by the Alexander von Humboldt Foundation.

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Lutz, S. Carnap on empirical significance. Synthese 194, 217–252 (2017). https://doi.org/10.1007/s11229-014-0561-8

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  • Empirical significance
  • Carnap
  • Logical empiricism
  • Carnap sentence
  • Verifiability
  • Falsifiability