Carnapian and Tarskian semantics

Abstract

Many papers have been devoted to the semantic turn Carnap took in the late 1930s after Tarski had explained to him his method for defining truth and his work on the establishment of scientific semantics. Commentators have often argued that the major turn in Carnap’s approach to languages had already been taken in the Logical Syntax of Language, but they have usually assumed that Carnap was happy to subsequently follow Tarski and adopt Tarskian semantics. In this paper, it is argued that this assumption needs to be qualified and that Carnap was actually far from following Tarski when he decided to complement his syntactic method with a semantic one. Carnap and Tarski had different goals, divergent programs, and dissenting views on truth and semantics. After exploring several possible methods for the explication of logical concepts such as L-truth and L-implication, Carnap opted for definitions based on concepts he had found in Wittgenstein’s Tractatus and Waismann’s work on a logical interpretation of probability. Carnap’s reasons for taking a semantic turn are to be understood in the context of his principle of tolerance. He first hoped he could use Tarski’s technique to recover in the semantic setting the completeness result he had tried to establish in the Logical Syntax. His eventual adoption of non-Tarskian semantics can be accounted for by his program of a logical analysis of science—including empirical science—and his ambition to elaborate a unified logical framework for deductive, inductive, and modal logic, and for explicating analyticity.

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Notes

  1. 1.

    LSL” from now on, unless a distinction needs to be made between the German original and the English translation of 1937.

  2. 2.

    A brief summary of Carnap’s technique is given in Niiniluoto (2003, pp. 5–6).

  3. 3.

    See Carnap (1942, p. vii).

  4. 4.

    Another place where the issue is touched upon is a series of conversations on logic, mathematics, science, and especially on finitism, that Carnap, Tarski, Quine, and others had at Harvard during the academic year 1940–1941. The notes Carnap took during these conversations have been preserved and they have recently been translated, published, and studied in Frost-Arnold (2013). Mancosu (2005) also provides an important analysis of these notes. Although the main topics discussed in these notes are finitism, nominalism, and the language of science, issues related to the semantic method are mentioned at some points, such as the relationship between the notions of state of affairs and model. In the whole, these notes also help us have a better understanding of Carnap and Tarski’s disagreements about the topics discussed and more generally about their philosophical ideas.

  5. 5.

    In earlier papers such as Carnap (1936), he had defended the idea of defining truth (as opposed to “confirmation”) against verificationist objections of various stripes, but he had not explained his personal views about semantics yet.

  6. 6.

    It may be remarked that Carnap had already used some concept of model in his Investigations on General Axiomatics, a manuscript which goes back to the late 20s, before Carnap’s so-called “syntactic period”. This manuscript has been posthumously published as Carnap (2000).

  7. 7.

    It is well-known that in the second half of the twentieth century, Tarskian semantics has become standard and this historical point is assumed here as background knowledge. But this by no means implies any Whiggish bias in our analysis of Tarskian and Carnapian semantics. In other words, we of course do not assume that model theory based on Tarski’s work is the correct way of doing semantics and that Carnap was wrongheaded in following his own non-Tarskian way.

  8. 8.

    To give just a very brief sample of these numerous studies, we can mention Feferman and Feferman (2004), Mancosu (2006, 2008, 2010), Patterson (2008, 2012), and Wolenski and Köhler (1998).

  9. 9.

    I am indebted to the anonymous reviewer who pointed out to me that this remark is of crucial importance for understanding the reasons why Carnap does not follow Tarski and chooses one particular “non-Tarskian” semantic method, at the time he decides to take the semantic turn. More on this point below.

  10. 10.

    The issue of fixed- versus variable domain interpretation has been the focus of many recent studies related to Tarski’s notion of logical consequence. For an overview, see Mancosu (2010). This issue has also been raised about the interpretation of Carnap’s so-called “early semantics”; see Schiemer (2013). This particular issue will not be discussed in this paper.

  11. 11.

    For example, here is what Tarski writes about the possibility of drawing a sharp boundary between logical and extra-logical terms: “Perhaps it will be possible to find important objective arguments which will enable us to justify the traditional boundary between logical and extra-logical expressions. But I also consider it quite possible that investigations will bring no positive results in this direction so that we shall be compelled to regard such concepts as ‘logical consequence’, ‘analytical statements’, and ‘tautology’ as relative concepts which must, on each occasion, be related to a definite, although in greater or less degree arbitrary division of terms into logical and extra-logical” Tarski (1936c/1956, p. 420). As for Carnap, he acknowledges that “Tarski seems to doubt whether there is an objective difference [between factual truth and logical truth] or whether the choice of a boundary line is not more or less arbitrary” Carnap (1942, p. vii). See also another quotation from Tarski (1936c) given below (§4, first quotation).

  12. 12.

    I am indebted to the anonymous reviewer who pointed out the importance of this quotation for the present paper.

  13. 13.

    Typically: the construction of languages for which a proper explication of the distinction between logical truth and factual truth can be provided, as in LSL, or the construction of languages for achieving a proper explication of the principles of empiricism, as in Carnap (1936-1937). (Although Carnap does not use the word “explication” in print before 1945, he himself later applies it to his previous work, for example in the preface to the second edition of Carnap (1928), published in 1961. More on “explication” below.)

  14. 14.

    It is not our business to set up prohibitions, but to arrive at conventions. (...) In logic, there are no morals. Everyone is at liberty to build up his own logic, i.e. his own form of language, as he wishes”, Carnap (1937, pp. 51–52).

  15. 15.

    See the quotation from Carnap’s intellectual autobiography given below (§3, third quotation).

  16. 16.

    On the issue of explication in Carnap, see Wagner (2012).

  17. 17.

    Here again, I am indebted to the anonymous reviewer who pointed out this quotation to me, and suggested to be more explicit about Carnap’s and Tarski’s views on explication.

  18. 18.

    “Tarski says further that the characterization given is also in agreement with the ordinary use of the word ‘true’. It seems to me that he is right in this assertion, at least as far as the use in science, in judicial proceedings, in discussions of everyday life on theoretical questions is concerned” (Carnap 1942, p. 29). Again, I am indebted to the anonymous reviewer who pointed out this quotation to me.

  19. 19.

    The connection of Tarski’s definition of truth with the ordinary use of this term has of course been the focus of many discussions, but this particular issue is beyond the scope of the present paper.

  20. 20.

    This paper was to be included in the English translation of Carnap (1934a).

  21. 21.

    For an analysis of Carnap’s criterion of logicality, see Bonnay (2009) and Creath (2015).

  22. 22.

    A definition of truth is materially adequate if all the T-sentences are consequences of the definition (a T-sentence is a sentence of the form “x is true if and only if p”, where “x” is the name of a sentence and “p” is its translation in the metalanguage).

  23. 23.

    In spite of the differences between Tarski’s notion of logical consequence in Tarski (1936c) and the now classical notion of logical consequence of model theory pointed out by commentators such as Etchemendy (1999).

  24. 24.

    For the languages Carnap considers in his (1947), an atomic sentence “holds in a given state description” iff it belongs to it; the negation of sentence S holds in a state description iff S does not hold in it; the conjunction of two sentences holds in a s.d. iff both of them hold in it; and a universal sentence such as (\(x){ Px}\) holds in a s.d. iff all substitutional instances of its scope (‘Pa’, ‘Pb’, etc.) hold in it. See Carnap (1947, § 2). What is striking here is that the definition of “holding in a state description” is actually syntactic: it does not depend on concepts such as truth, satisfaction, or reference at all.

  25. 25.

    This by no means implies that Wittgenstein and Waismann were the main source of inspiration for his program of inductive logic. Other major influences include John Maynard Keynes and Harold Jeffreys, among many others, as Carnap explains in Carnap (1963a, pp. 72–73).

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Wagner, P. Carnapian and Tarskian semantics. Synthese 194, 97–119 (2017). https://doi.org/10.1007/s11229-015-0853-7

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Keywords

  • Carnap
  • Tarski
  • Truth
  • Logical truth
  • Semantics