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Carnap’s Early Semantics

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Abstract

This paper concerns Carnap’s early contributions to formal semantics in his work on general axiomatics between 1928 and 1936. Its main focus is on whether he held a variable domain conception of models. I argue that interpreting Carnap’s account in terms of a fixed domain approach fails to describe his premodern understanding of formal models. By drawing attention to the second part of Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik, an alternative interpretation of the notions ‘model’, ‘model extension’ and ‘submodel’ in his theory of axiomatics is presented. Specifically, it is shown that Carnap’s early model theory is based on a convention to simulate domain variation that is not identical but logically comparable to the modern account.

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Notes

  1. Creath (1990), Awodey and Carus (2001), Awodey and Reck (2002), Bonk and Mosterin (2000), Reck (2007), and Goldfarb (2005).

  2. See Corcoran and Sangüillo (2009), Mancosu (2006) for further discussion of a similar distinction in Tarski’s work on the “methodology of deductive sciences” in the 1930s.

  3. For the readers convenience, a modernized version of Carnap’s original formulation of the syntax of a type-theoretic language is presented here. For Carnap’s original presentation of type theory, see also Carnap (1929).

  4. For a modern presentation of type theory see, e.g., (Andrews 2002, §50).

  5. See also Carnap (1929, 70–72). It should be noted that this convention of symbolizing axiomatic theories was common practice at that time. A similar treatment of axiomatic primitive terms as variables is present in the works of Frege, Russell, the American postulate theorists, and Tarski. See Mancosu (2006, 212–216) for a broader historical survey of this “widespread conception” of axiom systems.

  6. Apart from Hintikka (1991), Carnap’s extremal axioms have not been subject to extensive scholarly discussion. A short commentary can be found in Bonk and Mosterin (2000) where the projected structure of the second part of Untersuchungen is outlined. See also Awodey and Reck (2002) and Reck (2007).

  7. For a closer discussion of Carnap’s reception of Fraenkel’s axiom of restriction see Schiemer (2010).

  8. Obviously, “meta-axioms” like Hilbert’s axiom of completeness are, from a modern point of view, logically problematic since they “conflate formal languages with their model-theoretic semantics” (Shapiro 1991, 185). A main point in this paper will be to show precisely where Carnap’s account differs from modern model-theoretic semantics.

  9. Whitehead and Russell (1962, §21 and §30–38). See also Carnap (1929, 54). The notion also plays a central role (in the context of “structure descriptions” of relations) in Carnap’s Aufbau (Carnap 1928, §11).

  10. Compare Corcoran’s discussion of the historical notion of the structure of a model: “From a philosophical and historical point of view it is unfortunate that the term ‘mathematical structure’ is coming to be used as a synonym for ‘mathematical system’. In the earlier usage (…) two mathematical systems having totally distinct elements can have the same structure. Thus in this sense a structure is not a mathematical system, rather a structure is a ‘property’ that can be shared by individual mathematical systems. At any rate a structure is a higher order entity.” (Corcoran 1980, 188) For further history of the term ‘structure’ in modern mathematics and logic see also Mancosu (2006, 210).

  11. In Carnap’s notes in the Nachlass, the relation between the models of a theory and the structures described by it is specified in this way: “Given that a formal AS is satisfied by every model of a structure in case it is satisfied by one model of the structure, we say in this case: the AS is “satisfied” by the structure.” (RC 081-01-05)

  12. The \(\subseteq\)-sign in the definiens below expresses the standard submodel relation from model theory. See also Sect. 5.

  13. See Sect. 5 for a detailed discussion of Carnap’s understanding of the notions ‘submodel’ and ‘model extension’.

  14. The same definitions can be found in the notes for Part 2 of Untersuchungen (RC 081-01-05). A fuller discussion of Carnap’s notions of model structures and proper substructures and their implementation in a theory of extremal axioms will be given in a seperate paper.

  15. In Part 2 of Untersuchungen, cardinality axioms such as A4 are termed ‘absolute existence axioms’ that fix the Urklassen of a theory (RC 081-01-10).

  16. Compare Behmann (1927, 42) and Carnap (1929, 26–28) for further discussion of “arrow diagrams” for the illustration of relation structures.

  17. Two further aspects of Carnap’s notion of formal models that are clearly heterodox from the modern perspective should be mentioned here. First, note that, at least in Untersuchungen, functions and constant elements are not considered as possible constituents of Carnap’s models. Second, at several places in the manuscript, models are described not as sequences of relations but as sequences of relation constants of the type-theoretic background language. This ‘substitutional’ treatment of models will be further discussed in Sect. 4.1.

  18. Carnap presents a simple example of a transitivity axiom to illustrate this formalization:“If we express it with the help of the M-elements, the axiom is: \(\forall x \forall y \forall z ([M(x,y) \land M(y,z)] \rightarrow M(x,z)).\) Here x, y, z are not free variables, but bound by the universal operator in front. If M were now a constant, then the expression would be a sentence; it can only then be a propositional function if M is a variable.” (Carnap 2000, 89).

  19. In (Carnap 1930), the notion of ‘satisfaction’ is used instead of truth: “If fR is satisfied by the constant R1, where R1 is an abbreviation of a system of relations P1, Q1, …; R1 is called a “model” of f.” (Carnap 1930, 303).

  20. In a footnote, Carnap and Bachmann refer to Carnap (1935) for a formal definition of ‘analytic in LII’ (Carnap and Bachmann 1936, 84). For Carnap’s detailed definition of analyticity in the type-theoretic language LII in LSL, see Carnap (1934, §34).

  21. The universality of logic is generally considered as a premise in the logicist tradition. Compare Goldfarb’s well-known description of this conception of logic: “The ranges of the quantifiers - as we would say - are fixed in advance once and for all. The universe of discourse is always the universe, appropriately striated.” (Goldfarb 1979, 352) For a discussion of Carnap’s universal conception of logic in Untersuchungen see Awodey and Carus (2001, 159).

  22. Compare also Carnap (1931) for a more general characterization of the context-dependent range of quantification of STT: “To type 0 belong the names of the elements (“individuals”) of the domain of thought that are treated in the respective context (for instance \(\hbox{a},\hbox{b},\ldots\)).” (Carnap 1931, 96).

  23. See Andrews (2002, 185–186) for a detailed discussion of the semantics of type-theoretic languages.

  24. For instance, let φ be a sentence of the form ∀ X ψ. X is a monadic relation variable of type 〈i〉 and s an assignment from the set of monadic relation variables to \(\wp (D_{0})\). Then ∀ X ψ is analytic in LII iff for every s: \(\mathcal{V} \models \psi{\left[s\right]}\).

  25. The main difference from the first example is that Carnap treats higher-order quantification extensionally in LSL. Thus, instead of being restricted to a substitution class of predicate symbols in LII, the quantifier ‘\(\forall X \ldots \)’ is supposed to range over all relations or functions on the set of numerals of LII. For the historical details concerning Carnap’s move from a substitutional to an extensional treatment of higher-order quantification see Awodey and Carus (2007).

  26. For an early version of this approach see the first volume of Hilbert and Bernays’ Grundlagen der Mathematik, published in the same year as Carnap’s LSL (Hilbert and Bernays 1934, 8–12).

  27. The domain (Dom), range (Ran), and field (Fld) of a relation are understood here in the usual sense: let R(x, y) be a binary relation on set A. Then \(Dom(\hbox{R}) =_{df} \{ a \in A \mid \exists b \in A: (a,b) \in \hbox{R} \}\); \(Ran(\hbox{R}) =_{df} \{ b \in A \mid \exists a \in A: (a,b) \in \hbox{R} \}\); Fld(R) =  df Dom(R) ∪ Ran(R).

  28. For instance, let \({\mathfrak{M}}\) contain two first-order binary relations, R1 and S1, defined on D 0. Thus \(\hbox{R}_1, \hbox{S}_1 \subseteq D_{0} \times D_{0}\). The domain of \({\mathfrak{M}}\) is then simply Fld(R1) ∪ Fld(S1).

  29. This work, supervised by Heinrich Scholz, preceded the collaboration with Carnap. Nevertheless, Bachmann was well acquainted with Carnap’s published work on axiomatics at that time. He refers to Carnap (1929, 1930) repeatedly in the text.

  30. According to Bachmann, α allows different interpretations: as the “set of the natural numbers,” as the “set of natural numbers greater then 37,” as the “set of primes,” etc. (Bachmann 1934, 4–5).

  31. Perhaps the most explicit expression of this understanding can be found in a remark on “the values R (the models)” of an axiom system f(R) (RC 081-01-12).

  32. The relevant distinction between model domains and type domains at work here has a modern counterpart. There currently exist two definitions of the domain of a relation: (1) for a binary relation R, the domain and co-domain are simply taken to be the base sets A, B the relation is defined over, i.e. \(\hbox{R} \subseteq A \times B\); (2) the domain, range, and field of R are defined as subsets of A, B, and A × B respectively by the definitions stated in footnote 25. It is clear from the above that Carnap treats the domains of a particular model as the domains and ranges of its relations, i.e. in the sense of (2). In fact, one can find the modern definitions of what he terms “domains of relations” (“Relationsbereiche”), particularly of “Vorbereiche” and “Nachbereiche,” in his Abriss der Logistik (Carnap 1929, 36–37).

  33. See Mancosu (2010) for a recent overview of the debate and for references.

  34. Bays (2001) presents a similar interpretation of Tarski. See Goméz-Torrente (2009) for a detailed comparison.

  35. Compare Mancosu (2006), Bays (2001), and Goméz-Torrente (2009) for detailed discussions of Tarski’s use of domain predicates.

  36. Goméz-Torrente (2009) describes in detail “Tarski’s pluralism,” i.e. “his disposition to work within a variety of different frameworks” in his formalization of deductive theories at the time.

  37. Unlike Tarski, Carnap does not introduce a distinct variable specifically for the purpose of quantifier relativization to a particular domain of a model. However, at least one example in Tarski’s writings is discussed in Mancosu (2006) where a convention identical to Carnap’s is used. In describing an axiom system with only one primitive binary relation R, Mancosu holds that: “A model for such a theory is given by the finite sequence containing only the relation R (the ‘universe of discourse’ is implicitly defined by the field of R).” (Mancosu 2006, 224). This is precisely Carnap’s ‘domain-as fields’ account.

  38. At least in Carnap’s Nachlass, no documents can be found that suggest an influence on Carnap from Tarski’s side while working on Untersuchungen in 1928. This is not surprising since Carnap and Tarski met first in 1930. Compare Awodey and Carus (2001) and Reck (forthcoming) for historical details.

  39. The main mathematical influences for Carnap’s account of extremal axioms, in particular Fraenkel’s axiom of restriction, will be examined in a separate paper.

  40. Carnap and Bachmann also introduce a second, more restrictive notion of “structure extensions” where isomorphic extensions of a model are explicitly ruled out by definition, i.e. \(Erw_{s}(\hbox{N};\hbox{M})=_{df} \hbox{M} \subset \hbox{N} \land \neg Ism_{v}(\hbox{M}, \hbox{N})\). Given these two types of extensions, Carnap and Bachmann formulate two versions of extremal axioms: e.g. in the case of maximal axioms, “maximal model axioms” and “maximal structure axioms:” \(Max_{M}(F;M)=_{df} \neg (\exists N) (M \subset N \land M \not= N \land F(N)); Max_{S}(F;M)=_{df } \neg (\exists N) (M \subset N \land \neg Ism_{v}(M, N) \land F(N))\) (Carnap and Bachmann 1936, 77).

  41. See, e.g., Enderton (2001, 95).

  42. I translate here from the original formulation given in Carnap and Bachmann (1936). A similar version of the notion of a “submodel” (“Teilmodell”) can be found in his notes in the Nachlass. Here, Carnap states that for two models P: R, S, T and P′: R′, S′, T′ the submodel relation \(P \subset P'\) expresses that \(R \subset R', S \subset S', T \subset T'\) (see RC-081-01-19/119, notation changed).

  43. Note that the example is somewhat untypical for Carnap’s reconstruction of axiom systems. In the majority of cases, the level of individual expressions is simply not mentioned. A reason for this can be found in a document of Part 2 of Untersuchungen titled “Reduction of the primitive concepts” (“Reduktion der Grundbegriffe”) (RC 081-01-12). Carnap argues here that the primitive individual terms can usually be ‘eliminated’ from an axiom system by a “structural description” of it in terms of the other primitive signs. He refers to Principia Mathematica §122 for such an elimination in the case of basic arithmetic (Carnap 2000, 88–89).

  44. Compare also the following closely related passage: “Tm P (or Tm P,Q or Tm P, Q, R respectively) means: Tm, given that a proper partial relation exists of P (or of P, Q or of P, Q, R respectively), and for the other [relations] proper or improper [partial relations exist].” (ibid).

  45. The interpreted structures R a , R b , and R d are partial relations of R c , R e , R g , and R f in either one of the first two senses. For instance, in \(\hbox{R}_{a} \subset \hbox{R}_{e}\), R a is gained by restricting the range of R e , viz. {1, 3, 4} to the smaller class {1, 4}. The fields of R a and R e are identical.

  46. This is also reflected in Carnap’s distinction between “finite” and “infinite” axiom systems, i.e. axiom systems with models of “infinite structures (structures of models with an infinite number of immediate and mediate elements)” in Carnap (2000, 149–150).

  47. The theory is also discussed in (RC 081-01-10).

  48. Progressions are defined in Principia Mathematica as a specific type of series, i.e. as a set A with a “generating relation” R that satisfies the following conditions: (i) R is 1–1; (ii) there exists a base element not included in Ran(R); (iii) A is denumerably infinite and (iv) closed under R (Whitehead and Russell 1962, 245–248).

  49. In symbols: (b1) \((\forall x)(\forall y) [ R(x,y) \supset (\exists z(R(y,z))]\); (b2) \((\forall x)(\forall y)(\forall z) [(R(x,y) \land R(x,z) \supset x = y) \land (R(x,y) \land R(z,y) \supset x = z)]\); (b3) \(| {\it{Dom}}{\text{(R)}} - {\it{Ran}}{\text{(R)}} | = 1\); (b4) \(\neg(\exists N)(N \subset M \land Ism_{V}(M,N) \land F(N))\) (Carnap and Bachmann 1936, 79, notation changed).

  50. The concept of a “cycle” is defined by Carnap as follows: “By a cycle with n elements we understand a one-one relation whose field consists of a single closed R-family with n elements. (…) In the limiting case \(n = \infty\) there arises a one-one relation, with no first of last element, which consists of single open R-family.” (Carnap 1929, 178).

  51. If the base system b1-b3(R) is closed by a minimal structure axiom b4(R), the class of admissible interpretations is restricted to models, i.e. arithmetical progressions, of infinite ‘dividable structure’ (Carnap and Bachmann 1936, 80).

  52. A second algebraic model is used in Hilbert’s “model construction” (“Modellaufweis”) in the independence proof of the Archimedean axiom in §12. Here, the domain \(\Upomega(t)\) ranges over a denumerable set of algebraic functions (Hilbert 1910, §2). See Awodey and Reck (2002) for details.

  53. The relative semantic weakness of a language concerns the range of individuals of its interpretation here. Compare Carnap and Bachmann on this point: “It is to be sure quite possible to do without the greater richness afforded by admitting extensions of higher levels if the background language is rich enough, especially with respect to its domain of individuals. When, however, the language exhibits a certain poorness it is quite possible that extensions of higher level exist but none of the same level.” (Carnap and Bachmann 1936, 83).

  54. Two other options discussed in §6 to block the “possible objection” to their account can only be mentioned in passing here. The first approach is simply to stipulate that the language in use is sufficiently rich in order not to limit possible model constructions (Carnap and Bachmann 1936, 83). An explicit formulation of this can also be found in Bachmann (1936, 39). The second option exists in the specification of a flexible type theory, i.e. a higher-order language without “a rigid type structure” whose variables “have no definite type but run through a denumerably infinite sequence of types” (Carnap and Bachmann 1936, 85). Such a language would allow one to formalize “informal extensions” in mathematics based on the variation of type assignments (Carnap and Bachmann 1936, 84–85). The idea of a flexible type theory and its possible use in the formalization of axiomatic theories is elaborated on in detail in the second part of Untersuchungen (RC 081-01-01 to 081-01-33).

  55. Given the lack of further specification, one could be inclined to view this language “transition” as conceptually related to Carnap’s notion of a “translation” in LSL (Carnap 1934, §61). Briefly, a “transformance” is defined there as a correlation f between “expression classes” of two “isomorphic” language S 1 and S 2. f is called a “translation” of S 1 into S 3 if S 3 contains S 2 as a “sub-language” (see also Carnap 1934, §50). Note, however, that a comparison of “translation” and “transition” would overlook the different nature of language extension expressed in the two cases. In contrast to Carnap’s “sub-languages” in LSL, language transitions as outlined in Carnap and Bachmann (1936) concern primarily the extension of the range of individuals of a language. In the above case of a transition from S 1 and S 2 the crucial difference between the two languages is that S 2 is richer in terms of a “a larger domain of individuals.” In contrast, a translation in LSL also concerns languages with an identical interpretation of the individual variables (Carnap 1934, §50).

  56. “A model for a language (in the extensional sense of “model” customary in mathematics (…)) is an assignment of extensions of the following kind: To every type of variables a class of entities of this type is assigned as the range of values, and to every primitive constant of the type system an extension of the same type is assigned.” (Schilpp 1963, 902)

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Acknowledgments

Research on this article was partly funded by the Austrian Science Fund (FWF): J3158. Earlier drafts of this paper were presented to the SoCal HPLM Group at UC Irvine, in the Logisches Café - colloquium at the University of Vienna, at GAP.7 in Bremen as well as at Epsa2009 in Amsterdam. I thank the members of the respective audiences for useful comments. I am especially indebted to Michael Friedman, Erich Reck, Steve Awodey, Ilkka Niiniluoto, and Richard Heinrich for helpful discussions and valuable feedback. I also wish to thank two anonymous referees for their comments and suggestions that have substantially improved this paper.

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    Georg Schiemer

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Schiemer, G. Carnap’s Early Semantics. Erkenn 78, 487–522 (2013). https://doi.org/10.1007/s10670-012-9365-8

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