Abstract
In Untersuchungen zur allgemeinen Axiomatik (1928) and Abriss der Logistik (1929), Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap’s attempt shows, nevertheless, that his approach is much less confused and hopeless than it has often been made out to be. By providing such a reassessment, the paper contributes to a reevaluation of Carnap’s contributions to the development of modern logic.
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Notes
Carnap presented the main results of his Untersuchungen manuscript at the First Conference on Epistemology of the Exact Sciences, August 1929, in Prague. Already at the end of 1928/beginning of 1929, he considered submitting a version of the manuscript for publication, and circulated it to mathematicians and logicians, including Baer, Behmann, Gödel, Härlen, and Fraenkel; see Awodey and Carus (2001), note 1. Behmann provided extensive comments; some of his marginalia are preserved in one of the versions of the typescript in Carnap’s papers (ASP RC 080–34–02), and longer comments in Behmann’s papers (Staatsbibliothek zu Berlin, Nachlaß 335 (Behmann), K. 1 I 10). Behmann at first suggested the manuscript was not ready to be published in a pure mathematics journal, but withdrew these objections after closer reading in March 1929. However, Carnap delayed revision of the manuscript and eventually, in 1930, abandoned the plan to publish it, essentially after some conversations with Tarski during his first visit to Vienna; see Awodey and Carus (2001, pp.162–163) and Reck (2007). The edition (Carnap 1928/2000) includes items RC 080–34–02, 080–34–03, and 080–34–04 from Carnap’s papers, with uniformized page, section, and theorem numbering. These original items are now available online from the Archives of Scientific Philosophy (www.library.pitt.edu/rudolf-carnap-papers). We provide references to both the (out-of-print) edition and the original typescripts. Carnap’s work on general axiomatics is also documented in Carnap (1927, 1930) and Carnap and Bachmann (1936). For a general overview of logical work on axiom systems before Carnap, see Mancosu et al. (2009).
Carnap’s more general views on logic were in flux during the 1920s and early 1930s, especially after he encountered Hilbert (1928), Hilbert and Ackermann (1928), Gödel (1929), and Tarski’s early meta-logical work. Our discussion focuses on Carnap (Carnap 1928/2000) and Carnap (1929), two texts that were composed largely before those influences became dominant. It should be noted, however, that Carnap followed the developments of logic in Hilbert’s school closely even before the appearance of Hilbert and Ackermann’s textbook. In fact, the Abriss was originally conceived as a joint project with Hilbert’s student Heinrich Behmann in the early 1920s. However, Behmann came to prefer his own, more algebraic notation (see Mancosu and Zach 2015), while Carnap thought the Peano-Russell-Whitehead notation made popular by Principia was preferable for a textbook presentation (Carnap to Behmann, February 19, 1924, Staatsbibliothek zu Berlin, Nachlaß 335 (Behmann), K. 1 I 10). For additional historical background, see Awodey and Carus (2001), Awodey and Reck (2002a), and Reck (2004, 2007). A broader study of the evolution of Carnap’s metatheory up to and beyond the 1930s cannot be undertaken here; it deserves a separate paper.
Our discussion in this paper builds on existing scholarship. The first phase of engagement with Carnap’s “general axiomatics” project consisted in the pioneering but overly critical work by Coffa (1991) and Hintikka (1991, 1992). Both authors criticize the “monolinguistic approach” adopted by Carnap, i.e., his attempt to express axiomatic theories and their metatheory in a single type-theoretic language. A second phase set in with Awodey’s and Carus’ important paper on the logical and philosophical analysis of the main technical result in Untersuchungen, the so-called Gabelbarkeitssatz (Awodey and Carus 2001). That paper was followed by a number of articles aimed at a more balanced account of Carnap’s project. In them, not only the limitations of his approach were acknowledged, but also its innovative aspects and its significant influence on later developments in metalogic (Awodey and Reck 2002a; Reck 2004; Goldfarb 2005; Reck 2007). The third phase consists in fairly recent scholarship in which attention is drawn to previously neglected details of Carnap’s early model theory and in which its role in the development of metalogic is spelled out more (Reck 2011; Schiemer 2012b, a, 2013; Schiemer and Reck 2013; Loeb 2014a, b).
The content of this section will be familiar to the Carnap specialist. But it can serve as an introduction to the reader not yet familiar with Carnap’s general axiomatics program and the literature around it; we also expand on that literature with respect to some details. Similar remarks apply to the next two sections.
A detailed presentation of Carnap’s type-theoretic logic can be found in Carnap (1929, Sect. 9).
To be more precise, Carnap describes two different ways of formalizing axiom systems in Abriss. According to the first approach, primitive terms are expressed in terms of nonlogical constants that already have a predetermined (and fixed) meaning. According to the second approach (the one typical for formal axiomatics), the primitive terms defined by a theory have no predetermined meaning; they express, in Carnap’s terminology, only “improper concepts”. Such terms are symbolized by means of variables in the way described above. Compare Carnap (1929, Sect. 30b). In what follows, we will always assume this second approach to the formalization of theories.
Put informally, these axioms state that (1a) neighborhoods are classes of points, (1b) a point belongs to each of its neighborhoods; (2) the intersection of two neighborhoods of a point contains a neighborhood, (3) for every point of a neighborhood \(\alpha \) a subclass of \(\alpha \) is also a neighborhood, and (4) for two different points there are two corresponding neighborhoods that do not intersect; see Carnap (1929, Sect. 33a). Carnap’s own type-theoretic presentation of these axioms has been slightly modernized here, although we have also preserved some of his idiosyncratic notation.
See Awodey and Carus (2001), Reck (2004), and Carus (2007) for the broader philosophical context of Carnap’s metatheoretical work in Untersuchungen, including its connections to the Aufbau (1928). Carnap’s own views on the broader philosophical significance of his general axiomatics project are elaborated, in particular, in “Eigentliche und uneigentliche Begriffe” (Carnap 1927). Further exploring this part of the background deserves a separate investigation, one that may help to account more for some of the idiosyncracies in Carnap’s metalogical approach even beyond 1930. However, we cannot take on this additional task in the present paper.
It should be emphasized, however, that the early critical discussion of this monolinguistic character by Coffa and Hintikka does not do justice to the subtleties of Carnap’s approach. As was first analyzed in detail in Awodey and Carus (2001), Carnap presupposes a “basic system” (Grunddisziplin), that is, a theory of “contentful” logical and mathematical concepts, that forms the background for the logical study of axiomatic theories. In Carnap (Carnap 1928/2000), simple type theory is proposed as one possible choice for such an interpreted basic system. As we will show below, Carnap employed it similarly to the way in which we use axiomatic set theory as the background for the metatheoretic study of axiomatic theories today. Thus, while a clear-cut object/metalanguage is still missing in Untersuchungen, Carnap’s approach can be characterized as metatheoretic. Compare Schiemer and Reck (2013) for a further discussion of this point. In addition, it should be stressed that Carnap’s convention to formulate both axiom systems and their metatheoretic properties within a single and interpreted Grunddisziplin was not uncommon at the time. See, for instance, a related discussion of a necessary “absolute foundation” of axiomatic theories in Fraenkel’s third edition of Introduction to Set Theory (Awodey and Carus 2001, p.154). Finally, a similar idea is present in Carnap’s own subsequent work, in particular in his Logical Syntax: it is shown there that the “syntax language” for language LI can be formulated within LI itself. See (Carnap 1934, Sect. 18).
In Carnap’s own words: “g is called a ‘consequence’ of f, if f generally implies g: \(\forall \mathfrak {R}(f\mathfrak {R}\rightarrow g\mathfrak {R}\)), abbreviated: \(f \rightarrow g\). The consequence is, as is the AS, not a sentence, but a propositional function; only the associated implication \(f \rightarrow g\) is a sentence, namely a purely logical sentence, thus a tautology, since no nonlogical constants occur.” (Carnap 1930, p. 304)
In Untersuchungen Carnap argues that “g follows from f” and “g is derivable from f in TT”, while not identical, are equivalent. See (Carnap 1928/2000, p. 92); ASP RC 080–34–03, p. 41a–b. We will come back to this issue later.
As discussed by Awodey and Carus (2001) and Awodey and Reck (2002a), these correspond roughly to what would today be called categoricity, semantic completeness, and syntactic completeness. Then again, this correspondence must be viewed with caution, especially in the case of the latter two notions, as we will shown in Sect. 7.
In Carnap’s own words: “The models of an axiom system that is closed by a maximality axiom possess a certain completeness property in that they cannot be extended without violating some axioms of the axiom system.” (Carnap and Bachmann 1936, p. 82). Carnap’s main goal in part two of Untersuchungen is to give an explication of different types of extremal axioms and of corresponding notions of completeness. As pointed out in Schiemer (2013), the notes for it also contain results about the relationship between extremal axioms and the monomorphicity, or categoricity, of a theory.
The question of how this squares with the technique of model variation is discussed in Sect. 4.
Note here that for practical purposes the restriction to finite theories is largely irrelevant to the logicians of the 1920s. Theories we now think of as requiring infinite axiomatizations, such as Peano Arithmetic and ZFC, would have been given finite axiomatizations then. The induction schema, for instance, would be formalized in terms of a single sentence with a higher-order quantifier. In Hilbert’s axiomatics too, the schema would be formalized as a single axiom with a formula variable.
It is not specified in Untersuchungen which basic laws and inference rules are included in TT. But Carnap holds at one point that “the basic discipline has to contain theorems (Lehrsätze) about logical, set-theoretical, and arithmetical concepts” (Carnap 1928/2000, p. 61), RC 080–34–03, p. 6.
As Carnap writes: “Every treatment and examination of an axiom system thus presupposes a logic, specifically a contentual logic, i.e., a system of sentences which are not mere arrangements of symbols but which have a specific meaning. [Jede Behandlung und Prüfung eines Axiomensystems setzt also eine Logik voraus, und zwar eine inhaltliche Logik, d.h. ein System von Sätzen, die nicht bloße Zeichenzusammenstellungen sind, sondern eine bestimmte Bedeutung haben.]” (Carnap 1928/2000, p. 60); RC 080–34–03, p. 4. As noted by others, Untersuchungen does not contain an explanation of how the Grunddisziplin acquires its specific interpretation. Such an explanation is given, at least in outline, in Carnap’s sketch Neue Grundlegung der Logik written in 1929. See Awodey and Carus (2007) for further details.
Carnap remains neutral in Untersuchungen with respect to the specific choice of the signature of his background language. He holds that arithmetical and set-theoretical terms can either be understood as “logical” primitives or introduced by explicit definition in the (pure) type-theoretic language in a logicist fashion. See (Carnap 1928/2000, pp. 60–63); RC 080–34–03, pp. 4–8.
It is important to emphasize that this reconstruction is most likely not how Carnap understood the phrase “statement \(\varphi \) holds in the basic system” in 1928. As mentioned already, the current semantic notion of satisfaction was not part of his conceptual toolbox at the time. However, and as shown by Awodey and Carus (2007), this situation changed in the early 1930s, specifically after Carnap’s exchange with Gödel on the notion of analyticity. In particular, Carnap’s understanding of “metatheoretic” statements at that point becomes similar to the semantic reconstruction given here, as argued in Schiemer (2013).
This interpretive issue has been much debated in the secondary literature on Carnap’s early semantics. A central objection against his general axiomatics project, raised early on by Hintikka, was that, due to Carnap’s semantic universalism or “one domain assumption”, the idea of domain variation was simply inconceivable (Hintikka 1991). This view has recently been corrected in work by Schiemer, Reck, and Loeb. In that work, it is shown that Carnap was well aware of the importance of capturing the notion of domain variability for his metatheoretic work and was not without resources to do so. See Loeb (2014a, b), Schiemer (2012b, 2013), and Schiemer and Reck (2013).
As pointed out by Schiemer (2013), a very similar convention of type relativization can also be found in Tarski’s work on the “methodology of the deductive sciences,” from the same period. Compare Mancosu (2010b) for a detailed discussion of Tarski’s case and the other secondary literature on the topic.
This approach of specifying domain predicates was first discussed by Loeb (2014a).
See Schiemer (2013) for a more detailed discussion of Carnap’s “domain as fields” conception of models. Compare Loeb (2014a) for an alternative account of Carnap’s understanding of models. Loeb’s discussion focuses on examples of axiomatic theories where a domain predicate is introduced into the language in terms of an explicit definition from the theory’s primitive signs. These defined domain predicates provide further confirmation for the interpretation of Carnap’s conception of model given in Schiemer (2013), given that in most cases a model domain is explicitly specified as the domain or range or field of a given primitive relation. Nevertheless, Loeb holds that “the domains-as-fields conception is too strict to describe Carnap’s practice.” (Loeb 2014a, p. 427). The particular example she has in mind is an axiomatization of projective geometry with one unary primitive predicate ger (for the class of ‘lines’). The domain of a model of the theory (i.e., the class of ‘points’) is not defined as the field of the relations assigned to ger, but as the union of the elements of all lines (Carnap 1929, Sect. 34). We would like to stress that this and similar examples are fully in accord with the interpretation of model domains given in Schiemer (2013). For a more detailed analysis of model domains of lower types see, in particular, the discussion of Carnap’s notes on “domain analysis” from part two of Untersuchungen in (Schiemer 2013, pp. 506–508).
Carnap (1928/2000), p. 95; RC 080–34–02, Sect. 13, p. 42.
However, the usage of logical terminology was by no means settled then. For example, Skolem (1920) used the term “inconsistent [widerspruchsvoll]” to mean unsatisfiable; even Hilbert himself at times uses “A is a consequence of [folgt aus] B” to mean “\(A \rightarrow B\) holds.”
Carnap (1928/2000), p. 90ff; RC 080–34–03, Sect. 12, p. 39ff.
If the variables in \(\mathfrak {R}\) are at most second order, then the result holds. In that case, assuming soundness of type theory, \(f\mathfrak {R}\rightarrow g\mathfrak {R}\) is valid in first-order logic. Thus its provability in Hilbert’s system follows from Gödel’s completeness theorem, also from a proof-theoretic conservativity proof, neither of which were available to Carnap yet.
Sect. 2.4 of (Carnap 1928/2000, p. 96ff) is Sect. 14 in RC 080–34–03, p. 46ff; Theorem 2.4.1 is Satz 1 on p. 47; Theorem 2.4.8 is Satz 8 on p. 49.
Sect. 2.5 of Carnap (1928/2000) is Sect. 15 in RC 080–34–04, p. 52ff.
RC 080–34–03, p. 75.
Sect. 26, Satz 3, RC 080–34–03, p. 86.
Sect. 26, Satz 2, RC 080–34–03, p. 86.
Compare Awodey and Reck (2002a) and the references in it.
Typically isomorphisms of models was discussed only for models of a particular axiom systems, and often these only contained a single non-logical primitive. For instance, Veblen (1904) considered a version of geometry where the single primitive is a three-place collinearity predicate relating points. Neumann (1925) also discussed the (non)categoricity of his axiom system; but his axioms were formulated in German and both what counts as a model and what counts as an isomorphism was left vague.
It coincides, that is, except for a difference Tarski stresses: his definition requires the isomorphism to be a bijection of the entire domain of individuals, not of just of individuals “used” in the model.
It bears emphasizing here that, while in that paper Tarski alluded to Gödelian arithmetization of syntax, he did so only in order to explain how derivability from the axioms of a formal system according to given rules can be formalized in metalogic; but this did not play a role in his definition of consequence. In particular, Tarski did not give a uniform definition of “follows from” for an object language in a formal metalanguage which can refer to and quantify over all models and expressions, in contrast to his general definition of truth. Of course, Tarski’s discussion of consequence goes much beyond Carnap’s and is more nuanced in other respects, e.g., in its considerations of the distinction between derivability and consequence, in terms of which constants should count as logical, etc. He was able to accomplish that because Gödel had in the meantime identified the gap between what follows from and what can be derived from deductive theories. Tarski was also prompted to do it by Carnap’s Logical Syntax.
Tarski uses \(X \sim Y\) as an abbreviation for the the isomorphism relation.
We leave aside here, once again, the problems arising from taking \(\exists h\) to correctly express the existence of a formula (“propositional function” in that sense).
Behmann suggested that Carnap remain neutral between the absolutist and constructivist approaches (Behmann to Carnap, March 6, 1929, Staatsbibliothek zu Berlin, Nachlaß 335 (Behmann), K. 1 I 10). The first part (pp. 1–34) of Carnap’s typescript RC 080–34–02 apparently is a revised version dating from after Behmann’s letter: it does not bear Behmann’s marginal annotations which are present on the rest of the typescript, and the version to which Behmann responded had Sect. 10 covering pp. 31–35, while in the surviving typescript Sect. 10 covers pp. 30–35.
Carnap does not mention Langford’s work himself. It is mentioned by Fraenkel (1928), although not in the context of completeness or decidability of axiom systems.
Note that many of the axiom systems in Abriss involve higher-order axioms, even primitives or important defined concepts that are higher order. For instance, the axiom system for geometry in Sect. 34 has a second-order primitive ger, for the class of lines, which are themselves classes of points.
Sect. 29 of RC 080–34–03, p. 98ff.
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Acknowledgments
Georg Schiemer’s research was supported by the Austrian Science Fund, projects J3158–G17 and P–27718. Richard Zach’s research was supported by the Social Sciences and Humanities Research Council of Canada. We are grateful to A. W. Carus and an anonymous referee for their detailed and thoughtful comments on earlier drafts of this paper. We would also like to thank the audiences at the 2013 workshop Carnap on Logic at the Munich Center for Mathematical Philosophy, the Minnesota Center for Philosophy of Science, the 2014 Spring Meeting of the ASL, and the 2014 Society for the Study of the History of Analytic Philosophy Annual Meeting. Richard Zach acknowledges the generous support of the Calgary Institute for the Humanities and the Department of Philosophy at McGill University.
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Schiemer, G., Zach, R. & Reck, E. Carnap’s early metatheory: scope and limits. Synthese 194, 33–65 (2017). https://doi.org/10.1007/s11229-015-0877-z
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DOI: https://doi.org/10.1007/s11229-015-0877-z