Abstract
We shift attention from the development of model theory for demarcated languages to the development of this theory for fragments of a language. Although it is often assumed that model theory for demarcated languages is not compatible with a universalist conception of logic, no one has denied that model theory for fragments of a language can be compatible with that conception. It thus seems unwarranted to ignore the universalist tradition in the search for the origins and development of model theory. This point is illustrated by Carnap’s early semantics and model theory, which he developed within a type theoretical framework and which stand out both for their universalistic treatment and for certain idiosyncratic technicalities by which the construction is supported. One special property is that individuals are context relative in Carnap’s system. This leads to a model theory in which the model domains are more flexible than has been suggested in the literature.
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Notes
The dichotomy put forward by Van Heijenoort and Hintikka has often been challenged, especially in the last couple of years. According to some critics, Van Heijenoort has overdrawn the contrast between the the two traditions in the history of logic (Sluga 1987, p. 93; see also the discussion in Anellis 2012, p. 346, fn. 9). For example, they point out that the fact that Boole and Schröder had no quantification theory to speak of does not justify the claim that the absence of quantification theory is an essential feature for the whole of the algebra of logic (Peckhaus 2004, p. 11), which is seen as a characteristic school in the tradition of logic as a calculus. Similarly, the fact that Frege and Russell did not pursue metamathematical investigations does not necessarily lead to the conclusion that such investigations are at odds with a universalist approach to logic (Sluga 1987, p. 93), because that would ignore the many differences among its practitioners. Some claim specifically that the development of model theory is possible in the universalist approach to logic (de Rouilhan 2012). De Rouilhan gives a systematic argument for the claim that the universalist approach and model theory are not in disagreement.
It is exactly this claim that Frege never raises any metasystematic question that Tappenden rejects. See (Tappenden (1997), p. 201ff).
In Betti (2008) a further distinction of what we have called metasystematic investigations is made into metamathematics and metalogic. According to this distinction, metamathematics consists of “the investigation of logical or epistemological questions concerning logical or mathematical structures or methods with the aid of mathematical tools” (p. 49). Metalogic (metatheory), on the other hand, is considered a wider category that does not necessarily invoke mathematical means. In the current paper we use the notion of metasystematic investigations in a way similar to Betti’s use of the notion of metatheory. If the terms “metamathematics” and “metalogic” are used, we assume a difference in subject matter between the two (mathematics and logic, respectively), but, unlike Betti, no difference in method.
The precise extent to which Tarski’s Postscript to Tarski (1935) can be counted as universalist will be the subject of future research. I disagree with the very absolute claim that “the author of the postscript to Wahrheitsbegriff [Tarski] (\(\ldots \)) had abandoned universalism forever” (de Rouilhan 2012, p. 572, fn. 29). See also de Rouilhan (1998) and the discussion in Loeb (2014).
The fact that Tarski takes a language of infinite order as universal is indeed the reason why the construction of the definition of “true sentence” for such a language is not possible for him. For such languages there is no metalanguage that is stronger in the relevant sense.
See Mares (2011) for a historical and philosophical overview of the notion of propositional function.
In ähnlichem Sinne, wie wir hier die Voranstellung einer inhaltlichen Logik fordern, wird zuweilen die Voranstellung einer “inhaltlichen” oder “absoluten” Mengenlehre oder “absoluten Arithmetik” oder Kombinatorik gefordert.
Note that in his early work Carnap makes no essential distinction between concepts and objects. See also (Loeb (2013), p. 5, fn. 6).
Der Begriff “Individuum” ist ein relativer Begriff: derselbe Gegenstand, der in einem gewissen Zusammenhange Individuum ist, kann in einem erweiterten Zusammenhange Relation sein, indem hier die Glieder auftreten, während von solchen in dem ersten Zusammenhang nicht die Rede ist.
Die ganze Typentheorie besteht nun in der folgenden “Typenregel”: die zulässigen Argumentwerte einer bestimmten Stelle einer bestimmten Aussagefunktion, also auch die zulässigen Werte einer bestimmten Stelle einer Relation müssen isotyp sein.
Constitutional theory is the theory of constitutional systems, some of which Carnap explores in Carnap (1928). The common feature of all constitutional systems is “the circumstance that all scientific concepts are to be defined in a single system on the basis of a few fundamental concepts” (Friedman 1992, p. 15).
Man pflegt die Begriffe einzuteilen in Individualbegriffe und Allgemeinbegriffe: Der Begriff Napoleon ist ein Individualbegriff, der Begriff Säugetier ein Allgemeinbegriff. Vom Standpunkt der Konstitutionstheorie aus besteht diese Einteilung nicht zu Recht, oder Vielmehr: sie ist nicht eindeutig, jeder Begriff kann je nach dem Gesichtspunkt als Individualbegriff und auch als Allgemeinbegriff aufgefaßt werden.
Trotzdem kann auch die Klasse \(k\) als Klassenelement auftreten; freilich nicht als element ihrer selbst oder irgend einer andern Klasse, deren Elemente von der Gegenstandsart der \(a\), \(b\), \(c\) sind, sondern nur als Elemente zweiter Stufe. Die Gegenstände, die in einem bestimmten zusammenhang nicht als Klassen (oder andere Funktionen) auftreten, heißen “Individuen”; (dieselbe Gegenstände können aber ein anderesmal als Klassen auftreten; die Bezeichnung “individuum” gilt also nicht absolut, sondern nur in bezug auf eine Betrachtung). Es besteht dann eine Stufenreihe: Individuen, Klassen erster Stufe, Klassen zweiter Stufe. Dies sind die einfachsten “Typen”, deren vollständiger Aufbau (“Hierarchie der Typen”) sich später durch Hinzufügung der verschiedenen Typen von Relationen ergeben wird (\(\ldots \)).
On the other hand, this shows that Carnap does not adhere to the one domain thesis.
wir nehmen die Wörter “Zahl” und “Vorgänger” als neue, noch nicht mit Bedeutung versehene Termini and bestimmen, daß sie diejenigen Begriff bezeichnen sollen, die die im AS angegebene Beschaffenheit haben.
Die in einem Axiomensystem auftretenden Zeichen (z. B. Wörter) für die Grunbegriffe des Axiomensystems nennen wir die “Grundzeichen”. Sie haben nach der genannten Auffassung keine feste Bedeutung, sondern können je nach dem Anwendungsfall auf verschiedene Gegenstände bezogen werden. Die Grundzeichen sind demnach Variable, und eizelnen Axiome, sowie auch das ganze Axiomensystem sind Aussagefunktionen, nicht Aussagen.
Van Heijenoort has argued that “Hilbert’s position is somewhat between that of Frege-Russell and that of Peirce-Schröder-Löwenheim” (Van Heijenoort 1977, p. 185).
The influence of Hilbert’s school on Carnap’s early work extends further than this particular view of axiom systems: This early work is described elsewhere (Sandu and Aho 2009, p. 574) as “an attempt to reconcile the one-world and one-language view of the universalists with the metalinguistic views of Hilbert’s school.”
Absent in the German edition Carnap (1934).
Frege also comments on differences between the two kinds of variables in Hilbert’s work (see Frege 1906, pp. 388ff). Contrary to Carnap, Frege sees both kinds as variables expressing universality (“Allgemeinheiten”).
Recall that sentences in the basic discipline are meaningful, and therefore can be said to have a truth value.
[W]ir sprechen kurz von “Modellen” eines Axiomensystems und meinen damit logische Konstanten, also “Systeme von Begriffen der Grunddisziplin” (und zwar sind es meist Systeme von Zahlen).
Schreiben wir für \(f(R,S,T)\) kurz \(f\mathcal {R}\), und sind \(R_1\), \(S_1\), \(T_1\) bestimmte, etwa arithmetische Relationen, die zulässige Werte der Variabeln \(R,S,T\) sind, so können wir auch für das geordnete Relationensystem \(R_1, S_1, T_1\) eine abkürzende Bezeichnung einführen, etwa \(\mathcal {R}_1\). Das Modell \(\mathcal {R}_1\) ist dann ein Wert der Modellvariablen \(\mathcal {R}\). (\(\ldots \)) Ein zulässiges Modell \(\mathcal {R}_1\) von \(f\mathcal {R}\) ist nur dann auch Modell von \(f\mathcal {R}\), wenn \(f\mathcal {R}_1\) nicht nur sinnvoll, sondern wahr ist.
We can choose, for example, that the individuals will be the natural numbers in some model, and real numbers in the other. We will say more about this point below.
Schiemer’s domains as fields conception basically comes down to the idea that the model domain in Carnap’s early model theory is encoded as the intended fields of the basic relations of an axiom system. This conception enables a variable model domain, even if the domain of individuals (the range of the quantifiers) is fixed.
Loeb made clear the connection of the domains as fields conception with the relativisation of the domain by way of domain predicates, as known from the discussion of Tarski’s model domains. The key observation is that it is possible in type theory to define a predicate for the fields of the basic relations. This predicate then functions in the same way as \(N(x)\) in the passage above. It is argued that it is both conceptually easier and technically and historically more correct to explain Carnap’s early axiomatic practice by way of defined predicates, than by way of the domains as fields conception.
In the current paper we do not want to challenge either of the above views. Rather, we point out that yet another mechanism is at play that enables even more domain variability.
Jede Variable, die in irgendeinem Zusammenhang vorkommt, muß einen bestimmten Typus und eine Bestimmte Stufenzahl haben; (durch den Typus ist die Stufe bestimmt, aber nicht umgekehrt). So auch die Modellvariable, oder, bei anderer Schreibweise, die einzelnen Grundrelationsvariabeln eines Axiomensystems. (\(\ldots \)) Welche Gegenstände als “Individuen” inbezug auf eine bestimmte Variable anzusehen sind, ist durch den Typus der Variabeln nicht festgelegt; das eine Mal können Gegenstände dieser, das andere Mal Gegenstände jener Art als Individuen fungieren.
The precise implication of Carnap’s model domains for discussions concerning the informal notion of consequence is subject for future research.
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Acknowledgments
The author would like to thank Arianna Betti and Stefan Roski for valuable feedback on earlier versions of this paper. She is grateful for the opportunity to present her work at the Lunch Lecture series of the Theoretical Philosophy Group at Utrecht University, and would like to thank the audience for their helpful comments. The author was supported through ERC Starting Grant TRANH 203194 until September 2013. All passages from the Rudolf Carnap Papers are quoted by permission of the University of Pittsburgh. All rights reserved.
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Loeb, I. Uniting model theory and the universalist tradition of logic: Carnap’s early axiomatics. Synthese 191, 2815–2833 (2014). https://doi.org/10.1007/s11229-014-0425-2
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DOI: https://doi.org/10.1007/s11229-014-0425-2