Abstract
It is often argued that by assuming the existence of a universal language, one prohibits oneself from conducting semantical investigations. It could thus be thought that Tarski’s stance towards a universal language in his fruitful Wahrheitsbegriff (1933) differs essentially from Carnap’s in the latter’s less successful Untersuchungen zur allgemeinen Axiomatik (1927–1929). Yet this is not the case. Rather, these two works differ in whether or not the studied fragments of the universal language are languages themselves, i.e., whether or not they are closed under derivation rules. In Carnap’s case, axiom systems are not closed under derivation rules, which enables him to adopt a substitutional concept of models. His approach is directly rooted in the tradition of formal axiomatics, we argue, and in this contrary to Tarski’s. In comparing these works by Carnap and Tarski, our aim will be to qualify the connection between Tarski’s approach and the tradition of formal axiomatics, which has been overemphasized in the literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In Proops (2007) another five senses are identified in connection with Russell’s work.
Rodríguez-Consuegra does not claim, though, that this remark on ordinary language is based on Wb.
Although Tarski claims here that a complete system of mathematical logic is a universal language, it is important to maintain the distinction between these two notions. Nowadays one could argue that ZF set theory is a complete system of mathematical logic, but it is clear that we cannot straightforwardly claim that it is a universal language in the sense expressed in the above passage, because it is a first-order language.
This is not to say that De Rouilhan’s claim is correct. See Loeb (2014a).
However, unlike in Tarski’s division, neither of the fragments is in principle limited by any (finite) type.
Possibly extended with certain constants. See AA, p. 89.
The translations are the author’s, unless otherwise stated.
Rather than giving a somewhat vague list of some differences, as we did in the previous section.
Also, this condition can be found more explicitly in (Tarski 1930, p. 69/70).
This is only a rough distinction, according to Carnap, because it does not take into account “the three principle kinds of substitution”, namely substitution (1) for free variables, (2) for bound variables, and (3) for constants (!). (Carnap 1937, p. 190).
Note that the set of all propositional functions that are part of the language of an axiom system is thus a strict subset of its consequences in Carnap’s approach.
Note that not all terms were necessarily seen as placeholders.
This view that certain associations to formal representations are made at will is not restricted to Italian logicians, nor to the early period of formal axiomatics. For example, also in Kattsoff (1936), one of the American Postulate Theorists, we find the idea that in an (abstract) axiom system one may take the assumptions to be true (at will). In contrast to Pieri and Padoa, Kattsoff thus attaches a truth value at will to a whole assumption, and not a meaning to a primitive symbol that occurs within the assumptions.
Among the American Postulate Theorists are Huntington, Veblen, Langford, and Keyser (Scanlan 1991, p. 981; Scanlan 2003, p. 312ff). It is characteristic of the American Postulate Theorists that they axiomatised specific fragments of mathematics, such as geometry (rather than formalising all of mathematics), and did so in a way that was fully explicit with respect to the used (primitive) terms. They also investigated properties of their systems, such as independence and categoricity (Scanlan 1991, p. 981/982).
See Mares 2011 for a historical and philosophical overview of the notion of propositional function.
One reason to focus here on Huntington’s practice is that he is mentioned by Carnap in his discussion of axiom systems (Carnap 2000, p. 88). Carnap also mentions Hilbert (Carnap 2000, p. 87), but not in the context of a specific approach to axiomatics. Reck emphasises Hilbert’s influence on Carnap with respect to the approach to axiom systems, and consequentially characterises AA as the synergy of the influence of Frege and Russell on the one hand, and Hilbert on the other (Reck 2007, p. 2).
Huntington’s postulates correspond to Carnap’s axioms. This difference of terminology is not relevant to our purpose here.
In the current context it is relevant to consider Ajdukiewicz’s take on axiomatics, given his influence on Tarski (Mancosu et al. 2009, p. 134; Betti 2008, p. 61 ff). His take seems to go beyond what we have identified as formal axiomatics, because—much as in our current practice in logic—none of the terms he considers have meaning:
It is customary to say that the axioms of formalized, deductive sciences are statements or sentences, sentential functions etc. Our own view does not allow to say so. (\(\ldots \)) A symbol is a sentence (\(\ldots \)) if among its components there is an element which has intuitive sense and which expresses an assertion or denial. Since no such element with intuitive meaning is a component of formalized axioms, none of them may be regarded as a sentence in the intuitive sense. (I use the term “sentence” in its widest sense, in which it may be a statement, utterance, propositional function etc.). (Ajdukiewicz 1921, p. 14 (of Eng. trans.))
Thus Ajdukiewicz not only refrains from regarding axioms as propositions (“sentences”), but declines even to regard them as propositional functions (“sentential functions”). One of the reasons for this is that, according to Ajdukiewicz, the logical symbols are not endowed with meaning (Ajdukiewicz 1921, p. 20).
Betti’s suggestion (Betti 2008, p. 69) that by “meaningful sentence” one could read “well-formed sentence”, which she bases on a footnote in Tarski (1930), is not enough to enable a rejection of the claim that Tarski was working contentually. For Tarski’s contentual approach is indicated here not by his use of the words “meaningful sentence”, but rather by his remark that the notion of truth is not relevant for languages that are formal in the sense that no material sense is attached to signs and expressions (Tarski 1933, 1956b, p. 166).
Note therefore that for Tarski, contrary to Carnap’s practice, a model is formed not by the substituted terms, but rather by the objects they denote.
References
Ajdukiewicz, K. (1921). Z metodologii nauk dedukcyjnych. Lwow, Nakladam Polskiego Towarzystwa Filozoficznego. From the Methodology of the Deductive Sciences (Eng Trans: Giedymin, J.). Studia Logica, 19(1966), 9–46.
Betti, A. (2008). Polish axiomatics and its truth: On Tarski’s Leśniewskian background and the Ajdukiewicz connection. In D. Patterson (Ed.), New essays on Tarski and philosophy (pp. 44–71). Oxford: oxford University Press.
Carnap, R. (1937). The logical syntax of language. London: Kegan Paul, Trench, Trubner.
Carnap, R. (2000). Untersuchungen zur allgemeinen Axiomatik. Darmstadt: Wissenschaftliche Buchgesellschaft.
Coffa, J. (1991). The semantic tradition from Kant to Carnap: To the Vienna Station. Cambridge: Cambridge University Press.
De Rouilhan, P. (1998). Tarski et l’université de la logique: Remarques sur le post-scriptum au \(\langle \langle \) wahrheitsbegriff \(\rangle \rangle \). In F. Nef & D. Vernant (Eds.), Le formalisme en question: Le tournant des anneés 30 (pp. 85–102). Paris: Vrin, Problèmes et controverses.
de Rouilhan, P. (2012). In Defense of Logical Uiversalism: Taking Issue with Jean van Heijenoort. Logica Universalis. Special Issue: Perspectives on the History and Philosophy of Modern Logic: Van Heijenoort Centenary, 6, 553–586.
Hintikka, J. (1988). On the development of the model-theoretic viewpoint in logical theory. Synthese, 77, 1–36.
Huntington, E. (1913). A set of postulates for abstract geometry, expressed in terms of the simple relation of inclusion. Mathematische Annalen, 73(4), 522–559.
Jané, I. (2006). What is Tarski’s common Notion of Concequence? The Bulletin of Symbolic Logic, 12(1), 1–42.
Kattsoff, L. O. (1936). Postulational Methods. II. Philosophy of Science, 3(1), 67–89.
Keyser, C. J. (1918). Doctrinal Functions. The Journal of Philosophy, Psychology and Scientific Methods, 15(10), 262–267.
Korte, T. (2010). Frege’s begriffsschrift as a lingua characteristica. Synthese, 174(2), 283–294.
Loeb, I. (2014a). Towards Transfinite Type Theory: Rereading Tarski’s Wahrheitsbegriff. Synthese, 191(10), 2281–2299.
Loeb, I. (2014b). Uniting model theory and the universalist tradition: Carnap’s early axiomatics. Synthese, 191(12), 2815–2833.
Mancosu, P. (2006). Tarski on models and logical consequence. In J. Ferreirós & J. J. Gray (Eds.), The architecture of modern mathematics (pp. 209–237). Oxford: Oxford University Press.
Mancosu, P., Zach, R., & Badesa, C. (2009). The development of mathematical logic from Russell to Tarski, 1900–1935. In L. Haaparanta (Ed.), The development of modern logic (pp. 318–471). New York: Oxford University Press.
Mares, E. (2011). Propositional function. The Stanford Encyclopedia of Philosophy E. N. Zalta, Ed., fall 2011 ed.
Padoa, A. (1967). Logical introduction to any deductive theory. In J. van Heijenoort (Ed.), From Frege to Gödel: A source book in mathematical logic, (pp. 118–123). Cambridge: Harvard University.
Peckhaus, V. (2004). Calculus ratiocinator versus characteristica universalis? The two traditions in logic, revisited. History and Philosophy of Logic, 25(1), 3–14.
Pieri, M. (1900). Sur la géometria envisagée comme un système purement logique. Bibliothèque du Congrès international de philosophie, Paris, 1900, 3, 367–404, reprinted in Pieri (1980), pp. 235–271.
Pieri, M. (1980). Opere sui fondamenti della matematica. Collana: Opere grandi matematici italiani, Edizioni Cremonese.
Proops, I. (2007). Russell and the Universalist Conception of Logic. Noûs, 41(1), 1–32.
Reck, E. H. (2007). Carnap and Modern Logic. The Cambridge Companion to Carnap. Cambridge: Cambridge University Press.
Rodríguez-Consuegra, F. (2005). Tarski’s intuitive notion of set. In G. Sica (Ed.), Essays on the foundations of mathematics and logic (pp. 227–266). Monza: Polimetrica International Scientific Publishers.
Russell, B. (1903). The principles of mathematics. New York: Norton and Norton.
Scanlan, M. (1991). Who were the American postulate theorists? The Journal of symbolic logic, 56(3), 981–1002.
Scanlan, M. (2003). American postulate theorists and Alfred Tarski. History and Philosophy of Logic, 24(4), 307–325.
Sinaceur, H. (2001). Alfred Tarski: Semantic shift, heuristic shift in metamathematics. Synthese, 126(1), 49–65.
Sluga, H. (1987). Frege Against the Booleans. Notre Dame Journal of Formal Logic, 28(1), 80–98.
Tappenden, J. (1997). Metatheory and Mathematical Practice in Frege. Philosophical Topics, 25, 213–264.
Tarski, A. (1930). Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften, I. Monatshefte für Mathematik und Physik, 37, 361–404, reprinted in English as Tarski (1956b).
Tarski, A. (1933). Pojȩcie prawdy w jȩzykach nauk dedukcyjnych. Nakładem / Prace Towarzystwa Naukowego Warszawskiego, wydzial, III, 34.
Tarski, A. (1935). Grundzüge des Systemenkalkül, Erster Teil. Fundamenta Mathematicae, 25, 503–526.
Tarski, A. (1936). Grundzüge des Systemenkalkül, Zweiter Teil. Fundamenta Mathematicae, 26, 283–301.
Tarski, A. (1936). Über den Begriff der logischen Folgerung. Actes du Congrès Internationales de Philosophique Scientifiques, 7(394), 1–11.
Tarski, A. (1937a). Einführung in die mathematische Logik und in die Methodologie der Mathematik. Vienna: J. Springer.
Tarski, A. (1937b). Sur la méthode déductive. Traveaux du IX\(^{e}\) Congrès International de Philosophie, Tome 6. Herman et Cie, Paris, Actualités Scientifiques et Industrielles, 535, 95–103.
Tarski, A. (1941). Introduction to Logic and the Methodology of Deductive Sciences, third English edition. New York: Oxford University Press, 1966.
Tarski, A. (1956a). The concept of truth in formalized languages. Logic, semantics, metamathematics. Oxford: Oxford University Press.
Tarski, A. (1956b). Fundamental concepts of the methodology of the deductive sciences. Logic, Semantics, Metamathematics. Oxford: Oxford University Press.
Van Heijenoort, J. (1967). Logic as language and logic as calculus. Synthese, 17(1), 324–330.
Whitehead, A., & Russell, B. (1910). Principia Mathematica (Vol. I). Cambridge: Cambridge University Press.
Woleński, J. (1991). Semantic revolution: Rudolf Carnap, Kurt Gödel, Alfred Tarski. In J. Woleński & E. Köhler (Eds.), Alfred Tarski and the Vienna Circle: Austro-Polish connection in logical empiricism (pp. 1–15). Dordrecht: Kluwer Academic Publishers.
Acknowledgments
The author wishes to thank Arianna Betti for her comments on an earlier version of this paper. The author was supported by 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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Loeb, I. The role of universal language in the early work of Carnap and Tarski. Synthese 194, 15–31 (2017). https://doi.org/10.1007/s11229-014-0601-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-014-0601-4