Abstract
In his famous paper Der Wahrheitsbegriff in den formalisierten Sprachen (Polish edition: Nakładem/Prace Towarzystwa Naukowego Warszawskiego, wydzial, III, 1933), Alfred Tarski constructs a materially adequate and formally correct definition of the term “true sentence” for certain kinds of formalised languages. In the case of other formalised languages, he shows that such a construction is impossible but that the term “true sentence” can nevertheless be consistently postulated. In the Postscript that Tarski added to a later version of this paper (Studia Philosophica, 1, 1935), he does not explicitly include limits for the kinds of language for which such a construction is possible. This absence of such limits has been interpreted as an implied claim that such a definition of the term “true sentence” can be constructed for every language. This has far-reaching consequences, not least for the widely held belief that Tarski changed from an universalistic to an anti-universalistic standpoint. We will claim that the consequence of anti-universalism is unwarranted, given that it can be argued that the Postscript is not in conflict with the existence of limits outside of which a definition of “true sentence” cannot be constructed. Moreover, by a discussion of transfinite type theory, we will also be able to accommodate other of the changes made in Tarski’s Postscript within a type-theoretical framework. The awareness of transfinite type theory afforded by this discussion will lead, in turn, to an account of Tarski’s Postscript that shows a gradual change in his logical work, rather than any of the more radical transitions which the Postscript has been claimed to reflect.
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Notes
For a comparison between the Polish, German and English versions see Gruber (2012).
It may be argued that the choice of the abbreviations “Wb” and “Ps” is not very consistent, language-wise. Yet we will keep to these in order to facilitate the discussion of de Rouilhan (1998), who uses the same abbreviations.
Although by abuse de langage Tarski also applies the notion of semantical categories to objects (Tarski 1956, p. 219), and although the situation in Russell is hard to assess unambiguously.
A sentential function is an open or closed formula. This notion is used to differentiate those expressions from, for example, terms of the language.
Functors are signs denoting functions.
Tarski also defines the notion of semantical type (Tarski 1956, p. 219), which is based on the number of free variables and the semantical categories of those variables given some sentential function. Tarski’s notion of semantical types is thus very different from the notion of type in Russell’s type theories.
It is, of course, quite hard to imagine what such a language would look like. See also the remark on the idealisation of languages in (Linnebo and Rayo (2012), p. 277).
E.g. Sundholm (2003) as quoted above.
Note that Rodríguez-Consuegra distills a different, but related, notion of universality from the same passages of Wb (Rodríguez-Consuegra 2005, p. 229). What he calls mathematical universality concerns the capacity of expressing the whole of mathematics, whereas de Rouilhan’s notion revolves concerns the incorporation in a language of elements of all categories/orders. Moreover, Rodríguez-Consuegra considers Tarski’s formalisation of set theory to be universal, because it additionally uses a fixed, single universe of discourse, instead of one for every type (p. 230/232). Because in this latter understanding of the notion of universality one could even say that type theory is not universal, it already shows that it is quite far from our use of universality in the current paper.
Le cadre de la théorie des types simple avait quelque chose de rigide que celui de la théorie des ensembles n’a pas. L’idée d’un cadre supérieur à celui de la théorie des types simples, mais que l’on pût encore qualifier de “théorie des types simples” ne pouvait même venir à l’esprit: si l’on changeait de cadre, on en changeait radicalement. Maintenant, au contraire, il semble que l’on puisse toujours changer le cadre tout en préservant le caractère esembliste. L’universalité putative d’un cadre esembliste semble pouvoir toujours être supplantée par celle d’un autre de même genre, et l’universalisme perdre toute vraisemblance.
De Rouilhan, like Field, holds that a conclusion analogous to conclusion \(C\) of Wb can easily be provided for Ps. Yet contrary to Field’s proposal, de Rouilhan proposes that this conclusion runs somewhat as follows (de Rouilhan 1998, p. 98):
- \(\overline{C}\) :
-
On the other hand, even if the order of the metalanguage is at most equal to that of the language itself, the consistent and correct use of the concept of truth is rendered possible by including this concept in the system of primitive concepts of the metalanguage and determining its fundamental properties by means of the axiomatic method.
In particular it is always possible to construct the metalanguage in such a way that it contains variables of higher order than all the variables of the language studied. The metalanguage then becomes a language of higher order and thus one which is essentially richer in grammatical forms than the language we are investigating. (Tarski 1956, pp. 271–272)
Du point de vue de cette dernière théorie, les variables qui représentent des noms d’ensembles sont les mêmes (et sont donc de même categorie et de même ordre au sense syntaxique) que celle qui représentent des noms d’éléments éventuels de ces ensembles, etc., et, finalement, que celles qui représentent des noms d’individus. C’est dans cette mesure précisément que les objets d’ordre supérieur au sens ontologique méritent pleinement le nom d’“ensembles”. Du point de vue de la théorie de types simple, au contraire, la difference d’ordre au sens ontologique implique une différence d’ordre au sens syntaxique. Les variable qui représentent des noms d’objets d’ordre supérieur au sense ontologique sont elle-même d’ordre supérieur au sens syntaxique.
In fact, de Rouilhan distinguishes three notions of order: the syntactical and the ontological which are orders given to languages, and additionally an order given to theories.
Russell’s and Tarski’s solution differ on other points as well, such as on the intentionality/ extensionality issue. See (Church (1976), p. 756).
It may be exactly because the real variables are not restricted to a certain type that Russell did not make his types visible in the syntax of the ramified theory of types (see Church 1976, fn 9, p. 750).
Intuitively, given our modern understanding of logic and logical notions, one could say that apparent variables are bound whereas real variables are free (see e.g. Church 1976, p. 750). This identification seems, however, not only slightly inaccurate (the real variables are in fact bound, to wit by “any”), but also neglects a very important difference between apparent and real variables: the former are typed, whereas the latter are not.
Note that giving type \(\omega \) to the predicate “\(x\) is false” would not solve anything, though.
Especially relevant is Tarski and Corcoran (1986). In that paper Tarski compares type theory and first-order set theory with respect to the question whether or not the membership relation is a logical notion. At least it shows that Tarski had not completely abandoned research to type theory in 1966.
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Acknowledgments
I would like to thank Monika Gruber for discussions on the Wahrheitsbegriff that have been very helpful to me, in particular because in general we seem to disagree. Furthermore I like to thank the participants of the “Tarski Seminar”, which took place in fall term 2011 and made reading Tarski’s paper a very pleasant activity. Arianna Betti, Hein van den Berg, Lieven Decock, Wim de Jong, Jeroen de Ridder, Stefan Roski and Jeroen Smid have read an earlier version of this paper and given me valuable feedback. I thank the anonymous referees for their suggestions. Work on this paper was made possible by ERC Starting Grant TRANH 203194.
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Loeb, I. Towards transfinite type theory: rereading Tarski’s Wahrheitsbegriff . Synthese 191, 2281–2299 (2014). https://doi.org/10.1007/s11229-014-0399-0
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DOI: https://doi.org/10.1007/s11229-014-0399-0