Abstract
Tarski’s postscript (Ps) to the second, German edition of his famous work on the concept of truth (Wb, for “Wahrheitsbegriff”), began by a recantation about the theory of semantical categories. In Wb he had unreservedly supported this theory, in Ps he rejected it. But there is strong evidence that something deeper had happened, beyond this avowed change of opinion and irreducible to it, something unavowed, perhaps unthought. I identify this deeper change as an abandonment of the universalism inherited from the founding fathers of modern logic, to which Tarski had previously remained faithful. I argue that this abandonment was, beyond all mathematical proof, a matter for philosophy.
The present paper is a revised edition of [20]. The general line is the same as before, but some substantial changes appeared to be needed for the sake of clarity and persuasiveness. Thank you to Serge Bozon and François Rivenc for their stimulating comments. Thank you also to Claire O. Hill for her linguistic help at an earlier stage of the work. All awkward turns of phrase are of my own invention.
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Notes
- 1.
The French edition of Tarski’s work, in [28], vol. I, provides a seemingly questionable translation of the original, Polish edition [25], but usefully points out the changes introduced in the subsequent, German [26] and English [27] editions. There is also the second, revised edition [29] of [27], with the same pagination and a wonderful analytical index.
- 2.
I correct Finlay’s “logico-grammatical laws”, which fail to allow for the “rein” of “reinlogisch grammatische [Gesetze]”.
- 3.
- 4.
- 5.
- 6.
- 7.
- 8.
See [25] as translated in [28, vol. I, p. 216]: “Du point de vue intuitif la réponse est indubitable”. In the subsequent editions, the formulation would be more cautious, see [26, p. 336], [27 and 29, p. 216]: “From the standpoint of the ordinary usage of language, [the first principle] seems much more natural [than its negation].”
- 9.
- 10.
I borrow this term (“Bruchstück”) from Tarski (see below, § 6, first quotation), who, if I am not mistaken, never explained what, exactly, he meant by it. In the present case, I think that a fragment of the language of the theory of types is a (sub)language obtained by removing all variables of finitely or infinitely many types.
- 11.
Whether in Wb or in Ps, the constraint put by Tarski on extra-logical constants, that their semantical category be represented by variables, seems uselessly restrictive in the case of predicates and functors. It could be replaced by the same constraint put only on their arguments.
- 12.
One will perhaps feel like protesting that this embedding of the intended universe of the language of type theory into that of the language of set theory ZFC is but a takeover by force, which reduces the infinite multiplicity of empty classes of the former universe to the one and only empty set of the latter. To which I will respond that it is rather the usual description of the intended universe of the language of type theory that distorts the situation when one thinks to have to distinguish empty classes themselves in parallel with type distinctions of expressions that refer to them. (On this kind of illusion, which dates back to Frege, see Rouilhan [18, Chap. 5] and [21].)
- 13.
The two hierarchies would become identical, if we identified individuals of usual set theory and their singleton. This move, however unusual, would be legitimate, and even advisable, for none of the objections put forward since Frege and Peano against identifying an object to its singleton applies to individuals, and thus, until further notice, Occam’z razor commends doing it. I will perhaps be told that all this is grist to Quine’s mill for his well known explication of the notion of individual in set theory as object identical with its singleton (first published in [16]). Well, let me insist that, in contradistinction with Quine’s, my explication gets cleared of all artificiality. Quine got his explication all by a trick made to comply with the maxim of simplicity; mine is the fruit of a correction made to comply with the maxim of parsimony.
- 14.
I.e. Skolem [24], mentioned by Tarski, but also Frænkel, Neumann, etc.
- 15.
ZFC may taken to be of first order relatively to its intended universe and there is nothing wrong about that. It is often taken to be of first order in the sense of first-order logic, but it is a mistake, for its intended universe should then be a set, while actually it is a proper class.
- 16.
This notion of type must not be confused here with that of one or another theory of types. Set theory is not what one calls a “theory of types”.
- 17.
It is supposed, here, that individuals are intended to form a set.
- 18.
- 19.
Admittedly, the language of ZFC could be reinterpreted so as to appear as a first-order language grounded on type theory, but this reinterpretation would not be the originally intended interpretation, whose relation to type theory was exposed in section 4.
- 20.
One will think here of the influence of Kotarbinski and Lesniewski on the young Tarski, but also of the astonishing profession of faith cheerfully made by Tarski twenty years later in a cenacle of logicians: “I happen to be, you know, a much more extreme anti-Platonist. […] However, I represent this very rude, naïve kind of anti-Platonism, one thing which I could describe as materialism, or nominalism with some materialistic taint, and it is very difficult for a man to live his whole life with this philosophical attitude, especially if he is a mathematician, especially if for some reasons he has a hobby which is called set theory.” (Transcript of remarks, ASL meeting, Chicago, Ill., April 29, 1965, Bancroft Library, p. 3, quoted by Mancosu [15])
- 21.
The term (“vollständig”) is obviously not to be understood in the model-theoretical sense that it took on in Hilbert and Ackermann in 1928, then in Gödel in 1929 [5] and 1930 [6]. One could at the very most understand it in the “experimental” sense, as Herbrand said, that it could have from the point of view of the founding fathers. But, as the rest of the quotation shows clearly, it was first and above all here a matter of completeness of the expressive power, not of the proof-theoretical one [7, 8].
- 22.
In other words, which I am used to using, the given language is logically universal in the sense that everything expressible in any of the formalized languages envisaged is expressible, not necessarily within the given language itself, but within the framework of it, meaning here (to follow Tarski, but see above, n. 11) in some extension of it obtained by adding extra-logical constants whose semantical category is already represented by certain expressions of it.
- 23.
In my favorite terminology, to which, here as everywhere else, I strive to remain faithful, in no way is this theory a theory of sets, it is a theory of classes. I discussed the relationship between the theory of types and set theory and the proper use of set-theoretical terminology at length in [18, chap. VI]; see also [19, chap. II, sect. B].
- 24.
- 25.
Cantor, 1899, called certain “multiplicities”, such as that of ordinals, “absolutely infinite” [1].
- 26.
“[A] theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations of the theory be true.” [17, p. 33]
- 27.
For a brief description and a defense of the version in question, see [21].
- 28.
In Wb, about “expressions of ‘infinite order’” Tarski could not but say: “[N]either the metalangage which forms the basis of the present investigations, nor any other of the existing languages, contains such expressions. It is in fact not at all clear what intuitive meaning could be given to such expressions.”
References
G. Cantor, Letters to Dedekind dated July 28 and August 31, 1899. In: [2], pp. 447–448 (1932)
G. Cantor, in Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, ed. by E. Zermelo (Springer, Berlin, 1932)
R. Carnap, Abriss der Logistik (Springer, Vienna, 1929)
L. Chwistek, The theory of constructive types. Annales de la Société Polonaise de Mathématiques 2, 9–48 (1924)
K. Gödel, Über die Vollständigkeit des Logikkalküls, Dissertation, University of Vienna, 1929. In: [7], vol. I, pp. 60–101 (1986)
K. Gödel, Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematitik und Physik 37, 349–360 (1930)
K. Gödel, in Collected Works, 5 vols, ed. by S. Feferman et al. (Oxford University Press and Clarendon Press, New York and Oxford, 1986–2003)
D. Hilbert, W. Ackermann, Grundzüge der theoretischen Logik (Springer, Berlin, 1928); 2d edition 1938; 3d, English, edn 1950, see [9]
D. Hilbert, W. Ackermann, Principles of Mathematical Logic (revised and annotated edn by Luce, R. E.; translated by Hammond, L. M., Leckie, G. G., Steinhardt F.) (Chelsea, New York, 1950)
E. Husserl, Logische Untersuchungen, 2 vols (M. Niemeyer, Halle, 1900–1901); 2d edition, 3 vols, 1913–1921
E. Husserl, Logical Investigations, two vols (2d edn of [10], translated by Finlay, J. N.) (Routledge, London and New York, 1970)
D. Mirimanoff, Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles. L’Enseignement Mathématique 19, 37–52 (1917)
S. Lesniewski, Grundzüge eines neuen Systems der Grundlagen der Mathematik. Fundamenta Mathematicae 14, 1–81 (1929)
S. Lesniewski, Ueber die Grundlagen der Ontologie. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III 23, 111–1332 (1930)
P. Mancosu, Harvard 1940–41: Tarski, Carnap and Quine on a finitistic language of mathematics for science. Hist. Philos. Logic 26, 327–357 (2005)
W.V.O. Quine, New foundations for mathematical logic. Am. Math. Mon. 44, 70–80 (1970)
W.V.O. Quine, On what there is. Rev. Metaphys. 2, 21–38 (1948)
Ph. de Rouilhan, Frege. Les paradoxes de la representation (Editions de Minuit, Paris, 1988)
Ph. de Rouilhan, Russell et le cercle des paradoxes (Presses Universitaires de France, Paris, 1996)
Ph. de Rouilhan, Tarski et l’universalité de la logique, in Le formalisme en question. Le tournant des années 30 (Vrin, Paris, 1998), pp. 85–102
Ph. de Rouilhan, In defense of logical universalism. Taking issue with Jean van Heijenoort. Logica Universalis 6, 553–586 (2012)
B. Russell, see [30], vol. I, 2d edn (1925)
Th. Skolem, Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Mathematikerkongressen i Helsingfors den 4–7 Juli 1922, Den femte skandinaviska mathematikerkongressen, Redogörelse (Akademiska Bokhandeln, Helsinki, 1923), pp. 217–232
T. Skolem, Über einige Grundlagenfragen der Mathematik, Sckrifter utgitt av Det Norske Videnskaps-Akademi i Oslo, I. Mat-nat. Klasse 4, 1–49 (1929)
A. Tarski, Projecie prawdy w jezykach nauk dedukcyjnych (The concept of truth in the languages of deductive sciences). Varsaw (1933)
A. Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica 1, 261–405 (1936)
A. Tarski, Logic, Semantics, Metamathematics – Papers from 1923 to 1938 (edited and translated by Woodger, J.) (Clarendon Press, Oxford, 1956)
A. Tarski, Logique, sémantique, métamathématique 1923–1944, 2 vols (edited and translated by Granger, G.-G., et al.) (A. Colin, Paris, 1972–1974)
A. Tarski, revised edition of [27], ed. by J. Corcoran (Hackett Publishing Company, 1983)
A.N. Whitehead, B. Russell, Principia Mathematica, 3 vols. (Cambridge University Press, Cambridge, 1910–1913); 2d edn, 3 vols (1925–1927)
E. Zermelo, Untersuchungen über die Grundlagen der Mengenlehre, I. Mathematische Annalen 59, 261–281 (1908)
E. Zermelo, Ueber Grenzzahlen und Mengenbereiche – Neue Untersuchungen über die Grundlagen der Mengenlehre. Fundamenta Mathematicae 16, 29–47 (1930)
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de Rouilhan, P. (2016). Tarski’s Recantation: Reading the Postscript to “Wahrheitsbegriff”. In: Abeles, F., Fuller, M. (eds) Modern Logic 1850-1950, East and West. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-24756-4_5
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