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Alfred Tarski and the "Concept of Truth in Formalized Languages" pp 9–116Cite as

Commentary

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Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 39))

Abstract

[152] At the very beginning of his monograph, Tarski puts forward his goal. It is to construct “a materially adequate and formally correct definition of the term ‘true sentence’”. The conditions for defining certain concepts are also given here. The bibliographical note is worth of notice, especially since in this form it only appears in the English version. Only the first paragraph of the note can be read in the German translation.

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Notes

  1. 1.

    “\(1^{\circ }\) Pragnąłbym (o ile to jeszcze możliwe), by tytuł mojej pracy brzmiał “Der Wahrheitsbegriff in den formalisierten Sprachen”. \(2^{\circ }\) Praca moja była napisana przed trzema laty; od tego czasu poglądy moje w kilku punktach uległy zmianie. Pragnąłbym, by ten stan rzeczy znalazł swój wyraz w pracy, Mógłbym w tym celu zmienić zakończenie. Bardziejby mi jednak odpowiadało inne załatwienie sprawy – chciałbym po “Zakończeniu” umieścić jeszcze dodatek (“Nachwort”), który by zajął w przybliżeniu 2 strony druku, Czy to możliwe i czy mógłbym nadesłać tekst po polsku?” Letter L. 105/35 archived in Polskie Towarzystwo Filozoficzne, Poznań.

  2. 2.

    Givant (1991, 18).

  3. 3.

    Patterson’s book version used here, is the version available online in 2011, therefore it may happen that the page numbers do not have a one to one correspondence with the edition printed in 2012.

  4. 4.

    Cf. Betti (2004).

  5. 5.

    Cf. Woleński (1999), p. 1.

  6. 6.

    E.g. Field (1972), McDowel (1978).

  7. 7.

    Rojszczak (1999).

  8. 8.

    All the references refer to the CTFL: [37] (Tarski 1933, pp. 12–13, 27–29, 63–65), (Tarski 2006g, pp. 162–164, 176–178, 212–214); [38] (Tarski 1933, p. 5), (Tarski 2006g, p. 156); [39] (Tarski 1933, p. 25), (Tarski 2006g, p. 174).

  9. 9.

    Cf. Woleński (1989), Sundholm (2003), Betti (2008), Patterson (2012).

  10. 10.

    “I have no predilection for various ‘mathematical games’ that consist in writing out, according to one or another conventional rule, various more or less picturesque formulae which need not be meaningful or even – as some of the ‘mathematical gamers’ might prefer – which should necessarily be meaningless, I would not have taken the trouble to systematize and to often check quite scrupulously the directives of my system, had I not imputed to its theses a certain specific and completely determined sense , in virtue of which its axioms , definitions, and final directives (as encoded for SS5), have for me an irresistible intuitive validity. I see no contradiction, therefore, in saying that I advocate a rather radical ‘formalism’ in the construction of my system even though I am an obdurate ‘intuitionist’. Having endeavoured to express some of my thoughts on various particular topics by representing them as a series of propositions meaningful in various deductive theories, and to derive one proposition from others in a way that would harmonize with the way I consider finally intuitively binding, I know no method more effective for acquainting the reader with my logical intuitions than the method of formalizing any deductive theory to be set forth. By no means do theories under the influence of such a formalization cease to consist of genuinely meaningful propositions which for me are intuitively valid. (Leśniewski 1992, p. 487)

  11. 11.

    Cf. Patterson (2012).

  12. 12.

    A thorough discussion regarding Intuitionistic Formalism goes beyond the scope of this project as well as beyond my competence. For an exhausting presentation on this matter the reader is referred to Patterson (2012).

  13. 13.

    “Perhaps, it is unnecessary to add, that we are not interested here in languages and sciences which are ‘formal’ in certain specific sense of this term, namely such sciences to the signs and expressions occurring in them no intuitive meaning is attached; in regard to such sciences the issue raised here ceases to apply and it becomes no longer intelligible. To the signs occurring in the languages considered here we shall always ascribe quite concrete and, for us, intelligible meaning;”(translation M.G.)

  14. 14.

    See the commentary to [153].

  15. 15.

    For further discussion on this topic see Woleński (2003).

  16. 16.

    Cf. Betti (2004), p. 280.

  17. 17.

    For further discussions on this topic see for example Tarski (2006f), Tarski (2006d), Coffa (1987, p. 556 ff.), Patterson (2012).

  18. 18.

    Cf. Woleński (2013).

  19. 19.

    For a detailed discussion regarding Polish notation see Woleński (2013).

  20. 20.

    As will be discussed later (within the commentary to [215] and to Postscript), there is an open debate regarding Tarski’s choice of logical framework and his alleged change of it from the one used in the main text to the one used in the postscript. Although his theory referred to expressions, the notion of order introduced on [218] is not without ambiguity and can be interpreted as applying to objects. On the other hand, Whitehead and Russell’s theory is also difficult to assess unequivocally.

  21. 21.

    For further discussion on Tarski’s axioms see for example Essler (1999).

  22. 22.

    Tarski does not define the property of being categorical himself but refers to works by Veblen (1904) and (cf. Fraenkel (1928), pp. 334–354) Tarski did consider the problem of categoricity, however, together with the problem of completeness of concepts in Tarski (2006f).

  23. 23.

    For a discussion on Tarski and physicalism see e.g. Field (1972), McDowel (1978), Rojszczak (1999).

  24. 24.

    Cf. Tarski (2006g), pp. 215–227.

  25. 25.

    Cf. Patterson (2012), p. 75.

  26. 26.

    Cf. Łukasiewicz (1929, p. 163).

  27. 27.

    There is a considerable amount of literature regarding Tarski’s definition of consequence, however, it concentrates mostly on his definition of logical consequence as presented in the article “On the concept of logical consequence” (Tarski 2006e). For detailed discussions on this topic see for instance (Etchemendy 1990; Gómez-Torrente 1996; Patterson 2012; Ray 1996; Sher 1991).

  28. 28.

    The formalization of this and of the following definitions, theorems and lemmata has its origin in a seminar: Alfred Tarski, Wahrheitsbegriff and is an intellectual property of Edgar Morscher, who held the mentioned seminar at the University of Salzburg in 2007. He kindly offered that I use his formalizations in my dissertation, on which this publication is based. The translation into English and the further dissemination of the formulas is my own responsibility.

  29. 29.

    Morscher (2007).

  30. 30.

    Tarski (1944, p. 342).

  31. 31.

    Tarski (1944, p. 344).

  32. 32.

    Cf. Tarski (1944, p. 344).

  33. 33.

    E.g. deflationists like Field or Horwich.

  34. 34.

    An important exception are the infinitary languages whose formulas are identified as infinite sets. They however, had not been investigated until the late 50’s for example in Scott and Tarski (1958) “The sentential calculus with infinitely long expressions”. See also Bell (2012).

  35. 35.

    Patterson (2012, pp. 124–6) presents a thorough analysis of Convention T, including its historical and philosophical origins, also commenting on the problematic translations.

  36. 36.

    The following discussion on the notion of satisfaction is based on Morscher (2007). For further literature on this topic see Fine and McCarthy (1984), Betti (2008).

  37. 37.

    Cf. Wagner (2009), p. 26.

  38. 38.

    Patterson (2012).

  39. 39.

    Tarski (2006f, p. 297, ftn.1).

  40. 40.

    Patterson (2012, p. 82) notes that the quotation marks around the terms “interpretations” and “realizations” have been stripped in the English version compared to the German one. In Polish, there are no quotation marks around the term “interpretation” and instead of “realizations” Tarski writes “relacja R przekształca \(x'\) w \(x''\)” (Tarski 2006f, p. 133), which could be translated as “relation R transforms \(x'\) in \(x''\) ”. This, however, is equivalent to the English translation.

  41. 41.

    For a detailed discussion on categoricity and completeness of terms in Tarski’s works see Patterson (2012, pp. 81–7), for a discussion on essential richness see Ray (2005).

  42. 42.

    Cf. Sundholm (2003).

  43. 43.

    Cf. Tarski (2006g), p. 215.

  44. 44.

    Loeb (2014).

  45. 45.

    Sundholm (2003).

  46. 46.

    Feferman (2002).

  47. 47.

    Carnap (1929, p. 31).

  48. 48.

    Cf. Gruber (2015).

  49. 49.

    For further discussion regarding Tarski’s usage of the notion of order see e.g. Coffa (1987), de Rouilhan (1998), Loeb (2014), Patterson (2012).

  50. 50.

    Cf. Tarski (2006), p. 219.

  51. 51.

    Cf. Gruber (2015).

  52. 52.

    Cf. Gruber (2015).

  53. 53.

    Cf. Morscher (2007).

  54. 54.

    Cf. Gruber (2015).

  55. 55.

    For further reading on this topic see e.g., de Rouilhan (1998), Hintikka (1988), Mancosu (2010), Rodríguez-Consuerga (2005), Van Heijenoort (1967).

  56. 56.

    Gödel (2006, pp. 174–5).

  57. 57.

    For a thorough discussion regarding the problems of indefinability and inconsistency see, for example, Patterson (2012, pp. 163–188).

  58. 58.

    Cf. Morscher (2007).

  59. 59.

    E.g. Putnam (1994), Davidson (1990).

  60. 60.

    For a detailed discussion see Patterson (2012), Chap. 1.

  61. 61.

    Gómez-Torrente (2015).

  62. 62.

    See also Sect. 2.1.

  63. 63.

    “W wielu miejscach miałem jedank wątpliwości, czy prof. Adjukiewicz obrał przy przekładzie najtrafniejszy zwrot lub też czy zupełnie wienie oddał tekst oryginału. Rzecz jasna, przy mojej znajomości niemieckiego nie mogłem nawet myśleć o wprowadzeniu na własną rękę jakichkolwiek istotniejszych zmian stylistycznych, Z drugiej strony, ponieważ Wielce Szanowny Pan Profesor życzył sobie bezzwłocznego zwrotu manuskryptu, nie mogłem go przesłać do nikogo z tych znajomych wiedeńskich, którzy mi uprzednio przy korekcie pomagali, Dlatego też zadowoliłem się skontrolowaniem Nachwort’u pod względem rzeczowym, uzgodnieniem go pod względem terminologicznym, symbolicznym i częściowo ortograficznym z poprzedniemi arkuszami pracy oraz wprowadzeniem kilku mało istotnych zmian.” Letter L. 224/35 archived in Polskie Towarzystwo Filozoficzne, Poznań.

  64. 64.

    “Przesyłam równocześnie “Nachwort”; zajął on, niestety, więcej miejsca niż przypuszczałem. Bardzo się będę cieszył, jeśli zostanie on wydrukowany wraz z całą pracą; gdyby brak miejsca stał na przeszkodzie, gotów byłbym nawet zgodzić się na opuszczenie jakiegoś innego fragmentu pracy”. Letter L. 115/35 archived in Polskie Towarzystwo Filozoficzne, Poznań.

  65. 65.

    The discussion in the postscript is based on an article I’ve been working on simultaneously to this monograph. It appeared in Journal for the History of Analytical Philosophy, Vol. 3, No. 10 (2015); cf. Gruber (2015).

  66. 66.

    Carnap (1937, p. 188).

  67. 67.

    See for example Field (2008, pp. 23–41).

  68. 68.

    “Co do spraw, poruszonych w liście Wielce Szanownego Pana Profesora a dotyczącego tekstu “Nachwort’u”, to nie mogłem oczywiście sam rozstrzygnąć, który ze zwrotów “Nachfolger” czy “Fortsetzer” lepiej się nadaje – może raczej ten drugi (inne wydają się mniej odpowiednie). Sprawę użycia terminów “unendlich” i “transfinit” załatwiłem w ten sposób, że używam wyłącznie (tak jak i dotąd w pracy) “Sprachen unendlicher Ordnung”; slowo “transfinit” musiało jednak pozostać dla oznaczenia liczb porządkowych”. Letter L. 224/35 archived in Polskie Towarzystwo Filozoficzne, Poznań.

  69. 69.

    See e.g. Ray (2005).

  70. 70.

    Cf. Tarski (2006g), p. 271, ftn.1.

  71. 71.

    Cf. Patterson (2012), p. 191.

  72. 72.

    For a detailed discussion on this topic see de Rouilhan (1998) and Loeb (2014). Loeb also presents an interesting argument on a possible interpretation of Tarski’s choice of a logical framework in the postscript.

  73. 73.

    Cf. Feferman (2002, pp. 2–3).

  74. 74.

    “Otrzymałem już odpowiedź odd Carnapa, ale nie znalazłem w niej niestety żadnej rozsądnej rady. Carnap zastrzegł się tylko przeciw użyciu w tej sytuacji, o która mi chodzi, słowa “Name” (=“Eigenname”) i przysłał mi korektę swej nowej książki “Die logische Syntax der Sprache”, proponuję dostasować się do przyjętej tam terminologii”. Letter L. 149/34 archived in Polskie Towarzystwo Filozoficzne, Poznań.

  75. 75.

    Cf. Tarski (2006g), p. 218.

  76. 76.

    Tarski (2006g, p. 270, ftn. 2).

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    Monika Gruber

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Gruber, M. (2016). Commentary. In: Alfred Tarski and the "Concept of Truth in Formalized Languages". Logic, Epistemology, and the Unity of Science, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-319-32616-0_2

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