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Leśniewski on metalogic and definitions

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Abstract

Leśniewski’s metalogic is often considered to be difficult to understand because it differs greatly from its standard formulation. In this paper I try to explain the reasons of these idiosyncrasies. I claim that they have mainly two sources. First of all there is Leśniewski’s conviction that a formal system should be conceived as a set of concrete marks that can always physically and syntactically be expanded by the addition of new theses. Secondly there is Leśniewski’s conviction that definitions should neither be formulas belonging to the metalanguage, nor deduction rules, but formulas belonging to the object-language and expressed with the help of the biconditional functor. The realisation of the first point is linked to the second one in so far as the metalinguistic rule for the writing out of definitions has to be formulated in a way that makes it possible to build the formal system in agreement with Leśniewski’s conception. While explaining these points I give an overview of the main peculiarities of Leśniewski’s metalogic.

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Notes

  1. In the English translation of this text the temporal information is omitted. In the Polish text the word “kilkanaście” can be found which means, in this context, between 20 and 10 years before the publication of the present paper (1936).

  2. See Betti (2004).

  3. See Luschei (1962, pp. 167–88). I should also mention the following works: Simons (2002), Miéville (2009) and Betti (2010).

  4. Mereology is an extralogical theory because it does not need new metalinguistic rules to be built and does not introduce new semantic categories. In other words its grammar is given by the grammar of Protothetic and Ontology (Betti 2010, p. 300).

  5. The general ideas are indeed the same for Protothetic and Ontology, and the metalogic does not directly apply to Mereology, since this last system is extra-logical.

  6. See the formulas (1)–(5) in Leśniewski (1929, p. 76; Eng. trans. 1992, p. 485).

  7. For instance in the §11 of Leśniewski (1929, pp. 63–75; Eng. trans. 1992, pp. 472–484).

  8. See the formula (3) in Leśniewski (1929, p. 76; Eng. trans. 1992, p. 485).

  9. Luschei made a wonderful job trying to clarify Leśniewski’s T.E.’s in his 1962 monograph.

  10. See Leśniewski (1929, pp. 63–75; Eng. trans. 1992, pp. 472–484). The system of Ontology needs seven supplementary directives and 26 supplementary T.E.’s. See Leśniewski (1930, pp. 116–127; Eng. trans. 1992, pp. 610–624).

  11. Since in Leśniewski’s perspective formal systems are literally composed of their expressions, it is important to be able to speak about them in mereological terms.

  12. See Rickey (1972, 1973).

  13. He did not define the expressions coming from already existing formal systems, such as his own Ontology.

  14. Leśniewski did not try to axiomatize his metalanguage. In fact the first attempt to formalize a metalanguage was due to his student, Tarski, in his celebrated paper on truth. See Tarski (1933, pp. 41–42; Eng. trans. 1956, pp. 173–174).

  15. Joray speaks in this respect of a “genetic” conception of formal systems. See (2005b, p. 42).

  16. In fact Russell and Whitehead wrote the definitions in the following manner: ‘\(\ldots =\ldots \) Df’. I slightly modified their notation which I found sometimes ambiguous.

  17. See Łukasiewicz (1963, pp. 33 and 40).

  18. See Łukasiewicz (1963, p. 41).

  19. See Łukasiewicz (1963, p. 37).

  20. See Sobociński (1955/56).

  21. See (Leśniewski 1929, p. 33; Eng. trans. 1992, p. 441).

  22. On this definition see Joray (2011, pp. 57–83).

  23. This is the definition for the sentential constant of falsity.

  24. See the IV Investigation in Husserl (1970, pp. 39–62).

  25. See Joray and Godart-Wendling (2002, p. 40).

  26. If Leśniewski did not know how to offer a compositional treatment of the semantic category of the quantifier, this does not mean that such a treatment did not exist. Ajdukiewicz offered such a treatment of the quantifier with the help of Russell and Whitehead’s abstractor ‘⌃’. See Ajdukiewicz (1967, pp. 207–231) and also Joray and Godart-Wendling (2002, pp. 44–47).

  27. An extended sentential calculus enriched with quantifiers over sentential variables is discussed in Łukasiewicz (1963, pp. 92–102).

  28. See for instance Joray and Godart-Wendling (2002, pp. 42).

  29. See the requirement IV.a in Sobociński (1955/56, p. 61).

  30. Once such a definition has been written down, variables belonging to the semantic category so introduced may be bound by the quantifier.

  31. See the axiom \(A_n\) in Sobociński (1960, p. 67).

  32. In fact there can be more than one pair of such symmetric symbols, but I will not consider this case here.

  33. Leśniewski spoke of “parenthemes”. I preferred to use the expression “context”, whose meaning is less obscur.

  34. If this idea dates from 1924, it appeared in print only in Łukasiewicz (1929). See Łukasiewicz (1970, note 3, p. 180). Woleński remarks furthermore that the idea of prefixing the functors comes from Chwistek. He would have got this idea at the beginning of the years 1920 but did not see that he could then get rid of the parentheses. See Woleński (1989, p. 97).

  35. See the explanations in Łukasiewicz (1963, pp. 23 ff).

  36. In fact this is true only when the functor is applied to its arguments. When a functor is the argument of another functor, the context of the first functor may be omitted, since the context of the second functor is sufficient to determine the category of the first functor. I thank Pierre Joray for this insightful remark.

  37. In the English translation, the condition (mentioned in the original paper in German) according to which A has to be a symbol in order to a be a term is missing.

  38. In his paper of 1929 Leśniewski used three axioms to build his system of Protothetic, instead of (13′′′). Nevertheless the ideas exposed here can be easily transposed to his system of axioms. For these axioms see Leśniewski (1929, p. 33; Eng. trans. 1992, p. 441).

  39. This is what Suppes calls the “criterion of eliminability”.

  40. Suppes maintained that Leśniewski also formulated a criterion of non-creativity. This is false as Urbaniak and Hämäri rightly showed it (see 2012, pp. 159–189; and also Urbaniak (2014), chap. 6).

  41. See (Leśniewski 1929, pp. 70–72; Eng. trans. 1992, pp. 479–481).

  42. I omit here the complications due to functions with parameters. In the next explanations I relied heavily on Miéville (2001, in particular pp. 76–77).

  43. As I stated it in the previous footnote, I omitted here the case of functions whose values are functors.

  44. See T.E. 33.

  45. See Suppes (1999 [1957], p. 157).

  46. For an example see Suppes (1999 [1957], p. 157).

  47. On this topic, see the papers by Joray (2005a, b).

  48. This is the reason why it has sometimes been called an “inscriptional syntax”. See for instance Rickey (1972, pp. 1–33) and Simons (2002, p. 103).

  49. The first two in fact: Protothetic and Ontology.

  50. On this see Betti (2008, pp. 44–71).

  51. In private Tarski seems to have professed a philosophical preference for nominalism until the end of his life.

  52. See in particular Tarski (1933, note 5, p. 19; Eng. trans. 1956, note 1, p. 156).

  53. See for instance Tarski (1933, note 60, pp. 86–87). This passage does not appear in Woodger’s English translation of this text.

  54. See Patterson (2012, pp. 65–67).

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Acknowledgements

I would like to thank Pierre Joray for the numerous suggestive comments that he made on a first draft of this paper.

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    Sebastien Richard

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Richard, S. Leśniewski on metalogic and definitions. Synthese 195, 2649–2676 (2018). https://doi.org/10.1007/s11229-017-1343-x

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