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Paradigms and Self-reference: What Is the Point of Asserting Paradoxical Sentences?

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Abstract

A paradox, according to Wittgenstein, is something surprising that is taken out of its context. Thus, one way of dealing with paradoxical sentences is to imagine the missing context of use. Wittgenstein formulates what I call the paradigm paradox: ‘one sentence can never describe the paradigm in another, unless it ceases to be a paradigm.’ (PG, p. 346) There are several instances of this paradox scattered throughout Wittgenstein’s writings. I argue that this paradox is structurally equivalent to Russell’s paradox. The above quotation is Wittgenstein’s version of the vicious circle principle which counteracts the paradox. The prohibition Wittgenstein describes is, however, limited to a certain language-game. Finally, I argue that there is a structural analogy between a noun being employed as a self-membered set and a paradigmatic sample being included in or excluded from the set it generates. Paradoxical sentences are not prohibited forever; they can indicate a change in our praxis with a given paradigm.

Keywords

  • Paradox
  • Self-reference
  • Paradigm
  • Yardstick
  • Rule
  • Ludwig Wittgenstein
  • Gottlob Frege

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Notes

  1. 1.

    Cf. Kutschera (1989) for an overview.

  2. 2.

    Cf. Diamond (1995, p. 184): ‘The predicate “concept” cannot be predicated of any concept: we think we want it in order to say things about concepts, but to think that it will enable us to do so is confusion, confusion about what a concept is. Wittgenstein thought that the whole philosophical vocabulary reflected confusion.’

  3. 3.

    Cf. Wittgenstein’s remark that a paradigmatic sample (yardstick) alone is a dead object: ‘Lay the yardstick alongside a body; it doesn’t say that the body is such-and-such a length. Rather, I want to say, in and of itself it is dead, and it achieves nothing of what thought achieves’ (BT, pp. 70e–71e). On the other hand, the materiality of a paradigmatic sample must not be taken as its imperfection: ‘In the case of the measuring stick you can’t say: “Yes, the measuring stick measures length in spite of its corporality; but a measuring stick that had only length would be the ideal, would be the pure measuring stick”. No, if a body has length, then there can’t be any length without a body—and even if I do understand that in a certain sense only the length of the measuring stick does the measuring, yet what I put into my pocket is still the measuring stick—the body and not the length.’ (BT, p. 352e).

  4. 4.

    Every rule and concept has a pictorial (iconic, natural) component and a conventional (habitual, customary) component (the praxis).

  5. 5.

    Though there are other objects of comparison that do not involve rules; cf. Kuusela (2019, §§5.5 and 6.1) for details.

  6. 6.

    Cf. the following very early remark from 1914: ‘Proposition and situation are related to one another like the yardstick and the length to be measured’ (NB, p. 32). This idea is developed in the Tractatus (2.1512) and further in Wittgenstein’s later thought. For instance, ‘der Satz wäre wie ein Maßstab an die Wirklichkeit angelegt’ (Ms-109, 272), or in the Big Typescript: ‘When I compared a proposition to a yardstick, then strictly speaking all I did was to take a proposition that relies on a yardstick to state a length and use it as an example of a proposition./If one conceives of propositions as instructions for making models, their picture-like quality becomes even clearer’ (BT, p. 67e). In Ms-109, 176, Wittgenstein says: ‘“In der gleichen Sprache ausdrücken” heißt mit dem gleichen Maßstab messen.’ Or simply: ‘Vergleiche die Grammatik von “Satz”, “Maßstab” und “Gedanke”.’ (TS-219, 3).

  7. 7.

    Here is another formulation from the early 1930 s period: talking about the equation a + (b + c) = (a + b) + c, Wittgenstein says: ‘For if the proposition is a rule, a paradigm, which every calculation has to follow, then it makes no more sense to talk of working out the equation, than to talk of working out a definition’ (PG, p. 395, my italics).

  8. 8.

    This precise idea is expressed in the Big Typescript: ‘in an ostensive definition I do not state anything about the paradigm (sample), but only use it to make statements; that it belongs to the symbolism and is not one of the objects to which I apply the symbolism’ (BT, p. 408e). Or we can refer to §1 of the Investigations: the shopkeeper possesses paradigmatic samples for ‘apple’, ‘red’ and numbers, and uses them as described in the passage. ‘It is in this and similar ways that one operates with words.’

  9. 9.

    Cf. BT, p. 43; Ms-107, 264[3]; Ms-109, 244[2]; Ms-110, 280[2].

  10. 10.

    There is an analogous idiom in English: to eyeball something means to judge it by eye. Here, however, the intimate connection to a measurement is lost.

  11. 11.

    See Diamond (2001, pp. 125–132) for another elaboration of this parallel between the understanding of measurement and of rules.

  12. 12.

    ‘Und ein Satz kann das Paradigma im andern nie beschreiben, sonst ist es nicht das Paradigma.’ From the German wording it is clear that the pronoun in the second clause ‘es’ refers to the paradigm from the first clause.

  13. 13.

    This is, of course, a crude simplification of the actual praxis, which, however, does not impact on the overall argument.

  14. 14.

    This account follows Irvine and Deutsch's (2016) entry on Russell's paradox.

  15. 15.

    Russell’s theory of types is a more sophisticated version of this prohibition.

  16. 16.

    This could be seen as a general characterization of the later Wittgenstein’s method.

  17. 17.

    Cf. PI, §127: ‘The work of the philosopher consists in assembling reminders for a particular purpose.’

  18. 18.

    This rule is rather complex. There are differences between particular languages, such as English and German. Moreover, there are languages that do not use articles, such as Latin and Slavic languages.

  19. 19.

    The construction ‘X of Xs’ (‘song of songs’, ‘holy of holies’, ‘king of kings’, etc.) is common in the Old Testament. It expresses a superlative. Although it is in the singular, it does not necessarily express a single element (cf. Keel 1994, pp. 38–39). Today, there is a similar idiomatic construction: ‘the mother of all somethings’, e.g. Douglas Engelbart’s ‘The Mother of All Demos’. This refers to something’s being superlative not in the sense of having the greatest degree of a certain characteristic, but rather in having the greatest number of characteristics of X. There is a single thing that has the greatest number of characteristics of X, the most X, the paradigmatic sample of X.

  20. 20.

    Cf. Antonio Gramsci’s famous remark: ‘the old is dying and the new cannot be born; in this interregnum a great variety of morbid symptoms appear.’ (Gramsci 1971, pp. 275–276).

  21. 21.

    ‘The sign is the part of the symbol perceptible by the senses.’ (TLP 3.32).

  22. 22.

    This idea is captured by Agamben (1993, p. 72): ‘Actually, since every term refers by definition to every and any member of its extension, and can, furthermore, refer to itself, one can say that all (or almost all) words can be presented as classes that, according to the formulation of the paradox, both are and are not members themselves.

    It is not worth objecting against this that one never mistakes the term “shoe” for a shoe. Here an insufficient conception of self-reference blocks us from grasping the crux of the problem: What is in question is not the word “shoe” in its acoustic or graphic form (the suppositio materialis of medieval logicians), but the word “shoe” precisely in its signifying the shoe (or, a parte objecti, the shoe in its being signified by the term “shoe”). Even if we can completely distinguish a shoe from the term “shoe,” it is still much more difficult to distinguish a shoe from its being-called-(shoe), from its being-in-language. Being-called or being-in-language is the non-predicative property par excellence that belongs to each member of a class and at the same time makes its belonging an aporia.’

  23. 23.

    I have benefited from discussions with Wolfgang Kienzler, Andrew Napthine, Rupert Read and Oskari Kuusela. This work has been supported by the Czech Science Foundation, project no. GA19-16680S.

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Mácha, J. (2020). Paradigms and Self-reference: What Is the Point of Asserting Paradoxical Sentences?. In: Wuppuluri, S., da Costa, N. (eds) WITTGENSTEINIAN (adj.). The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-030-27569-3_9

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