Abstract
Like ambiguity, self-reference is a language phenomenon. Unlike it, however, self-reference is not a logical error. A sufficiently rich language should admit the possibility of introducing self-referent constructions. A natural language, as the richest of all that we have at our disposal, should make such constructions possible. Some of them generate contradictions, some other do not. It happens sometimes that we are too hasty in acknowledging contradiction of a self-referent construction. Showing that this is the case of the Liar Antinomy is the principal task of this chapter.
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Notes
- 1.
Russell’s Antinomy can be presented as follows: Let Z be the set of all sets, which are not their own elements. Is the set Z its own element? Let us assume it is. Then the set Z is outside the set Z, since it contains all and only those sets, which are their own elements. Consequently, the set Z does not belong to itself. Thus we have proved that if the set Z is its own element, it is not its own element. Then the set Z belongs to the set Z, because it contains all sets, which are not their own elements. Consequently, the set Z is its own element. Thus we have proved that if the set Z is not its own element, it is its own element. Both of the proved inverted implications give an equivalence: the set Z is its own element if and only if the set Z is not its own element. This is the result of Cantor’s accepting the so called axiom of abstraction, which identifies the existence of a property φ with the existence of a set \( A = \{ x: \varphi \left( x \right)\}, \) of all and only these objects x, which possess this property. Contradiction is generated by the so called Russell’s set \( R = \{x : x \notin x\} . \) Cantor’s set theory paradoxes go beyond the scope of this book and are, therefore, not included.
- 2.
Following tradition we accept that everything, whose existence does not imply contradiction, can exist.
- 3.
It is clear that this paragraph could be a part of a chapter devoted to paradoxes resulting from our imperfect intuition. However, the essence of Möbius’s ribbon is so close to the Liar’s Paradox that presenting this self-referent construction in the chapter on paradoxes of self-reference seems sufficiently justified.
- 4.
August Ferdinand Möbius (1790-1868), a German mathematician and astronomer, professor of the University of Leipzig, laid foundations for projective geometry and topology. He won the greatest renown for his 1858 discovery of a ribbon that has only one side, later known as “Möbius ribbon”.
- 5.
Felix Klein (1849–1925), a German mathematician, professor of the University of Leipzig, fellow of the Berlin Academy of Sciences. He specialized in non-Euclidean geometries, group theory, theory of algebraic equations and in the theory of elliptic and automorphic functions.
- 6.
A more precise description in topological categories of both Möbius ribbon and Klein’s bottle can be fund in mathematical monographs devoted to topology.
- 7.
Tarski [32].
- 8.
Borkowski [4], p. 356.
- 9.
Our opinion is completely different. The Liar Antinomy, a great paradox that it is, must cede its priority to such profound paradoxes as the heap, the bald, etc., which are discussed in the final chapter devoted to ontological paradoxes.
- 10.
Diogenes Laertios, Lives and Opinions of Great Philosophers, I, 13, pp. 13–14; I, 41–42, p. 31.
- 11.
Diogenes Laertios, Lives and Opinions of Great Philosophers, I, 111–112, p. 69.
- 12.
Plato, Complete Works, ed. J. M. Cooper, Hackett Publishing Company 1997, Euthydemus, 283E–286E. Cf. also Bocheński [3], p. 131.
- 13.
Aristotle, Sophistical Refutations (transl. W. A. Pickard-Cambridge), in: The Complete Works of Aristotle. The Revised Oxford Translation, ed. J. Barnes, Princeton University Press 1984, vol. I, 180b. Cf. also, Bocheński [3], p. 132.
- 14.
As can be seen, there is no visible reference to the Liar’s Paradox here. Moreover, Bocheński rightly observes that one can hardly accept that Aristotle presented any solution of the paradox there. Bocheński [3], p. 132.
- 15.
Diogenes Laertios, Lives and Opinions of Great Philosophers, II, 108, p. 137.
- 16.
Bocheński [3], pp. 132–133.
- 17.
Bocheński [3], p. 131.
- 18.
The Bible, Titus, 1, 12–13.
- 19.
Martin proves that both the sentence asserting its own falseness and the sentence asserting its truthlessness merit being called Liar sentences, since they lead to a contradiction with the Tarski convention (reminded later). Moreover, he also proves that a sentence predicating about its own falseness is a Liar sentence no matter whether we assume a two-valued logic or not. Such a sentence is sometimes called ordinary Liar, while the sentence predicating its truthlessness is called Strengthened Liar. Martin [20], pp. 1–2.
- 20.
Tarski [32].
- 21.
Tarski [32], p. 19.
- 22.
- 23.
Naturally, the sentence T is understood as generally quantified with respect to p. Thus, T has actually the form: ∀p (v(p) = 1 if and only if p).
- 24.
Tarski [32], p. 18. This form of the definition is known as “Tarski biconditionals”.
- 25.
Feferman [9], pp. 240–241.
- 26.
Martin [20], pp. 3–4.
- 27.
Every object, predication of which produces a true sentence, belongs to extension of a predicate. Every object, predication of which produces a false sentence, belongs to anti-extension of a predicate. If a predicate is an n-argument one, its extension contains ordered ns, predication of which produces true sentences, and its anti-extension contains ordered ns, predication of which produces false sentences. In this chapter, extension is called positive extension, while anti-extension is called negative extension.
- 28.
Tarski [32], p. 165.
- 29.
Russell [30].
- 30.
Church [7], pp. 301–302.
- 31.
Field [10], p. 374.
- 32.
- 33.
Parsons (1974).
- 34.
Parsons (1974), pp. 15–16.
- 35.
Burge [5].
- 36.
Burge [5], pp. 93–95.
- 37.
Burge [5], pp. 96–97.
- 38.
Martin [20], p. 7.
- 39.
Burge [5], p. 102.
- 40.
Burge [5], pp. 101–105.
- 41.
Burge [5], pp. 114–115.
- 42.
Martin and Woodruuff [21].
- 43.
As is known, in this logic, apart from truth (1) and falsity (0) there is the third logical value (u), which intuitively corresponds to the unknown logical value. Thus, if v(p) = 1 and v(q) = u, then the logical value of the conjunction p ∧ q (i.e., \( v(p \wedge q) \, = u \)) is unknown, since everything depends on what is the unknown to us value of the sentence q: if q is true, then the conjunction is true, but if q is false, then the conjunction is false.
- 44.
Martin [20], p. 7.
- 45.
Kripke [17].
- 46.
If P is an unspecified predicate, neither its extension nor anti-extension is a set. As can be seen, Kripke tacitly assumes this idealization, mentioned by Parsons. According to this idealization, extension and anti-extension is assigned in a specified way in case of every predicate, also the unspecified one—cf. Parsons’s Proposal above.
- 47.
Kripke [16], p. 64.
- 48.
Martin (1982), pp. 119–131.
- 49.
Herzberger [15].
- 50.
Gupta [14].
- 51.
Gupta [14], p. 181.
- 52.
Gupta [14], pp. 181–183.
- 53.
Gupta himself admits that in this way he makes use of symbols characteristic for the way of distinguishing the expressions of the language and those of metalanguage [14], p. 184f.
- 54.
Gupta [14], pp. 180–181.
- 55.
Gupta notices that a possibility of extending the model M to a standard model is guaranteed also by conditions weaker than (i)–(iv).
- 56.
Gupta [14], p. 186.
- 57.
Gupta [14], p. 191.
- 58.
Feferman [9].
- 59.
Feferman [9], pp. 250–252.
- 60.
Gumański [13].
- 61.
- 62.
This reasoning does not have to be accepted. \( L \leftrightarrow \neg L \) is, namely, a conjunction of two implications: \( L \to \neg L \) and \( \neg L \to L .\) From the first implication follows the rejection of the truthfulness of L, from the other, the rejection of the truthfulness of ¬L. Such an interpretation leads to the justification of a proposition that the Liar sentence L is an example of a sentence that has no logical value, so it illustrates the truth value gap. Yet, in the light of the first reasoning, which proves that L is a sentence that is true and false at the same time, it is clear that the argument leading to a conclusion that L has no logical value or is not based on the classical logic.
- 63.
Sometimes it is pointed out that such justifications are provided by sentences with vague terms. This opinion, however, seems difficult to accept. Cf. this chapter.
- 64.
If this sentence is true, it is false or has the third logical value, so it is untrue. If it is false, then things are as it predicates, so it is true and thus not false. If it has the third logical value, things are as it predicates, so it is true and thus does not have the third logical value. Naturally, this reasoning is yet another versions of the Revenge Paradox, mentioned above.
- 65.
Barwise and Etchemendy [2].
- 66.
We summarise Barwise’s and Etchemendy’s proposal following its clear and elegant presentation by Devlin [8], pp. 338–343.
- 67.
Devlin [8], p. 342.
- 68.
This solution is more precisely presented in [17].
- 69.
Woleński [33], pp. 91–97.
- 70.
A summary proof of this theorem is presented in [17], p. 72.
- 71.
Buridan [6], p. 200.
- 72.
Cf. Priest’s Proposal in the previous paragraph.
- 73.
Naturally, the sentence calculus with the conjunction : may also be applied to the original form of Buridan’s Paradox; however, this would have no sense, for in its original form Buridan’s problem is not a logical paradox.
- 74.
It should be addend here that there is no other solution, in which A′ would be a true sentence. For then the sentence P would be true, and that implies the falsity of the sentence A′.
- 75.
Skyrms (1984), pp. 119–131.
- 76.
It is easy to notice that there are many other, possible forms of this paradox.
- 77.
The expression “L 1. L 2” means that a sentence L 2 is called L 1.
- 78.
Quine [27], p. 131.
- 79.
Borkowski [4], p. 275.
- 80.
Richard [28], p. 143.
- 81.
Naturally, a sentence corresponding to the set G should be understood as a sentence in French and such expressions as “+”, “1”, “8”, “9”, “1.”, “2.” find their French word substitutes in the set G, i.e., “plus”, “un”, “huit”, “neuf”, “première”, “seconde”.
- 82.
Richard [28], p. 143.
- 83.
Poincaré [24], p. 145.
- 84.
Richard [28], pp. 143–144.
- 85.
Marciszewski, Antynomie w logice [w:] [19], pp. 20–21.
- 86.
Cf. van Heijenoort’s Introduction to Richard [28], p. 142.
- 87.
Naturally, a strict and formal adherence to Tarski’s principle of separation of language orders is an effective protection against Richard’s Antinomy.
- 88.
Russell [29].
- 89.
Krajewski, Antynomie, [in:] Marciszewski [18], pp. 177–178.
- 90.
Grelling and Nelson [11].
- 91.
Borkowski [4], p. 355.
- 92.
Quine [27], p. 133.
- 93.
Sainsbury [31], pp. 93–106.
- 94.
Naturally, for our problem it is irrelevant whether the number of days is expressed by 5, 10, or 365. Yet, shortening the period to two days may significantly change the conditions of the problem. We only signalize this reservation without going deeper into its analysis.
- 95.
Sainsbury [31], pp. 97–98.
- 96.
Sainsbury [31], p. 99.
- 97.
Sainsbury [31], p. 99.
- 98.
Sainsbury [31], p. 99.
- 99.
Sainsbury [31], pp. 100–101.
- 100.
Cf. Sect. 2.1.
- 101.
Ajdukiewicz [1], pp. 141–142.
- 102.
Ajdukiewicz [1], p. 142.
- 103.
Grzegorczyk [12], pp. 122–127.
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Łukowski, P. (2010). Paradoxes of Self-Reference. In: Paradoxes. Trends in Logic, vol 31. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1476-2_4
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