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A Pragmatic Dissolution of Curry’s Paradox

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Abstract

Although formal analysis provides us with interesting tools for treating Curry’s paradox, it certainly does not exhaust every possible reading of it. Thus, we suggest that this paradox should be analysed with non-formal tools coming from pragmatics. In this way, using Grice’s logic of conversation, we will see that Curry’s sentence can be reinterpreted as a peculiar conditional sentence implying its own consequent.

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Notes

  1. A similar paradox in physics is posed by the question: Could an unstoppable force (i.e. one that nothing can resist) move an immovable object?

  2. As for its philosophical relevance, it is worth noting that this paradox is Paul Kabay’s [33, ch. 2] main argument for trivialism, the thesis that all sentences are true.

  3. In Piscoya’s proposal, a paradox is technically defined as ‘a special type of contradiction constituted by a proposition P whose truth implies its falsity and whose falsity implies its truth’ [47, p. 205] (cf. [24, p. 233]).

  4. We justify the equivalence between this sentence and Curry’s expressions in Sect. 6

  5. Incidentally, my bachelor’s thesis deals with the Liar’s paradox [40] and my master’s with Russell’s [41]. This paper corresponds to my doctoral dissertation [43].

  6. This obviously raises the question of whether we can apply classical logical rules to self-referential statements, but this is beyond the scope of this investigation.

  7. Restall and Rogerson [53] play with the logical form of such expressions. For instance, let \(C \rightarrow _{0}A\) be equivalent to C, and \(C \rightarrow _{n + 1}A\) to \((C \rightarrow _{n}A) \rightarrow A\). Hence, whenever \(n = 0\), the expression \(C \rightarrow _{1}A\) would be equivalent to \((C \rightarrow _{0}A) \rightarrow A\), i.e., to \(C \rightarrow A\). Moreover, whenever \(n = 1\), the expression \(C \rightarrow _{2}A\) would be equivalent to \((C \rightarrow _{1} A) \rightarrow A\), i.e., to \((C \rightarrow A) \rightarrow A\). If we go on with this process, can make Curry’s conditional larger and larger, for we can indefinitely substitute C by \(C \rightarrow A\). This procedure is already found in Moh Shaw-Kwei [55], only with a different notation. The notation we use here is an adaptation of Moh Shaw-Kwei’s made by Robles and Méndez [54].

  8. A proof analogous to this is made by Meyer et al. [37].

  9. In Beall’s formalisation [5, p. 30], the set of normal worlds is defined as follows: \(1 \in v(A \rightarrow B,w) \leftrightarrow \) for ever world \(w' \in W,\) if \(1 \in v(A,w')\), then \(1 \in v(B,w')\). The set of non-normal worlds, instead, is defined as follows: \(1 \in v(A \rightarrow B,w) \leftrightarrow \) for every pair of worlds \(\langle w',w' \rangle \) such that \(wR \langle w',w' \rangle \), if \(1 \in v(A,w')\), then \(1 \in v(B,w')\).

  10. For instance, the following ones: verum sequitur ex quodlibet, which means that a true proposition is implied by any other: \(A \rightarrow (B \rightarrow A)\); ex falso quodlibet, which means that a false proposition implies any other: \(\lnot A \rightarrow (A \rightarrow B)\); necesarium sequitur ex quodlibet, which means that a necessary proposition is strictly implied by any other: \(\Box A \Rightarrow (B \Rightarrow A)\); and ex impossibile quodlibet, which means that an impossible proposition strictly implies any other: \(\lnot \Diamond A \Rightarrow (A \Rightarrow B)\).

  11. For an analysis of Curry’s paradox from a relevance logic framework, see Foukzon [20].

  12. A comparison is possible between this paradox and the Banach-Tarski paradox [3]. In the latter, from a sphere it is possible to obtain two of the same size as the original one just by rearranging its component parts in a special way. The Banach-Tarski paradox is a geometrical theorem and does not represent a logical contradiction, although it does contradict our common sense. It is in this sense that it is very similar to Curry’s paradox.

  13. The maxims of mode (it clearly increases the obscurity of the conversation by stating, ‘If I am right, ...’) and relevance (the conditional with self-reference in the antecedent is simply not relevant in a conversation) are also breached by Curry’s sentence. For the purposes of this paper, though, it is enough to show how at least one of Grice’s maxims is breached.

  14. This Cartesian expression has been the subject of much discussion. In this paper, we assume that it cannot be a conditional since, in order to be, one needs not to think first. Rather, both occur at the same time. However, it is recognised that the expression, ‘It is not true that I think and that I am not’, is a better formulation than the original, ‘If I think, then I am’, even though they are logically equivalent.

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Acknowledgements

I would like to sincerely thank the members of the jury of the Francisco Miró Quesada Logic Prize 2021: Verónica Borja Macías, Walter A. Carnielli, Aldo Figallo-Orellano, and David Villena Saldaña. I would also like to thank Professor Óscar García Zárate, who was the advisor of the doctoral thesis on which this article was based [43]. Finally, I would like to thank Luis Felipe Bartolo Alegre, organiser of the prize, for having translated this document and converted it into LaTeX.

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  1. Universidad Nacional Federico Villarreal (UNFV), Av. Nicolás de Piérola No 351, Lima, CP 15001, Peru

    Rafael Félix Mora Ramirez

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Correspondence to Rafael Félix Mora Ramirez.

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To the young students of the humanities that strive every day to show the world that we can have a pure vocation for knowledge beyond profit, and also to those who hope that one day we can live in peace, without privilege, without hatred, and, above all, in harmony with nature.

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Translated by Luis Felipe Bartolo Alegre (UNMSM)

Winner of the First Edition of the Francisco Miró Quesada Cantuarias Logic Prize (Peru 2021) and contestant of the 2nd World Logic Prizes Contest at UNILOG’2022.

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Mora Ramirez, R.F. A Pragmatic Dissolution of Curry’s Paradox. Log. Univers. 16, 149–175 (2022). https://doi.org/10.1007/s11787-022-00294-9

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