On Paradoxes in Normal Form

Abstract

A proof-theoretic test for paradoxicality was famously proposed by Tennant: a paradox must yield a closed derivation of absurdity with no normal form. Drawing on the remark that all derivations of a given proposition can be transformed into derivations in normal form of a logically equivalent proposition, we investigate the possibility of paradoxes in normal form. We compare paradoxes à la Tennant and paradoxes in normal form from the viewpoint of the computational interpretation of proofs and from the viewpoint of proof-theoretic semantics.

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Notes

  1. 1.

    Actually, in Tennant (1982) a more general notion of reduction, called shrinking, is considered, rather than the reductions \(\rightarrow _{\lambda }\). We address shrinking in Sect. 5.

  2. 2.

    In Tennant’s original account condition iii is stronger, as it is required that \(\mathscr {D}\) has a looping reduction; in Tennant (1995) the criterion was weakened: a non-terminating reduction, rather than a looping reduction, is required, in order to account for paradoxes without self-reference, such as Yablo’s.

  3. 3.

    In the examples considered by Tennant, paradoxes display exactly one redundancy.

  4. 4.

    We recall that an isomorphism between two types/formulas AB is a pair made of a derivation of B from A and a derivation of A from B such that both their mutual compositions reduce to the identity derivation (on A and B, respectively).

  5. 5.

    More precisely, to obtain an isomorphism, one must add to the reduction rule \(\rightarrow _{\lambda }\) a dual rule

  6. 6.

    Indeed, it can be easily verified that \(\Delta K I\Delta\) reduces to \(\Delta K I \Delta\).

  7. 7.

    Recall that any T-paradox can be transformed into a closed derivation of any propositional constant.

  8. 8.

    This family of computational interpretations include: BHK, realizability, Dialectica, Curry–Howard. See Sorensen and Urzyczyn (2006).

  9. 9.

    To be more precise, the derivations of the formula \(N=(X\rightarrow X)\rightarrow (X\rightarrow X)\) are seen as representations \(\underline{n}\) of positive integers \(n\in \mathbb {N}\) and the proofs of the formula \(\underbrace{N\rightarrow \dots \rightarrow N}_{n {\text { times}}} \rightarrow N\) are seen as representations \(\underline{f}\) of (n-ary) computable arithmetical functions \(f(x_{1},\dots , x_{n})\). Hence, if all derivations of N are the same, one has that \(\underline{0}\) is the same as \(\underline{1}\) and that, for all computable function f and integers pq, \(\underline{f}( \underline{p})\) is the same as \(\underline{f}(\underline{q})\).

  10. 10.

    For an exposition of these rules, see e.g. Negri and Plato (2001).

  11. 11.

    The fact that Russell’s paradox can be reformulated in purely semantical terms casts further doubt on the viability of a distinction between semantical and mathematical paradoxes.

  12. 12.

    Among the many existing graphical formalisms proposed to represent derivations we mention geometry of interaction (strongly related to proof-nets), Girard (1989), Kelly–MacLane graphs (Kelly and MacLane 1971; Blute 1993), string diagrams (Melliès 2006).

  13. 13.

    More precisely, an essential net, a graphical formalism for Intuitionistic Linear Logic, Lamarche (2008).

  14. 14.

    Actually, formulas with polarities \(+,-\), corresponding to the fact that the node correspond to a positive or negative occurrence of the formula (see Lamarche 2008).

  15. 15.

    Reduction in proof-nets is just path composition, see Lamarche (2008).

  16. 16.

    Actually, the linear\(\rightarrow\)L rule, corresponding to the \(\otimes\)-rule of Multiplicative Linear Logic.

  17. 17.

    A typical elimination based clause is: “a non canonical derivation \(\mathscr {D}\) of \(A\rightarrow B\) is valid if, when modus ponens is applied to \(\mathscr {D}\) and a valid derivation \(\mathscr {D}'\) of A, the resulting derivation can be reduced into a valid derivation of B”. As this example shows, elimination-based semantics clauses recall the realizability semantics.

  18. 18.

    However, a difference between some T-paradox and the corresponding N-paradox appears when considering, rather than validity, the computational behavior of the associated \(\lambda\)-terms, reflected in how such terms are interpreted in Scott Domains. For instance, in the original model \(D^{\infty }\) proposed in Scott (1976), only solvable \(\lambda\)-terms have a non-empty interpretation. This fact reflects the intuition that solvable terms are the only to have a non trivial computational behavior, while non solvable terms correspond to everywhere undefined partial functions. Now, while the \(\lambda\)-term arising from the T-paradox Rus is non-solvable, the one arising from the N-paradox \(Rus^{*}\) (when the latter is considered as a derivation of \((\lambda \rightarrow \lambda )\rightarrow \bot\)) is solvable: its operative content is to send any function from \(\lambda\) to \(\lambda\) into the empty denotation. One can argue in a similar way in the case of N-paradoxes of conclusion \(\lambda \wedge \lnot \lambda\). Roughly speaking, both T-paradoxes and N-paradoxes correspond to partial functions, the former being, so to say, “more partial” than the latter.

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Acknowledgements

The funding was provided by Agence Nationale de la Recherche (Grant No. ANR-14-FRAL-0002).

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Correspondence to Paolo Pistone.

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This work has been carried out as part of the ANR-DFG Project “Beyond Logic” (ANR-14-FRAL-0002). We wish to thank Luca Tranchini, Peter Schroeder-Heister, Volker Halbach and Marianna Antonutti Marfori for fruitful comments and an anonymous referee of the TOPOI journal for helpful suggestions and remarks on a first version of the paper.

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Petrolo, M., Pistone, P. On Paradoxes in Normal Form. Topoi 38, 605–617 (2019). https://doi.org/10.1007/s11245-018-9543-7

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Keywords

  • Proof-theoretic Semantics
  • Paradoxical System
  • Valid Derivation
  • Introduction Rule
  • Paradoxical Language