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

- 4.
We recall that an isomorphism between two types/formulas

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

- 6.
Indeed, it can be easily verified that \(\Delta K I\Delta\) reduces to \(\Delta K I \Delta\).

- 7.
Recall that any T-paradox can be transformed into a closed derivation of any propositional constant.

- 8.
This family of computational interpretations include: BHK, realizability, Dialectica, Curry–Howard. See Sorensen and Urzyczyn (2006).

- 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*p*,*q*, \(\underline{f}( \underline{p})\) is the same as \(\underline{f}(\underline{q})\). - 10.
For an exposition of these rules, see e.g. Negri and Plato (2001).

- 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.
- 13.
More precisely, an essential net, a graphical formalism for Intuitionistic Linear Logic, Lamarche (2008).

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

- 16.
Actually, the

*linear*\(\rightarrow\)L rule, corresponding to the \(\otimes\)-rule of Multiplicative Linear Logic. - 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.
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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## Additional information

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