## Abstract

Nontransitive responses to the validity Curry paradox face a dilemma that was recently formulated by Barrio, Rosenblatt and Tajer. It seems that, in the nontransitive logic ST enriched with a validity predicate, either you cannot prove that all derivable metarules preserve validity, or you can prove that instances of Cut that are not admissible in the logic preserve validity. I respond on behalf of the nontransitive approach. The paper argues, first, that we should reject the detachment principle for naive validity. Secondly, I show how to add a validity predicate to ST while avoiding the dilemma.

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## Notes

- 1.
See Fig. 1 below for a formulation of ST. The logic is called “ST” because its semantics can be described as “strict-tolerant” in a three-valued setting. It says that an argument is valid iff, in every model in which all premises are strictly true (have truth value 1), at least one conclusion is at least tolerantly true (has truth value \(\frac{1}{2}\) or 1).

- 2.
Advocates of other approaches to the semantic paradoxes have voiced various worries about the nontransitive approach (e.g. Shapiro 2013; Zardini 2013). Others have argued that the v-Curry isn’t as troubling as it may seem (Field 2017; Cook 2014; Ketland 2012). I will not discuss any of these issues here.

- 3.
If \(\varGamma \) is a singleton, like \(\{A\}\), I will write \(\left\langle A\right\rangle \) instead of \(\left\langle \left\{ A\right\} \right\rangle \) to avoid clutter.

- 4.
- 5.
Here is some intuition pumping in favor of VD: If \(\varGamma \vdash \varDelta \), then \(\varGamma ,\textit{Val}(\langle \varGamma \rangle ,\langle \varDelta \rangle )\vdash \varDelta \) follows by weakening. If \(\varGamma \not \vdash \varDelta \), then either \(\textit{Val}(\langle \varGamma \rangle ,\langle \varDelta \rangle )\) is incoherent and we get \(\varGamma ,\textit{Val}(\langle \varGamma \rangle ,\langle \varDelta \rangle )\vdash \varDelta \) by explosion or

*Val*\((\langle \varGamma \rangle ,\langle \varDelta \rangle )\) holds only in worlds (models, contexts, or what have you) in which either one element of \(\varGamma \) is false or one element of \(\varDelta \) is true. - 6.
I will follow Ripley and deal with this issue by fiat. Our meta-language terms “\(\pi \)” and “\(\textit{Val}(\left\langle \pi \right\rangle ,\left\langle A\right\rangle )\)” pick out the same formula in the object language, i.e., \(\pi \) = \(\textit{Val}(\left\langle \pi \right\rangle ,\left\langle A\right\rangle )\). I will assume that “\(\vdash \)” is extensional and, hence, that \(\pi \) and \(\textit{Val}(\left\langle \pi \right\rangle ,\left\langle A\right\rangle )\) are intersubstitutable on both sides of the turnstile. For a discussion of genuine self-reference in formal languages see Heck (2007). We could also add Peano Arithmetic (PA) (see Cobreros et al. 2013) or Robinson arithmetic (by using a technique from Negri and Plato 1998) to get the diagonal lemma. It is not obvious, however, that in a nontransitive setting, the diagonal lemma suffices to guarantee the intersubstitutability of \(\pi \) and \(\textit{Val}(\left\langle \pi \right\rangle ,\left\langle A\right\rangle )\). I am ignoring such issues here.

- 7.
I assume that we have \(\perp \) and \(\top \) in the object language. We could also use the name of the empty multiset to play the roles of \(\left\langle \perp \right\rangle \) and \(\left\langle \top \right\rangle \).

- 8.
As an anonymous referee points out, one might worry that this means that the nontransitive approach’s solutions to paradoxes aren’t as uniform as sometimes advertised. I will discuss some worries regarding the rejection of VD in the next section.

- 9.
Internalizing all admissible metarules would force us to go non-classical in our meta-theory. To see this, suppose we had an object language predicate

*Adm*such that where is an object language name of the sequent \(\varGamma \vdash \varDelta \textemdash \) just in case if \(\varTheta \vdash \varXi \), then \(\varGamma \vdash \varDelta \), i.e., just in case the move from \(\varTheta \vdash \varXi \) to \(\varGamma \vdash \varDelta \) is admissible. Call this biconditional ADM. Now, let , and call this identity CU. That trivializes our consequence relation if our meta-theory is classical. To see this, suppose that \(\vdash \kappa \). By CU, . By ADM, if \(\vdash \kappa \), then \(\vdash A\). By modus ponens, in the metalanguage, \(\vdash A\). Discharging our assumption by conditional proof in the metalanguage, if \(\vdash \kappa \), then \(\vdash A\). By ADM, . By CU, \(\vdash \kappa \). By modus ponens in the metalanguage, \(\vdash A\). Since*A*was arbitrary, that means that everything is provable. This is a variant of what Wansing and Priest (2015) call an “external Curry.” All we need to trivialize our consequence relation is a classical metalanguage, the internalization of all and only the admissible metarules (given by ADM), and self-reference (given by CU). Supposing that rejecting CU is not an option, this means that internalizing admissible metarules requires going non-classical in the meta-theory. While I am in principle open to this possibility, discussing it would lead us too far afield. - 10.
In order to be as ambitious as I can, I will aim to make the derivability relation pretty strong. That is why I used double-line rules in my formulation of ST below. These double-line rules make many metarules derivable that would be merely admissible if we used single-line rules. For example, the rule that allows us to move from \(\varGamma ,\lnot A\vdash \varDelta \) to \(\varGamma \vdash A,\varDelta \) is derivable with the double-line rules. With single-line rules, it would be admissible but not derivable in ST.

- 11.
I will call instances of that sequent schema the internalization of the corresponding instance of the metarule.

- 12.
Ripley has responded to Barrio, Rosenblatt and Tajer in talks and in an unpublished manuscript (Ripley 2018, Uncut). Barrio, Rosenblatt and Tajer argue that if we use rules for validity that are strong enough to internalize all derivable metarules, we can also derive: \(\textit{Val}(\left\langle \top \right\rangle ,\left\langle \lambda \right\rangle ),\textit{Val}(\left\langle \lambda \right\rangle ,\left\langle \perp \right\rangle )\vdash \textit{Val}(\left\langle \top \right\rangle ,\left\langle \perp \right\rangle )\). According to Ripley’s bilateralist interpretation of the turnstile, this sequent tells us that it is incoherent to assert \(\top \vdash \lambda \) and \(\lambda \vdash \perp \) while also denying that \(\top \vdash \perp \). But it can seem that this is precisely what Ripley actually does by endorsing ST. He seems to assert that we cannot assert the Liar and that we cannot deny the Liar, but he also seems to deny that all positions are incoherent.

Ripley’s solution is to distinguish between strict and tolerant assertion and denial. What is tolerant assertion? It is coherent to assert

*A*tolerantly just in case*A*cannot be coherently strictly denied. Similarly for denial. Now, Ripley says that his assertions of \(\top \vdash \lambda \) and \(\lambda \vdash \perp \) are not strict assertions but tolerant assertions. His denial of \(\top \vdash \perp \) is a strict denial. If, as his collateral view, Ripley asserts everything in \(\varGamma \) and denies everything in \(\varDelta \), then we can formulate his view on the liar as: \(\varGamma \vdash \varDelta ,\textit{Val}(\left\langle \top \right\rangle ,\left\langle \lambda \right\rangle )\) and \(\varGamma \vdash \varDelta ,\textit{Val}(\left\langle \lambda \right\rangle ,\left\langle \perp \right\rangle )\) but \(\varGamma \not \vdash \varDelta ,\textit{Val}(\left\langle \top \right\rangle ,\left\langle \perp \right\rangle )\). Thus, when Ripley formulates ST, what he expresses by the line in the sequent calculus is the preservation of undeniability among statements about positions.I think it is fair, at this point, to ask Ripley whether we can express this relation in the object language. After all, we would like to formulate our sequent calculus within our object language, in order to show that our semantic theory offers a treatment of the concepts it uses. I am skeptical about that, but I won’t investigate here whether preservation of undeniabilty can be expressed in an extension of ST. That is because Ripley’s response strikes me as radical. I would like to strictly assert my preferred semantic theory.

- 13.
This conclusion is similar to Shapiro’s (2013) view that every solution of the v-Curry must block the derivation of \(\kappa \vdash \perp \). But my reasons are very different.

- 14.
Ripley argued in a 2014 talk that all plausible arguments for transitivity also speak in favor of Contraction. Hence, he thinks that giving up Contraction without giving up Cut is not an option (see http://davewripley.rocks/docs/whynot-slides-silfs.pdf, accessed on April 13, 2017). It would be disappointing, from the nontransitive perspective, if the converse also held and giving up Cut while keeping Contraction wasn’t a stable position.

- 15.
For bilateralists like Ripley, this means that the v-Curry sentence is not undeniable. Nor is it unassertable. After all, \(\kappa \vdash \perp \) implies \(\vdash \kappa \) via faithfulness if \(\kappa =\textit{Val}(\left\langle \kappa \right\rangle ,\left\langle \perp \right\rangle )\).

- 16.
As Field (2017) has pointed out, giving up VD is what Peano Arithmetic (PA), as it were, does naturally with respect to the predicate that expresses that

*A*is derivable in PA from premise*B*, i.e. \(\textit{Deriv}_{\textit{PA}}(\ulcorner B\urcorner ,\ulcorner A\urcorner )\), where \(\ulcorner \phi \urcorner \) is \(\phi \)’s Gödel number. For, by Gödel’s second incompleteness theorem, for every sentence*A*whose negation is provable in PA, if PA is consistent, then it doesn’t prove that \(\textit{Deriv}_{\textit{PA}}(\ulcorner 1=1\urcorner ,\ulcorner A\urcorner )\vdash _{PA}A\). So it doesn’t prove \(1=1,\textit{Deriv}_{\textit{PA}}(\ulcorner 1=1\urcorner ,\ulcorner A\urcorner )\vdash _{\textit{PA}}A\) (otherwise we would get \(\textit{Deriv}_{\textit{PA}}(\ulcorner 1=1\urcorner ,\ulcorner A\urcorner )\vdash _{\textit{PA}}A\) by Cut). In effect, I argue in the text that the nontransitive approach should follow PA’s example.Despite the similarity, there is also an important difference between \(\textit{Deriv}_{\textit{PA}}\) and

*Val*, namely that \(\textit{Deriv}_{\textit{PA}}\) distributes over the conditional but*Val*doesn’t. That is, we have \(\textit{Deriv}_{\textit{PA}}(\ulcorner A\urcorner ,\ulcorner B\urcorner )\vdash _{\textit{PA}}{} \textit{Deriv}_{\textit{PA}}(\ulcorner \top \urcorner ,\ulcorner A\urcorner )\rightarrow \textit{Deriv}_{\textit{PA}}(\ulcorner \top \urcorner ,\ulcorner B\urcorner )\). But we don’t generally have \(\textit{Val}(\left\langle A\right\rangle ,\left\langle B\right\rangle )\vdash \textit{Val}(\left\langle \top \right\rangle ,\left\langle A\right\rangle )\rightarrow \textit{Val}(\left\langle \top \right\rangle ,\left\langle B\right\rangle )\). That is a reflection of the non-transitivity of the consequence relation*Val*expresses. After all, if the conditional obeys the deduction theorem, this means that \(\textit{Val}(\left\langle A\right\rangle ,\left\langle B\right\rangle ),\textit{Val}(\left\langle \top \right\rangle ,\left\langle A\right\rangle )\vdash \textit{Val}(\left\langle \top \right\rangle ,\left\langle B\right\rangle )\) can fail. That is how it should be. For, given faithfulness, that sequent expresses that the sequent \(\top \vdash B\) follows via a derivable metarule from \(A\vdash B\) and \(\top \vdash A\). But that is false. In moving from \(A\vdash B\) and \(\top \vdash A\) to \(\top \vdash B\) we are cutting on*A*. It is essential to the nontransitive approach that this fails, for example, when*A*is a Curry sentence and*B*is an absurdity.It is a consequence of this dissimilarity that we cannot give a proof that parallels the proof of Gödel’s second incompleteness theorem for

*Val*. It seems to me, however, that even though we cannot give a proof that parallels a proof of the second incompleteness theorem, the situation in PA should nevertheless make us reluctant to say that obeying VD is essential to any predicate that captures what follows from what. After all, \(\textit{Deriv}_{\textit{PA}}\) captures, in some respectable sense, what follows from what in PA. Otherwise, the significance of the second incompleteness theorem would be unclear. Thanks to an anonymous referee for raising this issue. - 17.
Moving up to meta-meta-theoretic reasoning may suggest that we should try to create a higher-order sequent calculus in the spirit of Kutschera (1968), similar to what Wansing and Priest (2015) have presented for the noncontractive approach. A higher-order sequent calculus that internalizes all its derivable rules while rejecting Cut would indeed allow the nontransitive approach to give a more general and powerful reply to Barrio et al.’s dilemma than what I can offer here. Unfortunately, I must confess that I am unable to construct such a system. As far as I can see, Wansing and Priest are going non-classical in their meta-theory (e.g. by disallowing multiple uses of an initial sequent in proof-trees, e.g., for their \(\vdash _{l}\)). The issue is complex and I cannot pursue it here. But I admit that if such a solution can be worked out, it may be superior to what I am offering here.

It is worth noting, however, that as long as we focus only on derivable rules, the idea that we must internalize meta-meta-rules cannot get any grip. The reason is that there simply are no derivable meta-meta-rules in ST. In order to derive a meta-meta-rule, we would need primitive rules that operate on metarules, and ST doesn’t contain any such rules.

- 18.
Obviously, if the noncontractive theorist can present strong cases for faithfulness and VD, Proposition 2 can be used as an argument against Contraction. It would be interesting to pursue this line of thought. Here, however, I want to stick to the perspective of the nontransitive approach.

- 19.
From now on, \(\left\langle \varGamma \right\rangle \) will be the object language name of the set \(\varGamma \). As before, I omit set brackets for singletons.

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## Acknowledgements

I am very grateful for invaluable conversations with and/or comments from Jaroslav Peregrin, Robert Brandom, Daniel Kaplan, Shawn Standefer, Ori Beck, David Ripley, Katharina Nieswandt, Stephen Mackereth, Shuhei Shimamura and two anonymous referees.

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This work was supported by the research project ‘Excellence at the University of Hradec Králové.’

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Hlobil, U. Faithfulness for naive validity.
*Synthese* **196, **4759–4774 (2019). https://doi.org/10.1007/s11229-018-1687-x

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### Keywords

- Naive validity
- Nontransitive logic
- V-Curry paradox
- Substructural approaches to paradox