Abstract
It is not uncommon among theorists favoring a deviant logic on account of the semantic paradoxes to subscribe to an idea that has come to be known as ‘classical recapture’. The main thought underpinning it is that non-classical logicians are justified in endorsing many instances of the classically valid principles that they reject. Classical recapture promises to yield an appealing pair of views: one can attain naivety for semantic concepts while retaining classicality in ordinary domains such as mathematics. However, Julien Murzi and Lorenzo Rossi have recently suggested that revisionary approaches to truth breed revenge paradoxes when they are coupled with the thought that classical reasoning can be recaptured in certain circumstances. What’s novel about the paradoxes they put forward is that they cannot be dismissed so easily. The concepts used to generate these paradoxes—those of paradoxicality and unparadoxicality—are concepts that non-classical theorists need in order to offer a diagnosis of the truth-theoretic paradoxes. My goal in this paper is to argue that non-classical theorists can represent the concept of paradoxicality without falling prey to revenge paradoxes. In particular, I will show how to provide a formal fixed-point semantics for a language extended with a paradoxicality predicate that adequately expresses the non-classical logician’s notion of paradoxicality.
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Notes
Their main argument is presented in Murzi and Rossi (2020a), although they have developed similar objections against non-classical accounts of the paradoxes elsewhere, cf. Rossi (2019), Murzi and Rossi (2020b). They articulate their favored view on the semantic paradoxes—a form of contextualism that retains classical logic by postulating a context shift in paradoxical arguments—in Murzi and Rossi (2018).
Murzi and Rossi are not the only ones that have criticized the recapturing strategy. In fact, it wouldn’t be too exaggerated to say that this idea has come under heavy fire in the last few years. See, for example, Woods (2019), Williamson (2018), and Halbach and Nicolai (2018) for various related objections. I’ve addressed Williamson’s charges in Rosenblatt (xxxx) and Woods’ criticism in Rosenblatt (2021). A reply to Halbach and Nicolai’s challenge can be found in Hartry Field’s Field (2021).
Prominent supporters of paracompleteness include Field (2008) and Horsten (2011). I hasten to say that some of the arguments to be offered below—though not all of them—are to some extent independent of the specific approach under consideration, and I suspect that the modifications that would be needed to accommodate some of the other approaches criticized by Murzi and Rossi shouldn’t be too substantial. At any rate, I will briefly return to this issue below (see Footnote 24).
Murzi and Rossi actually rely on the stronger assumption that for any open formula \(\phi (x)\) there is a term t such that \(t = \ulcorner \phi (t)\urcorner \) is provable, but the interderivability between \(\psi \) and \(\phi \ulcorner \psi \urcorner \) is enough for their purposes.
Murzi and Rossi employ multisets—rather than sets—because non-contractive approaches are among their targets. A multiset is just like a set except for the fact that it is sensitive to different occurrences of the same member.
To be sure, other paracomplete logicians might jettison some of these rules. For example, relevant logicians endorsing a paracomplete theory will balk at the rule \(\rightarrow \)-in\(_{-}\), since it embodies a form of weakening.
I am understanding \(\bot \) as a trivializing nullary connective. This means that, by definition, \(\bot \) entails any statement. Given that the consequence relation is transitive, it follows that for any \(\Gamma \), if \(\Gamma \vdash \bot \), then \(\Gamma \vdash \phi \) for any statement \(\phi \).
Note that in Murzi and Rossi’s derivation, as well as in the derivation below, only \(\rightarrow \)-in\(_{W}\) is used.
Cf. (Murzi and Rossi 2020a, p. 162).
What is typically meant by a notion being ‘merely instrumental’ is that the notion shouldn’t be seen as carving out a bona-fide semantic category, but rather as a tool that allows the non-classical theorist to prove that her theory is not trivial.
There are yet other versions of classical recapture that are in principle available to her, cf. Rosenblatt (2020).
In fact, Murzi and Rossi have suggested this idea in conversation.
A similar argument is usually offered to show that one also needs an unrestricted introduction rule licensing the step from \(\psi \) to \(Tr\ulcorner \psi \urcorner \).
I owe thanks to Julien Murzi and Lorenzo Rossi for raising this objection.
The contents of this section partially overlap with Sect. 4 of another paper of mine, Rosenblatt (xxxx). However, the emphasis there was not on the notion of paradoxicality. My aim was to show how to provide a formal fixed-point semantics for a language containing a predicate standing for the concept of groundedness. The construction to be offered below for the paradoxicality predicate results from a straightforward adaptation of the semantics I gave in that paper for the groundedness predicate.
Of course, min and max stand for the minimum and the maximum operations, respectively.
Cf. his Yablo (2003). Yablo’s paper is a critique of Field’s paracomplete approach to paradoxes.
The idea of applying Yablo’s technique to provide a model-theoretic treatment of a sentence-classifying (unary) operator in the context of a (broadly) Kripkean theory of truth is not new. In a previous piece I’ve suggested that it is possible to define a ‘pathologicality’ operator along these lines (see Rosenblatt and Szmuc 2014) and in another paper I show that one can have a ‘hypodoxicality’ predicate alongside Par(x), which is supposed to apply to statements like the truth-teller and its ilk (cf. Gallovich and Rosenblatt, 2021) Also, Roy Cook and Nicholas Tourville have recently provided a much more systematic study of intensional operators within the fixed-point approach (see Cook and Tourville (2018)). The main philosophical idea underpinning their account—namely, that a theory of truth capable of making intensional distinctions is desirable—is one that I am completely on board with.
Rest assured, I am not claiming that this is the only notion of validity available, but this is arguably the notion that best fits the paracomplete account.
According to the definition of paradoxicality provided above, there are unparadoxical statements \(\phi \) such that \(\lnot Par\ulcorner \phi \urcorner \) has value \(\frac{1}{2}\) at \(v^{FP}\). The truth-teller—a statement \(\tau \), saying of itself that it is true, \(Tr\ulcorner \tau \urcorner \)—is a case in point. Even thought it is not the case that \(\tau \) is \(\frac{1}{2}\) at every interpretation extending \(v^{FP}\), \(v^{FP}(Par\ulcorner \tau \urcorner ) =\frac{1}{2}\). But this ought not be too surprising. In this setting one shouldn’t expect the paradoxicality predicate to behave bivalently.
The proofs of all these facts are straightforward applications of the definition of validity and of the definitions of the logical connectives, so I omit them.
For example, I haven’t considered how paradoxicality interacts with disjunction, the conditional, or the quantifiers. In addition, I haven’t analyzed how far they can be iterated, nor do I know if some of the validities that fail could be recovered in some special cases. All these questions are interesting and deserve to be investigated; unfortunately, they are beyond the scope of this paper.
Before moving on, let me stress that the fixed-point semantics offered in this section was only intended to show how to characterize a paradoxicality predicate in a paracomplete setting. So a natural question is whether it is possible to offer a similar construction for the other approaches criticized by Murzi and Rossi. The answer to this is not at all obvious, and it will likely depend on the specificities of the account under discussion. In the case of non-reflexive accounts (see French 2016; Negri 2018, Fjellstad (2015).) the modifications that would be needed to offer a similar characterization of paradoxicality are, as far as I can see, fairly minimal. As for paraconsistent and non-transitive accounts (see e.g. Beall 2009; Ripley 2012, respectively), the crucial notion—i.e. the notion that allegedly cannot be expressed on pain of triviality—is not paradoxicality, but unparadoxicality. So the challenge, in those cases, would be to provide a fixed-point semantics for a language containing an unparadoxicality predicate. Lastly, non-contractive accounts present an additional complication. Some of these accounts (cf. Zardini 2011) lack a suitable semantic characterization, and thus a Kripke-like construction is not at present available for them (though I hasten to say that other non-contractive theories of truth do enjoy a model-theoretic characterization, see Rosenblatt 2019a and Rosenblatt (2019b)). To be sure, these remarks are somewhat superficial, but there is no space here to dig deeper. The task of rigorously adapting the fixed-point semantic to these other approaches must be left for another day.
As Murzi and Rossi (2020a, p. 154) put it: “From a revisionary perspective, the most natural way out of the problem is to treat the new paradoxes in the same way as the paradoxes of truth”.
Cf. Priest’s 1994.
One attempt at making this notion precise is given by Priest himself. His well-known ‘Inclosure-Schema’ can be used to suggest that if two paradoxes fit the schema, then they should receive the same type of solution. However, the jury is still out on whether this is a good way of capturing the thought underpinning the Principle of Uniform Solution. For a discussion of this issue and an interesting objection to Inclosure, see Beall (2014).
I am indebted to Julien Murzi for discussion of this issue.
In connection to this, see the final chapter of Gupta and Belnap (1993). I owe thanks to Pepe Martínez for a useful conversation about this.
This is what (Beall 2009, p. 66) calls the ‘exhaustive characterization project’. In his words, this project consists in “explain[ing] how, if at all, we can truly characterize—specify the ‘semantic status’ of—all statements of our language (in our language)”.
Of course, these replies are only tentative and merely scratch the surface of the matter, but the issue is large and goes beyond the scope of the paper.
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A talk based on an earlier version of this paper was given in 2020 at the SeloI online seminar. I am grateful to all its participants for a very stimulating discussion. Special thanks go to Eduardo Barrio, Jonas Becker Arenhart, Pablo Cobreros, Bruno Da Ré, Pepe Martínez, Federico Pailos, and Damián Szmuc. I also owe thanks to Camila Gallovich for many useful conversations about paradoxicality. Finally, I am deeply indebted to Julien Murzi and Lorenzo Rossi not only for numerous comments and various illuminating exchanges on the contents of the paper, but also for their encouragement and open-mindedness throughout the process of writing it. Financial support for this work was provided by the project “Logic and Substructurality” (FFI2017-84805-P), funded by the Spanish MINECO (Ministerio de Economía, Industria y Competitividad)
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Rosenblatt, L. Paradoxicality Without Paradox. Erkenn 88, 1347–1366 (2023). https://doi.org/10.1007/s10670-021-00405-w
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DOI: https://doi.org/10.1007/s10670-021-00405-w