Abstract
The idea of classical recapture has played a prominent role for non-classical logicians. In the specific case of non-classical theories of truth, although we know that it is not possible to retain classical logic for every statement involving the truth predicate, it is clear that for many such statements this is in principle feasible, and even desirable. What is not entirely obvious or well-known is how far this idea can be pushed. Can the non-classical theorist retain classical logic for every non-paradoxical statement? If not, is she forced to settle for a very weak form of Classical Recapture, or are there robust versions of classical recapture available to her? These are the main questions that I will address in this paper. As a test case I will consider a paracomplete account of the truth-theoretic paradoxes and I will argue for two claims. First, that it is not possible to retain the law of excluded middle for every non-paradoxical statement. Secondly, that there are no robust versions of classical recapture available to the paracomplete logician.
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Notes
I have argued in favor of this claim elsewhere, so here I will only develop the argument in outline. Cf. Rosenblatt (2020).
The set is said to be maximal because if any instance of (T) is added to it, the resulting set is inconsistent.
S can be any recursively axiomatizable theory entailing the axioms of Robinson’s arithmetic. For a modern presentation of Robinson’s arithmetic, cf. (Smith 2007).
Moreover, two interesting corollaries of McGee’s result are (i) that these maximal sets are complete and thus not effectively axiomatizable, and (ii) that there are non-enumerably many of them. I will ignore these facts since they are not relevant for my purposes, but more details can be found in McGee (1992), (Halbach 2011, ch. 19), and Rosenblatt (2020).
Some of these theories of truth are based on a non-classical logic that fails to have a conditional capable of proving (T). But they typically prove that the truth predicate is transparent, in the sense that, for every statement \(\phi\), ‘\(\phi\)’ and ‘\(\phi\) is true’ are everywhere (intentional contexts excluded) intersubstitutable.
See e.g. (Smith 2007) for one particular such theory and a proof of the diagonal lemma therein.
The locus classicus is (Tarski 1956).
There is another version of the liar paradox that avoids relying on (LEM) by appealing to the rule of reductio. However, approaches that address the truth-theoretic paradoxes by rejecting (LEM) typically reject reductio as well.
Of course, there is nothing special about \(\delta _{1}\), it can be shown that the instance of (LEM) for \(Tr\ulcorner \delta _{2}\urcorner\) is inconsistent on its own as well.
A KK-interpretation is a fixed-point in (Kripke 1975)'s sense.
Although we know that there are many consistent sets of instances of (LEM), I have not discussed what the ‘structure’ of these sets is. We have seen that there are many maximal sets. In addition, assuming, as usual, that we are considering, say, an arithmetical language and that we are focusing on the standard model of arithmetic, it is clear that there is a unique minimal set that contains all and only the instances of (LEM) for statements that are grounded in Kripke’s sense. Now, obviously, the minimal set and the maximal sets are not the only sets. An interesting question, then, is exactly what sets lie between them. My guess is that the structure of these sets mimics, so to speak, the structure of the sets of fixed-points studied by Kripke (for the cognoscenti: it is a coherent complete partial order) and, moreover, that one can obtain all the consistent sets of instances of (LEM) as the quotient of the set of Kripke fixed-points. For example, if \(\phi\) is a contingent statement, there is just one way to settle the statement \(\phi \vee \lnot \phi\), but there are two ways to settle the statement \(\phi\). Thus, whereas one can obtain two different Kripke fixed-points by deciding \(\phi\) in one way or another, only one set of instances of (LEM) can be obtained in this way. Of course, these observations are very sketchy and I am afraid that I have to leave a careful analysis of this for some other time. I would like to thank Dave Ripley for some useful comments on this matter.
Full proofs of these claims are available in Rosenblatt (2020).
The details can be found in the appendix of Murzi and Rossi (2020).
The simplification is a very common one in the literature on theories of truth and paradoxes; typically, one takes a language that is strong enough to talk about its own syntax, like the language of arithmetic, and one uses the standard model of arithmetic as one’s base model.
The term ‘hypodox’ was coined in Eldridge-Smith (2007) to characterize statements that “might consistently take either truth-value but we have no basis for determining which” (p. 178).
Kripke considers these types of statements in his classic paper when he discusses so-called ‘intrinsic’ fixed-points. As far as I know, the terms ‘semi-true’ and ‘semi-false’ were coined in Cook and Tourville (forthcoming). Cook and Tourville’s goal is, roughly, to obtain a framework that avoids revenge paradoxes by being capable of expressing every semantic concept, even those that, on the face of it, are intensional, like paradoxicality, semi-truth and semi-falsity.
I hasten to add that I am under no illusion about the force of this argument. It rests on the contentious premise that it is somewhat artificial to focus on a certain set of interpretations to determine whether (LEM) should hold for a statement. This makes it tentative at best. Nevertheless, I do think that the onus is on the theorist who intends to endorse this form of classical recapture to show that it can be motivated in a convincing way.
Here I am setting aside all worries springing from the fact that the set of grounded statements is highly complex from a computational point of view and thus that it is not possible to provide a complete axiomatization for it.
On account of this, WKK-interpretations are sometimes said to be ‘infectious’. If a statement has the value \(\frac{1}{2}\), then any statement containing it as a sub-statement is infected, in the sense that it will also have the value \(\frac{1}{2}\).
That none of the latter sets are included in the former can be seen by considering the following instance of (LEM): \((\lambda \vee (\tau \vee \lnot \tau ))\vee \lnot (\lambda \vee (\tau \vee \lnot \tau ))\). Every maximal set of instances of (LEM) generated by a KK-interpretation contains this statement, since they all contain \(\tau \vee \lnot \tau\), but due to the presence of \(\lambda\), the statement cannot be accepted by the weak Kleene theorist.
I owe thanks to Bruno Da Ré and Damián Szmuc for discussion of this issue.
As one reviewer pointed out, it would be interesting to see how the present discussion relates to non-classical logics that are naturally paired with a different approach to classical recapture. Although a full treatment of this matter lies beyond the scope of this paper, I think that it is useful to distinguish at least two different (though probably related) questions that one might pose in connection to the idea of classical recapture. First, one might ask about the extent to which classical principles can be retained. Second, one can ask how it is that (i.e. by means of which method) one can recover classical principles. In this paper my sole concern was with the first question, and my aim was to consider various types of statements for which classical reasoning might be allowed. The second question pertains to the method by means of which the non-classical logician seeks to preserve classical logic in certain circumstances. By a ‘method’ of classical recapture I roughly mean the sort of resources that the non-classical theorist needs to bring into play to show that classical logic can be recovered. With regard to this there are various options available. Priest’s preferred way of carrying out classical recapture employs (the non-monotonic logic) minimal LP (see (Priest 2006, ch. 16)). Beall favors an alternative account in terms of ‘shrieked’ and ‘shrugged’ predicates (see e.g. Beall 2018). Logicians in the Brazilian tradition of logics of formal inconsistency typically suggest that it is convenient to introduce a consistency operator into one’s theory. I take no stand here on which method should be chosen; my own preferred approach is developed in Gallovich and Rosenblatt (2020). For some discussion, although not in the context of theories of truth, see Tajer (forthcoming).
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Acknowledgements
I would like to express my gratitude to an anonymous referee of this journal for her excellent comments and suggestions. I am also indebted to Bruno Da Ré, Jonathan Dittrich, Andreas Fjellstad, Camila Gallovich, Ulf Hlobil, Bruno Jacinto, Francesco Paoli, Dave Ripley, Lorenzo Rossi, Shawn Standefer, Damián Szmuc, Paula Teijeiro, the Buenos Aires Logic Group, and the participants of the VI Jornadas de Lógica y Argumentación, which was held in Buenos Aires in 2019. 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. Classical recapture and maximality. Philos Stud 178, 1951–1970 (2021). https://doi.org/10.1007/s11098-020-01517-9
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DOI: https://doi.org/10.1007/s11098-020-01517-9