Abstract
We cast doubts on the suggestion, recently made by Graham Priest, that glut theorists may express disagreement with the assertion of \(A\) by denying \(A\). We show that, if denial is to serve as a means to express disagreement, it must be exclusive, in the sense of being correct only if what is denied is false only. Hence, it can’t be expressed in the glut theorist’s language, essentially for the same reasons why Boolean negation can’t be expressed in such a language either. We then turn to an alternative proposal, recently defended by Beall (in Analysis 73(3):438–445, 2013; Rev Symb Log, 2014), for expressing truth and falsity only, and hence disagreement. According to this, the exclusive semantic status of \(A\), that \(A\) is either true or false only, can be conveyed by adding to one’s theory a shrieking rule of the form \(A \wedge \lnot A \vdash \bot \), where \(\bot \) entails triviality. We argue, however, that the proposal doesn’t work either. The upshot is that glut theorists face a dilemma: they can either express denial, or disagreement, but not both. Along the way, we offer a bilateral logic of exclusive denial for glut theorists—an extension of the logic commonly called \(\mathsf {LP}\).
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Notes
We assume, here and throughout, that the disagreement in question isn’t of the faultless kind: at most one between Marc and Lisa can be correct as to whether Dover is North of London.
As is well known, the logic of nontrivial theories containing gluts must be paraconsistent, i.e. it may not validate the principle of explosion \(A \wedge \lnot A \vdash B\). More on this in Sect. 1.
Ripley (2014) calls this the denial equivalence.
For general background on denial in nonclassical theories, see Ripley (2011a, Sect. 3).
Our arguments apply, mutatis mutandis, to paracomplete theories whose logic at least includes the dual of \(\mathsf {LP}\), viz. the Strong Kleene logic \(\mathsf {K3}\) (Kripke 1975; Field 2008). Just like paraconsistent theories postulate (at least) the possibilities of an overlap between truth and falsity, and thus reject \(A \wedge \lnot A \vdash B\), paracomplete theories postulate a gap between truth and falsity, and reject the Law of Excluded Middle \(B \vdash A \vee \lnot A\). Accordingly, paracomplete logicians reject the lefttoright direction of Classical denial: in a paraconsistent framework, the denial of \(A\) is weaker than the assertion of \(\lnot A\) (see Ripley 2014). This raises the problem of expressing disagreement over a ‘gappy’ \(A\). For suppose (1) and (2) are ‘gappy’. Then, by denying (2) Lisa does not express disagreement with Marc’s denial of (2), for both denials can be correct. This raises the question how disagreement, and rejection, can be expressed in a paracomplete setting. For reasons of space, we exclusively focus on the paraconsistent case, although we briefly comment on paracompleteness in the parenthetical remark of Sect. 3.1 below.
The material conditional in \(\mathsf {LP}\) is not detachable (counterexample provided in the main text below). Indeed, Beall et al. (2012) show that \(\mathsf {LP}\) is entirely detachmentfree, i.e. it cannot define a detachable connective. Detachmentfree gluttheoretic approaches to semantic paradox are discussed in Goodship (1996), and have been recently revamped in Beall (2011).
We make the simplifying assumption that every object on the domain serves as a name of itself. Conjunction, material implication and the universal quantifier are defined the standard way.
More specifically, the set \(\{Tr(\ulcorner {L}\urcorner ) \rightarrow L\), \(Tr(\ulcorner {\lnot L}\urcorner )\), \(\lnot L\}\) may fail to imply \(\lnot Tr(\ulcorner {L}\urcorner )\).
Clauses C1 and C2 involve a a rather objective sense of correctness: the truth (falsity) of a proposition suffices for the correctness of its assertion (denial). The reader is free to interpret C1 and C2 evidentially, e.g. taking evidence for \(A\)’s truth (untruth) to be sufficient for the correct assertion (denial) of \(A\), and modify the logic accordingly (e.g. invalidate the Law of Excluded Middle and other classical principles).
Glut theorists may avoid the problem by restricting some of the standardly accepted structural rules, such as \(\mathsf {Contraction}\) (that if \(\Gamma , A, A \vdash B\), then \(\Gamma , A \vdash B\)). For reasons of space, however, we cannot consider this option here. For some recent substructural approaches to the semantic paradoxes, (see e.g. Shapiro 2011; Zardini 2011; Ripley 2011b). For a recent attempt to create a revenge problem for the substructural theory presented in Zardini (2011), see Murzi (2014).
Accordingly, the obvious interpretation of \(\bot \) is as follows:

(\(\bot \)) \(v(\bot ) = 0\).

We’ll return to this point in Sect. 4 below; see especially Sect. 4.1 and fn. 16.
We note in passing that the role of shriek rules is similar to that of the coordination principle \(\mathsf {Coord}_2\) in \(\mathsf {LP}^*\). The latter ensures that \(A\) may not be coherently both asserted and denied, on pain of triviality.
Similarly, asserting \(A \wedge (\lnot A \rightarrow \bot )\) ensures, up to triviality, that it is true only.
There is a second problem. Consider a Curry sentence \(C\) such that \(Tr(\ulcorner {C}\urcorner ) \leftrightarrow (C \rightarrow \bot )\), where \(Tr(x)\) satisfies at least \(Tr\)I and \(Tr\)E. If \(C\) were true then \(C \rightarrow \bot \) would also be true; but then, by modus ponens (which holds for Priest’s conditional), it would be possible to derive \(\bot \). The glut theorist is thus forced, on pain of trivialism, to reject \(C\). If \(C\) is rejected, however, \(C \rightarrow \bot \) should also be rejected. But then \(A \rightarrow \bot \) cannot be used to express in general the rejection of \(A\): in order to express the rejection of \(A\), \(A \rightarrow \bot \) must be true (and correctly denied sentences must be false only). As far as we can see, this effectively undermines Priest’s proposal.
It might be thought that a similar objection might be levelled against \(\mathsf {Coord_2}\) within the \(\mathsf {LP}^*\) framework we introduced in Sect. 2. That is, one might argue that, contrary to what \(\mathsf {Coord_2}\) effectively states, there shouldn’t be a logical connection between the joint assertion and denial of \(A\) and \(\bot \). However, it is not obvious that the objection would apply: unlike Priest’s entailment connective, which is intended to express logical consequence, \(\mathsf {LP}^*\)’s consequence relation need not be logical.
The Curry problem we mentioned in footnote 16 above doesn’t obviously arise in the present context, for at least two reasons. First, in order to replicate the relevant Currylike reasoning, one would have to define a sentence \(Q\) in some sense equivalent to \(Q \vdash \bot \). But, it would seem, such a sentence would be bound to be ungrammatical. To be sure, one could define a sentence \(Q\) equivalent to a sentence to the effect that the argument \(\langle Q \therefore \bot \rangle\) is valid. This would give rise to a validityinvolving version of Curry’s Paradox, the vCurry Paradox (Beall and Murzi 2013). However, Beall rejects one of the key semantic ingredients of the paradox, a rule to the effect that \(A\) and the the claim that the argument \(\langle A \therefore B\rangle\) is valid jointly entail \(B\). This would effectively block the relevant Currylike reasoning. Second, and relatedly, Beall’s logic \(\mathsf {LP}\) doesn’t enjoy a detachable conditional (see fn. 7 above), which, again, suffices to block the Currylike reasoning sketched in fn. 16.
To be sure, unlike glut theorists, classical logicians may not express naïve truth, i.e. a notion of truth satisfying (at least) \(Tr\)I and \(Tr\)E. But this is how it should be: classical logicians and glut theorists negotiate the trade between consistency (nontriviality) and expressibility at different points.
Beall (2009) argues that truth and truth only really are the same notion, so that both can overlap with falsity and falsity only. But I take the present shrieking proposal to be a change of heart.
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Acknowledgments
The first author is grateful to the British Academy, the Alexander von Humboldt Foundation, the School of European Culture and languages at the University of Kent, and the University of Padua for generous financial support during the time this paper was written. We wish to thank Salvatore Florio, Graham Priest, and the audience of a workshop at the University of Padua for helpful feedback and discussion, and Jc Beall, Pablo Cobreros, Dave Ripley, and two referees for very helpful comments on earlier drafts. Special thanks to Enrico Martino for many enjoyable conversations on denial, dialetheism and semantic paradox.
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Murzi, J., Carrara, M. Denial and Disagreement. Topoi 34, 109–119 (2015). https://doi.org/10.1007/s112450149278z
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DOI: https://doi.org/10.1007/s112450149278z