Modeltheoretic semantics and revenge paradoxes
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Abstract
Revenge arguments purport to show that any proposed solution to the semantic paradoxes generates new paradoxes that prove that solution to be inadequate. In this paper, I focus on revenge arguments that employ the modeltheoretic semantics of a target theory and I argue, contra the current revengetheoretic wisdom, that they can constitute genuine expressive limitations. I consider the antirevenge strategy elaborated by Field (J Philos Log 32:139–177, 2003; Revenge of the Liar, Oxford University Press, Oxford, pp 53–144, 2007; Saving truth from paradox, Oxford University Press, Oxford, 2008, §§21–23) and argue that it does not offer a way out of the revenge problem. More generally, I argue that the difference between ‘standard’ and ‘revenge’ paradoxes is illconceived and should be abandoned. This will contribute to show that the theories that provide a uniform account of truth and other semantic notions are the ones best equipped to avoid the paradoxes altogether—‘standard’ and ‘revenge’ alike.
Keywords
Modeltheoretic semantics Semantic paradoxes Revenge paradoxes Modeltheoretic instrumentalism1 Introduction
Prima facie, revenge paradoxes can be characterized as arguments to the effect that any proposed solution to the semantic paradoxes generates new paradoxes that prove that solution to be inadequate. Such paradoxes often make use of notions employed in the theories they are directed against, and are argued to be similar to the ‘standard’ semantic paradoxes, such as the Liar and Curry’s paradox (see e.g. Field 2007).^{1} Revenge arguments are typically used to conclude that revengeprone theories do not solve the semantic paradoxes in general: even though they avoid the ‘standard’ semantic paradoxes, they suffer from new, structurally similar antinomies, that can only be avoided at the cost of significant expressive limitations.
A straightforward revenge strategy involves the modeltheoretic (henceforth ‘MT’) semantics of a theory of truth. MTrevenge arguments point to the inexpressibility in a theory T of some notion that is definable in or justified by the MTsemantics for T. For instance, let \(\lambda \) be a sentence equivalent to ‘\(\lambda \) is not true’, and let T be a nonclassical theory of truth in which \(\lambda \) is not in the extension of the truth predicate, nor in the extension of the negated truth predicate.^{2} In the MTsemantics for T, it is typically possible to define a predicate for ‘determinateness’ that captures the status of \(\lambda \), declaring it to be not determinate. However, expressing such a predicate in T makes the theory Ttrivial—i.e. it forces T to contain every sentence of its language—and hence such a predicate is inexpressible in T. The inexpressibility of the modeltheoretical notion of ‘determinateness’ provides an example of an MTrevenge paradox. As we will see below, the MTrevenge paradox just sketched can be adapted to essentially all nonclassical theories of truth.^{3}
The importance of MTrevenge derives from its scope. Most theories of truth come with an MTsemantics, which is used to provide an interpretation for them, or to prove them nontrivial. However, MTrevenge paradoxes are often considered to be as easy to construct as they are to defuse, and ultimately unproblematic.^{4} In this paper, I challenge this view. In order to do so, I consider the vigorous defence against MTrevenge elaborated by Field (2007, 2008, §§21–23). Field’s antirevenge strategy can be applied to every theory for which an MTsemantics can be given: if successful, it promises to shield every theory of truth from revenge attacks based on MTsemantics. In a nutshell, Field argues that MTrevenge arguments depend on a confusion between modelindependent and modelrelative semantic notions: MTrevenge paradoxes employ the latter, but only the former can be used to characterize genuine semantic notions, and hence to give rise to genuine semantic paradoxes. However, I will argue that this distinction does not provide an acceptable ground to distinguish between genuine and nongenuine semantic notions, and thus it does not offer a way out of the revenge problem. More generally, I will argue that the difference between ‘standard’ and ‘revenge’ paradoxes is illconceived and should ultimately be abandoned. This will contribute to show that the theories that provide a uniform account of truth and other semantic notions are the ones best equipped to avoid the paradoxes altogether.^{5}
The purpose of this paper is not to argue that every MTrevenge argument is successful. As Beall (2007) has warned, MTrevenge paradoxes can be very simple to formulate, and this can make the resulting arguments ‘too easy’, and ultimately uninteresting. In order to work, revenge arguments must have a proper justification, and their success can only be determined on a casebycase basis. What I will show is that MTrevenge arguments do not fail qua modeltheoretic: if they fail, they do so so for specific reasons, such as an unconvincing justification or lack of importance.
The paper is structured as follows. In Sect. 2 I introduce some representative ‘standard’ and ‘revenge’ semantic paradoxes. In Sect. 3 I present Field’s antirevenge strategy, and in Sect. 4 I argue that it fails to neutralize MTrevenge arguments. In Sect. 5 I discuss a possible rejoinder, namely the idea that the MTsemantics is a mere tool to prove nontriviality results and, as such, it does not characterize genuine semantic notions; I argue that this reply also fails to neutralize MTrevenge arguments. Section 6 concludes.
2 Paradoxes and revenge
There are several options to restrict classical logic in order to nontrivially admit some form of naïveté. Semantically, this corresponds to adopting nonclassical evaluation functions. Classical evaluations assign to every sentence either value \(\mathbf {1}\) or value \(\mathbf {0}\), and no sentence is assigned the same classical value as its negation. This sits poorly with naïveté: the Liar paradox features a sentence \(\lambda \) that is forced to have the same value as its negation. However, several nonclassical evaluations feature three or more semantic values: they behave as classical evaluations on classical values, but \(\lambda \) and \(\lnot \lambda \) can be assigned the same nonclassical value. In this way, nonclassical evaluations can assign the same value to \(\varphi \) and \({\mathsf {True}}(\ulcorner \varphi \urcorner )\) (and thus validate intersubstitutivity or even the tschema).^{8}
In order to make the following discussion more precise, I will now introduce a family of nonclassical logics that can be used to formulate several theories of naïve truth. Moreover, I will recall a few basic facts about one specific theory of naïve truth, the one developed in Field (2003, 2007, 2008), which will make it easier to discuss Field’s antirevenge strategy. However, both Field’s antirevenge strategy and my arguments against it are completely independent from the revengebreeding notions and the specific theories being considered.

The value of \(\lnot \varphi \) is \(\mathbf {1}\)minus the value of \(\varphi \).

The value of \(\varphi \wedge \psi \) is the minimum of the values of \(\varphi \) and \(\psi \).

The value of \(\forall x \varphi \) is the infimum of the values of its instances \(\varphi (t)\).
\({\mathsf {K3}}\) is a very weak logic: on the one hand, several classically valid inference rules turn out to be invalid in it (notably, the classical rules for introducing negation and the conditional); on the other, \({\mathsf {K3}}\) does not have any logical laws: not even the principle \(\varphi \rightarrow \varphi \) is \({\mathsf {K3}}\)valid.^{10} Several theories have been developed that strengthen \({\mathsf {K3}}\) without losing intersubstitutivity. In particular, a series of theories developed in recent years by Field (2002, 2003, 2008, 2013) succeeded in equipping \({\mathsf {K3}}\)based theories of truth with primitive, strong conditional connectives, not equivalent to \({\mathsf {K3}}\)’s conditional, that validate several classically valid principles.
\({\mathsf {F}}\) provides a solution to the Liar and all the other paradoxes that can be formulated in its language: not only does it validate intersubstitutivity and a form of the tschema, it also provides a treatment of intuitively paradoxical sentences such as \(\lambda \) via (iterations of) the determinateness operator. Does this show that \({\mathsf {F}}\), and similarly expressive theories of naïve truth more generally, solve all the semantic paradoxes? Revenge arguments aim to answer negatively to this question. More precisely, MTrevenge arguments aim at showing that theories of naïve truth, despite solving the truththeoretic paradoxes, fall short of solving other semantic paradoxes closely related to the ‘standard’ ones, involving other semantic notions closely related to naïve truth, definable in their MTsemantics. The upshot of revenge arguments is clear: revengeprone theories suffer from crucial expressive limitations: they avoid triviality only because they cannot express the semantic notions that could trivialize them, just like classical theories cannot express naïve truth.
Nonclassical theorists typically deny the legitimacy of the above paradox, and relevantly similar ones, arguing that revengebreeding notions such as bivalent determinateness are not genuine semantic notions.^{14} In the next section, I will discuss an antiMTrevenge strategy elaborated by Hartry Field in a series of works (2003, 2007, 2008). Even though Field’s strategy is articulated in the context of his theory \({\mathsf {F}}\), it can be applied to every theory for which an MTsemantics can be given.^{15}Bivalent Determinateness Let T be a theory of truth that validates intersubstitutivity (but a similar argument applies for the tschema), and let v be a nonclassical evaluation for T. T cannot contain any operator \({\mathsf {BDet}}\) with the following semantics:$$\begin{aligned} v({\mathsf {BDet}}(\varphi )) = {\left\{ \begin{array}{ll} \mathbf {1},&{} \text{ if } v(\varphi ) = \mathbf {1},\\ \mathbf {0},&{} \text{ if } v(\varphi ) \ne \mathbf {1} \end{array}\right. } \end{aligned}$$To see this, let \(\lambda _{\mathsf {d}}\) be intersubstitutable for \(\lnot {\mathsf {True}}(\ulcorner {\mathsf {BDet}}(\lambda _{\mathsf {d}})\urcorner )\) in T, and apply v to \(\lambda _{\mathsf {d}}\):
Suppose \(v(\lambda _{\mathsf {d}}) = \mathbf {1}\). By definition of \({\mathsf {BDet}}\) and naïveté, \(v({\mathsf {BDet}}(\lambda _{\mathsf {d}})) = \mathbf {1} = v({\mathsf {True}}(\ulcorner {\mathsf {BDet}}(\lambda _{\mathsf {d}})\urcorner ))\). No evaluation assigns the same classical value to a sentence and its negation, but \(\lambda _{\mathsf {d}}\) is by definition intersubstitutable for \(\lnot {\mathsf {True}}(\ulcorner {\mathsf {BDet}}(\lambda _{\mathsf {d}})\urcorner )\), and therefore \(v(\lnot {\mathsf {True}}(\ulcorner {\mathsf {BDet}}(\lambda _{\mathsf {d}})\urcorner )) \ne \mathbf {1} \ne v(\lambda _{\mathsf {d}})\), against our supposition.
Suppose \(v(\lambda _{\mathsf {d}}) \ne \mathbf {1}\). By definition of \({\mathsf {BDet}}\) and naïveté, \(v({\mathsf {BDet}}(\lambda _{\mathsf {d}})) = \mathbf {0} = v({\mathsf {True}}(\ulcorner {\mathsf {BDet}}(\lambda _{\mathsf {d}})\urcorner ))\). Nonclassical evaluations are classical on classical values, and \(\lambda _{\mathsf {d}}\) is intersubstitutable for \(\lnot {\mathsf {True}}(\ulcorner {\mathsf {BDet}}(\lambda _{\mathsf {d}})\urcorner )\), therefore \(v(\lnot {\mathsf {True}}(\ulcorner {\mathsf {BDet}}(\lambda _{\mathsf {d}})\urcorner )) = \mathbf {1} = v(\lambda _{\mathsf {d}})\), against our supposition.
Whether \(\lambda _{\mathsf {d}}\) is assigned \(\mathbf {1}\) or a different value, \(\lambda _{\mathsf {d}}\) and \(\lnot {\mathsf {True}}(\ulcorner {\mathsf {BDet}}(\lambda _{\mathsf {d}})\urcorner )\) are impossibly forced to have the same semantic value, since \({\mathsf {BDet}}\) works as a ‘classicalizer’ for nonclassical evaluations. Since T and v are arbitrary, no theory of naïve truth can feature an operator for bivalent determinateness.
3 Field on MTrevenge
Field views MTrevenge arguments as the result of a confusion between modelrelative and modelindependent notions. Modelrelative notions are essentially defined via reference to some model or evaluation function: ‘having semantic value x’ is a case in point. By contrast, truth is modelindependent: it is characterized by principles such as the tschema or intersubstitutivity that involve no modeltheoretic reference. Field argues that genuine semantic notions are modelindependent, and insofar as MTrevenge paradoxes resort to modelrelative notions, they are not genuine paradoxes.
Since modelrelative notions are relativized to some models, they always capture incorrectly the extension of modelindependent notions. Models have sets as domains. As a consequence, a modelrelative semantic notion \({\mathsf {N}}\), defined relatively to a model \({\mathscr {M}}\) whose domain is M, can only have a subset of M as its extension. By contrast, modelindependent notions are not relative to a particular model, and therefore their extension is not restricted to any particular set. This is exactly the divergence in extension between modelrelative and modelindependent truth that Field’s quote points to. And it is because of such a divergence, Field argues, that the understanding of modelindependent notions is not mediated nor conveyed by MTdefinitions: one cannot ‘extrapolate an understanding of a modelindependent notion like truth or determinate truth from the modelrelative notions’ (Field 2007, p.105). But genuine semantic notions are not modeltheoretically restricted, and are therefore modelindependent, or so Field’s argument suggests.But now consider the sentence ‘There are inaccessible cardinals’: it’s true, but false in \({\mathscr {M}}_1\), i.e. has semantic value \(\mathbf {0}\) in \({\mathscr {M}}_1\); its negation is false, but has value \(\mathbf {1}\) in \({\mathscr {M}}_1\). Having semantic value \(\mathbf {1}\) in \({\mathscr {M}}_1\) doesn’t correspond to truth, or to determinate truth, or anything like that [\(\ldots \)]. The point made here for \({\mathscr {M}}_1\) applies to any other model that can be defined within set theory, by Tarski’s Theorem, and this includes all models of set theory that are at all ‘natural’. (Field 2007, p. 104)
The modelindependent notion of bivalent determinateness (obeying D1–D5), Field argues, is only motivated by his modelrelative counterparts: but how can the latter justify the former, given that modelrelative notions fall inevitably short of determining the extension of the modelindependent ones?The proponent of the [MT]revenge problem doesn’t intend [bivalent determinateness] to be understood as modelrelative. The question then arises, how is it to be understood. I do not deny that it is possible to introduce into the language an operator [\(\ldots \)] with many of the features that the proponent of revenge wants [i.e. D1–D4], and which is not modelrelative. [\(\ldots \)] But such [operators] only breed paradox if they satisfy all the assumptions used in the [revengeparadoxical] derivations [\(\ldots \)]; the one place they fail is that excluded middle [i.e. D5] cannot be assumed for them. So there is a revenge problem [\(\ldots \)] only if there is reason to think that we can understand a notion of [bivalent determinateness] that obeys those other assumptions plus excluded middle.
And why assume that? I think what underlies the [MT]revenge problem is the thought that the modelrelative [bivalent determinateness operators] all obey excluded middle, so there must be an absolute [operator] that does too. But this assumption seems to me completely unwarranted: one just can’t assume that one can extrapolate in this way from the case of modelrelative predicates, which make sense only by virtue of ‘misinterpreting’ the quantifiers as having restricted range, to the unrelativized case where no such ‘misinterpretation’ is in force. (Field 2007, pp. 108–109)
 (extension)

A sufficiently clear understanding of a modelindependent notion \({\mathsf {N}}\) entails the existence of a criterion to determine the extension of \({\mathsf {N}}\).
Field’s argument does not rely on the specificities of his theory, or of bivalent determinateness: if successful, it would defuse every MTrevenge argument. However, in the next section I’ll argue that Field’s argument must ultimately be rejected: MTsemantic notions, with their potential for revenge, still stand.^{19}
4 Intelligibility and extensions
extension effectively extends Field’s argument to modelindependent notions motivated by the MTsemantics. However, it is too weak. The schemata D1–D5 that characterize bivalent determinateness modelindependently provide criteria to determine its extension just like the tschema or intersubstitutivity does for truth. Therefore, extension does not suffice to conclude that naïve truth is a genuine semantic notion, while D1D5determinateness is not, which is what Field’s strategy aims to accomplish.
But D1–D5 can also be read as criteria to determine, perhaps partially but at least not incorrectly, the extension of \({\mathsf {Det}}\):^{20}
‘\(2+2 = 4\)’ is in the extension of \({\mathsf {True}}\) if and only if \(2+2 = 4\),
‘there are inaccessible cardinals’ is in the extension of \({\mathsf {True}}\) if and only if there are inaccessible cardinals,
 D1

 From a derivation of ‘if \(2+2 = 4\) then \(3+2 = 5\)’, infer that if ‘\(2+2 = 4\)’ is in the extension of \({\mathsf {Det}}\) then ‘\(3+2 = 5\)’ is in the extension of \({\mathsf {Det}}\),
 From a derivation of ‘if there are inaccessible cardinals then there are large cardinals’, infer that if ‘there are inaccessible cardinals’ is in the extension of \({\mathsf {Det}}\) then ‘there are large cardinals’ is in the extension of \({\mathsf {Det}}\),
 D2

 From a derivation of ‘\(2+2 = 4\)’, infer that ‘\(2+2 = 4\)’ is in the extension of \({\mathsf {Det}}\),
 From a derivation of ‘there are inaccessible cardinals’, infer that ‘there are inaccessible cardinals’ is in the extension of \({\mathsf {Det}}\),
 D3

 if ‘\(2+2 = 4\)’ is in the extension of \({\mathsf {Det}}\), then \(2+2 = 4\),
 if ‘there are inaccessible cardinals’ is in the extension of \({\mathsf {Det}}\), then there are inaccessible cardinals,
 D4

 From a derivation of ‘if \(2+2 = 4\) then it is not the case that \(2+2 = 4\)’, infer that ‘\(2+2 = 4\)’ is in the extension of the negated \({\mathsf {Det}}\),
 From a derivation of ‘if there are inaccessible cardinals then there are not inaccessible cardinals’, infer that ‘there are inaccessible cardinals’ is in the extension of the negated \({\mathsf {Det}}\),
 D5

 Either ‘\(2+2 = 4\)’ is in the extension of \({\mathsf {Det}}\) or ‘\(2+2 = 4\)’ is in the extension of the negated \({\mathsf {Det}}\),
 Either ‘there are inaccessible cardinals’ is in the extension of \({\mathsf {Det}}\) or ‘there are inaccessible cardinals’ is in the extension of the negated \({\mathsf {Det}}\),
 (cextension)

A sufficiently clear understanding of a modelindependent notion \({\mathsf {N}}\) entails the existence of a criterion that determines the extension of \({\mathsf {N}}\), and it requires \({\mathsf {N}}\)’s extension to be nontrivial.
Incompatible negations Let us consider again the logic \({\mathsf {K3}}\) (see page 4). \({\mathsf {K3}}\)logical validity is defined as preservation of value \(\mathbf {1}\) in every partial evaluation. However, other logics can be defined from partial evaluations that are weak enough to support forms of naïveté: the logic of paradox, \({\mathsf {LP}}\), is a case in point.^{22} A sentence \(\varphi \) is a \({\mathsf {LP}}\)consequence of a set of sentences \(\Gamma \) if, for every partial evaluation p, if \(p(\Gamma ) = \mathbf {1}\) or 1/2, then \(p(\varphi ) = \mathbf {1}\) or 1/2. \({\mathsf {K3}}\) and \({\mathsf {LP}}\)based theories are dual in several respects. In particular, \({\mathsf {K3}}\)based theories do not satisfy all instances of the law of excluded middle (\({\mathsf {LEM}}\)) \(\varphi \vee \lnot \varphi \), but satisfy all instances of ex falso quodlibet (\({\mathsf {EFQ}}\)) \( \varphi \wedge \lnot \varphi \vdash \psi \), while \({\mathsf {LP}}\)based theories validate all the instances of \({\mathsf {LEM}}\), but fail to validate all the instances of \({\mathsf {EFQ}}\). Since the \({\mathsf {K3}}\)negation trivialize \({\mathsf {LP}}\)based theories (and vice versa), accepting cextension would force advocates of \({\mathsf {K3}}\)based theories to declare the \({\mathsf {LP}}\)negation unintelligible (and vice versa), even though both excluded middle and ex falso quodlibet are classically valid and customarily employed in (formalized) mathematical reasoning.
Can cextension be defended even after one realizes its implications for the comparisons of theories of truth? The foregoing considerations should give the advocate of cextension pause. However, there are far more conclusive reasons to reject this principle: in fact, cextension seems to clash directly with the intelligibility of naïve truth. If cextension is correct, it applies across the board, and not just to revengebreeding semantic notions. Its application to naïve truth shows that this notion was not sufficiently understandable before the development of suitable nonclassical theories in relatively recent times. At the very least, cextension entails that naïve truth was not sufficiently understandable when it was first formulated as a limitative result (see e.g. Tarski 1936) in that it yields triviality once added to the accepted formal systems of firstorder arithmetic (or some other sufficiently expressive theory) formulated in classical logic.
One might try to save cextension postulating two senses of ‘understanding’. In a ‘shallow’ sense, one understands semantic notions irrespectively of their extensions and of whether they trivialize other notions. In a ‘deep’ sense, understanding semantic notions requires cextension to be satisfied. Some form of shallow/deep distinction is customary in scientific domains: nonspecialists understand shallowly notions such as computable, black hole, or DNA, while specialists understand these notions properly. Understanding semantic notions properly might require to consult the relevant experts, while the shallow sense ‘saves the appearances’ according to which everyone can understand naïve truth and bivalent determinateness. However, in order to use cextension to escape revenge paradoxes, one must suppose that only the deep sense is relevant to determine which semantic notions and paradoxes there are. But there are at least two problems with this supposition. First, the community of truththeorists is split over which theory of truth is correct. Therefore, one accepting cextension has to conclude that there are experts lacking a deep understanding of genuine semantic notions just because they disagree (for discussion, see Williamson 2007, Chapter 4). Second, this supposition is problematic for the reasons highlighted above. Before nonclassical theories were available, naïve truth was only shallowly understandable but it generated genuine paradoxes, such as the Liar paradox. Why, then, should bivalent determinateness fail to generate genuine paradoxes?
One could further object that the understanding of naïve truth is and has always been (at least implicitly) coupled with some restriction of classical logic that avoids triviality. From this perspective, the understanding of naïve truth is and has always been perfectly in line with cextension. Now, this objection might in effect provide a construal of cextension that does not make it immediately implausible, although independent support would be needed to show that the understanding of naïve truth is indissolubly coupled with a suitable nonclassical logic. However, this objection fails to vindicate Field’s antirevenge argument, for it is equally applicable to bivalent determinateness. If naïve truth trivializes classical theories but it is to be understood as an essentially nonclassical concept, then the same could be said for bivalent determinateness: this notion trivializes several currently available nonclassical theories but it is also to be understood as a (super)nonclassical concept that requires extremely weak nonclassical theories. Therefore, this objection does not separate naïve truth from bivalent determinateness: it shows that, in both cases, a nonclassical logic that makes these notions nontrivial is required to claim an understanding of them. But this does not make bivalent determinateness any less acceptable than naïve truth.^{23}
In conclusion, Field’s argument does not offer an acceptable way to distinguish between naïve truth and bivalent determinateness so that the former can be claimed to generate genuine paradoxes while the latter can’t. Since nothing in the foregoing arguments relies on the specificities of bivalent determinateness or Field’s theory, I conclude that Field’s argument does not offer a way out of revenge paradoxes motivated by the MTsemantics. In the next section, I consider a possible criterion to rule out MTsemantic notions, that implicitly informs several antirevenge strategies, and argue that it also fails to provide a convincing base to distinguish ‘revengebreeding’ from ‘standard’ paradoxical notions.
5 Modeltheoretic instrumentalism
As Field’s argument makes clear, naïve truth and D1–D5bivalent determinateness are separated by their conceptual justification: the latter is motivated by the MTsemantics of an objecttheory, while the former isn’t. If one could show that the MTsemantics cannot motivate genuine semantic notions, one could argue that bivalent determinateness does not yield a genuine paradox. Some authors view the MTsemantics as a mere instrument to prove the objecttheory nontrivial; as such, the MTsemantics might not allow one to carve out genuine semantic notions (see e.g. Beall (2007, pp. 10–11) and Yablo (2003, pp. 328–329)). Call this position MTinstrumentalism. One possible motivation for it is that different MTsemantics can be given in different metatheories. Some MTsemantic notions definable in a classical metatheory might not be definable in a nonclassical metatheory, including revengebreeding notions such as the one employed in the paradox of Bivalent Determinateness.
Nevertheless, MTinstrumentalism does not provide a way out of the MTrevenge paradoxes. In short, even if the MTsemantics is regarded as a mere tool to prove the nontriviality of a theory T, there are crucial facts about the logical behaviour of T’s sentences that can be captured in any MTsemantics for T. More precisely, every MTsemantics for T can distinguish sentences that respect all the principles of classical logic from sentences that obey distinctively nonclassical principles. This shows that MTsemantics track genuine semantic distinctions, irrespectively of the specific models being employed, and thus irrespectively of the restrictions imposed on the extension of semantic notions from the settheoretic nature of models.
Another way to discern the nonclassical logical behaviour of some sentences in \({\mathsf {F}}\) uses Field’s very notion of D1D4determinateness: in any evaluation function \(v^*_{\mathscr {F}}\) that defines \({\mathsf {F}}\), \(v^*_{\mathscr {F}}\big ({\mathsf {Det}}(\forall x (x=x))\big ) = \mathbf {1}\) and \(v^*_{\mathscr {F}}\big (\lnot {\mathsf {Det}}(\lambda )) = \mathbf {1}\). A moment’s reflection shows that some grasp of the nonclassicality of sentences such as \(\lambda \) is at least presupposed for Field’s theory. For, it if weren’t, what could have motivated a determinateness operator that describe its ‘indeterminateness’ in the first place?^{26}
The foregoing observations show that any MTsemantics tracks crucial objecttheoretic facts. This does not depend on whether one attributes an instrumental role to the MTsemantics, and it is also independent on the specific evaluation functions employed, and thus on the specific models on which they are defined. Therefore, even the MTinstrumentalist has to concede that the MTsemantics, whether developed in a classical or a nonclassical metatheory, can be used to articulate genuine semantic distinctions. This lesson finds a natural application to revenge paradoxes: they make explicit, via paradoxical notions, some semantic distinctions that can be made, if only partially, in the MTsemantics. For instance, bivalent determinateness makes explicit the different logical behaviour of sentences such as \(\lambda \) and \(\forall x (x=x)\).
An advocate of Field’s theory (or of another sufficiently expressive theory) might object at this point that the different logical behaviours of \(\lambda \) and \(\forall x (x=x)\) can already be captured by a determinateness operator obeying D1D4 alone, since both \({\mathsf {Det}}(\forall x (x=x))\) and \(\lnot {\mathsf {Det}}({\lambda })\) are in \({\mathsf {F}}\). If one then tries to formulate revengeparadoxical sentences, the objection continues, their nonclassical behaviour can be captured iterating D1D4determinateness. Recall the sentence \(\lambda ^{*}\) equivalent to \(\lnot {\mathsf {True}}(\ulcorner {\mathsf {Det}}(\lambda ^{*})\urcorner \), mentioned on page 5: its nonclassical behaviour cannot be captured by declaring it ‘not determinately true’, since it is equivalent to \(\lambda ^{*}\). However, Field’s theory declares \(\lambda ^{*}\) to be ‘not determinately determinately true’, since \(\lnot {\mathsf {Det}}({\mathsf {Det}}({{\mathsf {True}}({\urcorner\lambda \urcorner ^{*}})}) )\) is in \({\mathsf {F}}\). And so on. However, Welch (2014) has shown that if the iteration is continued long enough, ‘ineffable Liars’ can be defined in \({\mathsf {F}}\), namely sentences whose nonclassical behaviour cannot be captured by any iteration of determinateness that can be constructed in \({\mathsf {F}}\) (see also Rayo and Welch 2007). Yet, the nonclassical behaviour of Welch’s ineffable Liars can be characterized very easily in several MTsemantics for \({\mathsf {F}}\), simply using the notion of bivalent determinateness.
In conclusion, the MTsemantics might be thought to have primarily an instrumental role, but it also provides crucial information about the behaviour of sentences of the objecttheory. For this reason, the MTsemantics can be used to carve out genuine semantic notions, that make such behaviour explicit. These notions may be paradoxical, but cannot be thought of as the byproduct of a mere instrument.^{27}
6 Concluding remarks
Piet Hein famously quipped that ‘problems worthy of attack prove their worth by hitting back’. The growing attention which truth theorists devote to revenge paradoxes witnesses that they are considered to be worthy of attack. I hope to have shown that they do hit back.
In particular, Field’s antirevengestrategy does not offer a way out of the MTrevenge paradoxes. The relevance of Field’s strategy is hardly overestimated, since it is applicable to every theory of truth that has an MTsemantics, and it threatens to undermine the coherence of MTsemantic notions, or notions motivated by the MTsemantics, more generally. However, Field’s defence, once fully articulated, offers no acceptable reason to think that revengebreeding notions are not genuine semantic notions. MTinstrumentalism is equally unsuccessful as a strategy to reject revengebreeding MTnotions, for they can be seen to carve out genuine semantic distinctions about the objecttheories.
As emphasized in Sect. 1, the foregoing arguments do not show that every MTrevenge paradox is successful; they show that MTrevenge paradoxes are not to be dismissed just because they involve the MTsemantics. However, this modest conclusion has implications for our understanding of semantic paradoxes, and ultimately for deciding amongst approaches to treating truth and other semantic notions in the objectlanguage. One such implication is that there is no wellmotivated ground to distinguish between ‘standard’ and ‘revenge’ paradoxes. Cogently motivated revengebreeding notions are legitimate semantic notions, just like naïve truth. Revenge paradoxes are not ex post facto antinomies, that only arise ‘after’ a solution to the ‘standard’ paradoxes has been found, and target that specific solution: they are expressive limitations that have been affecting the language all along, exactly as the ‘standard’ paradoxes. The whole distinction between ‘revenge’ and ‘standard’ paradoxes should be abandoned.
But if there is no such a thing as a ‘standard’ versus ‘revenge’ distinction, then a theory of truth that treats successfully the ‘standard’ paradoxes but fails to treat the ‘revenge’ ones is just a theory that solves the paradoxes only partially. For example, if naïve truth and bivalent determinateness are equally legitimate notions, the very logical revision that is operated to express the former is not fully justified if it cannot also be used to express the latter. So, abandoning the ‘standard’ versus ‘revenge’ distinction can help decide among theories of truth and treatments of paradoxes.^{28}
However, there is no a priori reason to think that no theory can express all the semantic notions in the objectlanguage. The semantic paradoxes—‘standard’ and ‘revenge’ alike—simply show that deep expressive limitations stand on the way to this goal. What cannot be reasonably hoped for is a theory of truth and other semantic notions that is completely free from expressive limitations. What can be reasonably hoped for is a theory that addresses such expressive limitations uniformly—thus applying the same solution to ‘standard’ and ‘revenge’ paradoxes. From this point of view, some classical theories seem promising: in particular, contextualist theories can be given that offer the very same solution to all semantic paradoxes, while retaining the strength of full classical logic.^{29}The strong revenge worry is that adding such an operator [bivalent determinateness] to the language would produce a new paradox that requires giving up the truth schema. Substantiating this would be a fatal blow to any claim that a Gsolution [the kind of solutions he favours] adequately resolves all the paradoxes. (Field 2007, p. 120)
Footnotes
 1.
For simplicity, I will only consider semantic notions formalized as firstorder predicates and operators.
 2.
 3.
Revenge paradoxes also exist for classical theories of truth (see e.g. Bacon 2015): my arguments for MTrevenge generalize to classical theories as well, but I will not explicitly discuss classical approaches, in the interest of space.
 4.
 5.
My focus in this paper is on MTrevenge and on Field’s treatment thereof. Several general treatments of revenge paradoxes can be found in the literature, see e.g. Beall (2006, 2007, 2007b), Cook (2007), Eklund (2007), Maudlin (2007), Priest (2007), Restall (2007), Scharp (2007, 2013), Simmons (2007, Shapiro 2011 and Scharp (2013). In particular, Ketland (2003) and Beall (2007) discuss MTrevenge paradoxes in connection with classical metatheories, and Leitgeb (2007) focuses on revenge paradoxes affecting the theory developed in Field (2003, 2007, 2008) in classical and nonclassical metatheories. In this paper, I focus on Field’s argument, since it is meant to apply to every MTsemantics, irrespective of whether they are defined in a classical or nonclassical metatheory, and it applies to every theory of truth that has been given an MTsemantics.
 6.
Actually, semantic paradoxes arise also in sufficiently strong, but nonclassical, logics, such as intuitionistic logic. I will focus on classical logic for simplicity.
 7.
In keeping with the modeltheoretic focus of this paper, I present the Liar and other paradoxes semantically, and I only consider theories of truth for which an MTsemantics has been developed. While these restrictions rule out theories that have only been developed axiomatically (e.g. Zardini 2011), nothing prevents the arguments in this paper from applying to MTsemantics that have yet to be developed.
 8.
Several manyvalued logics can be given twovalued semantics: see Chemla et al. (2017) for a systematic study and Rosenblatt (2015) for applications to theories of truth. However, MTrevenge paradoxes can be reproduced in such alternative twovalued frameworks, and therefore I will not explicitly consider them.
 9.
 10.
The reason for this is easily stated: while \(\mathbf {1}\) is the only designated value of \({\mathsf {K3}}\), any formula whose subformulae are assigned value 1/2 by a partial evaluation p is itself assigned value 1/2 by p, so no formula receives value \(\mathbf {1}\) under all partial evaluations. For example, if \(p(\varphi ) = \)1/2, then \(p(\varphi \rightarrow \varphi ) = \)1/2 as well.
 11.
 12.
For simplicity, the notion of determinateness is treated as an operator \({\mathsf {Det}}\), namely as a syntactic connective applying to sentences. A determinateness predicate, applying to names of sentences, is however easily definable in \({\mathsf {F}}\), because \({\mathsf {F}}\) satisfies intersubstitutivity. So, a determinateness predicate \({\mathsf {DT}}\) obeying exactly the same principles of the operator \({\mathsf {Det}}\) can be defined putting \({\mathsf {DT}}(\ulcorner \varphi \urcorner ) := {\mathsf {True}}(\ulcorner {\mathsf {Det}}(\varphi )\urcorner )\).
 13.
As Field notes, D3 might be strengthened to the following condition: ‘From \(\varphi \rightarrow _{\mathsf {F}} {\mathsf {Det}}(\varphi )\), infer \(\varphi \vee \lnot \varphi \)’. I will ignore this possibility for the sake of simplicity.
 14.
The legitimacy of MTrevenge paradoxes cannot be questioned on purely formal grounds: \({\mathsf {BDet}}\) is definable in minimally strong metatheories (Leitgeb 2007). One could object that \({\mathsf {BDet}}\) is definable only because a classical metatheory is employed (Yablo 2003; Leitgeb 2007); a suitable nonclassical metatheory would make it undefinable. However, too little is known about (suitable) nonclassical metatheories to tell whether they would make all revengebreeding notions undefinable. In Sect. 5 I present an argument for MTrevenge that does not rely on the classicality of one’s metatheory. For a nonclassical metatheory for a relatively weak objecttheory, see Bacon (2013); for recent investigations on nonclassical metatheories, see Field et al. (2017).
 15.
Using bivalent determinateness as a representative revenge paradox in the context of Field’s theory may seem inappropriate, especially in view of verdicts such as the following one: ‘there can be no truthlike predicate for which excluded middle can be assumed’ (Field 2007, p. 89, emphasis in the original). Nevertheless, Field acknowledges that ‘the conviction that there must be truthlike predicates obeying excluded middle is one primary source of revenge worries’ (ibidem). Likewise, he concedes that an argument against his theory based on bivalent determinateness ‘is perhaps the one with most intuitive force: it is that we just need a unified [(i.e. bivalent)] account of determinacy or defectiveness’ (Field 2008, p. 140)—I thank an anonymous referee for pointing out the latter quote to me. In any case, in the context of Field’s theory, paradoxes structurally similar to the paradox of bivalent determinateness can be given without resorting to bivalence, e.g. using idempotent determinateness.
 16.
Field (2007, p. 104). Homophonic models ‘assign to a name its real bearer and analogously for function symbols, and […] in the classical case assign to a predicate those objects in its real extension that are also in the domain of the model.’ (Field 2007, ibidem) Field assumes for the sake of the example that there are inaccessible cardinals. If there are no inaccessible cardinals, the example can be altered accordingly.
 17.
This is a basic fact of ZermeloFraenkel Set Theory. See for example Jech (2002, pp. 167–168).
 18.
There are several options for such a criterion, e.g. Dummett’s criterion of applicability (Dummett 1993, pp. 75–77 and pp. 232–235). extension does not require a criterion that is effectively applicable, nor a criterion that determines completely the extension of a given notion. Field’s argument only requires the extension of a notion \({\mathsf {N}}\) to be determined not incorrectly: the sentence ‘there are large cardinals’ should not fall outside of the extension of \({\mathsf {True}}\) if there are large cardinals.
 19.
The relevance of Field’s argument goes well beyond revenge paradoxes: if it were successful, it would have a profound impact on most truthconditional semantic theories. On the one hand, Field’s argument tells against any approach to semantics that relies heavily on modeltheoretical resources, including several theories in the Montagovian tradition (see e.g. Chierchia and McCconnellGinet (2000); for an argument against Montagovian semantics based on their use of modelrelative notions, see Lepore (1983); for a recent reply, see Glanzberg (2014)). On the other, Field’s argument tells against several theories in the tradition of Davidson (1967), which are based on modelindependent notions motivated or justified by the MTsemantics (see e.g. Fischer et al. (2015, §3.1), Halbach (2014, Chapter 8)). Given its scope, one might worry that Field’s antirevenge argument rules out every approach to truthconditional semantics. However, the deflationary approach to truthconditions articulated in Field (1994) might be used to answer to this worry. See also Hill (2014, Part I) and McGee (2016) for a recent discussion.
 20.
Standardly, predicates have extension and operators don’t. However, I’m ignoring this fact, given that in the presence of intersubstitutivity one can equally treat bivalent determinateness, or other semantic notions, as predicates or operators. See footnote 12.
 21.
Besides being conceptually problematic, the idea that comparing theories of truth is a futile exercise is disproven by the truththeoretic literature. On the one hand, proponents of nonclassical theories devote considerable efforts in exploring and highlighting the advantages of their approach over their rivals, classical and nonclassical alike [see e.g. Beall (2009, Chapters 4–5), Field (2008, Chapters 7–8, 10–14, 24–26), Priest (2006, Chapter 20)]. On the other hand, advocates of classical theories of truth have sought to expose the limitations of nonclassical approaches, from mathematical and from broadly methodological standpoints [see e.g. Halbach (2014, Chapter 20), Halbach and Nicolai (2017), Williamson (2017), Murzi and Rossi (2018a)].
 22.
 23.
Alternatively, one might think that the classical logician has an understanding of naïve truth via her acceptance of a collection of instances of the naïve principles that does not yield triviality (see e.g. Horwich (1990), and McGee (1992), and Cieśliński (2007) for criticisms). However, this alternative objection would also fail to separate naïve truth from bivalent determinateness, thus failing to undermine the revenge problem posed by the latter notion.
 24.
In Murzi and Rossi (2018a), purely objectlinguistic revenge paradoxes are developed that exploit the characterization of sentences obeying, or failing to obey, all the principles of classical logic.
 25.
Were the metatheory in which \(v^*_{\mathscr {F}}\) is constructed sufficiently strong to prove \(v^*_{\mathscr {F}}(\lambda \vee \lnot \lambda ) \ne \mathbf {1}\), the nonclassical behaviour of \(\lambda \) would be more evident. However, that \(v^*_{\mathscr {F}}(\lambda \vee \lnot \lambda ) \ne \mathbf {1}\) is not a fact about \({\mathsf {F}}\), but about what is not in \({\mathsf {F}}\), and it might be out of reach for nonclassical metatheories. Nevertheless, the facts mentioned above suffice to conclude that \(\lambda \) and \(\forall x(x=x)\) have different logical behaviours.
 26.
Accepting some form of nonclassicality for \(\lambda \) seems to be a prerequisite to determine the attitudes a rational agent should have towards \(\lambda \) (see Caie 2012).
 27.
The MTinstrumentalist could object that the very idea of defining an objecttheory via evaluation functions, i.e. MTsemantic notions, is misguided. For instance, Field (2008, p. 277) argues that the MTsemantics could be seen just as offering ‘a proof that one won’t get into trouble operating with the inferences in it’, although what is validated by the MTsemantics ‘outruns ‘real validity”. However, such a position offers no way out of the foregoing arguments for the significance of MTnotions. Even if one could select a subset \({\mathsf {F}}_0\) of \({\mathsf {F}}\) which is simple enough to be characterized axiomatically, the distinctions concerning the logical behaviour of \({\mathsf {F}}\)’s sentences that can be made in the MTsemantics would apply to \({\mathsf {F}}_0\)’s sentences just as well.
 28.
 29.
Notes
Acknowledgements
Open access funding provided by Paris Lodron University of Salzburg. I am grateful to Volker Halbach and Hartry Field for helpful exchanges on revenge paradoxes and modeltheoretic semantics, and to an anonymous referee for useful comments. Special thanks are due to Julien Murzi for detailed feedback and extensive discussions that have led to significant improvements.
Funding
I am grateful to the Austrian Science Fund (FWF), Grant No. P2971G24, for generous financial support during the time this paper was written.
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