Wittgenstein and the liar

Abstract

Ludwig Wittgenstein’s remarks on contradictions and paradoxes have been met with incomprehension and have fueled the widespread and long-standing prejudice that his later thoughts on the foundations of logic and mathematics are the “surprisingly insignificant product of a sparkling mind” (Kreisel, British Journal for the Philosophy of Science 9:135–158, 1959, p. 158). This paper disagrees; it argues that Wittgenstein’s remarks on semantic paradoxes suggest an account of the Liar and its kin that is not only of historical interest but also represents a hitherto unnoticed paraconsistent alternative to established approaches to the Liar. In what follows, a reading of Wittgenstein’s remarks will be offered according to which Wittgenstein subscribes to a form of dialetheism (that is, the view that there are sentences that are both true and false). In contrast to modern dialetheist approaches to the Liar, however, some of Wittgenstein’s remarks suggest combining a dialetheist position with what is called ‘logical nihilism’ (that is, the view that there are no universally valid inference rules).

1 Introduction

The later Wittgenstein deals with the Liar and other semantic paradoxes in his Remarks on the Foundations of Mathematics (RFM), which were originally intended to become part of his Philosophical Investigations (PI), and in his Lectures on the Foundations of Mathematics (LFM), given in Cambridge in 1939. Wittgenstein’s remarks on paradoxes and on contradictions in general have been met with bewilderment, since he seems to propose a laissez-faire attitude according to which we can simply accept some contradictions and need not draw any conclusions from them. Michael Dummett (1959, p. 324) holds that particularly those passages in Wittgenstein’s Remarks “on consistency and Gödel’s theorem, are of poor quality or contain definite errors”, and Crispin Wright adds with special regard to Wittgenstein’s remarks on consistency that “the impression is not so much that of ordinary attitudes or assumptions questioned, as of good sense outraged” (Wright, 1980, p. 295). Kurt Gödel considered at least some of Wittgenstein’s remarks on contradictions to be “nonsense”,¹ and Wang (1996, p. 179) reports him going as far as to say: “Has Wittgenstein lost his mind? Does he mean it seriously? […] He has to take a position when he has no business to do so. For example, ‘you can’t derive everything from a contradiction.’ He should try to develop a system of logic in which that is true. It’s amazing that Turing could get anything out of discussions with somebody like Wittgenstein.”

According to a more charitable interpretation by Graham Priest (see, for example, Priest, 2006, p. 204), Wittgenstein, in his remarks on Gödel’s theorem, comes close to endorsing dialetheism—that is, the view that there are sentences which are both true and false—thereby rejecting the law of non-contradiction (Priest, 2004, p. 214). While Priest maintains that Wittgenstein in these passage “did not, it would seem, take the final step into dialetheism” (Priest, 2006, p. 204), it will be argued in Sect. 2 that there is other evidence that Wittgenstein does after all embrace a form of dialetheism. In addition to dialetheism, Wittgenstein contributes another thought to the debate on semantic paradoxes, namely that the involved paradoxical sentences are logical ‘dead ends’ that do not entail anything, which we will discuss in Sect. 3. It will be argued that this thought is the key to understanding Wittgenstein’s startling remark that we should simply not draw any conclusions from paradoxical sentences. Section 4 demonstrates how the two Wittgensteinian thoughts discussed in the preceding sections can be combined to form a new paraconsistent alternative to established approaches to the Liar. In contrast to modern dialetheist approaches to the Liar, Wittgenstein’s thought that the Liar is a logical dead end suggests combining a dialetheist position with what is nowadays called ‘logical nihilism’ (that is, the view that there are no universally valid inference rules), which will be discussed alongside other implications of the approach and arguments for it in Sect. 5. The closing Sect. 6 is devoted to the related issues of revenge paradoxes and semantic universality. The paper thus aims at sketching a Wittgenstein-inspired approach to the semantic paradoxes and outlining its merits. While the proposed reading of Wittgenstein’s remarks on the Liar and its kin claims to be the most charitable so far available, a detailed argument defending this exegetical claim as well as a word-for-word exegesis of Wittgenstein’s remarks must wait for another occasion.

2 Wittgenstein and dialetheism (i): ‘The contradiction is true’

According to Wittgenstein, our use of predicates is governed by ‘grammatical’ rules which specify in what kind of cases a predicate can be correctly attributed to or denied of an object. In some cases the rules refer to paradigmatic examples: “This and similar things are called ‘games’” (PI, §69; cf. §71). Wittgenstein points out that in such cases the giving of examples is “not an indirect means of explaining—in default of a better” (PI, §71). While Wittgenstein comes close to a variant of the so-called prototype theory of concepts in these cases,² he does not hold prototype theory to be true of all concept words. The rules governing, for example, the truth predicate seem to result in the transparency of truth, i.e., in the intersubstitutivity of p and “‘p’ is true” (in every extensional context): “For what does a proposition’s ‘being true’ mean? ‘p’ is true = p. (That is the answer.)” (RFM, I, Appendix III, §6, p. 117); cf. also PI, §136: “‘p’ is true = p / ‘p’ is false = not-p”. Because of these remarks, it is widely held that the later Wittgenstein adheres to a deflationary account of truth³ (this interpretation will also be taken for granted in what follows). According to this account, the rules governing the truth predicate comprise two inference rules, namely that it is admissible to infer “$\left\langle p \right\rangle$ is true” from p, and to infer p from “$\left\langle p \right\rangle$ is true” (the first of these is also known as Capture, the second as Release; the angle brackets represent a suitable naming device). Each proposition thus allows us to infer the claim that it is true, and vice versa.

Wittgenstein explicitly maintains that rules—for example, the rules of a formal calculus—might lead to contradictory results (cf., for example, RFM, VII, §12, p. 372; VII, §27, p. 394; cf. also PI, §125). He discusses the rules governing the truth predicate as a case in point: when applied to the Liar sentence, these rules “yield its contradictory, and vice versa” (RFM, I, Appendix III, §12, p. 120): “[I]f a man says ‘I am lying’ we say that it follows that he is not lying, from which it follows that he is lying and so on” (LFM, XXI, pp. 206–207). In the following, we will call a sentence P that, like the Liar, ‘yields its contradictory, and vice versa’ self-contradictory.⁴ Wittgenstein thus is one of the first philosophers to endorse the so-called inconsistency view of truth: the paradox of the Liar results from the correct application of the meaning-constitutive semantic rules that govern our use of the truth predicate.⁵ While these rules can be applied unproblematically in most cases, some applications of the rules lead to contradictory results.⁶

Another example of a rule that leads to a contradictory result is the one governing the predicate “heterological”, according to which a predicate is heterological if and only if it does not itself possess the property it expresses. As in the case of the truth predicate, this rule works fine for most cases and allows us, for instance, to correctly classify the predicate “monosyllabic” as heterological since it is not monosyllabic. As the predicate “polysyllabic” is polysyllabic, it can correctly be classified as not heterological. Just like the rules governing the use of the truth predicate, however, the rule governing the use of “heterological” can lead to a contradiction (RFM, VII, §27, p. 394). It gives rise to Grelling’s paradox: the predicate “heterological” itself is heterological if and only if it does not possess the property it expresses, that is, if and only if it is not heterological. The truth predicate and the predicate “heterological” thus seem to be overdetermined in the sense that there are (possibly sporadic) cases in which the semantic rules governing the use of these predicates lead us to inconsistent (though rule-conforming) categorizations, e.g., ‘The Liar is true iff it is not true’ or “Heterological” is heterological iff it is not heterological’.⁷ In the following passage Wittgenstein even explicitly speaks of ‘true contradictions’ (Wittgenstein here abbreviates “‘heterological’” by “‘h’”, “h” denotes the set of heterological predicates and “ε” denotes the membership relation):

‘h’ ε h ≡ ~(‘h’ ε h) might be called a ‘true contradiction’. […] ‘The contradiction is true’ means: it is proved; derived from the rules for the word ‘h’. (RFM, VII, §28, p. 396)

In the case of Grelling’s paradox, Wittgenstein thus holds that applying the meaning-constitutive rule for the concept word ‘heterological’ results in a sentence which, according to classical propositional logic, has the form of a logical falsehood: ‘h’ ε h ≡ ~ (‘h’ ε h).⁸ As with the truth predicate, Wittgenstein allows in this case that obeying the rules governing the use of a predicate can lead us to inconsistent results.⁹ Since ‘heterological’ is heterological if and only if it is not heterological, the proposition that ‘heterological’ is heterological if and only if it is not heterological is, according to the later Wittgenstein’s deflationary conception of truth, also true. While dialetheism proper is the view that there are sentences that are both true and false (i.e., also have a true negation), let us call LNC-dialetheism the view that the law of non-contradiction (LNC) is false, that is, the view that there are true contradictions (by a ‘contradiction’ we mean a sentence of the form of a logical falsehood by the lights of classical propositional logic, for example, sentences of the form ‘p and not-p’ or ‘p iff not-p’). While LNC-dialetheists thus hold that there are true contradictions, according to dialetheism proper there are pairs of contradictory sentences p, not-p, both of which are true. Usually, dialetheists not only embrace dialetheism proper but also LNC-dialetheism.¹⁰ According to the above, Wittgenstein seems to maintain that there are true contradictions and therefore seems to endorse LNC-dialetheism. Note, however, that Wittgenstein endorses LNC-dialetheism only hesitantly when he says that the sentence “‘h’ ε h ≡ ~ (‘h’ ε h) might be called a ‘true contradiction’” (RFM, VII, §28, p. 396; emphasis added).

At times, however, Wittgenstein’s remarks extend beyond LNC-dialetheism and he seems to embrace dialetheism proper too, which also becomes clear by his discussion of Grelling’s paradox:

We should like to say: “‘Heterological’ is not heterological; so by definition it can be called ‘heterological’.” And it sounds all right, goes quite smoothly, and the contradiction need not strike us at all. If we become aware of the contradiction, we should at first like to say that we do not mean the same thing by the assertion, ξ is heterological, in the two cases. […] We should then like to get out of the thing by saying: “~φ(φ) = φ₁(φ)”. But why should we lie to ourselves like this? Here two contrary routes really do lead—to the same thing. Or again:—it is equally natural in this case to say “~φ(φ)” and “φ(φ)”. (RFM, III, §79, pp. 206–207)

But if it is “equally natural” in this case to say “‘heterological’ is heterological” (which we will henceforth call the Grelling sentence) and “‘heterological’ is not heterological”, then—according to Wittgenstein’s deflationary account of truth quoted above (cf. PI, §136)—Wittgenstein also seems committed to the view that it is “equally natural” to say that the Grelling sentence is true and that it is false. This at least suggests that Wittgenstein subscribes to dialetheism proper, that is, the view that some sentences (like the Grelling sentence) are both true and false.

Another reason for ascribing dialetheism proper to Wittgenstein is that assuming paradoxical sentences like the Liar to be both true and false provides a neat explanation of what Wittgenstein seems to take to be distinctive of a paradoxical sentence like the Liar: that it is self-contradictory in the sense of ‘yielding its contradictory, and vice versa’ (op. cit.). Assuming that the Liar sentence is both true and false—i.e., assuming that it is a so-called truth-value glut—and that negation does not alter the truth values of truth-value gluts (i.e., the negation of a true-and-false sentence is true-and-false too), the Liar “yields its contradictory, and vice versa” in the sense that inferring the Liar from its negation (and vice versa) is truth-preserving. This peculiar logical status of the Liar sentence cannot be as neatly explained by assigning to the Liar sentence a classical truth value (i.e., only true or only false) or considering it to be a truth-value gap (i.e., a sentence that is neither true nor false).

Wittgenstein thus seems to endorse (at least hesitantly) LNC-dialetheism as well as dialetheism proper. Also, Wittgenstein’s claim that there are self-contradictory sentences like the Liar or the Grelling sentence that ‘yield its contradictory, and vice versa’ suggests, as we have seen, a dialetheist stance.

3 Wittgenstein and dialetheism (ii): ‘Don’t draw any conclusions from a contradiction’

What has upset readers and provoked the criticisms cited in the introduction, however, is not so much that Wittgenstein takes a dialetheist stance and accepts some contradictory sentences like the Liar and its negation as true, but how he recommends dealing with them: instead of proposing a paraconsistent logic in which a contradiction does not entail every proposition, Wittgenstein seems to give the advice not to draw any conclusions from (true) contradictory sentences:

One may say, “From a contradiction everything would follow.” The reply to that is: Well then, don’t draw any conclusions from a contradiction; make that a rule. You might put it: There is always time to deal with a contradiction when we get to it. When we get to it, shouldn’t we simply say, “This is of no use—and we won’t draw any conclusions from it”? (LFM, XXI, p. 209; cf. RFM, VII, §15, p. 376)

Moreover, Wittgenstein seems to hold that semantic paradoxes and the inconsistent rules that give rise to them are of only local interest; according to him, paradoxical sentences like the Liar are isolated phenomena that do not have any impact on the rest of our language:

Is there harm in the contradiction that arises when someone says: “I am lying.—So I am not lying.—So I am lying.—etc.”? I mean: does it make our language less usable if in this case, according to the ordinary rules, a proposition yields its contradictory, and vice versa?—the proposition itself is unusable, and these inferences equally; but why should they not be made?—It is a profitless performance! (RFM, I, Appendix III, §12, p. 120)

From the point of view of classical logic, however, Wittgenstein’s view that paradoxical sentences like the Liar sentence are harmless isolated phenomena is of course puzzling, to say the least. Given the ‘explosiveness’ of classical logic (i.e., that it validates ex contradictione quodlibet), dialetheism seems to imply trivialism, that is, the view that everything is true. Even if we do not de facto draw conclusions from the Liar sentence, it seems that we are de jure entitled to do so. Even worse, Wittgenstein seems to be aware of the validity of the ‘principle of explosion’ in classical logic and of its consequences.¹¹

However, in his Remarks on the Foundations of Mathematics Wittgenstein does not only assert that it is an empirical fact that de facto “no one does draw conclusions from the ‘Liar’” (RFM, VII, §15, p. 376); the Remarks also contain attempts to explain why we might de jure not be allowed to do so. Concerning the question whether we are de jure allowed to draw any conclusion whatsoever from inconsistent premises (like the Liar sentence and its negation), a range of different interpretations is conceivable. Wittgenstein’s remark “One may say, ‘From a contradiction everything would follow.’ The reply to that is: Well then, don’t draw any conclusions from a contradiction; make that a rule” (LFM, XXI, p. 209) could be interpreted along the lines of Charles Chihara (1984). Chihara holds that even though a contradiction entails every proposition, it is not reasonable to infer from a contradiction any proposition whatsoever. This approach, however, seems to create at least as many problems as it solves, since we now would have to explain what ‘reasonable inference’ is supposed to mean.¹² Another possible interpretation results from the assumption that Wittgenstein could have endorsed the so-called null account of negation (cf. Priest, 1999, pp. 141–142), according to which a contradiction has no content, in the sense that it does not entail anything. Priest (2004, p. 224, n. 11) mentions this as a possible interpretation but correctly points out that Wittgenstein “never seems to have pursued this idea in detail” (ibid.).

In what follows, an alternative reading will be outlined. The basic idea is that we do not draw consequences from self-contradictory (i.e., paradoxical) sentences like the Liar because such sentences constitute a kind of logical dead end so that nothing follows from them¹³: “Contradiction is to be regarded, not as a catastrophe, but as a wall indicating that we can’t go on here” (Z, §687). We are therefore not entitled to infer any sentence whatsoever from a (self-)contradictory sentence. According to this approach, none of our usual inference rules are applicable to such paradoxical sentences (in the sense of not being valid when applied to self-contradictory sentences). That rules are ‘made for’ the application to ordinary cases only (and not for every possible application) is a pervasive thought in Wittgenstein’s Philosophical Investigations. According to Wittgenstein, concepts are framed within a social context, viz., our so-called life-form. The rules governing our application of a concept cover only such cases that we usually encounter in our ‘form of life’ (and similar cases)—we should not expect the rules to cover all sorts of cases. In particular, the rules need not cover those cases that are radically out of the ordinary (like the contradictions resulting from overdetermined predicates, which hardly ever occur in ordinary life)¹⁴:

It is only in normal cases that the use of a word is clearly prescribed; we know, are in no doubt, what to say in this or that case. The more abnormal the case, the more doubtful it becomes what we are to say. (PI, §142)

Wittgenstein also explicitly suggests that it is not clear that ordinary inference rules are applicable to (in the sense of being valid for) unusual cases like paradoxical (i.e., true and false) sentences. In the following, Wittgenstein thus comments on the predicate “heterological”:

It is clear that if it does apply to itself, then it does not; and that if it does not, then it does. From this it presumably follows that it does and does not apply to itself. I would say, “And why not?” If I were taught as a child that this is what I ought to say, I’d gladly say so. (LFM, XXIII, p. 222; emphases added)

According to the definition of “heterological”, it is correct to say that if the predicate “heterological” applies to itself then it does not apply to itself, and vice versa. In classical logic this obviously entails that “heterological” applies to itself and does not apply to itself. Wittgenstein, however, is reluctant to draw this conclusion and expresses himself very cautiously when he says that this only “presumably” entails that “heterological” applies to itself and does not apply to itself. According to the interpretation advanced here, Wittgenstein is in doubt whether the inference rule (P ≡ ~ P / P ˄ ~ P) is applicable (i.e., is valid) in this case. By adding “If I were taught as a child that this is what I ought to say” he seems to stress that we have not been equipped with rules for such cases as the self-contradictory Grelling sentence and that it is not clear whether the inference rule is applicable here.

Before we analyze Wittgenstein’s position from a more systematic point of view, let us take stock of Wittgenstein’s two central claims on the Liar. Wittgenstein’s first central claim on semantic paradoxes is that the (correct) application of the meaning-constitutive rules of the predicates “true” and “heterological” leads us (in some cases) to contradictory results—either to outright (syntactical) contradictions or to ‘self-contradictory’ sentences that ‘yield their contradictory, and vice versa’. Since these contradictions ‘derive from the rules’ governing those predicates they can be called “true” according to the meaning-constitutive rules of the truth predicate (in particular, capture). Wittgenstein thus seems to endorse what we called LNC-dialetheism and also seems to be committed to dialetheism proper. What sets Wittgenstein’s approach apart from other dialetheist approaches to semantic paradoxes is his second claim, that true-and-false sentences like the Liar form a kind of logical dead ends in the sense that we are not entitled to draw any conclusions from them. The reason for this seems to be that our inference rules only hold for ‘usual’ sentences (with classical truth values) but need not be truth-preserving when applied to extraordinary sentences like the Liar, which are both true and false.

4 The Wittgenstein-inspired approach: a semantics for logical nihilism

In order to illustrate the above view, we will provide a formal model that is intended as a rational reconstruction of Wittgenstein’s central tenets on the Liar. From a formal point of view, Wittgenstein’s thought that the Liar forms a kind of logical dead end from which nothing follows entails what is nowadays called logical nihilism, that is, the view that there are no universally valid argument forms (i.e., the view “that there are no laws of logic”; Russell, 2017, p. 126): if there is a sentence with which counterexamples to every inference rule can be formed then, of course, every inference rule has a counterexample (or, to put it another way: if there were any universally valid inference rules, they would also be applicable to the Liar, resulting in valid consequences). In order to capture this idea, the following model provides a semantics for logical nihilism. Of course, Wittgenstein does not want to deny that there are logical inference rules that hold in ordinary contexts. Therefore, the following model also allows for using classical logic in all contexts in which only ‘usual’ sentences (with classical truth values) occur. Formally, the model is a variant of the subclassical logic FDE (‘first-degree entailment’) and will therefore be called FDE′.

Consider a first-order language S without identity. An FDE’ interpretation of S is an ordered pair M = ⟨D, d⟩, where D is a (non-empty) domain and d a denotation function that, as usual, assigns objects in D to names of S. Also, to each n-place predicate G is assigned an ordered pair consisting of its extension d⁺(G) and its anti-extension d^–(G) (where d⁺(G) and d^–(G) are subsets of Dⁿ, i.e., sets of ordered n-tuples of objects in D; d⁺(G) and d^–(G) need neither be jointly exhaustive of Dⁿ nor mutually exclusive). d is extended to an evaluation of all sentences that assigns to each closed well-formed formula one of the sets {1}, {0}, {1, 0} or Ø which represent the truth values (only) true, (only) false, true-and-false and neither-true-nor-false, respectively:

• 1 ∈ d(Gt₁…t_n) iff ⟨d(t₁), …, d(t_n)⟩ ∈ d⁺(G).

• 0 ∈ d(Gt₁…t_n) iff ⟨d(t₁), …, d(t_n)⟩ ∈ d^–(G).

• 1 ∈ d(¬P) iff 0 ∈ d(P).

• 0 ∈ d(¬P) iff 1 ∈ d(P).

• 1 ∈ d(P ˄ Q) iff 1 ∈ d(P) and 1 ∈ d(Q).

• 0 ∈ d(P ˄ Q) iff 0 ∈ d(P) or 0 ∈ d(Q).

The other truth-functional connectives can be defined in the usual way in terms of ¬ and ˄. In the case of quantified sentences, in order to avoid a detour through satisfaction, we extend the language and the interpretation d to ensure that every element e of the domain has a name n_e (such that d(n_e) = e). The truth conditions for quantified sentences are:

• 1 ∈ d(∀x R) iff for all e ∈ D, 1 ∈ d(R_x(n_e)).

• 0 ∈ d(∀x R) iff for some e ∈ D, 0 ∈ d(R_x(n_e)).

(where R_x(c) is the formula obtained by substituting the individual constant c for every free occurrence of x in R).

Furthermore, let S_T be S augmented with an additional unary (transparent truth-) predicate T and a name ⟨P⟩ for each sentence P of S. An FDE’ + T interpretation of S_T is an FDE’ interpretation whose domain contains every sentence of S_T and for which the following conditions hold (for every sentence P of S_T):

• 1 ∈ d(P) iff P ∈ d⁺(T).

• 0 ∈ d(P) iff P ∈ d^–(T).

In order to model the idea that what follows from what depends upon the kind of sentences involved, we will distinguish between different notions of logical truth and logical consequence for FDE’. In particular, these notions characterize the relation of logical consequence (a) as it holds (or not) between premise sets and conclusions where each sentence involved is classically valued, (b) as it holds between premise sets and conclusions where each sentence involved is classically valued or a truth-value gap, and (c) as it holds between any premise sets and conclusions whatsoever. Each of these notions, however, fully captures the intuitive concept of logical consequence according to which a sentence A follows from a set of premises Γ iff it is not possible that every premise is true while A is false (or not true). The latter relation (c) will be called logical consequence simpliciter (⊨_FDE’):

• Γ ⊨_FDE’ A iff there is no interpretation ⟨D, d⟩ such that 1 ∈ d(B) for all B ∈ Γ, but 0 ∈ d(A).¹⁵

Logical truth simpliciter is defined correspondingly:

• ⊨_FDE’ A iff ∅ ⊨_FDE’ A, i.e., iff there is no interpretation ⟨D, d⟩ such that 0 ∈ d(A).

In addition to logical consequence simpliciter we can distinguish between classical consequence (a) and three-valued consequence (b): Let a classical interpretation of the language S (without ‘T’) be a model that assigns to each predicate a classical extension, and the complement of this as its anti-extension (as in the case of classical logic, such that the extension and anti-extension of an n-ary predicate are disjoint and jointly exhaustive of Dⁿ). A classical interpretation assigns to every sentence exactly one of the values {1} or {0} (i.e., only true or only false). We can then define the notions of classical logical consequence (⊨_CL) and classical logical truth analogously to their above non-classical counterparts by restricting the respective interpretations to classical ones. Also, we can define the notions of three-valued logical consequence and three-valued logical truth (⊨_K3) on the basis of three-valued interpretations, where a three-valued interpretation is defined just like a classical interpretation except that it allows predicates to be assigned non-exhaustive pairs of extensions and anti-extensions (which, however, must be disjoint). Three-valued interpretations thus assign to every sentence one of the values {1}, {0} or $\varnothing$ (i.e., only true, only false or neither-true-nor-false), i.e., they leave truth-value gaps but don’t have truth-value gluts).¹⁶ As usual, we will call an inference rule Γ / A valid (relative to a notion of logical consequence ⊨_X) iff Γ ⊨_X A (where X is FDE’, CL or K₃, i.e., strong Kleene three-valued logic).

According to the above definition of logical consequence simpliciter, there are no valid inferences. This can be easily seen by considering a trivial interpretation which assigns ⟨Dⁿ, Dⁿ⟩ to every n-place predicate, so that, e.g., every one-place predicate will be true and false of every object in the domain. According to this interpretation, every sentence will be true and false, so that, in particular, in the case of an arbitrary inference Γ / A each of the premises in Γ will be true while A is false. This interpretation also shows that there are no logical truths simpliciter. In particular, FDE’ invalidates the principle of explosion (A, ¬A ⊭_FDE’ B) so that FDE’ is paraconsistent. In contrast to FDE, FDE’ does not result in a sophisticated paraconsistent logic that is applicable to reasoning in dialetheic contexts in which true-and-false sentences occur; FDE′s relation of logical consequence simpliciter is rather to be conceived of as a model of logical nihilism (i.e., the view that there are no universally valid inference rules).¹⁷ The above model thus captures Wittgenstein’s central intuition concerning a true-and-false sentence like the Liar, namely that it is a logical dead end and implies nothing logically. The Liar is thus inferentially sterile and, therefore, harmless. In classical contexts, however, in which all sentences have classical truth values, as, e.g., in mathematics or other unproblematic contexts, FDE’ allows for reasoning with classical logic. In these contexts, all classical inference rules are valid with respect to the concept of classical logical consequence (⊨_CL). In this way FDE’ captures Wittgenstein’s intuition that (inference) rules may hold only for ordinary cases of propositions (which are classically valued), while we are not equipped with valid logical inference rules that hold for contexts in which extraordinary propositions like the (true and false) Liar sentence occur. While there are no logical inference rules that hold for absolutely all propositions, the classical axioms and inference rules do hold in contexts (like that of classical mathematics) in which we can build on the assumption that every expression of our language is semantically evaluated in a classical way, i.e., in contexts in which we need to take into account classical possibilities (classical interpretations) only.¹⁸

The transparency of the truth predicate T that Wittgenstein takes to be constitutive of its meaning (see Sect. 2) is guaranteed by the above requirements for T. These requirements guarantee that T($\left\langle P \right\rangle$) has the same semantic status (truth value) as P. In particular, T($\left\langle P \right\rangle$) is true (or true and false) whenever P is (and vice versa), so that the inference rules of capture and release will hold in the sense that they are necessarily truth-preserving.¹⁹ If (and only if) we evaluate the Liar as being true and false, the model also captures Wittgenstein’s intuition that the Liar “yields its contradictory” in the sense that this inference is truth-preserving (we may think of it as an analytical, albeit not logical—i.e., formal—consequence; cf. fn. 4). This is how the model resolves the apparent discrepancy between Wittgenstein’s claims that the Liar is a ‘logical dead end’ and at the same time ‘yields its contradictory’.

FDE’ + T also captures Wittgenstein’s other central thoughts on the Liar. FDE’ + T allows for both forms of dialetheism we have discussed: besides capturing Wittgenstein’s intuition that the Liar “yields its contradictory” by evaluating the Liar L (i.e., ¬T(⟨L⟩)) as true and false (dialetheism proper), the model also allows for contradictory propositions like L ≡ ¬L that are true (and false, i.e., LNC-dialetheism). FDE’ thus allows for propositions which, as Wittgenstein describes them, ‘might be called true contradictions’.

Not every property of the model is intended to model the real phenomena, however. In view of the semantics of the logical constants and in view of logical consequence the model makes more precise predictions than Wittgenstein would have considered adequate. Wittgenstein probably would have considered the above model inaccurate for the reason that, according to the model, there are no valid inference rules for the Liar because the semantics is fixed in the way described by the model, while Wittgenstein seems to want to say that there being no such rules means that the semantics is not fixed.²⁰ This is because, according to the later Wittgenstein, the inference rules determine the meaning of the logical constants (not the other way round):

We can conceive the rules of inference—I want to say—as giving the signs their meaning, because they are rules for the use of these signs. So that the rules of inference are involved in the determination of the meaning of the signs. (RFM, VII, §30, p. 198)

The truth functions assigned to the logical constants above are thus not intended to represent the exact semantics of the logical constants but should rather be interpreted in an instrumentalist way: they enable the model to capture Wittgenstein’s thought that we do not have valid inference rules that might be applied to extraordinary propositions like the Liar but that there are valid inference rules that hold in the realm of classically valued sentences. From this point of view, the semantics of the logical constants of FDE’ is only partially determined. Bearing this difference in mind, we will refer to the above formal model as the “Wittgenstein-inspired” account of the semantic paradoxes.

From a technical point of view, the central idea behind FDE’ consists in combining an empty logic with the basic idea of adaptive logics, viz., that we should distinguish between a ‘lower limit logic’ which can be used in any context whatsoever and an ‘upper limit logic’ which can be used in contexts in which no abnormalities occur²¹ (however, an empty logic like FDE’ does not ‘officially’ classify as a lower limit logic in the sense of Batens, 2007, p. 223). There are, of course, different ways in which such a project might be realized. Another example of a ‘logical nihilist’ semantics, which could also have served to illustrate some of Wittgenstein’s ideas (though it is based on examples that Wittgenstein does not discuss), is given in Russell (2017, pp. 130–132). Russell argues that there are counterexamples to reflexivity (i.e., cases where A ⊭ A) and to the introduction and elimination rules for the logical connectives. Russell’s counterexamples comprise sentences like “This sentence is a premise”, “This sentence is the first part of a compound sentence” and “This sentence is unembedded”. Let Prem abbreviate “This sentence is a premise”; then Prem, like the true-and-false Liar sentence in FDE’, seems to provide a counterexample to reflexivity,²² i.e., Prem ⊭ Prem, since the inference has a true premise but a false conclusion. FDE’ accordingly provides an alternative semantics for logical nihilism that is not dependent upon the context-sensitive examples of Russell’s semantics.²³

As another possibility, FDE’s concept of logical consequence simpliciter could have been defined correspondingly to the respective concepts of a group of three-valued logics including TS and TS_ω that also abandon reflexivity and thus are to be counted among the substructural logics (see Cobreros et al., 2012, pp. 366–372, and, e.g., Pailos, 2022, p. 1394, on TS_ω).²⁴ Like these so-called mixed logics, FDE’s notion of logical consequence simpliciter makes use of what is known as a mixed consequence relation, according to which a consequence follows from a set of premises iff under each valuation in which the premises meet a certain standard S₁, the conclusion meets a (possibly different) standard S₂ (see Cobreros et al., p. 366). FDE’ differs, of course, fundamentally from these logics in its ‘adaptability’, i.e., that it allows to distinguish between different contexts like, e.g., contexts in which only classically valued sentences are involved and classical inference rules hold. Another difference to the abovementioned substructural logics consists in FDE’s being suited to model sets of sentences that include sentences with not just one but two different non-standard truth-values.²⁵

5 Arguments for and implications of the Wittgenstein-inspired approach

What speaks in favor of the Wittgenstein-inspired view? A major asset of the Wittgenstein-inspired approach consists in (i) its simplicity; it seems to result in an astonishingly simple way of (ii) dealing with a large range of paradoxical propositions that cannot be evaluated with a classical truth value, like the Liar and related paradoxes in which ungrounded sentences are involved (e.g., the postcard paradox, longer Liar circles or Yablo’s paradox²⁶) as well as Grelling’s paradox, in which ungrounded sentences do not seem to play a significant role (and which, therefore, seems to escape standard fixed-point approaches to semantic paradoxes like that of Kripke, 1975). There is reason to think that the solution also generalizes to Curry paradoxes.²⁷ The Wittgenstein-inspired approach thus nourishes the hope for (iii) a uniform solution of the semantic paradoxes: in each of these cases, the paradoxical sentences result from overdetermined predicates like “true” or “heterological”, which allow for some inconsistent classifications, e.g., sentences that ‘yield their contradictories’ and should, in the spirit of the Wittgenstein-inspired approach, be counted among the true-and-false sentences, which do not entail anything.²⁸ (Though these sentences may seem to imply false propositions if one applies inference rules to them that are not valid.) The account is also (iv) able to deal with so-called revenge paradoxes, which will be discussed in the following section. Moreover, (v) the Wittgenstein-inspired approach provides not only a formal model but has a principled story to tell that explains what goes wrong in semantic paradoxes (they result from semantically overdetermined predicates) and why we do not need to worry about them (inferential sterility of the resulting self-contradictory sentences).

Another major asset of the approach consists (vi) in its classicality: it allows for a uniform solution of many semantic paradoxes without abandoning classical logic. What needs to be given up is the assumption that logical rules hold without exception for every sentence. In particular, we have no rules for reasoning with extraordinary propositions like the Liar that are true and false. In classical contexts, however, e.g., in the case of classical mathematics, where every proposition involved has a classical truth value, the inference rules of classical logic will be valid and hold without exception. Of course, it can happen that we unknowingly reason with paradoxical sentences, e.g., in the case of contingent Liars as envisaged by Kripke (1975).²⁹ While we could rely on a paraconsistent logic in these cases, in order to prevent finding ourselves in a situation in which we seem to be licensed to derive anything, an alternative strategy consists in using classical logic as a default means of reasoning and trying to sort out the paradoxical culprit(s) after a contradiction has turned up, so that we can reason afterwards with classically valued sentences only. (This strategy seems to have been employed—in the absence of an alternative—in the historical case of Russell’s paradox; in this case, Frege postulated the truth of the unrestricted axiom schema of comprehension, which entails a contradictory classification in the case of the Russell set R (= {x | x ∉ x}), i.e., R ∈ R iff R ∉ R (which, of course, parallels Grelling’s ‘h’ ε h ≡ ~(‘h’ ε h)).

Another feature of the Wittgenstein-inspired approach is that it provides an elegant explanation of the perennial fundamental disagreements among many expert logicians on even the most basic facts concerning an adequate logic for the Liar and other paradoxical sentences (indeed, it might be argued that it provides the best explanation of this expert disagreement so far). Many questions concerning the Liar’s semantic status and its logical behavior have been answered in fundamentally different ways, e.g., what is its compositional impact to truth-functionally complex sentences and which inference rules hold for the Liar? From the point of view of the Wittgenstein-inspired approach, these disagreements result from the Liar being semantically and logically indeterminate in many ways: while the meaning-constitutive rules for the truth predicate (capture and release, see above) and for “heterological” lead to contradictory results when applied to the Liar or the predicate “heterological”, according to the Wittgenstein-inspired approach we do not have inference rules that allow us to draw any further conclusions from such paradoxical propositions. Such an indeterminacy might explain why so many approaches to the Liar paradox differ fundamentally with respect to their semantics of the logical constants: because the semantics of our logical vocabulary is not sufficiently fixed for such extraordinary cases, different semantic precisifications of the logical constants (in form of different truth tables or inference rules) might be proposed with roughly the same plausibility—just as the extension of, for example, the vague concept word “red” might be made precise in more than one way. The best explanation of the perennial expert disagreement in this case seems to be that there is no fact of the matter to be discovered and that the semantic rules of our natural language connectives are indeterminate with respect to the Liar.

Seen from this perspective, philosophical logics like FDE, Kleene’s weak and strong truth tables or Priest’s LP that were proposed in order to deal with the paradoxes are to be seen as attempts at a precisification of the semantic rules of natural language for truth-functional connectives with respect to the Liar—just as we can make vague predicates (more) precise if we wish to do so. From Wittgenstein’s point of view, however, proposals about how to precisify vague terms should be seen as pragmatic debates about which revision of the rules best serves practical purposes; they are not factual debates that can legitimately purport to yield discoveries about the nature of the original rules (since there is no fact of the matter to be discovered; cf., e.g., PI, §68). Precisifying a vague expression thus seems to be akin to the task of “having to sketch a sharply defined picture ‘corresponding’ to a blurred one”, which Wittgenstein calls ‘hopeless’ in his Philosophical Investigations: “Here I might just as well draw a circle or heart as a rectangle […]. Anything—and nothing—is right” (PI, §77).

The idea behind the Wittgenstein-inspired approach, that inference rules are valid only when applied to ordinary sentences (and not when applied to unusual cases like paradoxical sentences), has recently gained increasing attention. In particular, T. Hofweber (2008, 2010) argues that logical rules are ‘generic’ and hold under ‘normal’ conditions, but admit of exceptions and may fail under ‘abnormal’ conditions. Also, B. Whittle (2017, 2021) holds “that all—or almost all—logical rules have exceptions” (Whittle, 2021, p. 85). The relation of logical consequence thus can only be approximately described formally with the help of strictly valid inference schemes that abstract from the particular semantic content of the involved sentences.³⁰ What follows from what, thus, depends not only upon the content of the involved logical constants and the logical form of the propositions but upon the specific content of the propositions involved.

The view that logic is universal in the sense of having inference rules that are valid for every domain of discourse was already explicitly doubted in the middle ages; in medieval terms, the question was whether logic is ‘formal’. While the majority view was that classical logic (which then, of course, meant ‘Aristotelian logic’) is ‘formal’, i.e., universal, some philosophers disagreed. Robert Holcot and others, for instance,³¹ answered the question in the negative and held that Aristotelian logic could not be applied to the ‘paradoxes of trinity’; they called for a logic of faith for such cases. Wittgenstein thus is far from being the first philosopher to deny the universal applicability of logic, but he seems to be the first to tend to such a conclusion because of the Liar paradox.

Of course, this does not exclude that we can describe the relation of logical consequence for proper subsets of propositions, say the propositions of mathematics, in a purely formal manner. In such confined domains the relation of logical consequence could ‘behave orderly’ as expected and could adequately be described formally by inference schemes. It should be kept in mind, however, that the resulting logic is based on an idealization or simplification—like many other models in science—and is not applicable to any proposition whatsoever.

Wittgenstein also offers an explanation why logical laws admit of exceptions. According to him, classical logic is the result of observing and generalizing the logical behavior of a certain class of paradigmatic examples of propositions, namely “sentences about physical objects [and] sense datum propositions” (LFM, XXIV, p. 231), while ignoring propositions like the Liar that are radically out of the ordinary and do not behave in such a harmless manner. The inference rules of classical logic thus seem correct only because of such an “unbalanced diet” (PI, §593) of examples:

For one thing is clear. The law of contradiction is a result of continuing in a particular way the technique which we have in dealing with propositions. And by “propositions” we mean such things as “It rains”, “There are three chairs in the room”, etc. English sentences about physical objects, sense datum propositions; this forms the nucleus of what we call propositions; and it is the practice or techniques of using these expressions which is shown by the laws of logic. (LFM, XXIV, p. 231; emphasis added; cf. RFM, VII, §73, p. 436)³²

“[S]entences about physical objects, sense datum propositions” and similar sentences thus form, so to speak, the ‘intended domain of application’ of logical inference rules, where the rules hold without exception. However, these rules need not remain valid, and axiom schemes need not remain true, when we substitute extraordinary propositions like the Liar, which hitherto have not been taken into consideration in the course of our ‘unbalanced diet’. This seems to be Wittgenstein’s point in the following:

[T]he logical axioms are in fact not at all convincing if for the propositional variables we substitute structures which no one originally foresaw as possible values, when, that is, we began by acknowledging the truth of the axioms absolutely. (RFM, VII, §13, p. 372)

While logical axiom schemes are usually taken to be true for every substitution of the propositional variables because of the meaning of the logical vocabulary involved and the logical form of the schema, Wittgenstein seems to hold that the latter is not enough to determine the truth value of ‘unexpected’ substitution instances, e.g., when we substitute the Liar sentence. Just as it can be indeterminate whether a (‘family-resemblance’) concept applies to a given case or not (a so-called ‘borderline case’), Wittgenstein seems to hold that a similar form of semantic indeterminacy (comparable to vagueness³³) is also exhibited by sentence schemas like P ∨ ¬P (and thus by the involved logical vocabulary). Though Wittgenstein does not endorse this thesis explicitly, it is clearly Wittgensteinian in spirit, given the pervasiveness of the topic of semantic indeterminacy or vagueness in Wittgenstein’s later works: in the case of predicates, for example, semantic indeterminacy gives rise to cases where we are not equipped with rules for the application of the predicate, which Wittgenstein discusses at length in his Philosophical Investigations.³⁴

What particularly sets the Wittgenstein-inspired approach apart from more traditional ones is the way in which it logically isolates the Liar and other paradoxical sentences from non-paradoxical ones. Wittgenstein considers paradoxical sentences like the Liar to be “unusable” for practical purposes and the use of such sentences to be “a profitless performance” (RFM, I, Appendix III, §12, p. 120). In contrast to other approaches, which devise non-classical logics in order to explain how we reason with paradoxical sentences, Wittgenstein does not seem to consider the latter to be necessary. From the point of view of the Wittgenstein-inspired approach, it rather seems to be sufficient to be able to reason about paradoxical sentences with the help of non-paradoxical ones that have classical truth values, and to only mention the paradoxical sentences (e.g., “The Liar sentence consists of xy letters”) but not to use them in the sense that these paradoxical sentences occur, e.g., as truth-functional parts of other sentences. It is, however, not clear whether this strategy will work and offer a way out of the paradox. One problem that needs to be overcome (on which Wittgenstein is silent) consists in the so-called revenge paradoxes, to which we will turn now.

6 Revenge paradoxes

A general scepticism about there being a resolution of the Liar paradox is nourished by the so-called revenge paradoxes. Such paradoxes result from employing the semantic machinery that is used to solve a given paradox in order to construct similar paradoxes that cannot be handled like the original case and demand further semantic resources (which in turn threaten to engender new paradoxes, etc.). In the case of the Wittgenstein-inspired approach, such paradoxes might be thought to occur if we could express terms like “the extension of the truth predicate” (i.e., “d⁺(T)” in the above model) in a semantically universal language.³⁵ Wittgenstein does not address the problem of revenge paradoxes, and it seems fair to say that no approach to the Liar that has been proposed so far has been met with wide acceptance regarding its solution to revenge paradoxes. While a comprehensive discussion of the question whether and how semantic universality might be achieved is beyond the scope of this paper, in the remainder of this section we will outline an informal attempt at solving the problem.

In order to describe the Liar’s semantic status with classically valued propositions, the above model made use of the term “d⁺(T)”, i.e., “the extension of the truth predicate”. The account to be outlined below will make use of the natural-language equivalent of the functional term d⁺, i.e., the expression “the extension of”, which we will abbreviate by D. In addition, we will use Tr as a shorthand for the natural-language truth predicate (which we take to be transparent). For any unitary predicate φ of natural language, we will designate “the extension of ‘φ’” by Dφ. While Dφ is the natural-language analogue of d⁺(G) (where G is interpreted as φ), D can, in contrast to d⁺(G), be applied iteratively: Let D⁰φ be φ and let Dⁿ⁺¹φ be D(Dⁿφ), i.e., Dⁿ is the expression consisting of n iterations of “is an element of the extension of”, each followed by an opening quotation mark, followed by “φ”, which in turn is followed by n closing quotation marks. While there is, of course, room to debate the precise semantics of iterated applications of D, we will follow the intuition that each Dφ roughly relates to φ as d⁺(G) relates to G, although Dφ (unlike d⁺(G)) cannot have a classical semantics (as we will see below). With the help of such predicates we can express the semantic status of the Liar λ (i.e., not-Tr(λ)) by the only-true proposition: ‘DTr(λ) and D(not-Tr(λ))’, i.e., “The Liar is an element of the extension and of the anti-extension of ‘true’”. Just like the corresponding claim in our formal model, i.e., ‘L ∈ d⁺(T) and L ∈ d^–(T)’, this sentence seems to be only true, while the similar propositions ‘Tr(λ) and not-Tr(λ)’ and ‘Tr(λ) and Tr(not-λ)’ would, according to the Wittgenstein-inspired approach, be both true and false, like their formal equivalents in FDE’ + T.³⁶

The application of the natural-language operator the extension of φ to the truth predicate Tr (i.e., DTr) yields what Stephen Yablo calls a strong notion of truth³⁷: in contrast to the (weak) ‘transparent’ notion of truth Tr (or T in S_T), according to which P and Tr(P) have the same semantic status (truth value), the semantic values of P and “‘P’ is an element of the extension of ‘is true’” (i.e., DTr(P)) can differ (if P lacks a truth value and thus does not belong to the extension of “is true”, DTr(P) will simply be false; analogously, ‘P ∈ d⁺(T)’ will be simply false if 0 ∉ d(P) and 1 ∉ d(P)). While the extensions of “is true” and “is an element of the extension of ‘is true’” are the same, these predicates have different anti-extensions such that Tr(P) and DTr(P) can have different falsity conditions (if P is a truth-value gap, the same will hold of Tr(P), while DTr(P) will be false).³⁸ Moreover, given sentences like the Liar, which are both true and false, the extensions of the predicates “is an element of the extension of ‘is true’” (i.e., DTr) and “is an element of the anti-extension of ‘is true’” (i.e., D(not-Tr)) will not be disjoint (just as in the case of d⁺(T) and d^–(T)) and—given that there are sentences that are neither true nor false (e.g., sentences containing vague predicates)—also not jointly exhaustive.

While the transparent truth predicate is useful, e.g., as a device of disquotation, it is of little help when we are dealing with the semantics of propositions some of which do not have classical truth values. Thus Timothy Williamson argues, for example, that the principle of bivalence cannot be consistently denied when truth is expressed by a weak (transparent) truth predicate (cf. Williamson, 1994, Sect. 7.2). If we tried to deny the bivalence of a sentence P with the help of our transparent truth predicate, i.e., by ‘not-Tr(P) and not-Tr(not-P)’, the resulting claim would be equivalent to a logical contradiction, viz., ‘not-P and not-not-P’. It seems to be clear that this contradiction does not express the possibly only-true claim that P is not bivalent. Similarly, the assertion of the bivalence of a gappy sentence P with the help of our transparent truth predicate Tr would result in a gappy sentence ‘Tr(P) or Tr(not-P)’, while, intuitively, to assert bivalence of a gappy sentence should simply result in a false assertion.³⁹ However, we can assert the bivalence of a gappy sentence P with the proposition ‘DTr(P) or DTr(not-P)’, which, adequately, would be only false (this corresponds to our metalanguage proposition ‘P ∈ d⁺(T) or P ∈ d^–(T)’). For the same reason, the general principle of bivalence would have to be expressed with the help of DTr (or, in the case of our formal model, with d⁺(T) in the metalanguage).

Also, in the case of truth-value gluts like the Liar sentence, the predicate “is an element of the extension of ‘is true’” allows for an adequate (i.e., only true) description of their semantic statuses: the Liar sentence is an element of the extension of “is true” (DTr) (in the above formal model: d⁺(T)) as well as of its anti-extension D(not-Tr) (d^–(T)), but it is clearly not the case that it also falls under the predicate “is not an element of the extension of ‘is true’” (not-DTr) (not-d⁺(T) in the above model, which clearly differs from d^–(T)⁴⁰). Thus, the sentence “The Liar sentence is an element of the extension of ‘is true’ as well as an element of the extension of ‘is not true’” (i.e., ‘DTr(λ) and D(not-Tr(λ))’) adequately describes the Liar’s semantic status and will be only true (just like ‘L ∈ d⁺(T) and L ∈ d^–(T)’ in the above model).

In the case of natural languages, it clearly seems to be the case that we are equipped with such strong truth predicates as “is an element of the extension of ‘is true’”. As the above example by Williamson shows, it could otherwise hardly be explained how we manage to express coherent thoughts about truth-value gaps or gluts. Natural languages thus seem to be equipped with weak and strong truth predicates (though they often seem to be conflated).

It might be suspected, however, that the above expressive device allows for the formulation of revenge paradoxes along the following lines:

• $(\uplambda^{+})\quad \uplambda^{+}$ is not an element of the extension of “is true”,

  i.e.,

• $(\uplambda^{+})\quad Not-Dtr(\uplambda^{+})$.

From the point of view of natural language it seems clear that we can form such a self-referential proposition. However, λ⁺ does not seem to pose a more serious problem for the Wittgenstein-inspired approach than the Liar λ: If, on the one hand, λ⁺ is an element of the extension of “is true”, it is true (since “is true” and “is an element of the extension of ‘is true’” have the same extension) and, because of release, it is not an element of the extension of “is true”. If, on the other hand, λ⁺ is not an element of the extension of “is true”, λ⁺ is true (because of capture); it therefore is an element of the extension of “is true” (since “is true” and “is an element of the extension of ‘is true’” have the same extension). Just like the Liar sentence λ, λ⁺ thus seems to ‘yield its contradictory, and vice versa’ (op. cit.).

In the spirit of the Wittgenstein-inspired approach, λ⁺ should therefore be treated analogously to the Liar and should be counted as belonging to the extension, as well as to the anti-extension, of the predicate “is an element of the extension of ‘is true’”. The paradox can thus be taken to show that DTr is overdetermined analogously to Tr. Taking DTr to be overdetermined and λ⁺ to be true and false—just as in the case of the original Liar sentence—explains best why we take the above derivation to be convincing: it consists of steps that are truth-preserving either because of the semantic rules of the predicates involved or because of their consequences. Also, according to the above approach, λ⁺ would be a logical dead end and therefore as harmless as the original Liar. Moreover, the semantic status of λ⁺ can be adequately described by iteratively applying D, i.e., with the help of the predicates “is an element of the extension of ‘is an element of the extension of “is true”’” (D²Tr, i.e., DDTr) and “is an element of the extension of ‘is not an element of the extension of “is true”’” (i.e., D(not-DTr)). The semantic status of λ⁺ can thus be expressed by the (only true) proposition ‘DDTr(λ⁺) and D(not-DTr(λ⁺))’ (analogously to the above case of the original Liar). Since we can form similar Liar sentences for any number of iterations of the D-operator (and derive contradictions analogously), each of the DⁿTr predicates demands a non-classical semantics.⁴¹

As we have seen, the expressive resources of natural languages seem to comprise not only one strong truth predicate but a whole hierarchy of such predicates DⁿTr (for each n ∈ $\mathbb{N}$). Another worry this hierarchy of strong truth predicates gives rise to consists in the following ‘infinite Liar’ λ^∞:

• $(\uplambda^{\infty})$      There is no n such that λ^∞ is an element of DⁿTr.

Intuitively, we can reason in a similar way as in the case of λ⁺: If there is no n such that λ^∞ is an element of DⁿTr, then λ^∞ will be true (by capture). But then λ^∞ will be an element of D⁰Tr (since D⁰Tr is Tr) so that there is an n (= 0) such that λ^∞ is an element of DⁿTr. If, on the other hand, there is an n such that λ^∞ is an element of DⁿTr, λ^∞ will be true (since DⁿTr and Tr have the same extension). Therefore, by release, there is no n such that λ^∞ is an element of DⁿTr. It seems, therefore, that there is and is not an n such that λ^∞ is an element of DⁿTr.

Again, the Wittgenstein-inspired approach allows taking λ^∞ to be true and false and the predicate “There is no n such that x is an element of DⁿTr” to be overdetermined analogously to the simple (transparent) truth predicate Tr or the strong truth predicate DTr. It can thus be explained why we take the above derivation to be convincing: it consists of steps that are either truth-preserving because of the semantic rules of the predicates involved or valid in non-paradoxical contexts.⁴² Also, λ^∞ seems to be as harmless as the aforementioned cases: it seems to be true and false and, therefore, implies nothing (according to FDE’). Moreover, we can describe its semantical status by the (only-)true proposition “λ^∞ is an element of the extension of ‘There is no n such that x is an element of DⁿTr’ and λ^∞ is an element of the anti-extension of ‘There is no n such that x is an element of DⁿTr’”. This strategy for evading revenge paradoxes seems to be generalizable: Let φ be any unitary overdetermined predicate such that there is a corresponding paradoxical proposition λ^φ that “yields its contradictory, and vice versa” (op. cit.). Then λ^φ will be true and false and imply nothing⁴³; still, we can express λ^φ’s semantical status by the (only-)true proposition ‘λ^φ belongs to φ’s extension and λ^φ belongs to φ’s anti-extension’.⁴⁴

It is unclear whether (and if so how) these derivations can be formalized. The usual approach of a hierarchy of languages would render λ^∞ inexpressible since different DⁿTr’s would belong to different levels of the hierarchy. Another option consists in fixed-point constructions in the sense of Kripke (1975); so far, however, there are no formal languages that contain the expressive resources of the above D-operator and the resulting hierarchy of strong truth predicates. Whittle’s fixed-point approach, for instance, allows for the expression of the extension and the anti-extension of the truth predicate and the falsity predicate and of their complements in the metalanguage (see Whittle, 2021, p. 102). Such (possibly temporary) limitations of formalization should, of course, not be mistaken as in-principle limitations of the Wittgenstein-inspired approach.

While the above examples of revenge paradoxes are certainly not exhaustive, there remains hope that the Wittgenstein-inspired approach is not susceptible to revenge paradoxes (at least not to their most common variants). It thus nourishes the hope for a semantically universal formal language, though the above considerations suggest that it might still be a long way to go to reach this goal. Devising formal languages that at least approximate semantic universality without incurring semantic paradoxes (i.e., what Chihara (1979, pp. 590–591) calls the ‘preventative problem of the paradox’) is, of course, not a problem Wittgenstein is concerned with. While most contemporary approaches to the Liar paradox seem to be interested in this kind of question, Wittgenstein’s remarks on the Liar paradox are concerned with the ‘diagnostic problem of the paradox’ (ibid.), namely the problem of pinpointing the disease of which the semantic paradoxes are a symptom. Wittgenstein is thus concerned with the classical questions related to the semantic paradoxes, in particular questions concerning the Liar’s semantical and logical status. In view of the Liar’s semantical status, Wittgenstein holds, as we have seen, that the truth predicate is overdetermined, such that the rules governing its use can lead to contradictory results in exceptional cases like the Liar. This poses the threat of what Hofweber (2008, pp. 146–148) aptly calls ‘the Great Collapse’, that is, the truth of every sentence due to the explosiveness of classical logic. According to Wittgenstein’s diagnosis, the problem dissolves when we take into account that inference rules do not hold without exception and are not fixed for extraordinary cases like the—true and false—Liar sentence. In view of its logical status, the Liar thus turns out to be a kind of logical dead end in the sense that it is inferentially sterile and implies nothing logically. In addition, the approach seems to be able to cope with (at least some) ‘revenge paradoxes’, and it seems possible to describe the Liar’s semantic status in an adequate way, as demonstrated in this section; the Wittgenstein-inspired approach is hence not only of historical interest and well worth further exploration.