Substructural approaches to paradox: an introduction to the special issue

1 Volume topic

The idea of a “substructural approach to a paradox” is, naturally enough, the idea of an approach to a paradox that uses a substructural logic. While, contrary to what some might perhaps expect given also the formal sophistication of many such approaches, as it turns out no component of this complex idea lends itself to a very formal definition, it will be useful first to go through some informal elucidations of the idea’s two main components: paradoxicality (Sect. 2) and substructurality (Sect. 3). Against that background, I’ll then proceed to expound the philosophical significance of substructural approaches to paradox (Sect. 4), which will in turn serve as a springboard to introducing the papers of this volume (Sect. 5).

2 Paradoxicality

According to the traditional definition of paradoxicality (e.g. Sainsbury, 2009, p. 1), a paradox¹ is a situation where apparently²true premises apparently entail an apparently false conclusion.³ Let’s see some paradigmatic cases of paradox that typically motivate this definition (and that are extensively discussed in the papers of this volume):

The Liar paradox. For every circumstance that can hold or can fail to hold, there is a sentence (of some language or other) expressing that that circumstance holds, a sentence which is thereby true iff the circumstance holds. Letting T express truth, this entails the principle of correlation according to which, for every sentence $\varphi$, there is a sentence s such that $T(s) \leftrightarrow \varphi$ holds. Letting $\ulcorner \varphi \urcorner$ be a name for $\varphi$, for every $\varphi$ an optimal candidate for satisfying correlation is $\varphi$ itself, so that $T(\ulcorner \varphi \urcorner ) \leftrightarrow \varphi$ holds. Moreover, there is a sentence $\lambda$ identical with $\lnot T(\ulcorner \lambda \urcorner )$, so that $T(\ulcorner \lambda \urcorner ) \leftrightarrow \lambda$ holds. However, suppose that $T(\ulcorner \lambda \urcorner )$ holds. Then, by correlation and the entailment of modus ponens ($\varphi , \varphi \rightarrow \psi \vdash \psi$),⁴$\lambda$ (i.e. $\lnot T(\ulcorner \lambda \urcorner )$) holds, and so it seems that, by the entailment of adjunction ($\varphi ,\psi \vdash \varphi \& \psi$), $T(\ulcorner \lambda \urcorner ) \& \lnot T(\ulcorner \lambda \urcorner )$ holds. Suppose next that $\lnot T(\ulcorner \lambda \urcorner )$ (i.e. $\lambda$) holds. Then, by correlation and modus ponens, $T(\ulcorner \lambda \urcorner )$ holds, and so it seems that, by adjunction, $T(\ulcorner \lambda \urcorner ) \& \lnot T(\ulcorner \lambda \urcorner )$ holds. Therefore, discharging the suppositions, by the metaentailment of reasoning by cases (if $\Gamma _0,\varphi \vdash \Delta _0$ holds and $\Gamma _1,\psi \vdash \Delta _1$ holds, $\Gamma _0,\Gamma _1, \varphi \vee \psi \vdash \Delta _0,\Delta _1$ holds), $T(\ulcorner \lambda \urcorner ) \vee \lnot T(\ulcorner \lambda \urcorner )$—which, being the conclusion of the law of excluded middle ($\oslash \vdash \varphi \vee \lnot \varphi$), is a logical truth—entails $T(\ulcorner \lambda \urcorner ) \& \lnot T(\ulcorner \lambda \urcorner )$—which, being the premise of the law of noncontradiction ($\varphi \& \lnot \varphi \vdash \oslash$), is absurd. The argument is apparently valid, the premises (correlation, the existence of a sentence such as $\lambda$ and $T(\ulcorner \lambda \urcorner ) \vee \lnot T(\ulcorner \lambda \urcorner )$) are apparently true, but the conclusion ($T(\ulcorner \lambda \urcorner ) \& \lnot T(\ulcorner \lambda \urcorner )$) is apparently false (to the best of my knowledge, the first clear version of the paradox was given by Eubulides of Miletus).

Curry’s paradox. Letting $\chi$ be an absurd sentence, there is a sentence $\kappa$ identical with $T(\ulcorner \kappa \urcorner )\rightarrow \chi$, so that $T(\ulcorner \kappa \urcorner ) \leftrightarrow \kappa$ holds. However, suppose that $T(\ulcorner \kappa \urcorner )$ holds. Then, by correlation and modus ponens, $\kappa$ (i.e. $T(\ulcorner \kappa \urcorner )\rightarrow \chi$) holds, and so it seems that, by modus ponens, $\chi$ holds. Therefore, discharging the supposition, by the metaentailment of unipremise conditional proof (if $\varphi \vdash \psi$ holds, $\oslash \vdash \varphi \rightarrow \psi$ holds), $\kappa$ holds, and so, by correlation and modus ponens, $T(\ulcorner \kappa \urcorner )$ holds, and hence, by modus ponens, $\chi$ holds. The argument is apparently valid, the premises (correlation and the existence of a sentence such as $\kappa$) are apparently true, but the conclusion ($\chi$) is apparently false (to the best of my knowledge, the first clear version of the paradox was given by Juan de Celaya, though variations thereof (fn 51) appeared earlier in medieval times).

Russell’s paradox. For every way objects can be or can fail to be, there is a set of the objects that are that way, a set which thereby contains all and only those objects that are that way. Letting $\in$ express (set) containment, this is typically taken to entail the principle of comprehension according to which, for every formula $\varphi$, there is a set s (with ‘s’ ’s not occurring free in $\varphi$) such that $\forall \xi (\xi \in s \leftrightarrow \varphi )$ holds. Therefore, by comprehension, there is a set r such that $\forall x (x \in r \leftrightarrow \lnot (x\in x))$ holds, and so, by universal instantiation ($\forall \xi \varphi \vdash \varphi _{\tau /\xi }$), $r \in r \leftrightarrow \lnot (r\in r)$ holds. However, suppose that $r \in r$ holds. Then, by comprehension and modus ponens, $\lnot (r \in r)$ holds, and so it seems that, by adjunction, $r \in r \& \lnot (r \in r)$ holds. Suppose next that $\lnot (r \in r)$ holds. Then, by comprehension and modus ponens, $r \in r$ holds, and so it seems that, by adjunction, $r \in r \& \lnot (r \in r)$ holds. Therefore, discharging the suppositions, reasoning by cases, $r \in r \vee \lnot (r \in r)$—which, being the conclusion of the law of excluded middle, is a logical truth—entails $r \in r \& \lnot (r \in r)$—which, being the premise of the law of noncontradiction, is absurd. The argument is apparently valid, the premises (comprehension and $r \in r \vee \lnot (r \in r)$) are apparently true, but the conclusion ($r \in r \& \lnot (r \in r)$) is apparently false (to the best of my knowledge, the first clear version of the paradox was given by Russell, 1903, pp. 523–528).

The Sorites paradox. One does not stop being bald by the addition of one single hair. Keeping fixed the other dimensions of comparison relevant for baldness and letting B(i) express that a man with i hairs is bald, that entails the principle of tolerance according to which $B(i)\rightarrow B(i+1)$ holds. However, by tolerance, $B(1)\rightarrow B(2)$ holds, which, together with B(1), by modus ponens, entails that B(2) holds. Yet, by tolerance, $B(2)\rightarrow B(3)$ also holds, which, together with the previous lemma that B(2) holds, by modus ponens, entails that B(3) holds. With another 99, 997 structurally identical arguments, we reach the conclusion that B(100, 000) holds. It seems then that conclusion follows simply from tolerance and B(1). The argument is apparently valid, the premises (tolerance and B(1)) are apparently true, but the conclusion (B(100, 000)) is apparently false (to the best of my knowledge, the first clear version of the paradox was again given by Eubulides of Miletus).

The Preface paradox. One does not stop knowing something by the addition of one single known conjunct. Under suitable idealisations, that entails the principle of collection according to which, if one knows that P and that Q, one knows that [P and Q]. Moreover, let’s suppose that Greg only has the usual, nonentailing kind of evidence about history, and let’s take many, say 100,000, of his true epistemically best beliefs about it, where every belief is a belief in a proposition independent from the propositions that are the objects of the other beliefs (let these propositions be the propositions that $H_{1}$, that $H_{2}$, that $H_{3}$..., that $H_{100,000}$). Therefore, for every $1\le i\le 100,000$, we should plausibly accept that Greg knows that $H_{i}$, although he plausibly does not know that [$H_{1}$ and $H_{2}$ and $H_{3}$ ... and $H_{100,000}$]. However, by collection, if Greg knows that $H_1$ and that $H_2$, he knows that [$H_1$ and $H_2$], and so, since, by adjunction, the antecedent holds, by modus ponens he knows that [$H_1$ and $H_2$]. Yet, by collection, it is also the case that, if Greg knows that [$H_1$ and $H_2$] and that $H_3$, he knows that [$H_1$ and $H_2$ and $H_3$], and so, since, by the previous lemma and adjunction, the antecedent holds, by modus ponens he knows that [$H_1$ and $H_2$ and $H_3$]. With another 99, 997 structurally identical arguments, we reach the conclusion that Greg knows that [$H_{1}$ and $H_{2}$ and $H_{3}$ ... and $H_{100,000}$]. It seems then that that conclusion follows simply from collection and, for every $1\le i\le 100,000$, Greg’s knowledge that $H_i$. The argument is apparently valid, the premises (collection and, for every $1\le i\le 100,000$, Greg’s knowledge that $H_i$) are apparently true, but the conclusion (Greg’s knowledge that [$H_{1}$ and $H_{2}$ and $H_{3}$ ... and $H_{100,000}$]) is apparently false (to the best of my knowledge, the first clear version of the paradox was given by Makinson, 1965).

The Material-Implication paradox. ‘If Hellas Verona won the last Serie A, then they won exactly one Serie A’ is stronger than ‘Either it is not the case that Hellas Verona won the last Serie A or they won exactly one Serie A’ (for one thing, it is not the case that Hellas Verona won the last Serie A, and so, by the entailment of addition ($\varphi \vdash \varphi \vee \psi$ and $\psi \vdash \varphi \vee \psi$), ‘Either it is not the case that Hellas Verona won the last Serie A or they won exactly one Serie A’ holds, but ‘If Hellas Verona won the last Serie A, they won exactly one Serie A’ does not). However, suppose that either it is not the case that Hellas Verona won the last Serie A or they won exactly one Serie A and suppose further that Hellas Verona won the last Serie A. Then, by the entailment of disjunctive syllogism ($\varphi , \lnot \varphi \vee \psi \vdash \psi$), Hellas Verona won exactly one Serie A. Therefore, discharging the second supposition, by the metaentailment of multipremise conditional proof (if $\Gamma ,\varphi \vdash \psi$ holds, $\Gamma \vdash \varphi \rightarrow \psi$ holds), ‘Either it is not the case that Hellas Verona won the last Serie A or they won exactly one Serie A’ entails ‘If Hellas Verona won the last Serie A, they won exactly one Serie A’. The argument is apparently valid, the premise (‘Either it is not the case that Hellas Verona won the last Serie A or they won exactly one Serie A’) is apparently true, but the conclusion (‘If Hellas Verona won the last Serie A, they won exactly one Serie A’) is apparently false (to the best of my knowledge, the first clear version of the paradox was given by Faris, 1962, pp. 115–119).

There are many other paradoxes, but, as a matter of fact, these (possibly save for the Preface paradox) are the historically most salient ones where the correctness of classical logic has most severely been put into question.⁵ It is natural to group paradoxes in general in (natural) kinds, and ask whether in particular some of the paradoxes in our list are of the same kind. But, to discuss better this question and surrounding issues, we first need to understand better what it is for two paradoxes to be of the same kind. The traditional definition of paradoxicality would seem right in appealing to a subjective element (signalled by ‘apparently’), and etymology (Ancient Greek para doxan, beyond belief) concurs in revealing a perception of paradox as a moment of disconnection between appearance and reality. Paradoxicality consists in a certain general type of mistake (in the sense of an appearance [of something] which fails to hold).⁶ It is then natural to assume that two paradoxes are of the same kind iff they make the same kind of mistake, where, the notion of mistake plausibly having an aetiological element, sameness of kind of mistake implies sameness of kind of the cause that brings about the mistake, where in turn, at least for the philosophical purposes of this introduction, the most relevant cause can be taken to be the cause of the fact that is mistakenly represented (rather than e.g. the cause of the mistaken representation).⁷$^{,}$⁸ Notice that this natural conception straightforwardly implies that two paradoxes are of the same kind iff they have the same kind of solution:⁹ for they are of the same kind iff they make the same kind of mistake, where the latter is the case iff, in them, the same kind of fact having the same kind of cause is mistakenly represented by the same kind of representation—but the solution to a paradox is exactly the elimination of its mistaken representation (by replacing it with one that corresponds to the relevant fact) accompanied by an explanation of why it is mistaken (by individuating the cause for why what obtains is the relevant fact rather than the one the mistaken representation would correspond to), where both elimination and explanation are of the same kind iff the mistake is, in the way just spelt out, of the same kind. Relatedly, notice that, on this natural conception, while it is of course possible to know that two paradoxes are of the same kind without knowing what kind their solution is, it is not possible to know what kind a paradox is without knowing what kind its solution is, since what kind a paradox is consists in what kind of mistake it makes (which determines what kind of elimination of its mistaken representation its solution includes), which in turn depends on what kind of cause brings about the mistake (which determines what kind of explanation of why the representation is mistaken its solution includes).

Observe then that it is reasonable to suppose that at least some of the paradoxes in our list are of the same kind. This is immensely plausible for the Liar paradox and Curry’s paradox, as they both rely in the same way on the appearance of correlation plus selfreference (i.e. to make it appear that a proposition about truth is equivalent with a logical function of itself—in particular, with its own negation or with its own implication to a proposition respectively—in such a way that the logical interaction of the proposition with its logical function gives rise to an entailment that justifies the logical function)—it would then be immensely puzzling if, in the two cases, mistakes of two different kinds were linked with that same general type of appearance. Indeed, it is appealing to see a negation $\lnot \varphi$ as a special kind of implication (implication from $\varphi$ to the absurdity constant ${\mathsf {f}}$) or, vice versa, see an implication $\varphi \rightarrow \psi$ as a generalised kind of negation (which “$\psi$ises” $\varphi$ just as the negation $\lnot \varphi$ “${\mathsf {f}}$ises”—i.e. absurdises—$\varphi$).¹⁰ It is then possible to give very natural versions of the Liar paradox and Curry’s paradox that are totally analogous (see Zardini, 2015a for the details)—it would then be even more puzzling if, in the two cases, the mistakes occurred at different steps of the paradox, or if, even though occurring at the same step, they were nevertheless of two different kinds. It should be remarked though that this sameness of kind has recently been contested (by Priest, 1994; see Zardini, 2015a, p. 489, fn 43; Oms & Zardini, 2021, pp. 202–204 for some critical discussion).

It is also plausible that the Liar paradox and Russell’s paradox are of the same kind, since correlation and comprehension would seem supported by the same kind of idea (i.e. to connect a mundane state of affairs such as snow’s being white with a semantic state of affairs such as the sentence ‘Snow is white’ ’s being true or with a set-theoretic one such as the set of white objects containing snow, both in the ascending direction and in the descending direction) and since the Liar paradox and Russell’s paradox both rely in the same way on the appearance of correlation and comprehension respectively plus selfreference (i.e. to make it appear that a proposition about truth and containment respectively is equivalent with a logical function of itself—in particular, with its own negation in both cases—in such a way that the logical interaction of the proposition with its logical function gives rise to an entailment that justifies the logical function)—it would then be puzzling if, in the two cases, mistakes of two different kinds were linked with that same general type of appearance. Indeed, the Grelling-Nelson paradox can be got from Russell’s paradox by replacing sets with predicates and containing with being-true-of (see Grelling & Nelson, 1908 for the details). Presumably, the Liar paradox and the Grelling-Nelson paradox are of the same kind, but the Grelling-Nelson paradox and Russell’s paradox are totally analogous—it would then be even more puzzling if, in the latter two cases, the mistakes occurred at different steps of the paradox, or if, even though occurring at the same step, they were nevertheless of two different kinds.¹¹ It should be remarked though that this sameness of kind has long been contested (since Peano, 1906, p. 157 and then influentially by Ramsey, 1925, pp. 352–354; see Priest, 2003, pp. 155–157 for some critical discussion).

Using ‘semantic paradoxes’ as a label for the kind of paradox instantiated by the Liar paradox and Curry’s paradox (and ‘set-theoretic paradoxes’ as a label for the kind of paradox instantiated by Russell’s paradox), it also pays to ask what the cause of the mistake in the semantic paradoxes (and in the set-theoretic paradoxes, if these are the same kind as the semantic paradoxes) is. A popular diagnosis is that it is some sort of selfreference (Poincaré, 1906; however, the essence of the idea goes at least as back as Richard, 1905).¹² In one direction, such a diagnosis faces the challenge that there are selfreferential expressions that, even if expressing semantic properties, would seem unproblematic (e.g. ‘This sentence refers to itself’). In the other direction, such a diagnosis faces the challenge that there are semantic paradoxes that would seem not to involve any kind of selfreference, as in Yablo’s paradox (see Yablo, 1985, p. 340 for the details).¹³

Both these challenges motivate a different, equally popular diagnosis, according to which the cause of the mistake in the semantic paradoxes is some sort of ungroundedness (Herzberger, 1970; however, the essence of the idea goes at least as back as Langford, 1937). To exemplify with the paradigmatic case of truth, a sentence is grounded iff its truth or falsity can ultimately be traced back to the obtaining or not of nonsemantic facts by applications of the principle that the truth or falsity of ‘P’ depends on whether P. It’s easy to see that both the Liar sentence and the Yablo sentences generate a nonwellfounded dependence chain (circular in the former case and infinitely descending in the latter case) and are thus ungrounded. In one direction, such a diagnosis faces the challenge that there are ungrounded expressions that would seem unproblematic (e.g. ‘Every sentence is such that, if it is true, it is true’).¹⁴ In the other direction, such a diagnosis faces the challenge that there are semantic paradoxes that would seem not to involve any kind of ungroundedness. Consider, for example, a Curry sentence $\kappa '$ whose consequent is unproblematically true (say, ‘EZ likes pastissada’). Assuming that the truth of the consequent suffices for the truth of an implication,¹⁵$\kappa '$ would seem grounded, yet it gives rise to a version of Curry’s paradox: while ‘EZ likes pastissada’ is unproblematically true, it should not be provable by Curry-style reasoning (Zardini, 2015a, pp. 469–485 makes essentially the same point against views that postulate indeterminacy (Bočvar, 1938) or overdeterminacy (Priest, 1979) as an essential ingredient—let alone cause—of a semantic paradox). It therefore remains very much an open question what the cause of the mistake in the semantic paradoxes is (see Zardini, 2015a, p. 492; 2019a for a proposal that has at least the merit of being able to meet the challenges presented in this and the last paragraph).

Moving on to the other paradoxes in our list, it is much less plausible that the Sorites paradox is of the same kind as the semantic paradoxes: the crucial appearance in the former case would seem one to the effect that a big change comes about only through a series of small changes, whereas the crucial appearance in the latter case would seem one to the effect that a proposition is equivalent to a logical function of itself—it would then be puzzling if, in the two cases, the same kind of mistake were linked with such two different general types of appearances (however, see e.g. Priest, 2010 for a recent argument in favour of sameness of kind and Oms & Zardini, 2021 for a criticism).¹⁶ Similar considerations apply to the relation between the Preface paradox (where the crucial appearance would seem one to the effect that ignorance comes about only through a series of additions of known conjuncts) and the semantic paradoxes; it is however plausible that the former is of the same kind as the Sorites paradox (see my contribution to this volume for an argument in favour of sameness of kind). Moreover, it is not at all plausible that the Material-Implication paradox is of the same kind as the semantic paradoxes: the crucial appearance in the former case would seem one to the effect that a disjunction generates an implicational link from the negation of one of its disjuncts to its other disjunct—it would then be very puzzling if, in the two cases, the same kind of mistake were linked with such two different general types of appearances. Finally, it is not at all plausible that the Material-Implication paradox is of the same kind as the Sorites paradox—again, it would be very puzzling if, in the two cases, the same kind of mistake were linked with such two different general types of appearances.

Notice that already the less-than-paradigmatic $\kappa '$-version of Curry’s paradox belies the traditional definition of paradoxicality, since that is a situation where apparently true premises apparently entail an apparently true conclusion (López de Sa & Zardini, 2007, p. 246). Indeed, analogous points can be made for all the other paradoxes in our list. For the Liar paradox, consider the Liar sentence ‘This sentence is not true or EZ likes pastissada’ and the fact that Liar-style reasoning licences the true conclusion that EZ likes pastissada (mutatis mutandis for Russell’s paradox). For the Sorites paradox, consider a soritical series for baldness starting with Yul Brynner (in his fifties) and the fact that Sorites-style reasoning licences the true conclusion that Sean Connery (in his fifties) is bald (Oms & Zardini, 2019, p. 8, fn 14). For the Preface paradox, consider Greg’s situation and the fact that Preface-style reasoning licences the true conclusion that Greg knows that [$H_1$ and $H_2$ and $H_3$ $\ldots$ and $H_{10}$]. For the Material-Implication paradox, consider the circumstance that it is not the case that Hellas Verona won the last Serie A and the fact that [Material-Implication]-style reasoning licences the true conclusion that, if Hellas Verona won the last Serie A, they won at least one Serie A. The traditional definition breaks down across the board.¹⁷

Focusing now on the paradigmatic example represented by Curry’s paradox, one natural reaction to the point of the last paragraph is to say that what is paradoxical in Curry’s paradox is not that apparently true premises apparently entail an apparently false conclusion, but that apparently a priori (known, or knowable, or possibly justifiable etc.) premises apparently entail a conclusion that is apparently not a priori. However, even setting aside the fact that e.g. someone going through a transcendental proof of the external world based on the existence of a certain kind of content is almost invariably not subject to paradox-generating appearances, a version of Curry’s paradox with an a priori consequent such as e.g. Fermat’s Last Theorem belies this epistemological modification of the traditional definition of paradoxicality (which also would seem unable to cope with the point of fn 17).¹⁸ Another natural reaction to the point of the last paragraph is to say that what is paradoxical in Curry’s paradox is not that apparently true premises apparently entail an apparently false conclusion, but that everything (if anything) in the relevant range of propositions (possibly together with apparently true auxiliary premises) apparently entails everything (if anything) in the relevant range of propositions.¹⁹ Now, what is implicit behind this move is presumably that it is bad to entail everything, in turn presumably because not everything holds. If so, on this move, the paradoxicality of Curry’s paradox depends on the fact that not everything holds. However, the remarkable fact is that, even under the assumption of trivialism (the claim that everything holds, see Priest, 2000), Curry’s paradox is still paradoxical (another remarkable fact that can be used to the same effect is the one mentioned in the parenthetical observation at the end of fn 18).²⁰ Notice also that a shortcoming common to both this latter reaction and the modification of the former reaction discussed in fn 18—when one considers how they can be generalised beyond the case of Curry’s paradox—is that of failing to characterise correctly—of all things!—an archparadigmatic case of paradox such as the Liar paradox: while it is true that the paradox can be extended by applying ex contradictione quodlibet to conclude everything, paradox is hit as soon as contradiction is (Zardini, 2021d).

A better reaction to the point of the second last paragraph is to say that what is paradoxical in Curry’s paradox is that, apparently, even if the conclusion failed to hold,²¹all the elements of the putative proof of the conclusion would still be available. In other (more precise) words, apparently, even if the conclusion failed to hold, the premises would²²still be true and the argument would still be valid. Since it is known full well that one instance of the argument in question (fn 3) is the one featuring the conclusion and the premises in question, it is presumably known full well that that counterfactual implication fails to hold; yet, it does appear to hold, wherein lies the paradox.²³ For example, if the conclusion is ‘EZ likes pastissada’, what is paradoxical in the resulting version of Curry’s paradox is that, apparently, even if ‘EZ likes pastissada’ failed to hold, all the elements (correlation, the existence of the relevant Curry sentence, modus ponens, unipremise conditional proof etc.) of the putative proof of ‘EZ likes pastissada’ would still be available.

It’s important to realise that the proposed account relies on an understanding of counterfactual implication that does very substantial work. For we must be able to conceive of a situation where, apparently, the conclusion fails to hold, and so a situation where, apparently, also all unproblematic ways of getting to the conclusion involve either a premise or a logical principle that fails to hold (which, in the cases where the conclusion is a logical truth, makes the situation an apparently [impossible and indeed counterlogical] one). Moreover, we must be able to discriminate such unproblematic ways from the problematic one constituted by the putative proof, in such a way that we’re able so to conceive of the situation in question that, apparently, every premise and logical principle involved in the putative proof holds in the situation (which, given that one instance of the argument in question is the one featuring the conclusion in question, again makes the situation an apparently [impossible and indeed counterlogical] one).²⁴ Therefore, the proposed account is correct (if it is) because our ability to judge the crucial counterfactuals tracks our implicit grasp of the distinction between unproblematic and problematic ways of getting to a conclusion;²⁵ the proposed account is also illuminating (if it is) because our ability to judge the crucial counterfactuals at the same time (virtuously circularly) articulates our implicit grasp of the distinction between unproblematic and problematic ways of getting to a conclusion.

It is appealing to extend the proposed account from the particular case of Curry’s paradox to paradoxicality in general: a paradox is a situation where, apparently, even if the conclusion failed to hold, all the elements of the putative proof of the conclusion would still be available.²⁶ The proposed account of paradoxicality then fits nicely with the idea that paradoxicality consists in a certain general type of mistake, by providing a specification of exactly what that general type is: we’re mistakenly led to judge that, even if the conclusion failed to hold, all the elements of the putative proof of the conclusion would still be available. Fixing on a specific presentation of a paradox, this general type of mistake might be determined by the fact that, while we correctly judge that all the elements of the putative proof of the conclusion are available, we’re mistakenly led to judge that, even if the conclusion failed to hold, they would still be so, in which case we will after all have to come to terms with the fact that the premises really do entail the conclusion (so that—recalling from fn 22 that it is inessential that the elements of the putative proof include true premises—we must reject²⁷ one of the premises or accept the conclusion); alternatively (and more relevantly in the case of substructural approaches to paradox), the general type of mistake might be determined by the fact that we’re mistakenly led to judge that all the elements of the putative proof of the conclusion are available in the first place,²⁸ in which case—recalling again from fn 22 that it is inessential that the elements of the putative proof include true premises—we did after all rightly sense that the premises really do not entail the conclusion (so that we can accept the premises and reject the conclusion, and accept that, even if the conclusion failed to hold, all the valid elements in the vicinity of the elements of the invalid proof would still be available).²⁹ (Further, in either case, those still general types of mistake will in turn be determined by more specific types that are those that constitute the different kinds of paradox that have been discussed in this section.) The proposed account also fits nicely with the gloss on paradoxicality that (with Dan López de Sa) I gave in earlier works (López de Sa & Zardini, 2007, p. 246; 2011, pp. 472–473) to the effect that, in a paradox, despite the apparent validity of the argument, the premises apparently do not rationally support the conclusion, since the appearance of that epistemological fact is plausibly explained by the appearance of the metaphysical fact that, even if the conclusion failed to hold, the premises would still be true and the argument would still be valid.³⁰

3 Substructurality

A logic is structural iff it includes all the structural principles of classical logic, substructural otherwise. In turn, a principle is structural iff it does not concern particular object-language expressions.³¹ For example, adjunction is not structural in that it concerns the particular object-language expression &, whereas the metaentailment of contraction (see below) is structural in that it only concerns the metalanguage expression ‘, ’. There are infinitely many structural principles of classical logic (for example, for every i, consider the principle that there is an entailment with exactly i premises), but the most salient ones (in general but also for this introduction and this volume) are the entailment of reflexivity:

$\mathrm {(I)}$:

$\varphi \vdash \varphi$,

the metaentailment of monotonicity:

$\mathrm {(K)}$:

If $\Gamma _0 \vdash \Delta$ holds, $\Gamma _1,\Gamma _0 \vdash \Delta$ holds, and, if $\Gamma \vdash \Delta _0$ holds, $\Gamma \vdash \Delta _1,\Delta _0$ holds,

the metaentailment of transitivity:

$\mathrm {(S)}$:

If $\Gamma _0 \vdash \Delta _0,\varphi$ holds and $\Gamma _1, \varphi \vdash \Delta _1$ holds, $\Gamma _1,\Gamma _0 \vdash \Delta _0,\Delta _1$ holds,

the metaentailment of contraction:

$\mathrm {(W)}$:

If $\Gamma ,\varphi ,\varphi \vdash \Delta$ holds, $\Gamma , \varphi \vdash \Delta$ holds, and, if $\Gamma \vdash \Delta ,\varphi ,\varphi$ holds, $\Gamma \vdash \Delta ,\varphi$ holds

and the metaentailment of commutativity:

$\mathrm {(C)}$:

If $\Gamma _0,\varphi ,\psi ,\Gamma _1 \vdash \Delta$ holds, $\Gamma _0,\psi ,\varphi ,\Gamma _1 \vdash \Delta$ holds, and, if $\Gamma \vdash \Delta _0,\varphi , \psi ,\Delta _1$ holds, $\Gamma \vdash \Delta _0, \psi ,\varphi ,\Delta _1$ holds³²

(see Zardini, 2018, pp. 242–247 for a brief overview, for each of these principles, of the main philosophical reasons for denying the principle, whether or not those are related to the paradoxes in our list). The first three principles can be summed up by saying that logical consequence corresponds to a Tarski-Scott closure operation (Tarski, 1930; Scott, 1974), and, under extremely minimal assumptions, the last two by saying that premises and conclusions can be represented as forming sets, so that, restricting to the principles in our list, a logic is in effect structural iff it corresponds to a Tarski-Scott closure operation with sets of premises and conclusions, substructural otherwise.

I should stress that I’m adopting a literal—and so in a certain respect rather strict—understanding of what it takes for one of the principles in our list not to hold in a logic. In particular, let the implicational analogue of a metaentailment be the entailment where the premises and conclusion are got by replacing [the relevant entailment $\varphi _0,\varphi _1,\varphi _2\ldots , \varphi _i\vdash \psi _0,\psi _1,\psi _2\ldots , \psi _j$ in the metaentailment] with $\varphi _0 \rightarrow (\varphi _1 \rightarrow (\varphi _2\ldots \rightarrow (\varphi _i\rightarrow (\lnot \psi _0\rightarrow (\lnot \psi _1 \rightarrow (\lnot \psi _2\ldots \rightarrow \psi _j)))\ldots )$ (so that, for example, the implicational analogue of the metaentailment from $\varphi \vdash \psi$ to $\chi ,\varphi \vdash \psi$ is the entailment $\varphi \rightarrow \psi \vdash \chi \rightarrow (\varphi \rightarrow \psi )$). Then, quite a few logics star an implication for which, for some principles in our list, the implicational analogues of those principles do not hold while their consequence relation is so defined that the principles themselves do (literally) hold (for example, already in the 3-valued Łukasiewicz logic (Łukasiewicz, 1920), $\varphi \rightarrow (\varphi \rightarrow \psi )\vdash \varphi \rightarrow \psi$ does not hold while its consequence relation is so defined that (literally), if $\varphi ,\varphi \vdash \psi$ holds, $\varphi \vdash \psi$ holds). By my count, then, that is simply not a substructural logic (although one could of course use its materials to define a substructural logic in a natural way, for example by setting $\varphi _0,\varphi _1,\varphi _2\ldots , \varphi _i\vdash \psi _0,\psi _1,\psi _2\ldots , \psi _j$ to hold in the new substructural logic iff $\varphi _0 \rightarrow (\varphi _1 \rightarrow (\varphi _2\ldots \rightarrow (\varphi _i\rightarrow (\lnot \psi _0\rightarrow (\lnot \psi _1 \rightarrow (\lnot \psi _2\ldots \rightarrow \psi _{j})))\ldots )$ is a logical truth in the old structural one). In itself, that is of course a terminological point, but I’m making it because I want to focus on those logics that say something peculiar not simply about the rather lofty topic of (typically embedded) implications (we already had e.g. conditional logics (e.g. Nute, 1984) for that), but also about the much more down-to-earth topics of the combination of premises, the combination of conclusions and logical consequence.³³

An early indication of the superficiality of the notion of a substructural logic is given by how carefully phrased the characterisation given in the second last paragraph must be. As a first stab at defining substructurality, we could say that a principle is structural iff it does not concern particular logical operations, but that would be too relaxed, since e.g. identity is not an operation (i.e. a function whose codomain is identical with its domain) but the law of indiscernibility of identicals, which only concerns identity, is not structural.³⁴ As a reaction to that, we could say that a principle is structural iff it does not concern particular logical notions, but that would still be too relaxed, since e.g. the properties of being a bachelor and of being unmarried are not logical but the entailment from ‘x is a bachelor’ to ‘x is unmarried’, which only concerns the properties of being a bachelor and of being unmarried, is not structural. As a reaction to that, we could say that a principle is structural iff it does not concern particular notions, but that would now be too strict, since e.g. premise combination is a particular notion but (K), which concerns premise combination, is structural. As a reaction to that, we could say that a principle is structural iff it does not concern particular notions expressed by object-language expressions, but that would still be too strict, since e.g. the object language may contain an expression expressing premise combination but (K), which concerns premise combination, is structural. As a reaction to that, we could say (and have said) that a principle is structural iff it does not concern particular object-language expressions, but even that is not too strict only if ‘, ’ is not an object-language expression, with the result that there is a philosophically interesting distinction between structural and nonstructural principles only insofar as there is a philosophically interesting sense in which ‘, ’ is not an “object-language expression”. And that is in turn questionable on at least two counts. Firstly, ‘, ’ is certainly not part of the usual informal metalanguage employed in theorising about a logic—rather, it is part and parcel of the symbolism that is being theorised about, both at the semantic and at the proof-theoretic level, which makes it in a very natural sense an “object-language expression”. True, it does not come up in the definition of a well-formed formula, but that hardly marks a watershed of philosophical interest. Secondly, there are logics where ‘, ’ is an “object-language expression” even in the pedantic sense of coming up in the definition of a well-formed formula (von Kutschera, 1968), which then presumably makes the question whether a principle is structural logic-relative (and makes even, say, adjunction structural relative to some weirdly formulated logic where, analogously to the role played by ‘, ’ in a standardly formulated logic, & (alongside ‘, ’) is used to build up entailments but does not come up in the definition of a well-formed formula), thereby robbing it of much of its philosophical interest.

Even setting aside all these niceties, it is doubtful that anything like the notion of a structural principle and the subsequent notion of a substructural logic can mark a logically or philosophically interesting distinction. To take two paradigmatic examples, closure under uniform substitution (if $\Gamma _0 \vdash \Delta _0$ holds and $\Gamma _1 \vdash \Delta _1$ results from it by a uniform substitution, $\Gamma _1 \vdash \Delta _1$ holds) and compactness (if $\Gamma _0 \vdash \Delta _0$ holds, there are finite subcollections $\Gamma _1$ and $\Delta _1$ of $\Gamma _0$ and $\Delta _0$ respectively such that $\Gamma _1 \vdash \Delta _1$ holds) are presumably “structural principles” on any reasonable understanding of that notion. Moreover, they are usually supposed to be principles “of” “classical logic” and, in our context, the grounds for rejecting either part of that supposition are flimsy. Taking them in reverse order, could “classical logic” be taken to be something like standard second-order classical logic (which is not compact) rather than simply something like first-order classical logic? Hardly so: on a natural understanding of what it is to be a principle, very few logics—which do not include many logics paradigmatically considered to be structural, such as e.g. propositional classical logic—have all the structural principles of something like standard second-order classical logic (for example, think of the principle that, if a subset s of the consequence relation is recursively enumerable, for some $\Gamma$ and $\Delta$ it is the case that $\Gamma \vdash \Delta$ holds and does not belong to s). Further, could a principle “of” a logic be taken to be something like a principle constitutive (fn 33) of the logic (where presentations of classical logic usually do not feature closure under uniform substitution and compactness among its constitutive principles) rather than simply something like a principle holding in the logic? Hardly so: on many natural presentations of classical logic, very few principles—which do not include many principles paradigmatically considered to be structural, such as e.g. (W)—are constitutive of it (for example, think of a presentation of classical logic in terms of sets of premises and conclusions). It is therefore very hard to see how, in our context, closure under uniform substitution and compactness could fail to count as “structural principles of classical logic”, with the consequence that every logic that is either not closed under uniform substitution or not compact will count as substructural. That lumps together, say, Carnap-style modal logic (Carnap, 1946), infinitary logic (Henkin, 1955) and linear logic (Girard, 1987) as substructural logics. What kind of logical or philosophical insight can one expect to gain by thinking about such an unsavoury congeries of logics? It would seem more sensible to focus on a more unified, proper subset of structural principles (as is in effect done in research on substructural logics), like e.g. those principles concerning standard algebraic properties of premise combination and conclusion combination.³⁵

It is tempting to think that structural principles are located at a level different from and indeed prior to that of nonstructural principles (and to speculate that the advantages of substructural approaches to paradox that will be presented in Sect. 4flow from that putative fact), and, in fact, structural principles are commonly so thought of (and even occasionally so speculated about, see e.g. Ripley, 2015a, p. 310). For example, it is common to think that premise combination—the level at which a structural principle such as e.g. (K) operates—is different from and indeed prior to conjunction, in the sense that the logical properties of premise combination are constituted independently of those of conjunction and indeed help to constitute those of conjunction. However, a little reflection suffices to show how problematic the common thought is for virtually all the logics of interest for this introduction, and that, quite to the contrary, in these logics, it is conjunction that is prior to premise combination (in fact, premise combination consists in a certain kind of conjunction).³⁶

In spite of its usual representation as set (or multiset, or sequence, or whatnot, see fn 4) formation, premise combination as usually understood³⁷ is much more than that, as e.g.  $\varphi ,\psi$ has the force of representing $\varphi$ and $\psi$ as holding together (by contrast, possibly apart from irrelevant and obvious containment facts, $\{\varphi ,\psi \}$ neither represents anything as holding nor does it represent any things as doing something together). But that is exactly the kind of combination performed by the logical operation of conjunction, for ‘ ‘$\varphi$’ holds and ‘$\psi$’ holds’ has exactly the force of representing $\varphi$ and $\psi$ as holding together. The combination of premises $\varphi$ and $\psi$ consists in the conjunction that $\varphi$ holds and $\psi$ holds.³⁸ It might be tempting to reply to this point by going expressivist and say that to accept $\varphi ,\psi$ is simply to accept $\varphi$ and to accept $\psi$. However, since $\varphi ,\psi$ precisely represents $\varphi$ and $\psi$ as holding together, to accept $\varphi ,\psi$ must be equivalent with accepting that $\varphi$ and $\psi$ hold together, given which the expressivist temptation falls afoul of the fact that one may justifiedly accept $\varphi$ and justifiedly accept $\psi$ while justifiedly not accepting that $\varphi$ and $\psi$ hold together (for example, one may justifiedly accept that Benfica will win the next Portuguese Liga and justifiedly accept that Atlético will win the next Spanish Liga while justifiedly not accepting that Benfica will win the next Portuguese Liga together with Atlético’s winning the next Spanish Liga). Moreover, most of our thought about premise combination is in the context of thought about entailment and does not involve acceptance of $\varphi ,\psi$—the expressivist temptation is therefore subject to a particularly acute version of the Frege-Geach problem (Geach, 1965).³⁹

Notice that the conjunction that the premise combination $\varphi ,\psi$ consists in is not ‘$\varphi$ and $\psi$’ but ‘ ‘$\varphi$’ holds and ‘$\psi$’ holds’. This is crucial for the entailment $\varphi , \psi \vdash \varphi \& \psi$ to be correctly rendered as an entailment connecting $\varphi$’s holding and $\psi$’s holding with $\varphi \& \psi$’s holding instead of being incorrectly rendered as an entailment connecting (letting $\varphi$ mean that P and $\psi$ mean that Q) its being the case that P and Q with its being the case that P and Q or as an entailment connecting ‘$\varphi$ and $\psi$’ (synonymous with $\varphi \& \psi$) holding with $\varphi \& \psi$’s holding (Zardini, 2018, pp. 257–258).⁴⁰ Relatedly, notice also that this treatment does not assume that, when one competently infers from $\varphi$ and $\psi$, one must grasp what $\varphi ,\psi$ represents. Presumably, one can competently infer without grasping any sort of metalinguistic concept, or even without grasping any sort of conjunctive concept. In such cases, one displays a sensitivity to the state of affairs represented by $\varphi ,\psi$ without being able to grasp that state of affairs, and we should long ago have learnt to accommodate for this kind of sensitivity on account of the fact that one (think of higher-level animals and small children) can competently infer from $\varphi$ to $\psi$ without grasping any sort of metalinguistic concept (such as those of sentence and of entailment), or even without grasping any sort of implicational concept (such as the one expressed in ‘If $\varphi$, then $\psi$’).

The master argument in favour [of the claim that premise combination consists in a certain kind of conjunction] advanced in the second last paragraph is reinforced by three auxiliary arguments. A first auxiliary argument takes its lead from the fact that there is really nothing in the most general notion of premise combination that requires its usual understanding. The point is perhaps most direct in the dual case of conclusion combination. How should one understand the combined conclusions $\varphi ,\psi$? At such a level of abstraction, there is simply no (right) answer to this (wrong) question: $\varphi ,\psi$ could represent $\varphi$ and $\psi$ as holding alternatively (as is usually understood) or as holding together (as could equally naturally be understood) or goodness knows. There are therefore different kinds of conclusion combinations, and the most natural way of grounding their difference involves the corresponding logical operations: on such explanation, the first option is tantamount to what is expressed by ‘ ‘$\varphi$’ holds or ‘$\psi$’ holds’—thereby making conclusion combination a certain kind of disjunction—while (as we’ve already seen in the second last paragraph) the second option is tantamount to what is expressed by ‘ ‘$\varphi$’ holds and ‘$\psi$’ holds’—thereby making conclusion combination a certain kind of conjunction. The same point can then be made in the case of premise combination (see Zardini, 2021d for a style of presentation of a logic where both premise combination and conclusion combination can go both in conjunctive mode and in disjunctive mode, with arbitrary embeddings of one mode into the other one).⁴¹$^{,}$⁴²

A second auxiliary argument takes its lead from the fact that, in many nonclassical and substructural logics just as well as in classical logic, conjunction is fully intersubstitutable with premise combination, in the sense that $\Gamma _0, \varphi , \psi ,\Gamma _1 \vdash \Delta$ holds iff $\Gamma _0, \varphi \& \psi , \Gamma _1 \vdash \Delta$ holds (see Zardini, 2021d for a style of presentation of a logic that builds in that principle). Such full intersubstitutability would be a mystery if conjunction were not prior to premise combination—for, in that case, how could it be that one of the fundamental logical operations so perfectly matches independently constituted premise combination? One possible explanation would be given by the assumption (made e.g. by Beall & Ripley, 2018, p. 751) that conjunction expresses in the object language premise combination. However, it is not clear how such an assumption could be correct. Firstly, why should one of the fundamental logical operations be there to express premise combination—that would seem to make that logical operation capriciously redundant and make elementary logic weirdly reflexive. Secondly, it would not seem that a straightforward object-language sentence such as e.g. ‘Snow is white and grass is green’ represents what ‘Snow is white’, ‘Grass is green’ does: for one thing, the latter—whatever it exactly represents—is arguably about the sentences ‘Snow is white’ and ‘Grass is green’ (fn 40), whereas ‘Snow is white and grass is green’ is definitely not. Thirdly, the point made in the second last paragraph applies also to the assumption in question: if conjunction expressed premise combination, the entailment $\varphi \& \psi \vdash \varphi \& \psi$ would correspond to the entailment $\varphi , \psi \vdash \varphi \& \psi$, which it does not. Fourthly, a relative of the point made in fn 38 applies to the assumption in question: if conjunction expressed premise combination, it would be a mystery how a conjunction could be entertained independently of any possible inference that may be drawn from it—but that is most definitely not a mystery. Contrary to the formidable explanatory challenges thus faced by the assumption that conjunction is not prior to premise combination, the opposite assumption that premise combination consists in a certain kind of conjunction has an easy job at accounting for the full-intersubstitutability fact: that fact obtains precisely because premise combination consists in a certain kind of conjunction (and because ‘ ‘$\varphi$’ holds’ is fully intersubstitutable with $\varphi$)⁴³.

A third auxiliary argument takes its lead from the fact that many substructural logics enjoy a very natural model-theoretic semantics, and in such semantics premise combination is just defined in terms of conjunction, along the lines of something to the effect that $\varphi ,\psi \vdash \chi$ holds iff every model where $\varphi \& \psi$ has a designated value is a model where $\chi$ has a designated value. Given the naturalness of the semantics, that is strong evidence that premise combination consists in a certain kind of conjunction.⁴⁴

There are therefore strong reasons for thinking that premise combination consists in a certain kind of conjunction, and analogous reasons are available for thinking that conclusion combination consists in a certain kind of disjunction and that entailment consists in a certain kind of implication. I emphasise that such reasons rely on assumptions about the target logic that, while almost always unquestionable for virtually all the logics of interest for this introduction (and for many structural logics including classical logic),⁴⁵ might not be such for other (substructural or structural) logics, and the following conclusions about “logics” should accordingly be understood as implicitly so qualified. One can then understand the fact that certain structural principles hold or do not hold in a logic as the result of the fact that the corresponding principles for conjunction, disjunction or implication (specifically, the particular conjunction that underlies premise combination, the particular disjunction that underlies conclusion combination and the particular implication that underlies entailment) hold or do not hold in the logic. For one example, (K) holds in classical logic but not in nonmonotonic logics because $\varphi \& \psi$ entails $\varphi$ in classical logic but not in nonmonotonic logics (in the latter case, understanding & as expressing the particular conjunction that underlies premise combination in nonmonotonic logics). For another example, (S) holds in classical logic but not in nontransitive logics because $\varphi \rightarrow \psi$ and $\psi \rightarrow \chi$ entail $\varphi \rightarrow \chi$ in classical logic but not in nontransitive logics (in the latter case, understanding $\rightarrow$ as expressing the particular implication that underlies entailment in nontransitive logics)⁴⁶. For yet another example, (W) holds in classical logic but not in noncontractive logics because $\varphi$ entails $\varphi \& \varphi$ in classical logic but not in noncontractive logics (in the latter case, understanding & as expressing the particular conjunction that underlies premise combination in noncontractive logics).

Assuming that this is right, it implies the need for a reconceptualisation of substructural logics, not as logics that fundamentally deny some structural principle of classical logic, but as logics that fundamentally deny some principle of a certain specific kind that conjunction, disjunction or implication obey in classical logic—that is, the kind of principles that determine that classical logic has the structural principles it has. That arguably does make substructural logics less categorically different from structural nonclassical logics than is commonly assumed: they all fundamentally deny some principle of the logical operations. The real difference is in that they centre on logical operations other than negation.

Indeed, they typically do not centre on implication either and thus centre on logical operations (i.e. conjunction and disjunction) all of whose argument places are upwards monotonic (where, given an iary logical operation $\circ$, for every $j\le i$, its jth argument place (as occupied by $\varphi _{j}$ in $\circ (\varphi _0, \varphi _1,\varphi _2\ldots ,\varphi _{j}\ldots , \varphi _{i})$) is upwards monotonic iff, if $\psi$ is at least as strong as $\chi$, $\circ (\varphi _0, \varphi _1,\varphi _2\ldots ,\psi \ldots , \varphi _{i})$ entails $\circ (\varphi _0, \varphi _1,\varphi _2\ldots ,\chi \ldots , \varphi _{i})$).⁴⁷ That is clear for nonmonotonic, noncontractive and noncommutative logics. The situation is more nuanced for nontransitive logics. Under the assumption that the entailment-underlying implication of a nontransitive logic is reducible in the usual fashion to disjunction and negation,⁴⁸ (S) in its basic form without side premises and side conclusions ultimately boils down to the entailment from $\lnot \varphi \vee \psi$ and $\lnot \psi \vee \chi$ to $\lnot \varphi \vee \chi$ (see e.g. Weir, 2015 for a logic where (S) does not hold but its basic form does and Zardini, 2021b for its critical discussion). In turn, given those premises, a suitable version of the principle of selection of conjunction over disjunction (licensing the entailment $\varphi _0 \& (\varphi _1 \vee \varphi _2)\vdash (\varphi _0 \& \varphi _1)\vee \varphi _2$⁴⁹ also in the scope of a disjunction) yields $((\lnot \varphi \vee \psi ) \& \lnot \psi ) \vee \chi$, given which selection [of conjunction over disjunction] as applied to the first disjunct yields $\lnot \varphi \vee (\psi \& \lnot \psi ) \vee \chi$. If the second disjunct can be ruled out by a suitable version of the principle of noncontradiction, that yields $\lnot \varphi \vee \chi$, thereby verifying (S) in its basic form. Focusing on nontransitive logics that satisfy the reducibility assumption concerning their entailment-underlying implication, this analysis makes it clear that (S) in its basic form is a more complex principle than other ones in our list, involving as it does both a principle concerning the interaction between conjunction and disjunction (such as the suitable version of selection of conjunction over disjunction)—contrary to the way in which other structural principles like (K), (W) and (C) only involve principles concerning the internal properties of conjunction and the internal properties of disjunction—and even a principle concerning negation (such as the suitable version of the principle of noncontradiction). Relatedly, the analysis makes it clear that there are at least two very different ways in which a logic might be nontransitive: by denying the suitable version of selection of conjunction over disjunction or by denying the suitable version of the principle of noncontradiction, where only the former way conforms to the reconceptualisation of substructural logics as logics that fundamentally deny some of the principles that conjunction, disjunction and implication obey in classical logic and that determine that classical logic has the structural principles it has (that it does so conform is also confirmed in a particularly revealing way by the presentation of classical logic mentioned at the end of the fifth last paragraph).

4 Substructural approaches to paradox

There are indeed approaches to paradox that use a substructural logic:⁵⁰ the development and discussion of such approaches is the general topic of this volume. Indeed, not only are substructural approaches to paradox as prima facie viable as any—they enjoy several noteworthy advantages over more traditional structural nonclassical ones. Firstly, substructural approaches to paradox often revise classical logic without revising the fundamental principles governing logical operations. For the purposes of this introduction (see Zardini, 2021b; d for deeper levels of analysis), and restricting throughout to the sentential level, these can be taken to be the pairs of principles determining how weak and how strong a sentence with that operation as main operation is (Zardini, 2019b, pp. 172–173, 179): for negation, the law of excluded middle and the law of noncontradiction; for conjunction, adjunction and the entailment of simplification ($\varphi \& \psi \vdash \varphi$ and $\varphi \& \psi \vdash \psi$); for disjunction, addition and abjunction; for implication, unipremise conditional proof and modus ponens. But why is it a good thing to maintain the fundamental principles governing logical operations? Well, for example, the law of excluded middle and the law of noncontradiction would seem correct at least for an understanding of $\lnot \varphi$ as covering every way in which $\varphi$ fails, but structural nonclassical approaches to the semantic and set-theoretic paradoxes (Bočvar, 1938; Asenjo, 1966), as well as to the Sorites paradox (Tye, 1990; Ripley, 2005), typically do deny either of those principles, thereby implausibly committing themselves to the incoherence of the notion of failure. One might then wonder what the point is of vindicating the notion of truth at the cost of jettisoning the related notion of failure (Zardini, 2011, p. 514; 2014b, pp. 193–196). Similarly, unipremise conditional proof and modus ponens would seem correct at least for an understanding of $\varphi \rightarrow \psi$ as covering every way in which $\varphi$ suffices for $\psi$, but structural nonclassical approaches to the semantic and set-theoretic paradoxes typically do deny either of those principles (Priest, 2006; Goodship, 1996), thereby implausibly committing themselves to the incoherence of the notion of sufficiency. One might then wonder what the point is of vindicating the notion of truth at the cost of jettisoning the related notion of sufficiency (Zardini, 2011, p. 517).

Secondly, substructural approaches to paradox often provide a unified solution to the paradoxes of a certain kind. For one example, as I’ve argued in Sect. 2, the Liar paradox and Curry’s paradox are of the same kind. Yet, the former stars negation whereas the latter stars implication, and structural correlation-friendly nonclassical approaches to the semantic paradoxes block the Liar paradox by denying either the law of excluded middle or the law of noncontradiction, but block Curry’s paradox by denying either unipremise conditional proof or modus ponens. Pending an unlikely account explaining how these two denials flow from a common source, such approaches do not provide a unified solution to the semantic paradoxes (Zardini, 2015a). For another example, it is immensely plausible that the tolerance version of the Sorites paradox (the one presented in Sect. 2) is of the same kind as the lack-of-sharp-boundaries version of the Sorites paradox (which can be got from the one presented in Sect. 2 by replacing tolerance with the principle of lack of sharp boundaries according to which $\lnot (B(i) \& \lnot B(i+1))$ holds and modus (ponendo) ponens with the entailment of modus ponendo tollens ($\varphi , \lnot (\varphi \& \psi ) \vdash \lnot \psi$) together with the entailment of double-negation elimination ($\lnot \lnot \varphi \vdash \varphi$), see Oms & Zardini, 2019, pp. 6–7 for the details). Yet, the former stars implication whereas the latter stars negation and conjunction, and, for the implication and conjunction that arguably most adequately capture the spirit of tolerance and lack of sharp boundaries respectively, structural tolerance-friendly nonclassical approaches to the Sorites paradox (Goguen, 1969; Ripley, 2005) either [block the tolerance version of the Sorites paradox by accepting tolerance while denying modus ponens, but block the lack-of-sharp-boundaries version of the Sorites paradox by rejecting lack of sharp boundaries by means of disputing the idea that one between B(i) and $\lnot B(i)$ holds] or [block the tolerance version of the Sorites paradox by rejecting tolerance by means of disputing the idea that $B(i+1)$ follows from (a sentence that holds together with) B(i), but block the lack-of-sharp-boundaries version of the Sorites paradox by accepting lack of sharp boundaries while denying modus ponendo tollens]. In either case, pending an unlikely account explaining how the two moves in question flow from a common source, such approaches do not provide a unified solution to the Sorites paradox (Zardini, 2019b, p. 170, fn 5, p. 179, fn 23; Oms & Zardini, 2021, pp. 212–213, fn 16).

Relatedly, and by way of transitioning to the third point, some other times the problem for structural nonclassical approaches to paradox is not that there are two different logical operations at play—rather, there are none, only a notion of a kind such that nonclassical approaches typically vindicate its characterising principles. For one example (suggested by some of the considerations in the second last paragraph), just like $\varphi \vdash T(\ulcorner \varphi \urcorner )$ and $T(\ulcorner \varphi \urcorner ) \vdash \varphi$ are characteristic of the notion of truth, the laws $\oslash \vdash \varphi , F'(\ulcorner \varphi \urcorner )$ and $\varphi ,F'(\ulcorner \varphi \urcorner ) \vdash \oslash$ would seem characteristic of the notion of failure, yet they give rise to a variation of the Liar paradox with no logical operation at play. For another example (also suggested by some of the considerations in the second last paragraph), just like $\varphi \vdash T(\ulcorner \varphi \urcorner )$ and $T(\ulcorner \varphi \urcorner ) \vdash \varphi$ are characteristic of the notion of truth, the principle that, if $\varphi \vdash \psi$ holds, $S(\ulcorner \varphi \urcorner , \ulcorner \psi \urcorner )$ holds and the entailment $\varphi , S(\ulcorner \varphi \urcorner , \ulcorner \psi \urcorner ) \vdash \psi$ would seem characteristic of the notion of sufficiency,⁵¹ yet they give rise to a variation of Curry’s paradox with no logical operation at play.⁵²$^{,}$⁵³ For yet another example, just like $\varphi \vdash T(\ulcorner \varphi \urcorner )$ and $T(\ulcorner \varphi \urcorner ) \vdash \varphi$ are characteristic of the notion of truth, the material entailment $B(i)\vdash B(i+1)$ (a close relative of tolerance which would seem on equal footing with it) would seem characteristic of the vagueness of the notion of baldness (see Zardini, 2008b, pp. 27–28, 175–176; 2015b, pp. 221–222; 2019b, p. 169, fn 4 for more details on material validity), yet it gives rise to a variation of the Sorites paradox with no logical operation at play (Zardini, 2019b, p. 176).⁵⁴

Thirdly, substructural approaches to paradox often afford the only way to uphold certain compelling principles concerning the original notions with their intended force. For one example, even more compelling than the convergence version of correlation we’ve been working with is its divergence version according to which both $\lnot (\varphi \& \lnot T(\ulcorner \varphi \urcorner ))$ and $\lnot (\lnot \varphi \& T(\ulcorner \varphi \urcorner ))$ hold, and that is naturally understood as having the force of making $\varphi \& \lnot T(\ulcorner \varphi \urcorner )$ and $\lnot \varphi \& T(\ulcorner \varphi \urcorner )$ absurd—but virtually no structural approach can uphold the divergence version of correlation with such a force (Heck, 2012; Zardini, 2013a). For another example, tolerance is naturally understood as having the force of making $B(i+1)$ follow from $B(i)\rightarrow B(i+1)$ and B(i)—but virtually no structural approach can uphold all the instances of tolerance with such a force (Zardini, 2019b, p. 170).

As per the argumentation of Sect. 3, typically centring on conjunction and disjunction, substructural logics typically centre on logical operations whose arguments are all upwards monotonic—that is, in effect, logical operations of positive composition. In the framework of Sect. 2, approaches to a paradox that use any such logic thus individuate the mistakenly represented fact of the paradox in a peculiar behaviour of positive composition. While such a take on a paradox might initially come across as rather surprising and unlikely given the feeling of familiarity and obviousness that positive composition emanates as opposed to other kinds of logical operations, importantly, even with substructural logics so reconceptualised, substructural approaches to paradox retain all the advantages expounded in this section, which can then be understood as evidence for the idea that the paradoxes in our list are indeed rooted in mistakes that we’re led to make when (explicitly or implicitly) operating with conjunction and disjunction in the course of a paradoxical reasoning. Therefore, substructural approaches to paradox represent a powerful trend in contemporary philosophy of logic, which typically adopts a stimulatingly new attitude towards the paradox-monger: rather than fixing on his flamboyant nots, they try to unmask his trick by going after his nonchalant ands and ors.

5 Volume contents

The foregoing elucidations and expositions hopefully afford an interesting vantage point from which to appreciate the rich variety of developments and discussions of substructural approaches to paradox offered by the papers of this volume. More in detail, Ross Brady’s and Edwin Mares’ papers consider a type of approach to the semantic and set-theoretic paradoxes in the tradition of relevant logics (Orlov, 1928) that denies the implicational analogues of (K), (W) and (C). On the basis of general considerations concerning definitions, Brady pleads for a logic where implication represents a notion of meaning containment (under a specific understanding of what such containment amounts to) for which paradox-driving principles fail even though they hold for logical consequence and in which a disjunction is provable only if either disjunct is; he further takes note of the fact that several prima facie intelligible notions such as e.g. the one of failing cannot be added to the logic on pain of paradox. Mares extends the information-theoretic interpretation developed in Mares (2004) for stronger relevant logics to weaker relevant logics friendly to correlation and comprehension (the basic idea being that the implicational analogues of (K), (W) and (C) fail because implication represents a notion of information application that is sensitive to those structural features), submits that prima facie intelligible notions are not admissible on such an interpretation because they do not correspond to “positive” information conditions and demonstrates that, in the logics he considers, comprehension must be restricted to properties that correspond to such conditions. In the same tradition, Pilar Terrés’ paper considers an approach to the Material-Implication paradox and related paradoxes that denies (K). Terrés distinguishes between a minimal, classical notion of logical consequence and of its accompanying operations (which looks for truth preservation) and an enriched, relevant one (which looks for the reasons for accepting a sentence), with either being selectable depending on the features of context: she argues that, while the Material-Implication paradox and related paradoxes are sound arguments in classical logic, they usually have unsound readings (involving the occurrence of an intensional operator) in a relevant logic.

Neil Tennant’s and Peter Schröder-Heister and Luca Tranchini’s papers consider a type of approach to the semantic and set-theoretic paradoxes in the tradition of normal proofs (Prawitz, 1965) that denies (less crucially) (K) and (more crucially) (S). Replying to an overgeneralisation objection levelled by Schröder-Heister & Tranchini (2017) that relies on the addition of a certain intuitive reduction procedure to normalisation, Tennant refines his proof-theoretic criterion of paradoxicality (according to which a paradoxical derivation is one whose normalisation does not terminate) by adopting natural-deduction generalised elimination rules (Schröder-Heister, 1981) and shows that, on the refined criterion, derivations of Russell’s paradox that do not write comprehension into the rules of the system do not count as paradoxical, whereas derivations of the Liar paradox that write correlation into the rules of the system do count as paradoxical (since then there are normal proofs of the Liar sentence and of its negation, but no normal proof of absurdity). Schröder-Heister and Tranchini counterreply to Tennant by pointing out that generalised elimination rules call for adding yet a further intuitive reduction procedure to normalisation, one that however reinstates the overgeneralisation objection; they suggest that the general problem be at least partially tackled instead by imposing strict conditions on admissible reduction procedures in normalisation, requiring preservation of identity of derivations (under a specific understanding of what such identity amounts to).

Elia Zardini’s paper considers an approach to the Sorites paradox in the tradition of tolerant logics (Zardini, 2008a; b, pp. 93–174) that denies (S). Zardini observes that vagueness is also crucial in the situation of the Preface paradox and related epistemic and implicational paradoxes and that the Sorites paradox on the one hand and those paradoxes on the other hand are totally analogous, concluding that they are all of the same kind and proceeding to apply his favoured solution to the Sorites paradox also to those other paradoxes. In the same tradition, Pablo Cobreros, Paul Égré, Dave Ripley and Robert van Rooij’s paper considers a type of approach to the Sorites paradox that denies either (K) or (S) or both. Drawing on Cobreros et al. (2015), Cobreros, Égré, Ripley and van Rooij add to the basic framework of tolerant logics a further interpretation of sentences driven by speakers’ intuitions of assertability: they use that interpretation to define a nondeductive logic (which denies (K)) that allows arbitrary iterations of tolerant reasoning about similar objects (at least as long as false conclusions are not reached) and where tolerance itself is invalid, alongside a deductive logic (which denies (S)) that disallows such iterations but where tolerance itself cannot be used as a premise, alongside a nondeductive logic (which denies both (K) and (S)) that allows such iterations but where tolerance itself can be used as a premise (at least as long as it is not applied to contradictory cases such as borderline ones). Also in the same tradition, Eduardo Barrio, Lucas Rosenblatt and Diego Tajer’s paper considers an approach to the semantic paradoxes (due to Cobreros et al., 2013) that denies (S). Barrio, Rosenblatt and Tajer explore the addition to the object language of a predicate expressing the notion of validity: they uncover the incoherence in the combination of circumstances that, while (S) fails for the logic they consider, since the validity version of a Curry sentence is absurd in that logic, then, according to that logic, the paradoxical instances of (S) for that sentence hold for the notion of validity expressed in the object language (and that indeed, employing plausible stronger principles for validity in the style of Zardini, 2014a, according to that logic, every instance of (S) holds for the notion of validity expressed in the object language).

Zach Weber’s and Petr Cintula and Francesco Paoli’s papers consider a type of approach to the semantic and set-theoretic paradoxes in the tradition of BCK-logics (Tarski, 1936) that denies (W). Weber explores the addition to the object language of a predicate expressing the notion of provability (i.e. validity with no premises): assuming the same approach to the ensuing provability version of Curry’s paradox, he highlights how principles that one way or another force (W) to hold for sentences about provability (even for those that are not provable in the logic) are then not admissible. Replying to charges brought up by Ripley (2015a, pp. 322–325; 2015b), Cintula and Paoli first remark that there are several versions of (S) that, by not building in contraction, hold on a noncontractive approach but not on a nontransitive one and then, by employing a conjunctive mode of conclusion combination, formulate a notion of a multiset-based consequent relation and its corresponding notion of a multiset-based closure operation that are compatible with failure of (W).

Lionel Shapiro’s paper considers an approach to the semantic paradoxes in the tradition of dialetheism (Priest, 1979 and then especially the version of Goodship, 1996 taken up by Beall, 2015) but develops a related substructural logic that denies (S), (W) and (C). Shapiro proves that there is a mapping between what is valid in his substructural logic and what is valid in the logic that would seem to be used by Beall (2015)’s approach and contends on this basis that there is no fact of the matter as to which of the two logics Beall (2015)’s approach uses (and, more generally, that there is no fact of the matter as to whether an approach to a paradox uses a substructural logic). Ole Hjortland’s paper considers several types of substructural approaches to the semantic paradoxes that deny either (I) or (S) or (W). Focusing on the question of the extent to which all such approaches deviate from classical logic, Hjortland notes that every such approach cannot accept principles of classical logic that build in the structural principle it denies and, conversely, that certain types of structural approaches have a syntactic presentation (of the kind also used in Shapiro’s paper) under which what is denied is a structure-related feature that classical logic enjoys under that presentation (i.e. the absence in a sequent of an intermediate position between premises and conclusions). Julien Murzi and Lorenzo Rossi’s paper considers an approach to the semantic paradoxes in the tradition of groundedness (Herzberger, 1970) that denies (I). Replying to critical points raised by Zardini (2013b, pp. 636–638) and Field (2017), Murzi and Rossi defend the good standing of a notion of validity typically understood as obeying something like the analogues of the principles for sufficiency mentioned in Sect. 4, doing so mainly by providing a fixed-point construction for a [material-implication]-like predicate where (I) as well as the relevant analogue of the second principle for sufficiency mentioned in Sect. 4 fails.

6 Volume acknowledgements

This volume originates from the highly successful and productive workshop Substructural Approaches to Paradox, which I organised in 2013 at the University of Barcelona in the framework of my Marie Skłodowska-Curie Intraeuropean Research Fellowship 301493 A Noncontractive Theory of Naive Semantic Properties: Logical Developments and Metaphysical Foundations and in which the vast majority of the volume’s contributors took part. To the best of my knowledge, that was the first conference ever dedicated to the topic of substructural approaches to paradox—which has in the meanwhile proven to be one of the most fertile research areas in philosophy of logic in the last decade or so—and this is now the first volume ever dedicated to it.⁵⁵ I would like to thank all the contributors for enthusiastically accepting my invitation and for delivering such challenging pieces—it has been a great pleasure to engage in a philosophical dialogue with each of them. I would also like to thank all the anonymous referees for Synthese who helped with their careful and constructive comments to improve the papers even further. Special thanks go to the editors of Synthese for accepting my volume proposal, and in particular to Gila Sher, Catarina Dutilh Novaes and Otávio Bueno for their great guidance, great support and great patience at different stages of this project. Finally, thanks to Laura and Miguel for refreshingly not giving a damn about the topic: life doesn’t (wholly) consist in substructural approaches to paradox.