2 Mixed logic

When defining logical consequence model-theoretically, we usually stipulate that an inference is valid iff in all models where all of the premises are satisfied, at least one of the conclusions is satisfied too. Within this template, there is room for variation from logic to logic: which language we are defining validity for, what the class of models looks like, and when a model satisfies a sentence. But one common feature is that there is a single notion of satisfaction-in-a-model, which occurs twice in the template: once in relation to the premises, and again for conclusions. Mixed logics are what we get when we drop the assumption that the same satisfaction condition fulfills these two roles, and instead say that an inference is valid just in case every model which satisfies $_1$ all the premises, also satisfies $_2$ one of the conclusions. But satisfaction $_1$ and satisfaction $_2$ may be quite different.

Three-valued logic provides a notable example. Let $\mathcal {L}^{PL}$ be a propositional language, formed by closing a set of propositional atoms under negation, conjunction, disjunction and implication.Footnote ¹ A Strong Kleene valuation is a function from the sentences of $\mathcal {L}^{PL}$ to $\{0, \frac {1}{2}, 1\}$ which respects the Strong Kleene truth tables (Figure 1). Suppose we call an inference valid just in case every Strong Kleene valuation which assigns all the premises 1 assigns one of the conclusions at least $\frac {1}{2}$ . This is known as strict-tolerant logic, or ST [Reference Cobreros, Egré, Ripley and van Rooij8, Reference Cobreros, Egré, Ripley and van Rooij9, Reference Ripley33, Reference Ripley34]. Although the mixed definition looks odd, ST has something going for it. Ripley [Reference Ripley33] proved that ST can be conservatively extended with a fully transparent truth-predicate, remarkably—and much unlike other three-valued approaches to truth [Reference Field14, Reference Priest30]—without giving up any classically valid inferences.

Figure 1

The Strong Kleene truth tables.

Figure 1 Long description

Four truth tables for negation, conjunction, disjunction, and implication. Negation swaps 1 and 0 while leaving 1/2 unchanged. Conjunction returns the minimum value of inputs. Disjunction returns the maximum value. Implication returns 1 when the antecedent is 0, matches the consequent when the antecedent is 1, and yields 1, 1/2, 1/2 for an antecedent of 1/2.

As usual, there is a catch. Assuming some self-referential machinery, a Liar sentence $\lambda $ can be formulated in the extended language, and it must take value $\frac {1}{2}$ in every model to avoid contradiction. This means that $\lambda $ is a theorem of ST, since it always has value at least $\frac {1}{2}$ . At the same time, the inference from $\lambda $ to any atom $p$ is valid, because $\lambda $ never has value $1$ . But of course, atoms are not logical theorems. So here we have our first example of a logic seemingly disobeying its own logical laws: $\lambda $ is a theorem, and the inference from $\lambda $ to $p$ is ruled valid, yet $p$ is not a theorem. As we’ll see, this is but one instance of a much broader phenomenon.

Since we want to say some things about mixed logics in general, it will be useful to have language- and model-independent definitions. We only assume that we have some formal language $\mathcal {L}$ , with $\mathcal {L}_0$ its set of well-formed sentences, and some class of models $\mathcal {M}$ . In this abstract setting, a satisfaction standard for sentences is a relation $X \subseteq \mathcal {M} \times \mathcal {L}_0$ . We will call such relations 0-standards for short. When $\langle M, \varphi \rangle \in X$ , we say that $M$ $X$ -satisfies $\varphi $ , and write $M \Vdash _X \varphi $ . If $\Gamma $ is a set of sentences, we write $M \Vdash _X \Gamma $ to mean that $M \Vdash _X \gamma $ for every $\gamma \in \Gamma $ . Similar abbreviations will be used without further comment below. If $X$ and $Y$ are both 0-standards for $\mathcal {M}$ and $\mathcal {L}$ , then $X/Y$ denotes a 1-standard for $\mathcal {M}$ and $\mathcal {L}$ . If $\Gamma $ and $\Delta $ are sets of $\mathcal {L}$ -sentences, the inference from $\Gamma $ to $\Delta $ is $X/Y$ -valid when: $\forall M \in \mathcal {M}$ , if $M \Vdash _{{X}} \Gamma $ , then also $M \Vdash _{{Y}} \delta $ for some $\delta \in \Delta $ . For instance, take $\mathcal {L}^{PL}$ and the set of Boolean valuations, and define a 0-standard $C$ so that $v \Vdash _{C} \varphi $ iff $v(\varphi )=1$ . Then the validities of classical propositional logic are just the validities of the 1-standard $C/C$ . For Strong Kleene valuations, we can define two 0-standards $S$ and $T$ , such that $v \Vdash _{S} \varphi $ iff $v(\varphi )=1$ , while $v \Vdash _{T} \varphi $ iff $v(\varphi )\geq \frac {1}{2}$ . Then Strong Kleene logic K $_3$ corresponds to $S/S$ , the logic of paradox LP to $T/T$ , and our mixed logic ST to $S/T$ .

So much for plain inferences. We are interested not only in which sentences follow logically from which sentences, but also in which inferences follow from which inferences, which inferences between inferences follow from which inferences between inferences, and so forth. Such metainferences play important roles in inferential practice and logical theorizing, especially in the BA-plan. Starting with sentences, now thought of as $0$ -inferences, we define a hierarchy of inferential levels inductively: for any $n \in \mathbb {N}$ , an $n+1$ -inference of $\mathcal {L}$ is an expression of the form $\Gamma \Rightarrow _{n+1} \Delta $ , where $\Gamma $ and $\Delta $ are sets of $n$ -inferences of $\mathcal {L}$ .Footnote ² Thus regular inferences are $1$ -inferences, inferences between those are $2$ -inferences, and so on. All of them are called inferences.

Naturally, we can mix premise and conclusion standards also for higher levels. We’ll see below that, by doing so, we can enhance some of the desirable properties of ST. So we generalize the notion of a standard upwards. If $X$ and $Y$ are n-standards for $\mathcal {M}$ and $\mathcal {L}$ , then $X/Y$ is an $n+1$ -standard for $\mathcal {M}$ and $\mathcal {L}$ . Of course, $\Gamma \Rightarrow _{n+1} \Delta $ is $X/Y$ -satisfied in $M$ (written $M \Vdash _{X/Y} \Gamma \Rightarrow _{n+1} \Delta $ ) when, if $M \Vdash _X \Gamma $ , then also $M \Vdash _Y \delta $ for some $\delta \in \Delta $ . Finally, $\Gamma \Rightarrow _{n+1} \Delta $ is $X/Y$ -valid (written $\vDash _{X/Y} \Gamma \Rightarrow _{n+1} \Delta $ ) if it is $X/Y$ -satisfied in all models.

This is the so-called local definition of validity for metainferences. It contrasts with the alternative, global validity, which requires preservation of validity rather than of satisfaction. As I argue in [Reference Kortenbach21], global validity is at least as suitable as local, if not more so. Nonetheless, we only study local validity in the current paper, because it is the notion at play in the BA-plan.

Either way, a given $n$ -standard is only designed to tell us which inferences of level $n$ are valid. But we would like logics to decide on the validity of all inferences. Therefore, from now on, by a logic for $\mathcal {L}$ and $\mathcal {M}$ we mean a set of standards for $\mathcal {L}$ and $\mathcal {M}$ , containing precisely one $n$ -standard for every $n \in \mathbb {N}$ .Footnote ³ Equivalently, a logic can be thought of as an $\omega $ -sequence such that the $n$ th element is an $n$ -standard. Naturally, an $n$ -inference is satisfied/valid according to a logic $\mathbf {L}$ just in case it is so according to the unique $n$ -standard in $\mathbf {L}$ . Call a logic pure if it is of the form $\{X,\ X/X,\ X/X/X/X,\ \ldots \}$ for some 0-standard $X$ . Otherwise, it is mixed. Self-disobedience gives us some reason to worry that perhaps only pure logics are legitimate. In the sequel, we investigate under which circumstances mixed-ness produces self-disobedience (§3), and to what extent self-disobedience in turn renders a logic untenable (§4).

For now, it remains to introduce some important example logics. We may assume that the classical logician uses her own standards at every metainferential level, for premises as well as conclusions. Thus full classical propositional logic is simply the set $\mathbf {CL} = \{C,\ C/C,\ C/C/C/C,\ \ldots \}$ . Clearly, $\mathbf {CL}$ is pure. As for mixed logics: the $\mathbf {ST}$ -hierarchy consists of an infinite sequence of logics, one dubbed $\mathbf {ST}^n$ for each $n>0$ , plus a final logic $\mathbf {ST}^\omega $ . The exact definitions are left for the Appendix, since they aren’t needed for what’s to come. We only need to know that (i) the $\mathbf {ST}$ logics are all based on Strong Kleene valuations, (ii) each is a mix of $S$ ’s and $T$ ’s, and (iii) each starts out with the standards $T$ and $S/T$ for $0$ - and $1$ -inferences, respectively.

The $\mathbf {ST}$ -hierarchy has garnered a lot of attention, for the following reasons. Barrio et al. [Reference Barrio, Pailos and Szmuc4] showed that, for each $n>0$ , the logic $\mathbf {ST}^n$ has precisely the same validities as $\mathbf {CL}$ at all inferential levels $m \leq n$ , but is weaker than $\mathbf {CL}$ at every level $m>n$ . So each next logic in the sequence is a bit stronger and a bit more classical than those that came before. $\mathbf {ST}^\omega $ , finally, is classical all the way: it validates precisely the same $n$ -inferences as $\mathbf {CL}$ , for all $n \in \mathbb {N}$ . Moreover, Pailos [Reference Pailos26] proved that each $\mathbf {ST}^n$ , as well as $\mathbf {ST}^\omega $ , can be safely extended with transparent truth. Let us call the resulting logics $\mathbf {ST}^n_+$ and $\mathbf {ST}^\omega _+$ .Footnote ⁴ They are a series of fully truth-transparent, yet increasingly classical systems, topped off by a completely classical one. For this reason Barrio, Pailos, and Toranzo Calderón [Reference Barrio, Pailos and Calderón3] and Pailos and Da Ré [Reference Pailos and Da Ré27] among others view $\mathbf {ST}^\omega _+$ as the ideal logic of truth. This is the paradise promised by the BA-plan.Footnote ⁵ Self-disobedience threatens to expel us from it (other attacks have been launched [Reference Dicher and Paoli12, Reference Fjellstad17, Reference Golan19–Reference Kortenbach21, Reference Priest, Petrolo and Venturi31], but they lie beyond the present scope). It is time to get clear on precisely what self-disobedience is, and to demarcate the affected territory.

3 Self-disobedience in mixed logics

A few subtly different notions of self-obedience have appeared in the literature. Each is a measure for whether the behaviour of a logic is consonant with itself across adjacent inferential levels. Specifically, self-obedience properties check whether we could actually make the inferences declared valid by a logic, without ever passing from premises which are correct by the logic’s own lights, to conclusions which aren’t. This section elucidates the different self-obedience properties and their relations to each other and to mixed-ness.

The first notion of self-obedience was formulated by Scambler [Reference Scambler39] (though he referred to it as logics being ‘closed under their own laws’; the term ‘self-obedient’ originates with [Reference Ripley36]). I’ll call it (SM), for Scambler Multiple-conclusion. A logic has (SM) at a level $n \geq 1$ just in case every valid $n$ -inference with valid premises also has at least one valid conclusion. That is, $\mathbf {L}$ has (SM) at $n$ just in case, for all $n$ -inferences $\Gamma \Rightarrow _n \Delta $ :

(SM) $$ \begin{align} \text{If}\ \vDash_{\mathbf{L}} \Gamma \Rightarrow_n \Delta\ \text{and}\ \vDash_{\mathbf{L}} \Gamma,\ \text{then}\ \exists \delta \in \Delta\ \text{s.t.}\ \vDash_{\mathbf{L}} \delta. \end{align} $$

Moreover, if $\mathbf {L}$ has (SM) at every $n \geq 1$ , we’ll simply say that it has (SM) (likewise for the other criteria to be introduced). As Scambler points out, none of the $\mathbf {ST}^n_+$ , nor $\mathbf {ST}^\omega _+$ , has (SM). For instance, all of them start with $T$ and $S/T$ . Therefore they validate $\lambda $ , and $\lambda \Rightarrow _1 p$ for any atom $p$ , but not $p$ .Footnote ⁶

Ripley [Reference Ripley36] later argued that (SM) is too strong—for reasons we’ll see in a moment—and proposed a weaker alternative. Let’s call it (SS), for Scambler Single-conclusion, as it is exactly Scambler’s criterion limited to inferences with at most one conclusion.Footnote ⁷ That is, $\mathbf {L}$ has (SS) at $n$ just in case, for all $n$ -inferences $\Gamma \Rightarrow _n \varphi $ :

(SS) $$ \begin{align} \text{If}\ \vDash_{\mathbf{L}} \Gamma \Rightarrow_n \varphi\ \text{and}\ \vDash_{\mathbf{L}} \Gamma,\ \text{then}\ \vDash_{\mathbf{L}} \varphi. \end{align} $$

Of course, the example we gave of $\mathbf {ST}^n_+$ and $\mathbf {ST}^\omega _+$ failing (SM) was already single-conclusion, so it equally works as a counterexample to (SS).

Other notions of self-obedience can be gleaned from Porter’s [Reference Porter28]. We’ve just learned that in the $\mathbf {ST}$ -hierarchy logics, the set of validities at a given level doesn’t always obey the valid inferences of the level above. Porter points out that the same is true of theories. The $\mathbf {L}$ -theory of a set of inferences $\Gamma $ , denoted $Th_{\mathbf {L}}(\Gamma )$ , is the set of inferences which are $\mathbf {L}$ -satisfied in every model which $\mathbf {L}$ -satisfies $\Gamma $ . The idea being that $Th_{\mathbf {L}}(\Gamma )$ contains everything which follows from the assumption that we are in a model of $\Gamma $ . Porter asks whether reasoning according to the validities of $\mathbf {L}$ , starting from some $\Gamma $ , won’t ever take us beyond $Th_{\mathbf {L}}(\Gamma )$ . As before, we could require this in general, or restrict to validities with at most one conclusion (Porter himself doesn’t specify). So let’s define two properties, (PM) and (PS), following our previous naming convention. $\mathbf {L}$ has (PM) at $n$ just in case, for all $n$ -inferences $\Gamma \Rightarrow _n \Delta $ :

(PM) $$ \begin{align} \text{If}\ \vDash_{\mathbf{L}} \Gamma \Rightarrow_n \Delta,\ \text{then}\ Th_{\mathbf{L}}(\Gamma) \cap \Delta \neq \emptyset. \end{align} $$

Unpacking this a bit will help comparison with the other notions. So, equivalently:

(PM) $$ \begin{align} \text{If}\ \vDash_{\mathbf{L}} \Gamma \Rightarrow_n \Delta,\ \text{then}\ \exists \delta \in \Delta\ \text{s.t.}\ \forall M \in \mathcal{M},\ \text{if}\ M \Vdash_{\mathbf{L}} \Gamma,\ \text{then}\ M \Vdash_{\mathbf{L}} \delta. \end{align} $$

Restricting these conditions to single-conclusion inferences gives us, respectively:

(PS) $$ \begin{align} \text{If}\ \vDash_{\mathbf{L}} \Gamma \Rightarrow_n \varphi,\ \text{then}\ \varphi \in Th_{\mathbf{L}}(\Gamma). \end{align} $$

And equivalently:

(PS) $$ \begin{align} \text{If}\ \vDash_{\mathbf{L}} \Gamma \Rightarrow_n \varphi,\ \text{then}\ \forall M \in \mathcal{M},\ \text{if}\ M \Vdash_{\mathbf{L}} \Gamma,\ \text{then}\ M \Vdash_{\mathbf{L}} \varphi. \end{align} $$

Note that whereas Scambler-style properties record whether $\mathbf {L}$ -validities preserve $\mathbf {L}$ -validity, the Porter variants check that $\mathbf {L}$ -validities preserve $\mathbf {L}$ -satisfaction. Porter observes that none of the $\mathbf {ST}^n$ nor $\mathbf {ST}^\omega $ have (PS) nor (PM). For instance, recall that all the $\mathbf {ST}$ logics start with $T$ and $S/T$ . Then note that $p, \neg p \Rightarrow _1 q$ is $S/T$ -valid (as no model gives both $p$ and $\neg p$ value $1$ ), yet there are models which $T$ -satisfy both $p$ and $\neg p$ (by giving them both value $\frac {1}{2}$ ) without $T$ -satisfying $q$ .

Now why did Ripley deem (SM) too strong? The reason most relevant to our purposes is that even completely pure logics like $\mathbf {CL}$ don’t generally have (SM). For example, $\emptyset \Rightarrow _1 p, \neg p$ is classically valid, even though neither $p$ nor $\neg p$ is. I take this example to illustrate that, besides self-obedience, having (SM) also requires a type of generalized disjunction property, to the effect that whenever a set of $n$ -inferences is collectively valid (every model satisfies at least one of them), one among them is valid on its own. Perhaps constructivists would be happy to impose such a requirement on logics. But even so, that is a question independent from self-obedience, so we better keep the two requirements separate, rather than packaging one in with the other. By contrast, it is easy to see that every pure logic has (SS).

What about (PM)? Spelling out and comparing the definitions makes it obvious that every logic which has (PM) also has (SM). It immediately follows that $\mathbf {CL}$ does not have (PM) either, and that (PM) also includes the generalized disjunction property.Footnote ⁸ On the other hand, it is again simple to verify that every pure logic has (PS). In the remainder, then, I will largely leave (SM) and (PM) aside as unsuitable notions of self-obedience.Footnote ⁹

We may wonder, though, whether no plausible self-obedience property can take into account multiple-conclusion inferences at all. Prima facie there should be a sensible way to talk about validities being obeyed or not, even if they have multiple conclusions. I submit that indeed there is. Consider again the second definition of (PM). It is so strong because it requires that one and the same conclusion is satisfied in all models of the premises, rather than merely that each such model satisfies some potentially different conclusion. So I propose we flip the order of the quantifiers. Let’s call the result (PD), for Porter Distributed (as it ‘distributes’ the burden over the different conclusions)Footnote ¹⁰ :

(PD) $$ \begin{align} \text{If}\ \vDash_{\mathbf{L}} \Gamma \Rightarrow_n \Delta,\ \text{then}\ \forall M \in \mathcal{M},\ \text{if}\ M \Vdash_{\mathbf{L}} \Gamma,\ \text{then}\ \exists \delta \in \Delta\ \text{s.t.}\ M \Vdash_{\mathbf{L}} \delta. \end{align} $$

An analogous move can be applied to (SM), after rewriting $\vDash _{\mathbf {L}} \delta $ as $\forall M \in \mathcal {M},\ M \Vdash _{\mathbf {L}} \delta $ . Naturally, we call the result (SD):

(SD) $$ \begin{align} \text{If}\ \vDash_{\mathbf{L}} \Gamma \Rightarrow_n \Delta\ \text{and}\ \vDash_{\mathbf{L}} \Gamma,\ \text{then}\ \forall M \in \mathcal{M},\ \exists \delta \in \Delta\ \text{s.t.}\ M \Vdash_{\mathbf{L}} \delta. \end{align} $$

It is once more easy to check that every pure logic has (SD) and (PD). I regard them as the proper multiple-conclusion analogues of (SS) and (PS), respectively.

Altogether we have six notions of self-obedience. Figure 2 is the Hasse diagram of their ordering by strength. The Scambler- and Porter-style properties line the top left and bottom right, respectively. Within each group, the naive multiple-conclusion variant strictly entails the distributed version (because of quantifier scope), which strictly entails the single-conclusion one (as a special case). Moreover, every Porter-style criterion strictly entails the corresponding Scambler variant (because preserving satisfaction entails preserving local validity). All of the depicted entailments are immediate upon unpacking and comparing the definitions. Counterexamples to the non-entailments are left to the interested reader.

Figure 2

Self-obedience properties.

What about our example logics? We’ve already mentioned that $\mathbf {CL}$ , like all pure logics, has (PD), and hence all four of the properties forming the top right diamond in Figure 2. But like most logics, pure or mixed, it has neither (SM) nor (PM). We’ve also seen that $\mathbf {ST}^n_+$ and $\mathbf {ST}^\omega _+$ lack (SS), which means they lack all six. Lastly, while $\mathbf {ST}^n$ and $\mathbf {ST}^\omega $ themselves don’t have (PS) nor hence (PD), they actually do have (SS) and (SD), as is proven in the Appendix.

So if we only care about which properties are met fully, it doesn’t matter how high up in the $\mathbf {ST}$ -hierarchy we are— $\mathbf {ST}^1$ fares the same as $\mathbf {ST}^{500}$ and $\mathbf {ST}^\omega $ . Yet it is important to also check the properties per level. The self-disobedience of the different $\mathbf {ST}$ logics is spread across the levels to very different degrees. Namely: for each $n \geq 1$ , the corresponding logic $\mathbf {ST}^n_+$ fails (SS) and (SD) only at the levels $m \leq n$ , but has them at $m> n$ . $\mathbf {ST}^\omega _+$ fails these criteria at all levels. Similarly, $\mathbf {ST}^n$ and $\mathbf {ST}^n_+$ fail (PS) and (PD) at all the levels $m \leq n$ , but have them at every $m> n$ , while $\mathbf {ST}^\omega $ and $\mathbf {ST}^\omega _+$ don’t have them at any level. So the further up the $\mathbf {ST}$ -hierarchy we climb, the higher the inferential level to which self-disobedience has spread, until finally we are self-disobedient all the way.Footnote ¹¹ This observation will become crucial in §4.

One key issue for the current section remains: how does self-disobedience relate to mixed-ness? All pure logics have (PD), so mixed-ness is at least necessary for self-disobedience. But it is not sufficient, as we’ll see below. This raises the question: can we characterize, in terms of the structure of the sequence of standards, precisely which mixed logics are and aren’t self-obedient in each sense? It is unlikely that any elegant or informative characterization applies across the board. The reason is that we are working with an abstract and highly inclusive definition of ‘standard’. Because of this, the space of all logics includes many degenerate cases, and we can say relatively little with full generality. It will be more insightful to temporarily restrict our attention to logics based on a (relevant) class of well-behaved standards.

To this end, we introduce some terminology. A logic is compositional if its models are valuations assigning a range of semantic values to the sentences, such that the value of any complex sentence is determined by the values of its components (see the Appendix for the formal definition). $\mathbf {CL}$ and the $\mathbf {ST}$ logics are obviously compositional.Footnote ¹² Moreover, a logic is constant complete if the language contains a constant for each semantic value (see again the Appendix). Constants need not be logical primitives; $\mathbf {CL}$ is constant complete, though we haven’t included $\bot $ and $\top $ in $\mathcal {L}^{PL}$ , because $p\wedge \neg p$ and $p\lor \neg p$ suffice. The Strong Kleene tables don’t produce constants automatically, but we could simply add logical constants to make the $\mathbf {ST}$ logics constant complete. The family of logics we’re going to focus on is that of the compositional and constant complete ones.

Let’s also agree on some notation that will come in handy throughout the remainder. If $\mathbf {L}$ is a logic, we write $\mathbf {L}(n)$ for the unique $n$ -standard in $\mathbf {L}$ . Moreover, if $X$ is an $n$ -standard, ${{\uparrow }} X$ denotes the $n+1$ -standard $X/X$ . Also, if $X/Y$ is an $n$ -standard, ${{\swarrow }} (X/Y)$ denotes the $n-1$ -standard $X$ , and ${{\searrow }} (X/Y)$ the $n-1$ -standard $Y$ .Footnote ¹³ Observe that in this notation, $\mathbf {L}$ is pure iff $\mathbf {L}(n+1)={\uparrow }\mathbf {L}(n)$ for all $n \in \mathbb {N}$ iff ${\swarrow } \mathbf {L}(n+1)={\searrow } \mathbf {L}(n+1) = \mathbf {L}(n)$ for all $n \in \mathbb {N}$ . Finally, we define a relation on standards. If $X$ and $Y$ are both $n$ -standards for $\mathcal {M}$ and $\mathcal {L}$ , we write $X \leq Y$ when, for every $M$ and $\varphi $ , if $M \Vdash _X \varphi $ then $M \Vdash _Y \varphi $ . Our characterization of self-obedience will centre on the following property.

The term ‘quasi-pure’ indicates that if we replace both occurrences of ‘ $\leq $ ’ in the definition with ‘=’, we obtain a definition of purity.

It turns out that, for logics in our class of interest, the four plausible self-obedience criteria collapse into one, so we can speak of self-obedience simpliciter. Moreover, exactly what is needed for self-obedience in such logics is not purity per se, but quasi-purity. This fact is proven in the Appendix.

So, for instance, the fact that logics like $\mathbf {ST}^n$ and $\mathbf {ST}^\omega $ have (SD) and (SS) without having (PD) and (PS) is only possible because we didn’t include constants in $\mathcal {L}^{PL}$ . $\mathbf {ST}^n$ and $\mathbf {ST}^\omega $ are not quasi-pure, so once we add constants, they lose (SD) and (SS) too. For another quick application: the so-called $\mathbf {TS}$ -hierarchy is a dual of the $\mathbf {ST}$ -hierarchy, obtained by swapping every $S$ for a $T$ and vice-versa [Reference Scambler39]. It is easy to check that the $\mathbf {TS}$ -hierarchy logics are all quasi-pure, therefore Theorem 3.2 guarantees self-obedience when constants are present. Note that this is despite those logics being thoroughly mixed, so indeed purity is not necessary for self-obedience.

To close the section, let’s briefly look at the narrow gap between quasi-purity and purity proper. Call $\mathbf {L}$ reflexive if $\vDash _{\mathbf {L}} \varphi \Rightarrow _{n+1} \varphi $ for every $n$ -inference $\varphi $ of every $n \in \mathbb {N}$ .

The proof is in the Appendix. Combining with Theorem 3.2 produces the following.

Reflexivity, compositionality, and constant completeness are natural and common properties. Hence in many important cases, mixed-ness will immediately entail self-disobedience by Theorem 3.4. This shows that self-disobedience, though not universal among mixed logics, is at least far from peculiar to the $\mathbf {ST}$ -hierarchy. Next, we examine whether this is problematic for the logics affected.

4 Is self-disobedience acceptable?

To endorse a self-disobedient logic is to put oneself in the place of our interlocutor from the introduction, who refused to grant $A$ while acknowledging that it follows from premises which they maintain.Footnote ¹⁴ More precisely, to endorse a logic which lacks (SS) is to consider something invalid while acknowledging that it follows logically from things which one considers valid. Mutatis mutandis for (SD), (PS), and (PD).

On the face of it, such positions appear absurd. Scambler voiced this impression when discussing modus ponens, in particular the instance $\lambda , \lambda \supset \bot \Rightarrow _1 \bot $ , which is another example of an inference that is validated but not obeyed by the $\mathbf {ST}$ -logics.

The qualification at the end of this quote is prudent. As David Lewis [Reference Lewis24] famously noted, incredulous stares are not exactly conclusive objections. Yet they are not dialectically irrelevant either. Lewis [Reference Lewis25] interpreted the incredulous stares directed at his modal realism as expressions of the view that modal realism defies common sense to such an extent that it outweighs the theory’s advantages. Lewis for his part recognized the steep price, but, contrary to his critics, judged it worth paying. Moreover, he considered this disagreement to be at an impasse.

Likewise, if proponents of some self-disobedient logic were to simply insist that self-disobedience is a price worth paying in return for the virtues of their logic, I am not sure what we could say to prove them wrong. To my mind, self-disobedience strains credulity past the breaking point, hence I would remain unwilling to accept it regardless of the gains that might await, but I don’t quite see how this view can be pressed.

Instead, the purpose of this section is as follows. Whereas Lewis admitted to the cost of his theory and affirmed a willingness to pay, BA-planners have maintained that self-disobedience is no substantial cost in the first place. They have put forth three arguments to the effect that, upon closer consideration, self-obedience isn’t as obvious or desirable as it initially looks, and its failure is perfectly fine. This is the part of their story I want to actively dispute. In my view, each of those three arguments gets things wrong in one way or another, and self-disobedience is in fact just as bizarre as it looks at first glance.

In the remainder, then, I will go over the three arguments in turn and explain where I take issue with each. Finally, we will also explore a fourth argument for the acceptability of self-disobedience, but conclude that—insofar as it can work at all—it cannot be applied to $\mathbf {ST}^\omega $ and $\mathbf {ST}^\omega _+$ . For concreteness, I will focus chiefly on (SS), which is both the simplest and weakest self-obedience property.

4.2 Argument 2: The deduction theorem analogy.

The second type of argument is an analogy drawn by Barrio and Bezerra [Reference Barrio and Bezerra1, p. 18]. The deduction theorem for the material conditional ‘ $\supset $ ’ in (say) classical logic states that any $1$ -inference $\Gamma , \varphi \Rightarrow _1 \psi $ is valid if and only if $\Gamma \Rightarrow _1 \varphi \supset \psi $ is valid. In both LP and K $_3$ , this equivalence breaks down. One of the consequences is that, for example, LP doesn’t generally validate modus ponens: $\varphi , \varphi \supset \psi \Rightarrow _1 \psi $ . Or, to make the analogy fully explicit, that LP does not meet the following condition for all sentences $\varphi $ and $\psi $ :

(PS_⊃) $$\begin{align} \text{If}\ \vDash \varphi \supset \psi,\ \text{then}\ \forall M \in \mathcal{M},\ \text{if}\ M \Vdash \varphi,\ \text{then}\ M \Vdash \psi. \end{align} $$

Similarly, in LP extended with constants or transparent truth, we don’t always have that:

(SS_⊃) $$\begin{align} \text{If} \vDash \varphi \supset \psi\ \text{and} \vDash \varphi, \text{then} \vDash \psi. \end{align} $$

Of course, these properties are essentially (PS) and (SS), respectively, except they concern the object language connective ‘ $\supset $ ’ in place of the inferential arrow ‘ $\Rightarrow _n$ ’ of some $n$ .

Thus the analogy argument goes as follows. The first premise has it that (i) the failure of the deduction theorem in LP and K $_3$ is acceptable. And the second that (ii) failures of self-obedience are analogous to the failure of the deduction theorem in LP and K $_3$ . From (i) and (ii), it is then concluded that (iii) failures of self-obedience are acceptable as well.

I think there are multiple reasons to doubt this argument, starting with premise (i). The material conditional of LP is typically dismissed, even by supporters of LP, as untenable qua conditional, and for reasons essentially tied to the failure of the deduction theorem. For instance, here is LP’s most outspoken advocate, Priest:

[D]ialetheism [the stance underpinning LP] disposes, once and for all, of “material implication”. The reason is simple. Any conditional worth its salt, $\rightarrow $ , should satisfy the modus ponens principle: $\{\alpha , \alpha \rightarrow \beta \} \vDash \beta $ . This is, indeed, analytically part of what implication is. Yet this principle fails for material implication [in LP.] Hence material implication is not the conditional. [Reference Priest30, p. 83]

Much work on LP is devoted precisely to the study of alternative conditionals, which may be added to the language in order to take over the roles that we want a proper conditional to play (see [Reference Carrara and Martino6] and the references therein).

Premise (i) may be more plausible in the case of K $_3$ . Yet this is only because K $_3$ ’s failure of the deduction theorem is very different from LP’s. The deduction theorem requires a general equivalence between $\vDash \Gamma , \varphi \Rightarrow _1 \psi $ and $\vDash \Gamma \Rightarrow _1 \varphi \supset \psi $ . LP delivers the left-to-right direction but not the right-to-left one, while for K $_3$ it is exactly the other way around. One consequence is that unlike LP, K $_3$ does validate modus ponens for the material conditional. Also unlike LP, K $_3$ satisfies (PS⊃) and (SS⊃).Footnote ¹⁵ But then K $_3$ ’s failure of the deduction theorem is nothing like self-disobedience. The upshot is that the analogy argument is really two separate arguments: one about LP and one about K $_3$ . In the former, premise (i) fails, and in the latter, premise (ii) does.

Lastly, even if premises (i) and (ii) both held up, this wouldn’t lend support to conclusion (iii). When two problems are “analogous,” one of them might still be far more serious than the other, for example, by affecting a more critical part of the system. A fractured toe is analogous to a fractured spine. Likewise, a given issue might be tolerable when merely affecting some connective definable in the object language, yet disastrous when affecting the inferential arrow ‘ $\Rightarrow _n$ ’ itself—the logic’s very notion of validity.

Concretely, for one, the difference means that while LP and K $_3$ are often extended with alternative conditionals, so as to avoid using ‘ $\supset $ ’, the analogous maneuver is not available to advocates of self-disobedient logics. To adopt a different inferential arrow (for any level) would be to abandon ship and adopt a different logic altogether. If one merely modifies or extends the object language of a system, there is a meaningful sense in which we can still speak of the result as (another version of) the same logic. But the same cannot be said once one has altered the underlying notion of validity of some level by replacing one of the inferential arrows.

Summing up: the failure of the deduction theorem for ‘ $\supset $ ’ in LP is analogous to self-disobedience, and unacceptable, while that in K $_3$ is not analogous to start with. Moreover, even if some other object-language conditional turns out to fail the deduction theorem in a manner that’s both acceptable and analogous to self-disobedience, this still shouldn’t lead us to accept self-disobedience.Footnote ¹⁶

4.3 Argument 3: Mixed obedience.

The third argument is the one most emphasized by BA-planners. It is developed in [Reference Barrio, Pailos and Calderón3, pp. 12579–12581] and [Reference Roffé and Pailos37, pp. 748–750], and appealed to by [Reference Barrio and Bezerra1, p. 19]. The main claim is that, while we should demand some kind of self-obedience from logics, the specifics of (SS) and the like are only appropriate with respect to pure logics. The properties must be adapted before being demanded of mixed logics, in order not to distort the spirit of the original demand—or so the story goes. The resulting modified property, mixed obedience or mixed closure, is then shown to be satisfied by the $\mathbf {ST}$ logics.

The reason why requiring (SS) is supposedly inappropriate for mixed logics is as follows. Let us recall the definition of (SS), but this time, we replace each occurrence of ‘ $\vDash _{\mathbf {L}}$ ’ with a description of the specific standard at play in each case:

(SS) $$ \begin{align} \text{If}\ \vDash_{{\swarrow}\mathbf{L}(n)/{\searrow}\mathbf{L}(n)} \Gamma \Rightarrow_n \varphi\ \text{and}\ \vDash_{\mathbf{L}(n-1)} \Gamma,\ \text{then}\ \vDash_{\mathbf{L}(n-1)} \varphi. \end{align} $$

This is just (SS), rewritten to be explicit about the fact that an inference of a given level is valid in a logic just in case it is valid in that logic’s standard for the relevant level. So ‘ $\vDash _{\mathbf {L}} \Gamma \Rightarrow _n \varphi $ ’ was short for ‘ $\vDash _{\mathbf {L}(n)}\Gamma \Rightarrow _n \varphi $ ’, and $\mathbf {L}(n)={\swarrow }{\mathbf {L}(n)}/{\searrow }{\mathbf {L}(n)}$ , while ‘ $\vDash _{\mathbf {L}} \Gamma $ ’ and ‘ $\vDash _{\mathbf {L}} \varphi $ ’ were short for ‘ $\vDash _{\mathbf {L}(n-1)} \Gamma $ ’ and ‘ $\vDash _{\mathbf {L}(n-1)} \varphi $ ’, respectively.

The observation meant to motivate mixed obedience is this: when $\Gamma $ first appears in (SS), the standard to which it ends up being measured is ${\swarrow }{\mathbf {L}(n)}$ , the premise component of $\mathbf {L}(n)$ . Yet the second appearance of $\Gamma $ is measured according to $\mathbf {L}(n-1)$ . Similarly, the first occurrence of $\varphi $ is held to ${\searrow }{\mathbf {L}(n)}$ , while the second occurrence is assessed with respect to $\mathbf {L}(n-1)$ . Of course, if $\mathbf {L}$ is pure, then ${\swarrow }{\mathbf {L}(n)} = {\mathbf {L}(n-1)} = {\searrow }{\mathbf {L}(n)}$ . But in mixed logics these equalities might not hold, in which case there appears to be a mismatch in (SS): we first assume something about the way $\Gamma $ and $\varphi $ relate to ${\swarrow }{\mathbf {L}(n)}$ and ${\searrow }{\mathbf {L}(n)}$ , and from this we expect to draw conclusions about how they relate to ${\mathbf {L}(n-1)}$ . It’s because of this mismatch that [Reference Barrio and Bezerra1, Reference Barrio, Pailos and Calderón3, Reference Roffé and Pailos37] deem the original formulation unsuitable for mixed logics. As Roffé and Pailos put it:

Scambler’s Closure demands that if (i) if $\Gamma $ is satisfied by a valuation according to [ ${\swarrow }\mathbf {ST}^\omega _+(2)$ ], then $\Delta $ is satisfied by the valuation in [ ${\searrow }\mathbf {ST}^\omega _+(2)$ ], and (ii) $\Gamma $ is valid according to [ $\mathbf {ST}^\omega _+(1)$ ], then (iii) $\Delta $ is valid in [ $\mathbf {ST}^\omega _+(1)$ ]. But there is no reason why (iii) should follow from (i) and (ii). [Reference Roffé and Pailos37, p. 749]

Thus the supposed improvement, mixed obedience, is obtained by correcting the mismatch.

[F]or spirit of the critics’ demand to be adequately preserved in a mixed metainferential setting [ $\ldots $ ] one should read the premises in the same way that one understands the premise part of [ $\vDash _{\mathbf {L}} \Gamma \Rightarrow _n \varphi $ ]. [Reference Roffé and Pailos37, p. 749]

In case of (SS) this would result in the following:

(MSS) $$ \begin{align} \text{If}\ \vDash_{{\swarrow}\mathbf{L}(n)/{\searrow}\mathbf{L}(n)} \Gamma \Rightarrow_n \varphi\ \text{and}\ \vDash_{{\swarrow}\mathbf{L}(n)} \Gamma,\ \text{then}\ \vDash_{{\searrow}\mathbf{L}(n)} \varphi. \end{align} $$

It is easy to see that every logic in the $\mathbf {ST}$ -hierarchy has this property, and so it is concluded that they are self-obedient after all. In fact, every logic whatsoever has (MSS), so it might be argued by the same token that all mixed logics are self-obedient.Footnote ¹⁷

I take it this argument too is flawed. To understand why, it is essential to mind the distinction between the relativized claim that some inference is valid in a given standard/logic, and the absolute claim that it is valid, that is, valid simpliciter. A claim of the former kind is about a theory of logic, whereas the latter are the kinds of claims made by such a theory. Moreover, as Field has noted in a different context:

someone who thinks that “disjunctive syllogism” [ $\ldots $ ] is not a valid form of inference will, if she accepts a bare minimum of mathematics, agree that the inference is classically valid, and will say that that just shows that classical validity outruns genuine validity. Those who accept disjunctive syllogism don’t just believe it classically valid, which is beyond serious contention; they believe it valid. [ $\ldots $ ] Whatever logic L one advocates, one should recognize a distinction between the concept ‘valid-in-L’ and the concept ‘valid’. [Reference Field, Caret and Hjortland15, pp. 33–34]

The mixed obedience argument goes wrong by overlooking this distinction.

For suppose that the concern about self-disobedience was as follows. “Proponents of (say) $\mathbf {ST}^\omega _+$ are committed to the claim that $\lambda \Rightarrow _1 p$ is valid in $\mathbf {ST}^\omega _+$ . Moreover, they are committed to the claim that $\lambda $ is valid in $\mathbf {ST}^\omega _+$ . Therefore, they should be committed to the claim that $p$ is valid in $\mathbf {ST}^\omega _+$ .” Clearly, the conclusion of this objection doesn’t follow from its premises. In general, this reasoning pattern doesn’t go through for mixed logics, precisely because of the aforementioned mismatch. So if this had been the concern, then pointing out the mismatch and proposing a corrected pattern which accounts for mixed standards would have hit the nail on the head. But that mischaracterizes the concern. To see why, simply note that relativized validity claims, about what is and isn’t valid in $\mathbf {ST}^\omega _+$ , aren’t up for debate in the first place. That $\lambda \Rightarrow _1 p$ and $\lambda $ are valid in $\mathbf {ST}^\omega _+$ , and $p$ is not, are simple mathematical facts. There is no room for disagreement here.

Instead, there is disagreement about what is valid simpliciter, and it’s there that the worry arises. Proponents of $\mathbf {ST}^\omega _+$ are not only committed to the relativized claims that $\lambda \Rightarrow _1 p$ and $\lambda $ are valid in $\mathbf {ST}^\omega _+$ (as everyone is). By endorsing $\mathbf {ST}^\omega _+$ , they (unlike their critics) are also committed to the absolute claims that $\lambda \Rightarrow _1 p$ and $\lambda $ are valid. And from those commitments, it does seem to follow that they are committed to the absolute claim that $p$ is valid.

The mismatch observation doesn’t apply, because we are not talking about validity in certain standards. Invoking the mismatch observation would require first substituting out absolute validity claims for relativized ones. Such substitutions are made extensively and deliberately in [Reference Barrio, Pailos and Calderón3, Reference Roffé and Pailos37], with the idea that this merely makes things more explicit. But in fact, due to the distinction emphasized above, such substitutions don’t preserve meaning. As a consequence, applying them here leads to a major misrepresentation of the point.

Note finally that, contrary to (SS), (MSS) can’t express any requirement on absolute validity claims. Each logic $\mathbf {L}$ represents a body of absolute validity claims, namely to the effect that the validities simpliciter are precisely all and only the validities of $\mathbf {L}$ . But at each $n$ , the $n$ -inferences which $\mathbf {L}$ validates, and hence claims are valid simpliciter, are all and only those that are valid in $\mathbf {L}(n)$ . Therefore (SS) gets things exactly right by only considering validity in $\mathbf {L}(n)$ and $\mathbf {L}(n-1)$ , while (MSS) blunders by looking at ${\searrow }\mathbf {L}(n)$ and ${\swarrow }\mathbf {L}(n)$ , which simply do not correspond to the absolute validity claims actually made by $\mathbf {L}$ .Footnote ¹⁸

Mixed obedience was introduced with the aim of doing better justice to the idea underlying the demand for self-obedience. Unfortunately, it rests on a misunderstanding of said idea.