Metainferential Reasoning on Strong Kleene Models

Barrio et al. (Journal of Philosophical Logic, 49(1), 93–120, 2020) and Pailos (Review of Symbolic Logic, 2020(2), 249–268, 2020) develop an approach to define various metainferential hierarchies on strong Kleene models by transferring the idea of distinct standards for premises and conclusions from inferences to metainferences. In particular, they focus on a hierarchy named the ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {S}\mathbb {T}$\end{document}-hierarchy where the inferential logic at the bottom of the hierarchy is the non-transitive logic ST but where each subsequent metainferential logic ‘says’ about the former logic that it is transitive. While Barrio et al. (2020) suggests that this hierarchy is such that each subsequent level ‘in some intuitive sense, more classical than’ the previous level, Pailos (2020) proposes an extension of the hierarchy through which a ‘fully classical’ metainferential logic can be defined. Both Barrio et al. (2020) and Pailos (2020) explore the hierarchy in terms of semantic definitions and every proof proceeds by a rather cumbersome reasoning about those semantic definitions. The aim of this paper is to present and illustrate the virtues of a proof-theoretic tool for reasoning about the ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {S}\mathbb {T}$\end{document}-hierarchy and the other metainferential hierarchies definable on strong Kleene models. Using the tool, this paper argues that each level in the ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {S}\mathbb {T}$\end{document}-hierarchy is non-classical to an equal extent and that the ‘fully classical’ metainferential logic is actually just the original non-transitive logic ST ‘in disguise’. The paper concludes with some remarks about how the various results about the ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {S}\mathbb {T}$\end{document}-hierarchy could be seen as a guide to help us imagine what a non-transitive metalogic for ST would tell us about ST. In particular, it teaches us that ST is from the perspective of ST as metatheory not only non-transitive but also transitive.

1 Introduction [1] and [2] develop an approach to define various metainferential hierarchies on strong Kleene models by transferring the idea of mixed inferences to the metainferential level. In particular, their investigations reveal that a particular hierarchy named ST is definable where the inferential logic at the bottom of the hierarchy is the non-transitive logic ST advocated by [3] and [4] but where each subsequent metainferential logic 'says' about the former logic that it is transitive. While [1] suggest that this hierarchy provides metainferential logics where each subsequent level is 'in some intuitive sense, more classical than' the previous level, [2] proposes an extension of the hierarchy through which a 'fully classical' metainferential logic can be defined. Both [1] and [2] explore the hierarchies from a semantic perspective and every proof proceeds by a rather cumbersome reasoning about those semantic definitions.
The primary aim of this paper is to develop and illustrate the use of a prooftheoretic tool obtained by combining ideas from nested sequent calculi with labelled sequent calculi for reasoning about ST and the other metainferential hierarchies definable on strong Kleene models. To that purpose, Section 2 presents the approach to metainferential hierarchies on strong Kleene models developed by [1] and Section 3 presents a "labelled nested" sequent calculus based on the definitions provided in Section 2. This tool is then employed to make some remarks about [1]'s metainferential hierarchy ST and [2]'s 'fully classical' metainferential logic. In particular, it is shown in Section 4 that each level in the ST hierarchy is non-classical to an equal extent, a result which is extended in Section 5 to the 'fully classical' metainferential logic presented by [2]. Moreover, it is also shown that every metainference of the 'fully classical' metainferential logic is equivalent to an inference of the original non-transitive logic ST, and that the former is thus the latter 'in disguise'. Finally, the paper proposes in Section 6 that the hierarchy ST can fruitfully be understood as a tool to help us imagine what ST would tell us about ST if ST is used as metatheory where the most interesting observation being that ST is from the perspective of ST both transitive and non-transitive.

Language and Models
This section presents the language and models that will form the basis for the proof theory.
Definition 2.1 (The language) Let L be a propositional language based on a countable set of propositional variables, a nullary connective λ, a unary connective ¬ and the binary connective ∨. Let FORML L be the set of formulas of L.
We shall use upper case Latin letters A, B etc as metalinguistic variables for formulas in general and lower case Latin letters p, q etc as metalinguistic variables for propositional variables.
In addition to having formulas that are assigned values and can satisfy certain standards on strong Kleene models, we are interested in metainferences as objects that can satisfy certain appropriate standards on strong Kleene models, that is, as objects that can feature in a satisfaction relation. Following [1] we will define a hierarchy of metainferential objects as follows using the notation [. Metainferential objects can thus be seen as binary connectives that applies to sets of objects. They are however not part of L even if they contain objects from FORM L . Moreover, while one might be tempted to add numerals to the objects in order to identify its level, this is not necessary since X i in [X 1 , . . . , X n ⇒ Y 1 , . . . , Y m ] will be an object of the previous level if the level is > 0 or a formula if the level is 0. 1 2 , 0} is a strong Kleene valuation just in case V(λ) = 1 2 and the following conditions are satisfied for every complex formula:

Definition 2.3 (Strong Kleene valuations) A function
Following [5], a formula on trivalent models can be either strictly or tolerantly satisfied. This is made precise as follows: [1] extends the notion of satisfaction from formulas to metainferential objects using a hierarchy of metainferential standards based on the strict-tolerant distinction. Informally presented, the idea is as follows: -A formula can satisfy one out of two standards, s and t.
-A metainference of level 0 can satisfy one or more out of four standards, st, ts, ss and tt. -A metainference of level 1 can satisfy one or more out of sixteen standards: stst, tsst, ssst, ttst stts, tsts, ssts, ttts stss, tsss, ssss, ttss -sttt, tstt, sstt, tttt -A metainference of level 2 can satisfy one or more out of 256 standards which we shall not list.
An inductive definition can thus be given as follows: Definition 2.5 (The standards) -s and t are formula standards.
-if x and y are formula-standards, then xy is a metainferential standard of level 0.
-if x and y are metainferential standards of level n, then xy is a metainferential standard of level n + 1.
Following [1], the notion of satisfaction can now be extended as follows: Definition 2.6 (Satisfaction of metainferences) If [Γ ⇒ Δ] is a metainferential object of level n and xy a metainferential standard of level n, then, Finally, validity for the various inferential and metainferential logics is now defined as follows: Unsurprisingly, the various inferential and metainferential logics definable on strong Kleene models recently discussed in the literature fall out of this definition. We shall in general refer to a particular logic through its standard, e.g. the logic st or the logic tsst. There is however one logic definable on strong Kleene models that is not captured by this approach and which on occasion is discussed in the literature, e.g. by [6], namely that definable using ≤ as follows: Γ Δ iff every V is such that min(V(A) ∈ Γ ) ≤ max(V(B) ∈ Δ). This is an acceptable limitation considering the aim of this paper.

The HST Calculus
This section presents a sequent calculus representing metainferential hierarchies on strong Kleene valuations based on the definitions provided in the previous section. The hierarchical strict-tolerant calculus will be a labelled sequent calculus in the sense that the rules will not manipulate formulas directly as in the case of a standard sequent calculus, but rather labelled formulas and labelled metainferential objects. To that purpose we shall introduce one label for each standard to thereby obtain labelled formulas (e.g. s:A) and labelled metainferential objects of the form x:[Γ ⇒ Δ] where Γ and Δ will be formulas if it is a metainferential object of level 0 and metainferential object of level n if it is a metainferential object of level n + 1. Since the calculus will thus contain expressions that look like standard sequents nested within each others, it can also rightly be described as a sequent calculus for nested sequents. It is thus a labelled nested sequent calculus.
While the calculus is straightforwardly modified to also include the addition of socalled "antivalidities" as introduced into the debate by [7] and thus also capture the arguments presented by [7], such modifications are purposely left out to keep things simpler and more straightforward. The reader with an interest in such issues is invited to make the appropriate amendments. For a sequent expression of type (α, x), we refer to α as the level and x as the standard. We shall use x : X to refer to an arbitrary sequent expression of any type. Moreover, we let X (α,x) and Y (α,y) designate finite multisets of sequent expressions of type (α, x) and (α, y) respectively. We also let X designate the multiset obtained by removing labels from the members of X (α,x) , i.e. X = {X | x:X ∈ X (α,x) }. Note also that it follows from the notation that a metainference of level n is represented by a sequent expression of level n + 1.
The following are examples of typed sequent expressions.
As the examples suggest, it will be increasingly difficult to read the sequent expressions in order to decipher their level and thus their meaning. Luckily the construction is compositional and we are in general only interested in the inductive steps from level n to level n + 1.
Rules for sequent expressions of level > 0: This calculus is well-behaved from the perspective of structural proof theory as elucidated by [8] and [9]. In particular, we have the following lemmas and theorems where each proof is obtainable through a straight-forward adaptation of the corresponding proof in [9]. -If D is an initial sequent or conclusion of a zero-premise rule, then We'll say that a rule is height-preservingly admissible (HP-admissible) just in case whenever there is a derivation of the premise-sequent with height n then there is a derivation of the conclusion-sequent with height ≤ n.

Lemma 3.4 (Weakening and contraction) The following rules are HP-admissible in HST:
The proofs are obtained by slight modifications on the proofs of proposition 4.4 and theorem 4.12 in [9].

Lemma 3.5 (Inversion) The inversion of each primitive HST-rule is HP-admissible in HST.
Proof Proof is a modification of proposition 4.11 in [9]. Definition 3.6 (Formula complexity) The complexity of a L-formula A,|A|, is defined inductively as follows: Definition 3.7 (Expression weight) Suppose that x:X is a sequent expression. Then the weight of X, W(X), is defined as follows: if X is a formula A, then The following rule is admissible in HST: Proof By double induction on the weight of X and the sum of the heights of the derivations of the premise-sequents. See theorem 4.13 in [9].
Proof The right-to-left direction proceeds as usual by induction on the height of a derivation. The left-to-right direction proceeds as usual via the construction of a reduction tree for every underivable sequent from which a countermodel for that sequent is extracted. We present here a few details from the latter proof.
Assume that Γ ⇒ Δ is underivable. It follows that we can construct a tree above it by applying the rules of the HST calculus backwards until each branch ends with a sequent containing only labelled propositional variables and labelled λ's. At least one branch will be such that the leaf is not an initial sequent or a zero-premise rule of the HST calculus. We pick such a branch B and define a function V from the set of propositional variables of L to {1, 0, 1 2 } as follows where Γ ⇒ Δ is the leaf-sequent of B: The definition of V is extended to complex formulas and λ in accordance with definition 2.3. The satisfaction relation is defined in accordance with definitions 2.4 and 2.6. We can now show by induction on the complexity of a formula that for every It is left to show by an induction on the set of standards from definition 2.5 that also the following statements hold for every Γ ⇒ Δ ∈ B: -If xy: With Γ ⇒ Δ ∈ B, it follows that for every typed sequent expression x:X ∈ Γ , V x X and for every typed sequent expression y:Y ∈ Δ, V y Y . V is thus a countermodel for the sequent Γ ⇒ Δ.

Corollary 3.11 ⇒ xy:[Γ ⇒ Δ] is derivable if and only if Γ xy Δ
Before we dive into the perhaps more serious applications of this proof-theoretic tool, we shall first provide a few illustrations of its immediate usefulness. As a first curiosity we shall show that the inferential logic defined with the standard st is nontransitive using the admissibility of cut. Proof The empty sequent follows by inversion, but the empty sequent is excluded by design. Proof If that sequent is derivable then theorem 3.8 and lemma 3.13 together imply that ⇒ st:[ ⇒ ] is derivable but this is excluded by lemma 3.12.
With inspiration from [10], we have thus shown the nontransitivity of a logic typically defined proof-theoretically by rejecting cut using the admissibility of cut.
In fact, we actually have a tool which can be used as a "metasequent" calculus for the four logics st, ts, tt and ss. In the case of st, for example, the following sequents are derivable: To provide further familiar facts about the logics (and also for some propositions in the next section), the following lemma for transforming labels will be useful.

Lemma 3.15 (label transformation) (a) If
A is a formula, then following rules are HP-admissible: Proof The proofs straightforwardly follow the general strategy in structural proof theory to show the height-preserving admissibility of a rule.
Regarding (a), we prove t/sL and s/tR simultaneously. We here focus on t/sL. Base case: Assume 0 t:A, Γ ⇒ Δ. If A is a propositional variable and t:A ∈ Δ, then s:A, Γ ⇒ Δ is also an initial sequent. If Γ ⇒ Δ is an initial sequent or an instance of zero-premise rule, then s:A, Γ ⇒ Δ is also an initial sequent or an instance of a zero-premise rule and thus 0 s:A, Γ ⇒ Δ holds.
Inductive step: Assume n+1 t:A, Γ ⇒ Δ. If t:A is principal and of the form ¬B, it follows that n Γ ⇒ Δ, s:B. By the inductive hypothesis we obtain n Γ ⇒ Δ, t:B and by one application of ¬L we obtain n+1 t:¬B, Γ ⇒ Δ. The case for ∨L is similar. If t:A is not principal, the sequent is obtained with some kpremise rule R and it is thus the case that n t:A, Γ i ⇒ Δ i for every i < k. We apply the inductive hypothesis to obtain n s:A, Γ i ⇒ Δ i and one application of R delivers n+1 s:A, Γ ⇒ Δ.
Regarding ( This lemma has two immediate corollaries. The first concerns the relationship between valid inferences in the four logics as familiar from [5]. (Relationships between st, ts, ss and tt) The following sequents are derivable:

Corollary 3.16
ts:X ⇒ tt:X ts:X ⇒ ss:X tt:X ⇒ st:X ss:X ⇒ st:X The second concerns the relationship between inferences of st and ts on the one hand, and formulas that are tolerantly and strictly satisfied on the other hand.

Corollary 3.17
The following rules are admissible were x = y: Proof By inversion and straightforward applications of the relevant rules.
An immediate consequence of that corollary is the following result by [11] and [12]: The corresponding result for ts as presented by for example [13] is obviously also available: What is interesting here is not the fact that these propositions hold about the four logics st, ts, tt and ss, but the ease with which we have obtained them using proof analysis.

Approximating Classicality with the ST-Hierarchy?
The results in the previous section concerned only inferences and metainferences, not inferences of metainferences and so forth. Given the generality of our calculus, it should be clear that we can also use it to prove facts about "higher-order" metainferences. To illustrate that we shall have a look at what we can say about the ST-hierarchy of metainferential standards presented by [1]. The basic idea with the hierarchy is to "reproduce" the "st-phenomenon" at a metainferential level by defining a hierarchy of metainferential standards where the standard for being a premise in a sound inference is stricter than the standard for being a conclusion. For our purposes, we can replicate the hierarchy of standards with labels using the following definition: The standard for metainferences of level 1 is thus tsst as opposed to the standard stst, and for metainferences of level 2 we have the standard sttstsst as opposed to the standard tssttsst. We will use ST n to refer to the nth level in ST, so that ST 0 is st, ST 1 is tsst, and so on.
What is interesting about ST according to [1] is that "in some intuitive sense, TS/ST is classical to a greater degree than ST", and moreover that we obtain at each ST n+1 a "metainferential" logic which is supposedly more similar to classical logic than ST n because stage n is according to stage n + 1 transitive. To illustrate this, consider the following observation about tsst: Proposition 4.2 (tsst concerns a transitive logic) The following sequent is derivable:  Again, while these observations are already made by [1], our proofs thereof are obtained using proof analysis. In particular, the key ingredients are our cutelimination theorem in 3.8 and the label transformation lemmas. Our proofs are thus arguably more elegant and easier to read than those presented involving semantic reasoning by [1].
In addition to establishing the fact that each stage is a reflexive "metainferential" logic which concerns a transitive "metainferential" logic, we can also establish that each stage is inconsistent, again using proof analysis. Proof The following pieces of reasoning are admissible: With st being inconsistent, the following proposition follows:

Proposition 4.7 For every xy ∈ ST, xy is inconsistent.
Finally, we obtain thus the following: The conclusion should thus not be that each level is classical to a greater extent than the previous level as we transcend up in the hierarchy as if the next level takes us closer to classical logic (even if we never reach classical logic), but rather that each stage is non-classical to the same extent. This should not be too surprising considering how each stage in the hierarchy is a st-ish logic for the previous stage obtained by what amounts to a strict-tolerant standard for that stage.

A "fully classical" Metainferential Logic?
Following the observation that no stage in the ST-hierarchy is classical, [2] presents a way to "recovers every classically valid metainference of every level". This consists in defining a collection of metainferences STω of any level in such a way that we can understand it "as the union of" each x ∈ ST. In this way then, we are supposed to obtain a "fully classical" (metainferential) logic. In the concluding remarks in [2], it is observed that there still is plenty work to do in relation to these logics and truth theories. For example, it seems not easy to imagine a proof theory for them [2].
As it turns out, it is straightforward to extend the HST calculus to a sequent calculus for STω. Let us thus proceed with the definitions. In [2], a definition of STω is provided which is equivalent to the following: Suppose that X is a metainference of level j > 0. Then X ∈ STω if and only if for every V, V ST j X.

Theorem 5.7 (Cut) The following rule is admissible in EHST:
Γ ⇒ Δ, x:X x:X, Γ ⇒ Δ Γ, Γ ⇒ Δ, Δ As above in the HST-calculus, cut-elimination does not imply that a metainferential logic defined with the calculus is transitive. A reasonable question to ask now is thus whether ω really is a "fully classical" metainferential logic? Proposition 5. 8 Reasoning in ω about metainferential levels in ST is not transitive.
Proof By proposition 4.6 it follows that for any metainferential level, there is an X such that: ⇒ ω: By cut-elimination, it follows that the sequent is not derivable.
It follows that ω is not "more" classical than anything in ST. In fact, we can show that ω is st in disguise. Definition 5.9 (Notation) Let X ←→ Y mean that the following rules are admissible: With the rules for introducing a metainference having the same shape as the rules for introducing a formula of the form (A 0 ∧ . . . ∧ A n ) ⊃ (B 0 ∨ . . . ∨ B m ) where ∧ and ⊃ are defined as ¬(¬A ∨ ¬B) and ¬A ∨ B respectively, it is relatively evident that we can engage in a process of flatting metainferences. For example, it follows by corollary 3.17 that the following claims hold where Γ ⊃ Δ abbreviates With that established we proceed to observe the following: To make sense of this theorem, it is useful to observe that ω does not, despite its label, represent a limit. Instead, it is simply a collection of metainferences of various levels. Every metainference of ω will be of a particular finite level, and can thus be reduced according to proposition 5.11. Moreover, and with that in mind, this result shouldn't actually be particularly surprising considering how the ST-hierarchy and ω is defined and how for example tsst is the metainferential analogue of st since ts is a stricter standard than st in the same way as s is a stricter standard than t.

Imitating st as Metatheory
Since we have used the proof-theoretic tool developed in this paper to illustrate problems with the interpretation of the ST-hierarchy proposed by [1] and [2], it seems appropriate to use the concluding remarks of this paper to engage in some speculation about whether we can utilise the results obtained with the tool to provide an alternative interpretation of the ST-hierarchy, and whether this could be used to make sense of st.
The flatting of the ST-hierarchy into st suggests that ST-hierarchy is merely a metainferential twist on the st-phenomenon, as if the ST-hierarchy doesn't tell us anything that we couldn't already express within st, and one could thus argue that the hierarchy is somehow superfluous.
On With that in mind we can reason as follows. Under the assumption that the valid inferences about the material conditional in st represent the metainferences that hold of st within st, st is transitive according to st. In other words, according to st it is the case that the inferences from anything to the liar and from the liar to anything together imply that anything follows from anything. This observation corresponds to that made by [14] with regard to a validity predicate defined in st along the lines of the material conditional. From the perspective of a classical metatheory according to which st is non-transitive, that claim is false, and [14] considered their observation as presenting a problem for such an approach to defining a validity predicate in st. However, we are no longer supposed to think of st from the perspective of classical logic. Instead, we are considering st from the perspective of st, and what if st really is transitive when st is the metatheory for st? After all, st is non-transitive from within a classical metatheory because assuming otherwise leads to inconsistency. With st as metatheory, however, the inconsistency is not an issue, and it follows that st can be transitive according to st.
Continuing down the rabbit hole then, we also note that st tells us about st that anything implies the liar and that the liar implies anything. Now, does it follow, since those facts imply that anything implies anything, that anything implies anything? On the one hand, we can find formulas A and B such that ¬(A ⊃ B) follows from no premises, a fact which is reasonably interpreted as that it is not the case that anything implies anything. On the other hand, we have cases such as ¬(λ ⊃ λ) and (λ ⊃ λ) which are both valid according to st. Taking each statement to represent an inference as suggested above, there are thus inferences that are both valid and not valid in st according to st. From the perspective of st then, it seems reasonable to think of transitivity of entailment in the same way; that it is (metainferentially) valid, but there are counterexamples. While a classically minded referee would certainly protest at this point since being valid and having a counterexample are supposed to be 'at least contraries', such a protest would just illustrate the uphill battle faced by an advocate of paraconsistent metatheory as generalised to include metainferences. Indeed, could the moral be that you're free to apply transitivity when reasoning within st about st as long as you're willing to accept that the conclusion you draw is a dialetheia?