Relevant entailment and logical ground

Pierre Saint-Germier<fnref n="1" />,<fnref n="2" />*, Peter Verdée<fnref n="1" /> and Pilar Terrés
Villalonga<fnref n="1" />,<fnref n="3" />

Abstract

According to an intuitive picture of relevant entailment, an entailment
is relevant if all the formulas it contains contribute to its validity. In
this paper, we provide a ground-theoretic analysis of this notion of
contribution, and as a result of relevant entailment. We build a sys-
tem of bilateral logical grounding within which we can derive classical
entailment and analyze the contribution of premises and conclusions,
in terms of a certain type of connection between their respective
logical grounds. The resulting framework is shown to have a strong
unifying and explanatory power by bringing together the notions of
grounding and relevance and explaining the commonalities and dif-
ferences between alternative contribution-based accounts of relevance.

**Keywords:** relevance, logical grounding, entailment, inferentialism, bilateralism

<footnote n="1">¹*Institut Supérieur de Philosophie, Université Catholique de Louvain, Place Cardinal Mercier 14, Louvain-la-Neuve, 1348 , Belgium.</footnote>

<footnote n="2">²CNRS-IRCAM-Sorbonne Universités, UMR 9912, 1 place
Igor-Stravinsky, Paris, 75004, Paris.</footnote>

<footnote n="3">³Facultad de Filosofía, Universidad de Salamanca, Edificio
F.E.S. Campus Miguel de Unamuno, Salamanca, 37007, Spain.</footnote>

<footnote>*Corresponding author(s). E-mail(s): saintgermier.p@gmail.com;</footnote>

<footnote>Contributing authors: peter.verdee@uclouvain.be;</footnote>

<footnote>pilar.terres@gmail.com;</footnote>
# 1 Introduction

Entailment is traditionally introduced as the converse of the relation of deducibility (Moore, 1920). Since deducibility involves truth-preservation, it follows that a relation of entailment is usually taken to hold between a set Γ of sentences and a sentence A only if it is impossible that all the sentences in Γ are true while A is not. Whether truth-preservation is sufficient has been questioned, notably in light of infamous paradoxes of relevance. Consider the following arguments, followed by a straightforward formalization:

(1) a. The People's Republic of China is a democracy. The People's Republic of China is not a democracy. Therefore, Elvis is still alive.

b. $p, \neg p \supset q$

(2) a. Elvis is dead. Therefore, Donald Trump is a socialist or Donald Trump is not a socialist.

b. $p \supset q \lor \neg q$

(3) a. Elvis is dead. Donald Trump is not a socialist. Therefore, Elvis is dead.

b. $p, q \supset p$

Clearly, something sounds wrong with all these arguments, even though they are all (for all we know) truth-preserving.<fnref n="1" /> The consensual diagnosis is that in all such arguments, at least some premise is irrelevant to the conclusion, or the conclusion is irrelevant to the premises. And these irrelevancies are responsible for the oddity of the arguments.

Opinions differ about the theoretical conclusions to draw from such examples. Some deny that the arguments above instantiate the logical relation of entailment at all. For them, it is not enough that the truth of the premises is preserved in the conclusion. All the premises must be *relevant* to the conclusion for a relation of entailment to exist at all. In other words, entailments are relevant by nature. Let us call such theorists *entailment relevantants*.<fnref n="2" /> Others, e.g. classical logicians, accept that in all the arguments above, the premises entail the conclusion. The only reasons why they sound wrong are merely pragmatic: it would be misleading, unwise, or inefficient to advance such arguments in conversation. From this point of view, entailments may very well be relevant, but only contingently so. Let us call those theorists who admit irrelevant entailments *entailment irrelevantants*.<fnref n="3" /> Although classical logicians are entailment irrelevantants, they are not necessarily the only ones. Intuitionists, and

<footnote n="1">¹The first one is vacuously truth-preserving because the two premises cannot be true together. But it is truth-preserving nonetheless.</footnote>

<footnote n="2">²Among the most prominent entailment relevantants, one may cite Smiley (1959), Anderson and Belnap (1975), Tennant (1979), and Read (1988).</footnote>

<footnote n="3">³Among entailment irrelevantants who specifically addressed relevant entailments as a proper subset of entailments simpliciter, we may cite Gärdenfors (1976), Weingartner (1985), Weingartner and Schurz (1986), Schurz (1991) Verdée et al. (2019), and Brauer (2020).</footnote>
those dialetheists who take LP to be the one true logic, also count as entailment irrelevantists: the former reject (2), and the latter reject (1), but both at least accept (3), and this is enough to count them as entailment irrelevantists.

Whatever stand one takes on this issue, the general question

(QRE) What makes a relevant entailment relevant ?

remains to be answered. So there is a need for a general answer to QRE that, ideally, remains independent of the disagreement over entailment relevantism. Whether entailments are relevant by nature or not, all parties in the debate need an independent criterion for classifying the arguments (1), (2), and (3) above as irrelevant.

In this paper, we approach the problem from the standpoint of a classicist entailment irrelevantism, for the sake of simplicity. In other words, we shall only address the question

(QCRE) What makes a relevant classical entailment relevant ?

The central notion in terms of which our answer to QCRE will be elaborated, the notion of logical ground, is not particularly tied to classical logic, though. We see no a priori reason why the notion of logical ground may not be adapted to cover non-classical logical grounding explanations.<fnref n="4" /> But we will not address this issue directly here. So from now on, unless specified otherwise, “entailment” will mean “classical entailment”, and “validity” will mean “classical validity”.

In the most general setting where entailments connect a (possibly empty) plurality of premises with a (possibly empty) plurality of conclusions, we propose that what makes a relevant entailment relevant is the fact that all premises and conclusions *jointly contribute* to the entailment. This suggestive metaphor is then analyzed in terms of logical grounding: in order to determine how premises and conclusions jointly contribute to an entailment, we need to look at the way their logical grounds are connected to one another. Our ground-theoretical account provides a philosophically motivated and formally precise characterization of this sort of connection.

The architecture of the paper is the following. We first present the target of our analysis, which is the property, for a premise or a conclusion, of contributing to an entailment. Four ways to understand this notion are distinguished, and mapped to four accounts of relevant entailment (§1). We then describe the basic building block in terms of which the notion of contribution will be analyzed, namely the notion of logical ground, of which we give a bilateral account, formally explicated by the system GK (§3). The next step is to derive the notion of entailment from this bilateral account of ground. In order to do this, we build an integrated system GLK which extends GK and allows us to reason simultaneously about logical grounding and validity. Within GLK, premises and

<footnote n="4">⁴See for example Poggiolesi (2020) for a recent attempt to account for the logical grounding of relevant implication.</footnote>
conclusion can be ascribed logical grounds, and classical propositional entailments can be derived on the basis of appropriate relations between the logical grounds of premises and conclusions (§4). The final move is to explicate each of the various notions of premise- and conclusion-contribution distinguished in §1, in terms of connections between their logical grounds (§5). The proofs of all the technical results mentioned in the text without justification are gathered in the Appendix at the end of the paper.

## 2 Relevant classical entailment

In this section, we present our preferred characterization of the class of relevant entailments among classical propositional entailments. Following an intuition recently put forward and explored at length by Brauer (2020), we shall understand the relevant classical entailments to be those where all the formulas *contribute* to the entailment. Our modest aim in this section is to introduce and distinguish various ways to understand this general idea.

Before entering the analysis of this notion of contribution, let us introduce some terminology. We shall use the term *sequent* to refer to a certain class of strings of symbols namely collections of formulas (of an antecedently specified language) separated by the consecution symbol ">". The members of the leftmost collection are called premises and the members of the rightmost collection are called conclusions. We shall use "sequent element" to refer to formulas as playing the role of premise or conclusion in a sequent, when it is not necessary to specify which.<fnref n="5" /> Sequents may vary in format depending on the kind of collections separated by the consecution symbol. For instance, sequents in SET-FMLA format take a set of premises and a unique conclusion<fnref n="6" />. Sequents in MSET-MSET format take a multiset of premises and a multiset of conclusions<fnref n="7" />. *Valid* sequents are sequents that satisfy the requirement of truth-preservation. *Entailments* are traditionally thought as valid sequents in SET-FMLA format, although the notion is sometimes extended to the validity of SET-SET sequents—with the charge of providing a clear meaning to multiple conclusions. For specific reasons that will appear shortly, we shall however work by default with sequents in MSET-MSET format. Our general motivation for having multiple conclusions is that it will be useful to exploit the symmetry between premises and conclusions that this representation allows. Our general motivation for conceiving multiple premises and multiple conclusions as multisets is that the same sequent element may make two distinct contributions. Having multisets of premises and conclusions allows us to keep track

<footnote n="5">⁵Sequent elements are individuated by their specific role of premise or conclusion within a sequent. If we say that A and B are the same sequent element, we mean that A and B are the same formula occurring as a premise or the same formula occurring as a conclusion. If the same formula occurs both as a premise and as a conclusion, its premise-occurrence does not count as the same sequent-element as its conclusion-occurrence.</footnote>

<footnote n="6">⁶The notation for the various sequent formats is borrowed from (?, §1.21).</footnote>

<footnote n="7">⁷Informally speaking, a multiset is just like a set except that the same element may occur in it several times. A formal definition of multisets, relations and operations on multisets is given in section A.1 of the technical appendix</footnote>
of those various contributions. By default, then, upper case Greek letters will
represent multisets of formulas.

With these precisions in mind, let us go back to the general intuitions
shared by all contribution-based accounts of relevant entailment. Intuitively,
we consider a premise as irrelevant if it is so unrelated to the other premises
and the conclusion of the entailment in which it occurs that it does nothing to
guarantee the truth of the conclusion, assuming the truth of the other premises.
This is the case, for example, of the premise *p* in (2-b). In paradigmatic exam-
ples of irrelevant premises such as this one, an irrelevant premise owes its
presence in an entailment only to the monotonicity of truth-preservation, and
not to the logical work it does to ensure the validity of the entailment in which
it is involved.<fnref n="8" />

Contribution-based accounts of relevant entailment generalize this insight
to all sentences involved in an entailment, including conclusions. It says that
a sequent expresses a relevant entailment if all the premises *and all the
conclusions* it contains contribute to its validity.

The idea that the conclusion of a sequent contributes something to its
validity might seem obscure at first sight, but pairs of valid and invalid sequents
such as:

(4) a. $p, p \supset q > q$

b. $p, p \supset q > r$

suggest that a sequent may be made invalid by the inadequate contribution of
the conclusion, rather than of the premises<fnref n="9" />.

Furthermore, this symmetrical treatment of premises and conclusions pro-
vides a unified diagnosis of the irrelevance of sequents such as (1-b) and (2-b).
On the one hand, the entailment (2-b) is seen as irrelevant because the con-
clusion (i) is tautological and (ii) shares no variable with the premise. From
the standpoint of a contribution-based account of relevance, those symptoms
indicate that the premise *p* (i) need not and (ii) cannot possibly contribute
to the validity of (2-b). On the other hand, the entailment (1-b) is seen as
irrelevant because the premises (a) are inconsistent and (b) share no variable
with the conclusion. Symmetrically, those symptoms may be taken to indi-
cate that the conclusion *q* (a) need not and (b) cannot possibly contribute
to the validity of (1-b). Thus, by considering conclusions equally as potential
contributors to the validity of sequents, contribution-based accounts provide a
unified framework within which premise-relevance, conclusion-relevance, and
entailment-relevance may be defined.

What needs to be clarified, then, is this notion of *contributing* to the validity
of a sequent. As it turns out, there are various ways to unpack this notion.

<footnote n="8">⁸By the monotonicity of truth-preservation, we refer to the well-known property that if Γ > A preserves truth, then Π > A also preserves truth for all Π ⊇ Γ.</footnote>

<footnote n="9">⁹This point has been made by (Brauer, 2020, p. 342).</footnote>
**2.1 Contribution as difference-making**

A clearly *sufficient* condition for contributing to the validity of a sequent is to
make a difference to its validity in the following sense:

**Definition 1** If Γ > Δ is valid,

• the premise $A \in \Gamma$ makes a difference to the validity of $\Gamma > \Delta$ if $\Gamma - \{\{A\}\} > \Delta$ is invalid.

• the *conclusion* $B \in \Delta$ *makes a difference* to the validity of $\Gamma > \Delta$ if $\Gamma > \Delta - \{\{A\}\}$ is invalid.<fnref n="10" />

One obtains a first definition of relevance by requiring that all sequent
elements contribute by way of difference-making:

**Definition 2** Γ > Δ is a relevant<sub>DM</sub> entailment if:

1. Γ > Δ is valid, and

2. each premise $A \in \Gamma$ and each conclusion $B \in \Delta$ make a difference to the validity of $\Gamma > \Delta$.

This notion of difference-making is tightly linked to the notion of perfect validity.<fnref n="11" /> Let us say that a sequent $\Gamma > \Delta$ is a *subsequent* of $\Pi > \Sigma$ if $\Gamma \subseteq \Pi$ and $\Delta \subseteq \Sigma$. A sequent $\Gamma > \Delta$ is a *proper* subsequent of $\Pi > \Sigma$ if in addition $\Gamma \neq \Pi$ or $\Delta \neq \Sigma$. Then perfect validity is defined as follows:

**Definition 3** Γ > Δ is *perfectly valid* if:

1. Γ > Δ is valid, and

2. no proper subsequent of Γ > Δ is valid.

**Proposition 2.1** A sequent $\Gamma > \Delta$ is perfectly valid iff it is valid and all its premises and conclusions make a difference to the validity of $\Gamma > \Delta$.

It is debatable, however, whether making a difference is a necessary
condition for contributing to the validity of a sequent. According to the
difference-making account it is, but in general, one may genuinely contribute
to a collective achievement without making a difference. When a parliamentary
group wins a vote with a large majority, all votes contributed to the positive
outcome, even though no vote, on its own, made a difference. Yet, we consider
that the members of that group who took part in the vote jointly contributed

<footnote n="10">¹⁰This notion of difference-making is structurally similar to a notion of difference-making for grounding defined by Krämer and Roski (2017).</footnote>

<footnote n="11">¹¹The definition of perfect validity was first introduced by Tennant (1984) in the context of SET-SET sequents.</footnote>
to the positive outcome. For analogical reasons, we shall not consider that a sequent element needs to make a difference to an entailment in which it occurs in order to make a genuine contribution.<fnref n="12" />

The idea of non-decisive contribution can be developed in two orthogonal ways.

## 2.2 Non-decisive contribution I: Brauer

One way for a formula to contribute non-decisively to the validity of $Γ > Δ$ is to occur in some perfectly valid subsequent of $Γ > Δ$. By being part of a perfectly valid subsequent, it fulfills the sufficient condition for contributing. It need not however be part of every perfectly valid subsequent of $Γ > Δ$ in order to contribute. Consider for example the sequent:

$$ (5) \quad p \wedge q, q \wedge p > p $$

Each premise on its own is part of a perfectly valid sequent, and may be seen as a contributor for that reason, even though leaving it out would not destroy the validity of the initial sequent. This is the view explored by Brauer under the label of “relevance” in (Brauer, 2020).<fnref n="13" />

One then obtains a second definition of relevance:

**Definition 4** $Γ > Δ$ is a relevant<sub>B</sub> entailment if:

1. $Γ > Δ$ is valid, and

2. for all $A \in Γ + Δ$, there is a perfectly valid subsequent $Π > Σ$ of $Γ > Δ$ such that $A \in Π + Σ$.

An important property of this notion of relevance<sub>B</sub> is that it is closed under the rule of Amalgamation:

$$ \frac{\Gamma > \Delta \qquad \Pi > \Sigma}{\Gamma, \Pi > \Delta, \Sigma} \quad (\text{Amalgamation}) $$

**Proposition 2.2** If $Γ > Δ$ is relevant<sub>B</sub> and $Π > Σ$ is relevant<sub>B</sub>, then $Γ, Π > Δ, Σ$ is relevant<sub>B</sub>.

In fact, the notion of relevance<sub>B</sub> can be characterized in terms of relevance<sub>DM</sub>, Amalgamation, and Contraction:

<footnote n="12">¹²The idea that a contribution is not required to make a difference was put forward by Yablo (ming), and plays an important role in Brauer’s views on the contribution of formulas to the validity of sequents (Brauer, 2020, p. 342).</footnote>

<footnote n="13">¹³What we introduce below as a definition of relevance<sub>B</sub> is presented by Brauer only as a property that is provably equivalent to “relevance” as he defines it (Brauer, 2020, p. 445). That property being somewhat simpler to state, we decided to adopt it as our official definition.</footnote>
**Proposition 2.3** *The set of relevant<sub>B</sub> sequents is the smallest set comprising the set of relevant<sub>DM</sub> sequents and closed under (Amalgamation), and the following Contraction rules:*

$$ \frac{\Gamma, A, A \supset \Delta}{\Gamma, A \supset \Delta} (C \supset) \qquad \frac{\Gamma \supset A, A, \Delta}{\Gamma \supset A, \Delta} (\supset C) $$

## 2.3 Non-decisive contribution II: Tennant

The notion of non-decisive contribution can be approached from another angle, considering imperfectly valid sequents, like:

$$ (6) \quad p \supset p \lor \neg p $$

which are substitution instances of perfect sequents, like:

$$ (7) \quad p \supset p \lor q $$

This idea of closing a restricted notion of entailment under substitution seems to have first appeared in print with Smiley (1959), and was developed in the form we discuss here by Tennant (1984) and Verdée et al. (2019), with different tools and motivations. Since Tennant is, as far as we know, the first to have defended this account in print, we will call it “relevance<sub>T</sub>”.

One way to motivate this move is as follows. Since validity is closed under substitution, if *p* contributes to the validity of (7), it contributes also to the validity of (6) for the same reason, namely that a disjunction is entailed by its left disjunct.<fnref n="14" /> Of course, there is a sense in which *p* does not contribute exactly in the same way, because the contribution of *p* to the validity of (7) is decisive, whereas it is not decisive in (6). Yet, what matters here is that both do make a contribution that may, but need not, be decisive, and that both contributions can be subsumed under the same schematic principle, namely that the premise *A* contributes to the validity of *A* > *A* ∨ *B*.

**Definition 5** (Abstraction) By requiring all elements of a sequent to contribute in this way, one obtains another notion of relevance.<fnref n="15" /> When it comes to formulating a

<footnote n="14">¹⁴By substitution here we understand homomorphic substitution, i.e. the substitution of for-mulas for propositional variables. Insofar as classical validity is closed under homomorphic substitution, it is reasonable to assume the division of the logical labor directed towards validity to be so closed. We do not commit here to the debated view whether logical form in general is closed under homomorphic substitution (see Schurz (2001) for a discussion). In logics such as Carnap's modal logic where validity is closed under isomorphic substitution only (i.e., of propositional variables for propositional variables), the division of logical labor should be similarly closed under isomorphic substitution only. If one wishes to identify the class of relevant modal entailments, taking Carnap's modal logic as the standard, this would indeed be the right thing to do. We thank an anonymous reviewer for pressing this point.</footnote>

<footnote n="15">¹⁵Those relevance theorists who deny that relevant entailment in general is closed under homomorphic substitution will leave the boat here and say that this notion of relevance overgenerates (Brauer, 2020; Schurz, 1991; ?). Note however that our goal here is not to identify the one true notion of relevance, but to draw a coherent map of the space of possible contribution-based accounts.</footnote>
precise definition for MSET-MSET sequents, one needs to be careful to allow for con-
tractions. One would not want relevance to vanish after a contraction of premises or
conclusions.<fnref n="16" /> Therefore, following the method of Verdée et al. (2019), we introduce
the notion of abstraction, defined recursively as follows:

1. Γ, A, A > Δ is an abstraction of Γ, A > Δ

2. Γ > A, A, Δ is an abstraction of Γ > A, Δ

3. if Γ > Δ is a substitution instance of Π > Σ, then Π > Σ is an abstraction of Γ > Σ.

4. if Γ₁ > Δ₁ is an abstraction of Γ₂ > Δ₂, and Γ₂ > Δ₂ is an abstraction of Γ₃ > Δ₃, then Γ₁ > Δ₁ is an abstraction of Γ₃ > Δ₃.

Our third concept of relevance is then defined as follows:

**Definition 6** Γ > Δ is a relevant<sub>T</sub> entailment if:

1. Γ > Δ is valid,

2. there is a perfectly valid abstraction Π > Σ of Γ > Δ.

The valid sequents that are within the substitution closure of perfect
sequents have been sometimes called “perfectible” sequents (Burgess, 2009).
So the notion of relevance<sub>T</sub> can also be identified with the notion of per-
fectible entailment (modulo the necessity to allow for contraction in the current
MSET-MSET presentation.)

Although this notion of relevance is also parasitic on the notion of
difference-making, it exploits it in an orthogonal direction: not by way of
Amalgamation, but by way of Substitution. As a result, it retains the holistic
character of perfection: each sequent element plays a definite role in relation
to the role of the other sequent elements. In other words, all premises and con-
clusions need to cooperate in a definite way for relevance<sub>T</sub> to obtain. That is
why such a concept of relevance does not survive amalgamation.

**2.4 Non-decisive contribution III: Brauer-Tennant**

Since the relevance<sub>B</sub> and relevance<sub>T</sub> extend the concept of relevance<sub>DM</sub> in
orthogonal ways, it is natural to explore the combination of the two extensions
in order to obtain a fourth concept of relevance.

**Definition 7** Γ > Δ is a relevant<sub>BT</sub> entailment if:

1. Γ > Δ is valid,

2. there is a relevant<sub>B</sub> abstraction Π > Σ of Γ > Δ.

<footnote n="16">¹⁶For the skeptical or curious reader, we can give the following example due to Tony Dale. The sequent p ∨ q, ¬p ∧ ¬q > p ∧ q is neither perfectly valid nor a substitution instance of a perfectly valid sequent in MSET-MSET format. But it is, after contraction, a substitution instance of p ∨ q, r ∨ s, ¬t ∧ ¬q, ¬r ∧ ¬u > p ∧ s which is perfectly valid. See (Tennant, 1984, p. 199) for a discussion of this example.</footnote>
This hybrid account, for all we know, has not been defended in the literature (hence the composite name, *faute de mieux*).<fnref n="17" /> It is not totally surprising, since it combines two different notions of non-decisive contribution that push in different directions. When we apply Amalgamation, we follow a certain understanding of non-decisive contribution. When we apply Substitution, we follow another one. In any case, it is easy to define, and nicely completes the space of possible accounts of contribution.

## 2.5 What is it to contribute to the validity of a sequent ?

All those accounts are variations on the same theme of a premise or a conclusion contributing to the validity of a sequent. We will not argue here in favor of one account in particular over the others. We take all of them to be legitimate explications of the notion of contribution. Our main interest here is in understanding precisely what contribution means for each of them and how these meanings relate to one another.

<table><caption>**Table 1:** Four notions of relevant entailment</caption><thead><tr><th>Examples</th><th>relevant<sub>DM</sub></th><th>relevant<sub>B</sub></th><th>relevant<sub>T</sub></th><th>relevant<sub>BT</sub></th></tr></thead><tbody><tr><td>p, q &gt; p &amp; q</td><td>+</td><td>+</td><td>+</td><td>+</td></tr><tr><td>p, q &gt; p, q</td><td>-</td><td>+</td><td>-</td><td>+</td></tr><tr><td>p &amp; ¬p &gt; p</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>p &amp; ¬p, q &gt; p, q &amp; ∨¬q</td><td>-</td><td>-</td><td>-</td><td>+</td></tr></tbody></table>

While the idea of contribution is central to this family of accounts, it remains difficult to characterize positively and precisely the respective contributions of various elements of a valid sequent. A fully explanatory characterization would give us the means to individuate contributions and show how they help make sequents valid.

We propose to analyze the notion of contribution in terms of the more fundamental notion of logical ground. The blueprint of the analysis to come is the following. First, we define the notion of logical grounds for truths and falsities. Then, we show how to assign logical grounds to premises and conclusions, and to obtain entailments from a certain type of connections between the logical grounds of premises and conclusions. Once this is done, we will be in a position to describe the various types of contributions of premises and conclusions in terms of the logical grounds they contribute to entailments. At the end of the journey, we will have obtained a systematic ground-theoretic analysis of the notions of relevance belonging to the family of contribution-based accounts described in the current section.

To do so, we first need to explain what we mean by “logical grounds” for truths and falsities.

<footnote n="17">¹⁷It is briefly considered, and rejected, by Brauer (2020, p. 450). Brauer does not see why the class of relevant entailments in general should be closed under substitution, hence his coherent rejection of both relevance<sub>T</sub> and relevance<sub>BT</sub>.</footnote>
3 Bilateral logical ground

In this section, we present the account of logical grounding that constitutes
the basic building block in terms of which our analysis of relevance concepts
will proceed.

In contrast with standard presentations, e.g., Correia (2013), we adopt here
a *bilateral* approach to logical grounding.<fnref n="18" /> It is bilateral in that it takes truth
and falsity as primitive semantical notions that play a symmetrical role in
logical grounding explanations.<fnref n="19" /> For instance, we shall say that the truth of
*p* and the truth of *q* logically ground the truth of *p* ∧ *q*, while the falsity of *r*
logically grounds the falsity of *p* ∧ *r*.

The central motivation for this bilateral approach to logical grounding (in
addition to serving as a foundation for the analysis of relevance concepts to
come) is to provide a fully adequate explication of the Core Intuition spelled
out by Schnieder:

A sentence governed by a classical truth-functional connective has its truth-value because of the truth-values of the embedded sentences. (Schnieder, 2011, p. 448)

Such logical grounding explanations thus explain semantical facts (whether
a sentence is true or false) in terms of more fundamental semantical facts (the
truth or falsehood of a number of embedded sentences). Since the truth or fal-
sity of a sentence is always fact of a semantical nature, dealing explicitly with
falsity does not by itself force us to give up on the factivity of grounding expla-
nations. Yet, this bilateral approach diverges importantly both in motivation
and technical implementation from more familiar accounts in the literature.
We shall first present our account and then relate it to the existing literature.

Our formalization of grounding principles uses signed formulas, i.e. formu-
las followed by positive or negative signs. Intuitively, “$A^+$” means “the truth
of $A$” and “$A^-$” means “the falsity of $A$”. Logical grounding claims will be
expressed by *logical grounding sequents*. If $\Phi$ is a finite multiset of signed for-
mulas and $X$ is a signed formula, then $\Phi < X$ is a logical grounding sequent,
where $<$ expresses the notion of mediate strict full ground: the members of $\Phi$
together provide a complete explanation for $X$.

For reasons of technical convenience, the notion of mediate strict full ground
expressed by "<" will not be factive. We shall allow, for example the truth of
*A* and the falsity of *A* to ground together the truth of *A* ∧ ¬*A*, even though *A*
cannot be both true and false, as far as classical logic is concerned. However,
a corresponding factive notion of grounding can be characterized by accept-
ing only those of our grounding sequents for which the truths and falsities
mentioned on the left hand-side actually hold.<fnref n="20" />

<footnote n="18"><sup>18</sup> A similar bilateral account of logical ground has recently been proposed by Litland (2023), although the accounts differ both in philosophical motivation and technical detail. We compare our bilateral account with Litland’s in subsection E of the Appendix.</footnote>

<footnote n="19">¹⁹In a different context, Nils Kurbis has explored and defended a form of bilateralism in proof-theoretic semantics that also takes falsity as primitive on a par with truth. See Kürbis (2019).</footnote>

<footnote n="20">²⁰The adoption of a non-factive notion of ground for technical reasons has become standard in the literature. See e.g. (Correia, 2013, p.36).</footnote>
For instance, the grounding sequent

$$A_1^+, \dots, A_n^+, B_1^-, \dots, B_m^- < C^+$$

says that the truth of $A_1, \dots, A_n$ and the falsity of $B_1, \dots, B_m$ together ground the truth of $C$.

Symmetrically, the grounding sequent

$$A_1^+, \dots, A_n^+, B_1^-, \dots, B_m^- < C^-$$

says that the truth of $A_1, \dots, A_n$ and the falsity of $B_1, \dots, B_m$ together ground the falsity of $C$.

The system GK is a grounding sequent calculus that divides into a pure part and an impure part. The pure part deals only with the inferential profile of the notion of mediate strict full ground, while the impure part provides the grounding profile of the classical connectives.

Pure principles

$$\frac{X < X}{\perp} (\text{Irr<})$$

$$\frac{\Phi < X \quad X, \Psi < Y}{\Phi, \Psi < Y} (\text{Tran<})$$

Impure principles

$$\frac{A^+ < \neg A^-}{\neg} (< \neg^-)$$

$$\frac{A^- < \neg A^+}{\neg} (< \neg^+)$

$$\frac{A^-, B^- < A \lor B^-}{\lor^-} (< \lor^-)$$

$$\frac{A_i^+ < A_1 \lor A_2^+}{\lor_i^-} (< \lor_i^+)$$

$$\frac{A_i^- < A_1 \land A_2^-}{\land^-} (< \land^-)$$

$$\frac{A^+, B^+ < A \land B^+}{\land^+} (< \land^+)$$

$$\frac{A^+, B^- < A \supset B^-}{\supset^-} (\supset \supset^-)$$

$$\frac{A^- < A \supset B^+}{\supset_1^-} (< \supset_1^+)$$

$$\frac{B^+ < A \supset B^+}{\supset_2^-} (< \supset_2^+)$$

<caption>**Table 2:** The System GK of bilateral logical ground</caption>

To keep things as simple as possible, we did not provide an explicit formal treatment of the partial and immediate varieties of grounding in the pure part of GK. However, once the notion of mediate strict full ground is characterized via GK, it is easy to extract the remaining varieties.

**Definition 8** The notions of immediate strict full grounding ($<<$), mediate strict partial grounding ($<$), and immediate partial grounding ($<<$) are defined as follows:
• $\Phi \ll X$ if $\Phi < X$ is an axiom instance,

• $X < Y$ if $\Phi, X < Y$ for some $\Phi$,

• $X < < Y$ if $\Phi, X < Y$ is an axiom instance for some $\Phi$.

The impure part of the calculus provides a set of grounding axioms for each connective. One subset of this set shows how to ground the truth of a formula governed by the connective, while its complement shows how to ground the falsity of a formula governed by the connective.

It should be noted that the impure axioms are specified by means of schemata. As a result, our account of logical grounding is formal in the sense that logical grounding is preserved under the uniform substitution of arbitrary formulas to propositional variables. (This is also the reason why it is not factive.)

For each connective, there is a precise connection between the grounding axioms for that connective and the truth-functions associated to that connective. Informally, the truth-values of the grounds minimally determine the truth-value of the groundee: once the truth-value of the grounds are fixed, the truth-value of the groundee is fixed, and fixing the truth-value of no less than all the partial grounds is required to fix the truth-value of the groundee.<fnref n="21" />

A nice feature of the grounding axioms is that they are *separate*, in the sense that, for no connective, the grounding of the truth or the falsity of a formula governed by that connective essentially involves other connectives. This distinguishes the present approach from standard unilateral approaches, e.g. Correia (2013), where the grounding principles for negation are not separable: we need distinct grounding principles for grounding double negations, negations of conjunctions, and negations of disjunction. Here, the bilateral framework provides a treatment of negation that satisfies the separation requirement.

If we now look at the system GK as a whole, it is clear that the (Tran<) rule preserves the minimal truth-functional property informally described above (with the limitation observed in footnote 21). Furthermore, the system enjoys a version of the subformula property that guarantees the truth-values of logically complex sentences to be explained by *embedded* sentences: all the formulas that occur with a sign on the left of a derivable grounding sequent are subformulas of the formula that occurs with a sign on the right.

Before closing this section, let us briefly situate the current proposal in relation to the existing literature on logical ground. The main novelty consists

<footnote n="21">²¹This informal characterization is to be taken with a grain of salt. It holds strictly speaking only for “gaunt” derivable grounding sequents, (i.e. grounding sequents that cannot be obtained by uniform substitution from other GK-derivable grounding sequents, except by substituting propositional variables for propositional variables). For example, fixing the truth of both $p$ and $q$ is sufficient, and required, to fix the truth of $p \land q$. But such a simple characterization fails for other instances of ($< \land ^+$) such as $p^+, \neg p^+ < p \land \neg p$ or $p \lor \neg p^+, q \lor \neg q^+ < (p \lor \neg p) \land (q \lor \neg q)$. Furthermore, this minimal determination semantics would work primarily for non-strict grounding, for, trivially, the truth-value of $p$ minimally determines the truth-value of $p$. Then the strict notion would have to be defined in terms of the non-strict notion. For these reasons, a minimal truth-functional determination semantics for GK along the lines given above needs to be carefully worked out to take such complications into account. In the interest of space, we leave that work for another occasion.</footnote>
in considering logical grounding as a relation or operation on semantical facts
about truth and falsity. The standard view is that the facts that Socrates
was wise and that Socrates was ugly together logically ground the fact that
Socrates was wise and ugly (e.g., Correia 2013). In the current account, we only
consider the grounding of the truth of “Socrates was wise and ugly” on the
truths “Socrates was wise” and “Socrates was ugly” as an instance of logical
grounding. This bilateral move however allows us to give a precise explication
of Schnieder’s Core intuition, and to give a separate treatment of negation.
A similar bilateral approach to logical ground has been sketched recently by
Litland (2023), from an inferentialist point of view. We discuss some links with
this approach below in Section E of the Appendix.

4 From Ground to Entailment

The next step is to derive a relation of entailment from the formal theory of
logical grounding just presented.

It is hard to provide a ground-theoretic account of entailment without committing to a certain philosophical analysis of entailment. Given the account of logical ground given above in terms of truth and falsity, it is natural to adopt here a semantic analysis of entailment in the style of Bolzano and Tarski.<fnref n="22" /> Adapted to the general case of multiple-premise multiple-conclusion sequents, this approach leads to define a sequent as valid if for each interpretation $\mathcal{I}$, if all the members of $\Gamma$ are true according to $\mathcal{I}$, then some member of $\Delta$ is true according to $\mathcal{I}$.

In the case of classical propositional logic, interpretations are valuations
that assign the value true or false to propositional variables. Given an inter-
pretation, we can assign truth-values to all formulas of the language, by means
of the well-known truth-tables. Given a formula, we can identify the set of
interpretations according to which it is true or false. In particular, we shall
say that an interpretation satisfies $A^+$ if $A$ is true according to it, and that it
satisfies $A^-$ if $A$ is false according to it.

In order to facilitate the connection between logical ground and entailment, we shall however rely on a slightly different, but equivalent definition of validity:

**Definition 9** A sequent $\Gamma \supset \Delta$ is valid if for all interpretation $\mathcal{I}$, some member of $\Gamma$ is false according to $\mathcal{I}$ or some member of $\Delta$ is true according to $\mathcal{I}$.

In such a disjunctive reading, premises are to be interpreted as candidates
for falsity and conclusions as candidates for truth, given an interpretation. An
entailment holds if it is guaranteed that at least one such candidate will be
successful at what it aims to be, i.e. false if it is a premise, or true if it is a
conclusion, for any given interpretation.

<footnote n="22">²²For the classicists with inferentialist proclivities, we provide an alternative inferentialist interpretation of both logical ground an entailment in section E of the Appendix.</footnote>
In order to provide this guarantee, we propose to look at the logical grounds for premises qua falsities and the logical grounds for conclusions qua truths, and at the way they relate to each other. The central idea here is that exhibiting the possible grounds for the falsity of premises and for the truth of conclusions, and looking for appropriate connections between them, should lead us to see whether the falsity of at least one premise or the truth of at least one conclusion is being guaranteed, for all interpretations.

Let us illustrate this idea with a simple example:

$$ (8) \quad \neg p > \neg (p \land q) $$

To explain why this entailment holds, we look at the logical grounds for the premise, taken as a falsity, and the logical grounds for the conclusion, taken as a truth:

$$ (9) \quad p^+ < \neg p^- $$

$$ (10) \quad p^- < \neg (p \land q)^+ $$

Now we observe that the ground in the first grounding sequent and the ground in the second one have opposite signs. One consequence of $p^+$ logically grounding $\neg p^-$ is that each interpretation satisfying $p^+$ also satisfies $\neg p^-$, because it satisfies $p^+$. By the same token, one consequence of $p^-$ logically grounding $\neg (p \land q)^+$ is that each interpretation that satisfies $p^-$ also satisfies $\neg (p \land q)^+$ because it satisfies $p^-$. Now by definition, each interpretation either satisfies $p^+$ or $p^-$. So together, the two grounding sequents offer the guarantee that for each interpretation, $\neg p$ is false or $\neg (p \land q)$ is true. By Definition 9, we have secured the validity of the sequent (8).

Of course, this example is extremely simple. In the general case, we will need to cover premises (resp. conclusions) with full grounds consisting of a plurality of partial grounds such as, e.g.,

$$ (11) \quad p^+, q^+ < p \land q^+ $$

and of course sequents with a plurality of premises and a plurality of conclusions. However, iterating the same basic mechanism consisting in pairing logical grounds with opposite signs will allow us to derive the validity of any valid sequent from an appropriate collection of logical grounding sequents.

Such derivations are formalized within the system GLK which is an extension of GK. From a syntactic point of view, GLK is a grounding sequent calculus, where signed formulas and the logical grounding symbol retain the same intended meaning as in GK. The main syntactic innovation is the addition of a new operator [[·]] which takes multisets of signed formulas as inputs and yields in return what we shall call a sequent term, or an s-term for short. In the general case, a string of the form:

$$ (12) \quad [[A_1^-, \dots, A_n^-, B_1^+, \dots, B_m^+]] $$

is another way of representing the sequent
$$ (13) \quad A_1, \dots, A_n \succ B_1, \dots, B_m $$

understood as a purely syntactic object.

Such sequent terms will appear on the right-hand-side of grounding sequent and will play the role of groundee. Unlike formulas, which are semantically evaluated as true or false, sequents are evaluated as valid or invalid. So it is natural to expect grounding sequents of the form

$$ (14) \quad A_1^+, \dots, A_n^+, B_1^-, \dots, B_m^- < [[C_1^-, \dots, C_l^-, D_1^+, \dots, D_k^+]] $$

to ground the validity of the sequent expressed on the right on the truths and falsities expressed on the left. The intended interpretation of such sequents is however a little more complex and involves the notion of *enthymematic validity*. Traditionally, an enthymeme is defined as a valid argument with an unstated premise. Strictly speaking, enthymemes are invalid arguments. But they can be turned into valid arguments by making explicit the unstated information that is implicitly taken for granted. The enthymematic validity of a sequent can be explained by making explicit the implicit information that is required to turn the invalid sequent into a valid one. Since we operate in a bilateral framework, this missing information can be provided by means of falsities as well as truths.

The intended interpretation of grounding sequents with an *s*-term on the right of the form (14) is thus that the truth of $A_1, \dots, A_n$ and the falsity of $B_1, \dots, B_m$ together ground the enthymematic validity of the sequent

$$ (15) \quad C_1, \dots, C_l \succ D_1, \dots, D_k. $$

Here is an intuitive example, adapted from (Yablo, 2014, p. 60), that should convey the intuition behind the notion of grounding an enthymematic validity: there is a clear sense in which the implicit assumption of the truth of

(16) I'm ready.

logically grounds the enthymematic validity of

(17) You two are ready. So that makes three of us.

This is the same sense in which the truth of (16) and the truth of

(18) You are ready.

together ground the truth of

(19) We are ready.

Even though the truth of

(20) I'm ready and 2+2=4
guarantees the enthymematic validity of (17), it does not *ground* that enthymematic validity, for the same reason that (20) and (18) together guarantee but do not ground the truth of (19). Grounding the enthymematic validity of a sequent requires more than just providing missing elements: in order to ground, the missing elements should at the same time explain the enthymematic validity of the sequent, just like grounds explain the truths they ground.

If, in turn, enthymematic validity is analyzed in the style of Bolzano-Tarski by quantifying on all interpretations of the non-logical vocabulary, grounding sequents of the form (14) may be interpreted as follows: the truth of $A_1, \dots, A_n$ and the falsity of $B_1, \dots, B_m$ together ground the fact that for all interpretation $\mathcal{I}$ agreeing with the truth-values of $A_1, \dots, A_n, B_1, \dots, B_m$ at least one member of $\{C_1, \dots, C_l\}$ is false or at least one member of $\{D_1, \dots, D_k\}$ is true.

One special case is worth singling out. When an enthymematic validity is zero grounded:

$$ (21) \quad < [[A_1^-, \dots, A_n^-, B_1^+, \dots, B_m^+]] $$

the grounding sequent as a whole says that the sequent is enthymematically valid in virtue of no implicit truth or falsity, which means that the enthymeme is in fact a full-fledged entailment. Entailments, in other words, are *zero-grounded* enthymemes. Being zero-grounded, as standardly understood in the grounding literature, does not mean being ungrounded, but being grounded by nothing. Fine's illustrative example makes use of a variably-polyadic conjunction connective (Fine, 2012, p. 47). The truth of a statement governed by such a connective is then grounded by the truth of *all* its conjuncts. In the special case where it is formed from zero conjuncts, it will be vacuously true, and its truth will be zero-grounded, i.e., grounded by zero truths, rather than ungrounded. Here, similarly, zero truths or falsities are needed to restrict the set of interpretations and explain why, for all of them, some premise is false, or some conclusion is true. In other words, the enthymematic validity of the corresponding sequent is grounded by zero suppressed elements. Which is to say that the sequent is valid.

The calculus GLK is described in Table 3. Its purpose is to provide an integrated system within which one can derive entailments, step by step, from grounding principles. This requires the notions of grounding and entailment to communicate, and this is made possible by the sequent-forming operator [[·]]. In order to facilitate the description of those new grounding sequents, we shall describe signed formulas occurring on the left of the grounding symbol as *ground terms*, and signed formulas occurring on the right, outside the double brackets, as *groundee terms*. If $X$ is a ground term, $\bar{X}$ is the ground term formed with the same formula as $X$ but with an opposite sign. Expressions of the form [[Φ]] are called *sequent terms* ($s$-terms for short), negatively signed formulas within $s$-terms will be called *premise terms* (or $p$-terms), positively signed formulas within $s$-terms will be called *conclusion terms* (or $c$-terms).
Finally, sequent element term (e-terms) will be used as an umbrella term for both p-terms and c-terms.

As a proper extension of GK, GLK retains all its axioms and rules, and only adds axioms and rules to deal with s-terms. It should be noted that in the pure logic, the variable X stands for any term of the calculus, including s-terms.

The (Ent) axiom directly grounds single-premise and single-conclusion enthymemes on the corresponding truths and falsities. With our official reading in mind, the soundness of (Ent) is obvious: the truth (resp. falsity) of A grounds the fact that A is true (resp. false) for all interpretation compatible with the truth-value of A. Since those single-premise and single-conclusion enthymemes are the basic building blocks by means of which entailments will be generated, it is fair to describe the instances (Ent) as providing grounds for premise-roles or for conclusion-roles. The subtle but important conceptual difference between grounding the falsity (resp. truth) of A and grounding A as a premise (resp. conclusion), is that the latter makes A apt to combine with other sequent elements to generate entailments, by means of the (Bind) rule to which we now turn.

The (Bind) rule allows us to combine sequent elements together on the right and to remove partial grounds of opposite signs on the left. This is the mechanism that allows us to derive full-fledged entailments after all the partial grounds have been successively eliminated. Its soundness also follows from our official interpretation. It is a basic fact about classical interpretations that each interpretation either satisfies $A^+$ or $A^-$, for all formulae A. Now, suppose that for each interpretation $\mathcal{I}$ satisfying $\Phi + \{\{A^+\}\}$, some member of $\Pi_1$ is false or some member of $\Sigma_1$ is true. A fortiori, for each interpretation $\mathcal{I}$ satisfying $\Phi + \{\{A^+\}\}$, some member of $\Pi_1 + \Pi_2$ is false or some member of $\Sigma_1 + \Sigma_2$ is true. Suppose further that for each interpretation $\mathcal{I}$ satisfying $\Psi + \{\{A^-\}\}$, some member of $\Pi_2$ is false or some member of $\Sigma_2$ is true. A fortiori, for each interpretation $\mathcal{I}$ satisfying $\Psi + \{\{A^-\}\}$, some member of $\Pi_1 + \Pi_2$ is false or some member of $\Sigma_1 + \Sigma_2$ is true. By the basic fact mentioned above, we can conclude that, for all interpretations satisfying $\Phi + \Psi$, some member of $\Pi_1 + \Pi_2$ is false or some member of $\Sigma_1 + \Sigma_2$ is true.

The rule of Enthymeme Weakening ($<W$) reflects the monotonicity of entailment, and a fortiori of enthymematic validity. The rule of Ground Contraction ($<C$) reflects the fact that several full grounds for the same sequent element may be exploited to derive an entailment. The rule of Enthymeme Contraction ($<C$) reflects the fact that entailment, and a fortiori enthymematic validity, is insensitive to the repetition of premises and conclusions. The impure part of the calculus adds nothing more to the impure part of GK.

The following Enthymeme Rule is easily derivable and will often be convenient to shorten derivations:

$$ \frac{\Phi < \Psi}{\Phi < [[\Psi]]} \quad (\text{EntR}) $$

The fundamental connection between logical grounding and entailment is established by the following theorem:
Pure logic:

$$ \frac{X < X}{\perp} \text{ (Irr<)} $$

$$ \frac{\Phi_1 < X \quad X, \Phi_2 < Y}{\Phi_1, \Phi_2 < Y} \text{ (Tran<)} $$

$$ \frac{A^x < [[A^x]]}{A^x < [[A^x]]} \text{ (Ent)} $$

$$ \frac{\Phi_1, X < [[\Psi_1]] \quad \Phi_2, \bar{X} < [[\Psi_2]]}{\Phi_1, \Phi_2 < [[\Psi_1, \Psi_2]]} \text{ (Bind)} $$

$$ \frac{\Phi < [[\Psi]]}{\Phi < [[X, \Psi]]} \text{ (<W)} $$

$$ \frac{\Phi, X, X < \Psi}{\Phi, X < \Psi} \text{ (C<)} $$

$$ \frac{\Phi < [[X, X, \Psi]]}{\Phi < X, \Psi} \text{ (<C)} $$

Impure logic:

Each axiom of GK is an axiom of GLK of the same name.

<caption>**Table 3:** The system GLK</caption>

**Theorem 4.1** $< [[A_1^-, \dots, A_n^-, B_1^+, \dots, B_m^+]]$ is derivable in GLK iff $A_1, \dots, A_n > B_1, \dots, B_m$ is valid.

In other words, the foregoing analysis of entailment in terms of logical ground is both sound and complete with respect to classical entailment. This provides, as far as we know, a new way to implement a ground-first approach to logical consequence, which will be extended in the next section to derive the notion of relevant entailment.<fnref n="23" />

It is now time to come back to the issue of interpreting “<” as logical grounding in sequents with *s*-terms on the right. One might understand the notion of logically grounding truths and falsities but remain skeptical about the very idea of grounding *entailments* and a fortiori *enthymemes*. While entailments might be analyzed as certain kinds of truths (e.g., Definition 9), the logical form of such truths makes it difficult to isolate their logical grounds, at least until we have at our disposal a considerably sophisticated theory of logical grounding, capable of dealing not only with quantification, but also with quantification over interpretations of the non-logical vocabulary. Enthymematic validity is even more complex, since we need to make reference, in addition, to the omission of premises and conclusions. The theory of logical grounding developed in the previous section is way too rudimentary to allow such a sophisticated treatment.<fnref n="24" />

Instead, we propose to show that the notion expressed by “<” in GLK sequents is a natural generalization of the notion of logical grounding defined in the previous section, i.e., a generalization that preserves the central features of our prior concept of logical grounding. The basic idea is the following. We saw that our logical grounds for a logically complex truth *A* (resp. falsity *A*) could be seen as giving truths and/or falsities that together minimally determine the truth of *A* (resp. the falsity of *A*). Here, the logical grounds for

<footnote n="23">²³The idea of deriving logical consequence relations from a logic of ground has been explored in different ways by Correia (2013), Schnieder (2018), and Litland (2023).</footnote>

<footnote n="24">²⁴We however thank an anonymous reviewer for raising this issue in these terms.</footnote>
an enthymematically valid sequent give truths and/or falsities that together minimally determine the information that was missing for an enthymeme to become a genuine entailment. The Yablovian examples given above should make the parallelism clear.

Let us take a simple example of same logical form as Yablo's. From the standpoint of GLK, the truth of *q* grounds the enthymematic validity of the sequent

$$ (22) \qquad p \succ p \wedge q. $$

This is shown by the following derivation:

$$ \frac{\frac{p^+, q^+ < p \wedge q^+}{p^+, q^+ < [[p \wedge q^+]]} \quad (\text{Ent})}{\frac{p \wedge q^+ < [[p \wedge q^+]]}{q^+ < [[p^-, p \wedge q^+]]} \quad (\text{Tran<})} \quad \frac{p^- < [[p^-]]}{(\text{Bind<})} \quad (\text{Ent}) $$

The truth of *q* then minimally determines the information that needs to be added to turn the enthymeme into an entailment. This can be done simply by adding the premise *q*. To establish this, we may continue the derivation as follows:

$$ \frac{\frac{\begin{array}{c} \vdots \\ q^+ < [[p^-, p \wedge q^+]] \end{array}}{< [[p^-, q^-, p \wedge q^+]]}}{\begin{array}{c} \overline{q^- < [[q^-]]} \\ (\text{Ent}) \\ (\text{Bind}) \end{array}} $$

The reason why adding the premise *q* works is that the grounds for that premise, namely $q^-$, given by the axiom (Ent), is the opposite of the ground $q^+$ for the enthymeme. The mechanism of (Bind) completes the explanation of the validity of the sequent *p*, $q \succ p \wedge q$.

But this missing information $q^+$ could be exploited in various other ways to reach an entailment. Adding the premise $q \wedge r$ also works, for the same reason, namely that $q^-$ is a ground for the premise $q \wedge r$, as the following derivation shows:

$$ \frac{\frac{\overline{q^- < q \wedge r^-}}{q^- < [[q \wedge r^-]]} \quad (\text{Ent})}{\text{Tran<}} $$

Finally, for exactly the same reason, the addition of the conclusion $\neg q$ also suffices to create an entailment because $q^-$ is also a logical ground for the conclusion $\neg q$, as the following derivation shows:
$$ \frac{\frac{\overline{q^- < \neg q^+}}{\neg q^+ < [[\neg q^+]]} (\text{Ent})}{\overline{q^- < [[\neg q^+]]}} (\text{Tran}) $$

We now should have a clearer idea of what logical grounds for enthymemes are, and in what sense entailments are zero-grounded enthymemes. The final step of the analysis is to characterize the relevant entailments, i.e., those where all the formulas contribute, according to the various senses of contribution distinguished in section 2.

# 5 From Ground to Contribution

In order to provide a ground-theoretic analysis of relevant entailment in line with the various contribution-based accounts reviewed above, our next task will be to isolate the contributions of sequent elements in entailments. A nice feature of GLK is that the contribution of premises and conclusions to an entailment can be read off the derivations, by looking at the way their premise and conclusion *roles* are grounded, and the way those grounds interact by means of the (Bind) rule. However, in order to give a conspicuous and systematic representation of the contribution of sequent elements, we shall focus on a specific class of GLK-derivations that we call *analytic GLK-derivations*.

In order to cover all the accounts of relevance reviewed in section 2, we will work with the system GLKª which extends GLK with the following rule of Enthymeme Amalgamation:

$$ \frac{\Phi_1 < [[\Psi_1]] \qquad \Phi_2 < [[\Psi_2]]}{\Phi_1, \Phi_2 < [[\Psi_1, \Psi_2]]} \quad (\text{Amalg}) $$

Although it does not derive more entailments than GLK, it provides more ways to derive them. And the use of the (Amalg) rule will be crucial to differentiate various ways for sequent elements to contribute.

We define analytic derivations for GLKª, which is the most encompassing of our systems.

**Definition 10** (Analytic GLKª-derivation) A GLKª-derivation is analytic if:

1. no application of (Bind) follows an application ($<W$) or (Amalg),

2. no application of ($<W$) follows an application of any other rule.

In other words, analytic derivations can be decomposed into four successive phases, from leaf to root:

1. a *Logical Grounding* phase, where only axioms and the (Tran<) rule are applied, possibly with the Ground Contraction Rule (C<)

2. a *Binding* phase, where only the (Bind) rule is applied, possibly with the Ground Contraction (C<) and the Enthymeme Contraction rule (<C)
3. an Amalgamation phase, where only the (Amalg) rule is applied, possibly with the Ground Contraction (C<) and the Enthymeme Contraction rule (<C), and a

4. a Weakening phase, where only (<W) is applied.

The Grounding Phase provides the logical grounds for (at least some subset of) the premises and conclusions of the derived entailment. The Binding Phase shows how those premises and conclusions collaborate with each other in order to guarantee an entailment. The Amalgamation phase brings together derivations in which all sequent elements receive grounds and bind with each other. Finally, the Weakening Phase adds further premises that owe their presence only to the monotonicity of classical entailment.

Importantly, such a decomposition is available for all valid sequents.

**Theorem 5.1** If Γ>Δ is valid, then <[[Γ⁻, Δ⁺]] has an analytic GLKª-derivation.

There is no guarantee, however, that each valid sequent has a unique analytic derivation. As a matter of fact, this is not the case, and this will have important consequences later. But before we distill the philosophical import of this plurality of analyses, let us first see all we can do with just one analytic derivation.

Some notions will be useful to describe our derivations. In each rule of inference in GLK, we may distinguish the signed formulas in the input grounding sequent that are not preserved in the conclusion and those that are inherited in the conclusion. The former will be called *active*, and the latter *passive*. For example, in any instance of the (Tran<) rule, the signed formula instantiating *Y* and the signed formulas in the multisets instantiating $\Phi_1$ and $\Phi_2$ are passive, whereas the instances of *X* in each input are active. In the case of the contraction rules (C<) and (<C), we consider, by convention, the leftmost contracted signed formula as passive and the rightmost signed formula as active. Now we say that a signed formula occurrence in a derivation *D* is an *immediate descendent* of another occurrence of the same signed formula in *D* if the latter is passive, and the former is directly inherited from the latter in an inference. The relation *descendent* is the reflexive and transitive closure of the relation *immediate descendent*. The relations *immediate ancestor* and *ancestor* are the converses of the relations *immediate descendent* and *descendent*, respectively. An ancestor is *initial* if it occurs on a leaf of a derivation tree.

Given an analytic derivation, it is now possible to distinguish the sequent elements that make a genuine contribution, from those that do not.

**Definition 11** $X \in \Phi$ is a *contributor* relative to an analytic GLK-derivation $D$ of $< [[\Phi]]$ if its initial ancestor belongs to an instance of (Ent); otherwise it is a *non-contributor*.
If a sequent element occurs already in the Grounding Phase, it will neces-
sarily play a role in the Binding process, and thereby make a contribution to
validity. If it is introduced by Weakening at the end, after all the hard logical
work is done, we can be sure that it contributes nothing (at least relative to
that derivation).

Let us see how this analysis of contribution works with two simple
examples, starting with Disjunctive Syllogism:

$$
(23) \quad p \lor q, \neg p \succ q
$$

The Logical Grounding Phase comprises the following initial subderivations:

$$
D_1 := \frac{\frac{\overline{p^{-}, q^{-} < [p \lor q^{-}]} (< \lor^{-}) \qquad \overline{p \lor q^{-} < [[p \lor q^{-}]]} (\text{Ent})}{p^{-}, q^{-} < [[p \lor q^{-}]}} (\text{Tran<})}
$$

$$
D_2 := \frac{\frac{\overline{p^+ < \neg p^-} (< \neg^-)}{\overline{p^- < [[\neg p^-]}}} (\text{Ent})}{p^+ < [[\neg p^-]]} (\text{Tran<})
$$

$$
D_3 := \frac{\overline{q^+ < [[q^+]]}}{(\text{Ent})}
$$

Those three initial derivations then meet during the Binding Phase:

$$
\frac{\frac{\overline{p^{-}, q^{-} < [[p \lor q^{-}]]} \quad \overline{D_1} \quad \overline{p^{+} < [[\neg p^{-}]]} \quad \overline{D_2} \quad \overline{q^{-} < [[p \lor q^{-}, \neg p^{-}]]} \quad \text{(Bind)}}}{< [[p \lor q^{-}, \neg p^{-}, q^{+}]]} \quad \overline{q^{+} < [[q^{+}]]} \quad \overline{D_3} \quad \text{(Bind)}}
$$

The derivation stops there and shows that all premises and conclusions are
contributors.<fnref n="25" />

Our second example is the infamous classical principle of explosion:

$$
(24) \quad p \wedge \neg p > q
$$

The only analytic derivation of this sequent is the following:<fnref n="26" />

<footnote n="25"><sup>25</sup>This might seem like a wrong result from the standpoint of Relevance Logicians who abhor Disjunctive Syllogism. However, the simple instance of Disjunctive Syllogism given as an example is a perfect entailment and will be relevant according to all the accounts considered in section 2. So far, so good.</footnote>

<footnote n="26"><sup>26</sup>We use the derivable rule (EntR)—instead of the (Ent) axiom followed by an application of (Tran<)—in order to shorten the derivation. The resulting derivation is strictly speaking not analytic but may be understood as an abbreviation of the longer analytic derivation using (Ent) and (Tran<). The criterion of contribution spelled out in Definition 11 is meant to apply to the latter.</footnote>
$$ \frac{\frac{p^{-} < p \wedge \neg p^{-}}{\neg p^{-} < [[p \wedge \neg p^{-}]]} (< \wedge^{-})}{\frac{p^{+} < \neg p^{-}}{\text{(EntR)}}} \frac{\frac{p^{+} < p \wedge \neg p^{-}}{\neg p^{-} < p \wedge \neg p^{-}}}{\frac{p^{+} < [[p \wedge \neg p^{-}]]}{\text{(Bind)}}} (< \wedge^{-}) \text{(Tran<)} $$ $$ \frac{\frac{\neg p^{-} < p \wedge \neg p^{-}}{\neg p^{-} < [[p \wedge \neg p^{-}]]} (< \wedge)}{\neg p^{-} < [[p \wedge \neg p^{-}, p \wedge \neg p^{-}]]} (< \text{C}) $$ $$ \frac{\frac{\neg p^{-} < p \wedge \neg p^{-}}{\neg p^{-} < [[p \wedge \neg p^{-}]]} (< \wedge)}{\neg p^{-} < [[p \wedge \neg p^{-}, q^{+}]]} (< \text{W}) $$

What is notable here is that the premise $p \wedge \neg p$ makes two distinct contributions. In the Grounding Phase, two distinct logical grounds are assigned to the premise $p \wedge \neg p$:

$$ (25) \quad p^{+} < [[p \wedge \neg p^{-}]] $$

and

$$ (26) \quad p^{-} < [[p \wedge \neg p^{-}]] $$

So the premise binds with itself (or, rather, two occurrences of the premise bind with each other) to secure the entailment. The conclusion $q$ is here only introduced by Weakening. It does not bind with any other formula and thus fails to contribute to the entailment. So we can safely conclude that Explosion is irrelevant, something with which everyone agrees.

So far, our ground-theoretic account of contribution seems on the right track. By sharpening a little our tools, we can however go further and represent the differences between the various contribution-based accounts to relevance distinguished above.

## 5.1 Joint contribution

Once the notion of contributors and non-contributors are available, it is possible to provide fine-grained characterizations of various ways sequent elements can contribute *together* to the validity of a sequent.

**Definition 12** $X, Y \in \Psi$ are *immediate co-contributors* relative to an analytic GLK$^a$-derivation $\mathcal{D}$ of $< [[\Psi]]$ if $X$ has an ancestor in a sequent $\Phi_1, Z < [[X]]$ in the Grounding Phase of $\mathcal{D}$ and $Y$ has an ancestor in a sequent $\Phi_2, \bar{Z} < [[Y]]$ in the grounding phase of $\mathcal{D}$ such that some descendants of $Z$ and $\bar{Z}$ are cut off by an application of (Bind) in the Binding Phase of $\mathcal{D}$.

**Definition 13** $X, Y \in \Psi$ are *co-contributors* relative to an analytic GLK$^a$-derivation $\mathcal{D}$ of $< [[\Psi]]$ if they belong to the transitive closure of the relation of immediate co-contribution.
**Definition 14** The members of $\Phi \subseteq \Psi$ are joint contributors relative to an analytic GLKª-derivation $\mathcal{D}$ of $< [[\Psi]]]$ if any two members of $\Phi$ are co-contributors.

Now let us define GLK<sup>r</sup> from GLK by removing the rule of Weakening (<C). Analytic GLK<sup>r</sup>-derivations will simply be analytic GLK<sup>ra</sup>-derivations where both the Amalgamation Phase and the Weakening Phase are empty.

The following theorem then relates the calculus GLK<sup>r</sup> to the notion of joint
contribution.

**Theorem 5.2** In a GLK<sup>r</sup>-derivation of < [[Φ]]]:

1. all sequent elements are contributors, and

2. the contributors are joint contributors.

It turns out that GLK<sup>r</sup> derives all and only relevant<sub>T</sub> sequents

**Theorem 5.3** < [[Γ<sup>-</sup>, Δ<sup>+</sup>]] is derivable in GLK<sup>r</sup> if and only if Γ > Δ is relevant<sub>T</sub>.

So the notion of relevance<sub>T</sub> can be seen as capturing a situation where all premises and conclusions contribute, and contribute jointly to the validity of the sequent.

**5.2 Difference-Making**

We noted above that there is a many-to-one relation between analytic deriva-
tions and valid sequents. This means, potentially, that contributor status may
change from one analytic derivation to another.

For example, the sequent:

$$
(27) \quad p \wedge \neg p > p
$$

can be given an analytic derivation according to which the conclusion *p* does
not contribute: simply substitute *q* with *p* everywhere in the analytic derivation
of Explosion above. But there is also one according to which the conclusion *p*
does contribute:

$$
\frac{
  \frac{
    \frac{p^{-} < p \wedge \neg p^{-}}{p^{-} < [[p \wedge \neg p^{-}]]} (\text{EntR})
  }{\neg [p \wedge \neg p^{-}, p^{+}]} (\text{Ent})
}{
  \neg [[p \wedge \neg p^{-}, p^{+}]] (\text{Bind})
}
$$

What should we say in such cases?

We should remember that contribution, and joint contribution, on the
present account are dependent on a way to analyze a sequent, which is at the
same time a way to derive it. If one wants to use this criterion of joint con-
tribution to assess the relevance of entailment independently of the ways they
are analyzed or proved, one needs to face a choice between two policies.

The lax policy requires the existence of some analytic derivation where all premises and conclusions jointly contribute. This is the policy implicitly at work in the definition of relevance_T above. The strict policy requires instead that, for all analytic derivations, all premises and conclusion jointly contribute.<fnref n="27" /> As it turns out, the strict policy takes us to the difference-making account:

**Theorem 5.4** $\Gamma > \Delta$ is relevant<sub>DM</sub> if and only if, for each analytic GLK-derivation *D*, all the sequent elements are joint contributors.

This theorem brings a new insight regarding the difference between the
concept of contribution in the relevance<sub>DM</sub> and relevance<sub>T</sub> accounts. In the
Tennant account, it is only required that there is some way for all elements to
contribute. In the Difference-Making account, it is required that all elements
contribute anyway. Both accounts are joint-contribution accounts, though.
What forces all the contributions in the account of relevance<sub>DM</sub> to make a dif-
ference is that they have to contribute according to every possible analysis of
the entailment.

**5.3 Individual versus Joint Contribution**

In the absence of the Amalgamation rule, analytic derivations cannot distin-
guish between individual and joint contribution.

**Proposition 5.1** In all analytic GLK-derivations, the contributors are joint contributors.

Importantly, this property fails in GLKª-derivations, for possible applica-
tions of (Amalg) can introduce contributors that are not co-contributors. Here
is a simple counterexample:

$$
\frac{\frac{p^+ < [[p^+]]}{< [[p^-, p^+]]} \quad \frac{p^- < [[p^-]]}{< [[q^-, q^+]]} \quad \frac{q^+ < [[q^+]]}{< [[q^-, q^+]]} \quad \frac{q^- < [[q^-]]}{(\text{Amalg})}}{(\text{Ent})} \quad \frac{(\text{Bind})}{(\text{Ent})} \quad \frac{(\text{Bind})}{(\text{Bind})}
$$

Here, [[p<sup>-</sup>]] and [[q<sup>-</sup>]] are contributors but no co-contributors. As a result, the contributors in this derivation are not joint contributors. Only [[p<sup>-</sup>]] and [[p<sup>+</sup>]] co-contribute on the one hand, and [[q<sup>-</sup>]] and [[q<sup>+</sup>]] on the other.

<footnote n="27"><sup>27</sup>Schurz (? , p. 15) introduced a similar distinction between weakly proof-based and strongly proof-based relevance in his insightful study of proof-theoretic criteria of relevance.</footnote>
So taking into account analytic derivations in which the Amalgamation
phase is not empty allows us to characterize the notion of relevance<sub>B</sub> and
relevance<sub>BT</sub>.

**Theorem 5.5** $\Gamma > \Delta$ is relevant<sub>B</sub> if and only if for all analytic GLK<sup>a</sup>-derivation of
< [[$\Gamma^-$, $\Delta^+$]], all sequent elements are contributors.

**Theorem 5.6** $\Gamma > \Delta$ is relevant<sub>BT</sub> if and only if for some analytic GLK<sup>a</sup>-derivation of < [[$\Gamma^-$, $\Delta^+$]], all sequent elements are contributors.

We have now given a ground-theoretic representation of all the notions
of contribution corresponding to the four accounts of relevant entailment
identified in section 2.

In all cases, for a premise (resp. a conclusion) to contribute to the validity
of a sequent is for there to be a way for it to bind with some other sequent
elements via partial grounds of opposite signs. And in all cases, for a valid
sequent to be relevant, there has to be a way for all its elements to contribute
to its validity.

From this fundamental characterization, we obtain the four determinate
notions of relevance by answering further questions:

* whether all sequent elements need to jointly contribute,

* whether all sequent elements need to contribute for all the ways in which some element contributes.

If the questions are asked in this order, the answers:

* "yes" and "yes" give the relevance<sub>DM</sub> account

* "yes" and "no" give the relevance<sub>T</sub> account

* "no" and "yes" give the relevance<sub>BT</sub> account

* "no" and "no" give the relevance<sub>B</sub> account.

So the ground-first approach to entailment has given us new fruits, by
showing how to derive four notions of relevant entailment from our extended
theory of logical ground.

# 6 Conclusion

In this paper, we have provided a new characterization of relevant entailment in
terms of logical ground. The starting point was the idea that in relevant entail-
ments, all the premises and conclusions contribute to the entailment. The idea
of contribution can be specified in four different ways, and we gave a ground-
theoretic explication of these four notions of contribution, and consequently,
of the four notions of relevant entailment.

In addition to providing a detailed positive analysis of the intuitive notion
of contribution, our ground-theoretic approach has a strong unifying power.
On the one hand, it brings together the notions of grounding and relevance.
It is often remarked in general that grounds need to be relevant to what they
ground, and here we show in addition how a general approach to relevance
can be derived from the notion of grounding. We take that to be good news
for the theory of grounding, which gains new applications, and for the theory
of relevance, which gains new foundations. On the other hand, it provides a
new representation and a new foundation for various logics of relevant entail-
ment, approached independently by previous authors (Smiley (1959); Tennant
(1984); Verdée et al. (2019); Brauer (2020)), with different motivations and
different tools.

Some perspectives for future work naturally emerge. The ground-theoretic account developed here only covers classical propositional logic. As such, it fills its role of a proof of concept, but it would be desirable to cover quantifiers as well as other propositional connectives. Another, orthogonal, line of research would be to use this ground-theoretic approach to study the relevant entailments according to non-classical logics, such as intuitionistic logic and many-valued logics, in particular paracomplete and paraconsistent logics. The project of isolating the relevantly valid sequents among the merely valid ones makes no less sense than in the case of classical logic. Finally, it would be interesting to apply our ground-theoretic analysis to relevant logics themselves, e.g., R. If our approach really captures the core of relevance, and those logics are genuinely relevant, one expects the outcome of the ground-theoretic analysis to sanction every provable sequent as relevant. Such results would both further validate our ground-theoretic approach and provide a new philosophical explanation of what makes relevant logics relevant.

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# Appendix

We provide in this appendix the proofs for all the Propositions and Theorems announced in the main text. When necessary, we include Definitions and prove Lemmas not mentioned in the text.

## A Preliminaries

### A.1 Multisets

Multisets are collections of objects in which repetitions matter, even though order does not.

**Definition 15** (Multiset) A *multiset* over a set $X$ is a pair $(X, \lambda)$, where $\lambda: X \to N$ assigns a number of copies to each member of $X$.

We shall use double braces to describe multisets extensionally. The string `{{a, a, b}}` describes the multiset $\mathbf{X} = \langle X, \lambda_X \rangle$ such that $\lambda_X(a) = 2$, $\lambda_X(b) = 1$, and $\lambda_X(x) = 0$ for all member $x$ of $X$ distinct both from $a$ and $b$.

We shall use a number of relations and operations on multisets.
**Definition 16** (Multiset membership, inclusion, addition, difference)

When $\mathbf{X} = \langle X, \lambda_X \rangle$, $\mathbf{Y} = \langle Y, \lambda_Y \rangle$ and $x \in X$:

* $x \in \mathbf{X}$ if $\lambda_X(x) \ge 1$ (membership)

* $\mathbf{X} \subseteq \mathbf{Y}$ if $X \subseteq Y$ and for all $x \in X$, $\lambda_X(x) \le \lambda_Y(x)$ (inclusion)

* $\mathbf{X}+\mathbf{Y} = \langle X \cup Y, \lambda_{X+Y} \rangle$, where for all $x \in X \cup Y$, $\lambda_{X+Y}(x) = \lambda_X(x)+\lambda_Y(x)$ (addition)

* $\mathbf{X}-\mathbf{Y} = \langle X \cup Y, \lambda_{X-Y} \rangle$, where for all $x \in X \cup Y$, $\lambda_{X-Y}(x) = \lambda_X(x)-\lambda_Y(x)$ if $\lambda_X(x) \ge \lambda_Y(x)$, otherwise $\lambda_{X-Y}(x) = 0$ (difference)

## A.2 Syntax

**Definition 17** (*L*-formula) Given a set of propositional variables $V$, the set $\mathcal{L}$ of *L*-formulas is defined by the following grammar:

$$A := V \mid (\neg A) \mid (A \land B) \mid (A \lor B) \mid (A \supset B)$$

Since we shall not work with any other language, *L*-formulas will simply be called “formulas” hereafter.

We shall use the usual conventions for removing parentheses when no ambiguity results. For instance, we shall write “$\neg\neg p \supset p$” instead of “$((\neg(\neg p)) \supset p)$”.

We shall use upper Greek letters, e.g., $\Gamma, \Delta, \Pi, \Sigma, \dots$ as meta-variables for multisets of formulas and upper case Roman letters, e.g., $A, B, C, \dots$, as metavariables for formulas.

**Definition 18** (Sequent) When $\Gamma$ and $\Delta$ are multisets of formulas, the string “$\Gamma\triangleright\Delta$” is called a sequent.

We shall use commas for multiset-theoretic addition and have formulas stand for their own singleton multisets. Thus, “$\Gamma, A, A\triangleright\Delta, B\land C$” stands for “$\Gamma + \{\langle A, A\rangle\} \triangleright \Delta + \{\langle B \land C\rangle\}$”.

**Definition 19** (Substitution map) A substitution map on $\mathcal{L}$ is a map $\sigma : V \mapsto \mathcal{L}$ that assigns formulas to propositional variables.

**Definition 20** ($\sigma$-instance) Given a formula $A$, and an substitution map $\sigma$, the $\sigma$-substitution $A^\sigma$ is defined recursively as follows:

1. if $A \in V$, then $A^\sigma = \sigma(A)$,

2. if $A = \neg B$, then $A^\sigma = \neg B^\sigma$,

3. if $A = B \circ C$, then $A^\sigma = B^\sigma \circ C^\sigma$, for all $\circ \in \{\land, \lor, \supset\}$.
We shall also use the following handy abbreviation, when $\Gamma = \{A_1, \dots, A_n\}$:

$$ \Gamma^{\sigma} := \{A_1^{\sigma}, \dots, A_n^{\sigma}\} $$

**Definition 21** (Substitution instance)

* A sequent $\Pi \supset \Sigma$ is a $\sigma$-substitution instance of $\Gamma \supset \Delta$ if $\Pi = \Gamma^{\sigma}$ and $\Sigma = \Delta^{\sigma}$.

* A sequent $\Pi \supset \Sigma$ is a substitution instance of $\Gamma \supset \Delta$ if it is a $\sigma$-substitution instance of $\Gamma \supset \Delta$, for some substitution map $\sigma$.

# B Classical Relevance

We shall not rehearse here the standard definition of classical validity for classical propositional logic.<fnref n="28" />

## B.1 Classical Logic

We shall take the MSET-MSET sequent calculus LK (see Table 4) as our official proof-theoretic presentation of Classical Propositional Logic.

## B.2 Perfect Validity

**Proposition B.1** A sequent $\Gamma \supset \Delta$ is perfectly valid iff it is valid and all its premises and conclusions make a difference to the validity of $\Gamma \supset \Delta$.

*Proof* (left-to-right) Immediate. (right-to-left) Let $\Pi \supset \Sigma$ be any proper subsequent of $\Gamma \supset \Delta$ and let $A$ be some sequent element in $\Gamma + \Delta$ but not in $\Pi + \Sigma$. We only deal with the case where $A$ is a premise, the case where $A$ is a conclusion being symmetrical. By assumption, $\Gamma - \{A\} \supset \Delta$ is invalid. By the monotonicity of classical validity, $\Pi \supset \Sigma$ is also invalid. We conclude that every proper subsequent of $\Gamma \supset \Delta$ is invalid and thus that $\Gamma \supset \Delta$ is perfectly valid. □

## B.3 Perfectible validity

We shall make use of the system LK<sup>r</sup> as our official proof-theoretic presentation of the notion of perfectible validity in MSET-MSET format (see Table 5). This system is taken from Verdée et al. (2019) where it is named NTR.<fnref n="29" />

**Theorem B.1** $\Gamma \supset \Delta$ is LK<sup>r</sup>-derivable iff it is perfectly valid.

*Proof* See Theorems 1 and 2 of Verdée et al. (2019). □

<footnote n="28">²⁸See, e.g., (Negri et al., 2001, p. 58).</footnote>

<footnote n="29">²⁹See Tennant (1984) for an earlier SET-SET sequent calculus that is proved adequate for perfectible validity.</footnote>
Axiom:
$$ \frac{A > A}{A > A} \quad (\mathrm{Id}) $$

Operational Rules:
$$ \frac{\Gamma > A, \Delta \qquad \Gamma, A > \Delta}{\Gamma, \neg A > \Delta \qquad \neg \Gamma, A > \Delta} \quad (\neg \rightarrow) $$
$$ \frac{\Gamma, A > \Delta \qquad \Gamma, B > \Delta \qquad \Gamma > A_i, \Delta}{\Gamma, A \lor B > \Delta} \quad (\lor \rightarrow) $$
$$ \frac{\Gamma > A_i > \Delta \qquad \Gamma > A_1 \lor A_2 > \Delta \qquad \Gamma > A_1 \land A_2 > \Delta \qquad \Gamma > A > \Delta}{\Gamma, A_1 \land A_2 > \Delta} \quad (\land \rightarrow) $$
$$ \frac{\Gamma > A, \Delta \qquad \Pi, B > \Sigma}{\Gamma, \Pi \supset B > \Delta, \Sigma} \quad (\supset \rightarrow) $$
$$ \frac{\Gamma > A, \Delta \qquad \Gamma > A, \Delta}{\Gamma > A \supset B, \Delta} \quad (\supset \supset \rightarrow) $$

Structural rules:
$$ \frac{\Gamma > \Delta \qquad \Gamma > \Delta}{\Gamma, A > \Delta \qquad \Gamma > A, \Delta} \quad (\sim W) $$
$$ \frac{\Gamma, A > \Delta \qquad \Gamma > A, \Delta}{\Gamma, A > \Delta \qquad \Gamma > A, \Delta} \quad (\sim C) $$
$$ \frac{\Gamma > A, \Delta \qquad \Pi, A > \Sigma}{\Gamma, \Pi > \Delta, \Sigma} \quad (\mathrm{Cut}) $$

<caption>**Table 4:** The system LK</caption>

There is a close connection between LK<sup>r</sup> and LK. It can be shown that one recovers the full power of LK simply by adding the structural rules of Weakening (W>;) and (>W) to LK<sup>r</sup>. Let us call LK<sup>*</sup> the resulting sequent calculus.

**Proposition B.2** $\Gamma > \Delta$ is LK<sup>*</sup>-derivable iff it is LK-derivable.

*Proof* (left-to-right) It is enough to show that ($\lor \rightarrow$*^*$), ($\rightarrow \land$*^*$), ($\rightarrow \supset_1$*^*$) and ($\rightarrow \supset_1$*^*$) are derivable in LK. In all cases, the derivability is easily obtained by appropriately weakening the input sequents in order to apply the corresponding unstarred rule. We illustrate the procedure with ($\lor \rightarrow$*^*$), leaving the three remaining cases to the reader:

$$ \frac{\frac{\frac{\Gamma, A > \Delta \qquad (W >)}{\Gamma, \Pi, A > \Delta \qquad (>W)} \qquad \frac{\Pi, B > \Sigma \qquad (W >)}{\Gamma, \Pi, B > \Delta \qquad (>W)} \qquad \frac{\Pi, B > \Delta \qquad (W >)}{\Gamma, \Pi, B > \Delta, \Sigma \qquad (>W)}}{\Gamma, \Pi, A \lor B > \Delta, \Sigma} \qquad (\lor \rightarrow) $$

(right-to-left) In light of the well-known eliminability of (Cut) in LK (Gentzen, 1969), it is enough to show that ($\lor \rightarrow$), ($\rightarrow \land$), and ($\rightarrow \supset$) are derivable in LK<sup>*</sup>. In the first
Axiom:
$$ \frac{A > A}{A > A} \text{ (Id)} $$

Operational Rules:
$$ \frac{\Gamma > A, \Delta \qquad \Gamma, A > \Delta}{\Gamma, \neg A > \Delta \qquad \neg > \Delta} \text{ (}\neg > \text{)} $$
$$ \frac{\Gamma, A > \Delta \qquad \Pi, B > \Sigma \qquad (\lor > \text{*)}}{\Gamma, \Pi, A \lor B > \Delta, \Sigma} \text{ (}\lor > \text{)} $$
$$ \frac{\Gamma > A_i, \Delta \qquad \Gamma > A_1 \lor A_2, \Delta \qquad (\lor > \neg i \in \{1,2\})}{\Gamma, A_1 \land A_2 > \Delta} \text{ (}\land > \text{)} $$
$$ \frac{\Gamma > A, \Delta \qquad \Pi, B > \Sigma \qquad (\supset > \text{)}}{\Gamma, \Pi, A \supset B > \Delta, \Sigma} \text{ (}\supset > \text{)} $$
$$ \frac{\Gamma > A, \Delta \qquad \Gamma > A_i, \Delta \qquad (\lor > \neg i \in \{1,2\})}{\Gamma > A_1 \lor A_2, \Delta \qquad (\lor > \neg \Sigma)} \text{ (}\lor > \text{)} $$
$$ \frac{\Gamma > B, \Delta \qquad \Gamma > A, \Delta \qquad (\supset > \supset_1)}{\Gamma > A \supset B, \Delta \qquad (\supset > \supset_2)} \text{ (}\supset > \text{)} $$

Structural rules:
$$ \frac{\Gamma, A, A > \Delta \qquad \Gamma, A, A > \Delta \qquad (C > \Delta)}{\Gamma, A > \Delta \qquad \Gamma > A, A > \Delta \qquad (\sim C)} $$

<caption>**Table 5:** The system LK<sup>r</sup></caption>

two cases, the rule is easily derived by applying the corresponding starred rule, followed by appropriate contraction steps, as illustrated by the (\lor>) case, leaving the symmetrical derivation of (\supset \wedge) to the reader:

$$ \frac{\frac{\Gamma, A > \Delta \qquad \Gamma, B > \Delta \qquad (\lor > \text{*)}}{\Gamma, \Gamma, A \lor B > \Delta, \Delta \qquad (C > \Delta)} \qquad \Gamma > A, A > \Delta \qquad (\supset > \text{)} \qquad \Gamma > B, \Delta \qquad (\supset > \text{)}}{\Gamma, A \lor B > \Delta, \Delta \qquad (\supset > \text{)} \qquad \Gamma > A \supset B > \Delta \qquad (\supset > \text{)} \qquad \Gamma > A, A > \Delta \qquad (\sim C)} $$

The derivability of (\supset \supset) is only slightly more complex:

$$ \frac{\frac{\Gamma, A > B, \Delta \qquad (\supset > \supset_1)}{\Gamma > A \supset B, A, \Delta \qquad (\supset > \supset_2)} \qquad \Gamma > A, A > \Delta \qquad (\supset > \supset_1)}{\Gamma > A \supset B, A \supset B, \Delta \qquad (\supset > \supset_2)} \qquad \Gamma > A \supset B, \Delta \qquad (\sim C)} $$

□

## B.4 Relevance<sub>B</sub>

**Proposition B.3** If $\Gamma > \Delta$ is relevant<sub>B</sub> and $\Pi > \Sigma$ is relevant<sub>B</sub>, then $\Gamma, \Pi > \Delta, \Sigma$ is relevant<sub>B</sub>.

*Proof* See Proposition 3.2 in (Brauer, 2020, p.446).

□
**Proposition B.4** The set of relevant<sub>B</sub> sequents is the smallest set comprising the set of relevant<sub>DM</sub> sequents and closed under (Amalgamation), (C>.) and (>C).

*Proof* (Σ) That all relevant<sub>DM</sub> sequents are relevant<sub>B</sub> is immediate from the definitions. The closure of relevance<sub>B</sub> under (Amalgamation) follows from Proposition B.3. Contracting a relevant<sub>B</sub> Γ > Δ sequent preserves, for all A ∈ Γ + Δ, the existence of a perfect subsequent Π ⇒ Σ of Γ + Δ such that A ∈ Π + Δ. (⊤) Suppose that Γ > Δ is relevant<sub>B</sub>. For all A ∈ Γ + Δ, let Π<sub>A</sub> > Σ<sub>A</sub> be a perfectly valid subsequent of Γ > Δ such that A ∈ Π<sub>A</sub> + Σ<sub>A</sub>. Let Φ be the set of all sequents Π<sub>A</sub> > Σ<sub>A</sub> such that A ∈ Γ + Δ. One then recovers Γ > Δ by Amalgamating two by two the members of Φ and contracting the supernumerary copies of formulas. So Γ > Δ is part of the smallest set comprising the set of relevant<sub>DM</sub> sequents and closed under (Amalgamation), (C>.) and (>C). □

## B.5 Relevance<sub>BT</sub>

**Proposition B.5** Γ > Δ is relevant<sub>BT</sub> iff for all A ∈ Γ + Δ, there is a perfectly valid subsequent Π > Σ of Γ > Δ such that A ∈ Π + Σ.

*Proof* (left-to-right) By assumption, Γ > Δ is relevant<sub>BT</sub>. Then Γ > Δ has a relevant<sub>B</sub> abstraction Π > Σ. Let σ be the corresponding substitution map, and let A ∈ Γ + Δ. Then A = σ(B) for some B ∈ Π + Σ. By the definition of relevance<sub>B</sub>, there is a perfectly valid subsequent Π' > Σ' of Π > Σ such that B ∈ Π' + Σ'. Then from σ(Π') > σ(Σ'), after appropriate Contraction steps, one gets a perfectly valid subsequent of Γ > Δ and σ(B) = A ∈ Π + Σ. (right-to-left) For each A ∈ Γ + Δ, let Γ<sub>A</sub> > Δ<sub>A</sub> be any perfectly valid subsequent of Γ > Δ. By the definition of perfectibility, there is a perfectly valid sequent Π<sub>A</sub> > Σ<sub>A</sub> such that Γ<sub>A</sub> > Δ<sub>A</sub> can be obtained from Π<sup>σ<sub>A</sub></sup> > Σ<sup>σ<sub>A</sub></sup>, possibly after some Contraction steps, for some substitution map σ<sub>A</sub>. Let Φ = {Π<sub>A</sub> > Σ<sub>A</sub> : A ∈ Γ + Δ}. We will use these perfectly valid sequents to construct a relevant<sub>B</sub> abstraction of Γ > Δ. To do so, we reletter the members of Φ in such a way that distinct propositional variables are replaced with distinct propositional variables, and no two members of the resulting set of perfect sequents Ψ share a propositional variable. By so doing, we have defined for each A ∈ Γ + Δ an injective substitution map τ<sub>A</sub> from V to V, such that Ψ = {Π<sup>τ<sub>A</sub></sup> > Σ<sup>τ<sub>A</sub></sup> : A ∈ Γ + Δ}. Thus the inverse map τ<sub>A</sub><sup>-1</sup> exists and is also a substitution map, for all A. We define the substitution map σ as follows:

$$ \sigma(p) = \begin{cases} \sigma_A((\tau_A^{-1})(p)) & \text{if } p \text{ occurs in } \Pi_A^{\tau_A} > \Sigma_A^{\tau_A} \\ p & \text{if } p \text{ occurs in no member of } \Psi \end{cases} $$

Let Θ<sub>A</sub> = Π<sup>τ<sub>A</sub></sup> and Ω<sub>A</sub> = Σ<sup>τ<sub>A</sub></sup>. Now, for all A, Θ<sub>A</sub>>Ω<sub>A</sub> is obtained by the uniform substitution of variables to variables from the perfectly valid Π<sub>A</sub> > Σ<sub>A</sub>. Therefore, Θ<sub>A</sub>>Ω<sub>A</sub> is also perfectly valid. Let Θ > Ω be the result of Amalgamating all the Θ<sub>A</sub>>Ω<sub>A</sub> such that A ∈ Γ + Δ. By Proposition B.3, Θ > Ω is relevant<sub>B</sub>. By the construction of σ, suitable contractions of Θ<sup>σ</sup>>Ω<sup>σ</sup> give us back Γ > Δ. Thus Θ>>Ω is a relevant<sub>B</sub> abstraction of Γ > Δ, which makes the latter relevant<sub>BT</sub>. □
C Ground and Entailment

In this section, we prove the results connecting the notions of logical grounding with the notion of entailment. We adopt a synthetic approach here and prove the strongest and most general results available. As a result, the results announced in the body of the paper often come out as corollaries of the stronger results we prove here.

We have in total five systems for reasoning about non-strict mediate full
logical ground, represented by order of strength in the following diagram:

<figure><img src="image_1.png" /><figcaption>Fig. 1: Systems of logical grounding</figcaption></figure>

The system GK, described in Table 2, contains the grounding axioms for the
connectives and the principles of Reflexivity (Irr<) and Transitivity (Tran<).
The other systems can be obtained as follows:

• GLK<sup>r</sup> = GK + (Bind) + (C<) + (<C)

• GLK = GLK<sup>r</sup> + (<W)

• GLK<sup>ra</sup> = GLK<sup>r</sup> + (Amalg)

• GLK<sup>a</sup> = GLK + (Amalg)

**Proposition C.1** $\Phi < X$ is GLK-derivable iff it is GLK<sup>a</sup>-derivable.

*Proof* (left-to-right) Immediate. (right-to-left) The Amalgamation rule is derivable in GLK by (<W). □

**Definition 22** $X = A^x$ is an absolute subformula of $Y = B^y$ if $A$ is a subformula of $B$.
**Definition 23** A GLKª-derivation has the *absolute-subformula property* if each formula occurring in the derivation is an absolute subformula of some signed formula occurring as a groundee at the root of the derivation tree.

**Proposition C.2** All GLKª-derivations have the absolute-subformula property.

*Proof* An easy induction on the length of GLKª-derivations. □

The following abbreviations will prove handy. If $\Gamma = \{\langle A_1, \dots, A_n \rangle\}$, then:

$$ \bullet \Gamma^+ = \{\langle A_1^+, \dots, A_n^+ \rangle\} $$

$$ \bullet \Gamma^- = \{\langle A_1^-, \dots, A_n^- \rangle\}. $$

We define the translation function $\tau$ taking a grounding sequent into an ordinary sequent as follows:

$$ 1. (\Gamma^+, \Delta^- < A^-)^\tau = \Gamma, A > \Delta. $$

$$ 2. (\Gamma^+, \Delta^- < A^+)^\tau = \Gamma > \Delta, A. $$

$$ 3. (\Gamma^+, \Delta^- < [[\Pi^-, \Sigma^+]])^\tau = \Gamma, \Pi > \Delta, \Sigma. $$

**Lemma C.1** If $\Psi < X$ is derivable in GLK, then $(\Psi < X)^\tau$ is valid.

*Proof* By induction on the length of a GLK-derivation of $\Psi < X$. (Ent)-instances translate into (Id) instances. It is easily seen that translations of impure grounding axioms are derivable by applying the corresponding operational rule of LK* to appropriate instances of the (Id) axiom. We only give the derivation for the conjunction axioms as an illustration and leave the remaining cases to the reader. For axiom $(< \wedge_i^-)$, i.e., $A_i^- < A_1 \wedge A_2^-$, with $i \in \{1, 2\}$, the corresponding LK*-derivation is:

$$ \frac{\frac{\overline{A_i > A_i} \ (\mathrm{Id})}{A_1 \wedge A_2 > A_i} \ (\wedge > i)}{\overline{A_i > A_i} \ (\mathrm{Id})} $$

For axiom $(< \wedge^+)$, i.e. $A^+, B^+ < A \wedge B^+$, the corresponding LK*-derivation is:

$$ \frac{\frac{\overline{A > A} \ (\mathrm{Id})}{A, B > A \wedge B} \overline{B > B} \ (\mathrm{Id})}{(> \wedge^*)} $$

The next step is to show that the property is preserved by all rules of GLK. It is obviously the case for the (<W), (C<), (<C) rules, in virtue of the corresponding structural rules of LK*. The property is similarly preserved by (Tran<) in virtue of the admissibility of the (Cut) rule in LK*. (Bind) preserves the property also in virtue of the admissibility of (Cut) in LK*. Finally, (Amalg) preserves the property in virtue of the structural rules of Weakening in LK*. □

The proof also establishes the following:

**Corollary C.1** If $\Phi < X$ is derivable in GLK$^r$, then $(\Phi < X)^\tau$ is derivable in LK$^r$.
**Lemma C.2** If Γ > Δ is derivable in LK*, then < [[Γ⁻, Δ⁺]] is derivable in GLK.

*Proof* By induction on the length of a derivation $\mathcal{D}$ of Γ > Δ in LK*. The property holds if $\mathcal{D}$ reduces to an instance of the (Id) axiom:

$$ \frac{\frac{\overline{A^+ < [[A^+]]} \text{ (Ent)}}{\overline{< [[A^-, A^+]]} \text{ (Bind)}} \quad \overline{\frac{\overline{A^- < [[A^-]]} \text{ (Ent)}}{\overline{< [[A^-, A^-]]} \text{ (Bind)}}$$

To show that each rule of LK* preserves the property, it suffices to show how to simulate within GLK each rule of LK*. This is trivial for the structural rules of Weakening and Contraction. To simulate the operational rules, we need to insert the corresponding grounding axiom and an application of (Tran<>) when the e-term corresponding to the active formula is first introduced by the (Ent) axiom or an application of (<W>). We give a detailed treatment of the introduction of conjunction to the left (∧>), and leave the rest as exercises to the reader. Suppose we have a derivation $\mathcal{D}_0$ of < [[Ψ, Aᵢ⁻]]>, from which we want to obtain a derivation $\mathcal{D}$ of < [[Ψ, A₁ ∧ A₂⁻]]. There are two possibilities:

1. $\mathcal{D}_0$ is of the form:

$$ \frac{\overline{A_i^- < [[A_i^-]]} \text{ (Ent)} \quad \vdots \quad \mathcal{D}_1 \quad \vdots \quad < [[\Psi, A_i^-]]}{} $$

then $\mathcal{D}$ is:

$$ \frac{\frac{\overline{A_i^- < A_1 \wedge A_2^-} \text{ (< ∧⁻)} \quad \overline{A_1 \wedge A_2^- < [[A_1 \wedge A_2^-]]} \text{ (Ent)}}{\overline{A_i^- < [[A_1 \wedge A_2^-]]} \quad \vdots \quad \mathcal{D}'_1 \quad \vdots \quad < [[\Psi, A_1 \wedge A_2^-]]} \text{ (Tran<)} $$

where $\mathcal{D}'_1$ is obtained by substituting $A_1 \wedge A_2^-$ to all descendents of the occurrence of $A_i^+$ in the instance of (Ent) in $\mathcal{D}_0$ above.

2. $\mathcal{D}_0$ is of the form:

$$ \frac{\overline{\Phi_0 < [[\Psi_0]]} \text{ (<W)}}{\overline{A_i^- < [[\Psi_0, A_i^-]]} \text{ (<W)}} \quad \mathcal{D}_1 \quad \vdots \quad < [[\Psi, A_i^-]]} $$
then $\mathcal{D}$ is:

$$ \frac{\Phi_0 < [[\Psi_0]]}{A_i^- < [[\Psi_0, A_1 \wedge A_2^-]]} (<W) \\ \vdots \\ \mathcal{D}'_1 \\ < [[\Psi, A_1 \wedge A_2^-]] $$

where $\mathcal{D}'_1$ is obtained by substituting $A_1 \wedge A_2$ to all descendants of the occurrence of $A_i^+$ in the output of ($<W$) in $\mathcal{D}_0$ above.

□

The proof also establishes the following:

**Corollary C.2** If $\Gamma \supset \Delta$ is derivable in $LK^r$, then $< [[\Gamma^-, \Delta^+]]$ is derivable in $GLK^r$.

**Theorem C.1** $< [[\Gamma^-, \Delta^+]]$ is derivable in $GLK$ iff $\Gamma \supset \Delta$ is valid.

*Proof* By Lemmas C.1 and C.2, Proposition B.2, and the adequacy of $LK$ for classical validity. □

**Theorem C.2** $< [[\Gamma^-, \Delta^+]]$ is derivable in $GLK^r$ iff $\Gamma \supset \Delta$ is perfectly valid.

*Proof* By Corollaries C.1 and C.2, and Theorem B.1. □

# D Analytic derivations

In this section, we prove the results concerning analytic derivations.

We shall approach the problem of finding an analytic derivation for every $GLK^a$-derivable grounding sequent in a constructive way, by providing an effective method for transforming any given $GLK^a$-derivation into an analytic $GLK^a$-derivation of the same grounding sequent. The main steps of the transformation are broken down into the following Lemmas.

**Lemma D.1** If $\Phi < X$ is derivable in $GLK^a$, it has a derivation where no application of ($<W$) is followed by the application of any other rule.
*Proof* We consider any derivation $\mathcal{D}$ of $\Phi < X$ where ($<W$) is applied exactly once in last position. We consider all possible rule applications for the continuation of $\mathcal{D}$ and show how to construct a derivation $\mathcal{D}^\dagger$ of the same sequent where no application of ($<W$) is followed by the application of any other rule.

• $\mathcal{D}$ cannot continue with (Tran<).

• $\mathcal{D}$ continues with (Bind). Then $\mathcal{D}$ is of the form:

$$ \frac{\frac{\vdots \quad \mathcal{D}_1}{X, \Phi_1 < [[\Psi_1]]} (<W) \qquad \vdots \quad \mathcal{D}_2}{\Phi_1, \Phi_2 < [[\Psi_1, \Psi_2, Y]]} \quad (\text{Bind}) $$

and $\mathcal{D}^\dagger$ is:

$$ \frac{\frac{\vdots \quad \mathcal{D}_1}{X, \Phi_1 < [[\Psi_1]]} \quad \vdots \quad \mathcal{D}_2}{\frac{\Phi_1, \Phi_2 < [[\Psi_1, \Psi_2]]}{\Phi_1, \Phi_2 < [[\Psi_1, \Psi_2, Y]]} (<W)} \quad (\text{Bind}) $$

• $\mathcal{D}$ continues with ($<W$). Then $\mathcal{D}^\dagger = \mathcal{D}$.

• $\mathcal{D}$ continues with ($<C$). Then $\mathcal{D}$ is of the form:

$$ \frac{\vdots \quad \mathcal{D}_1}{\frac{X, X, \Phi_1 < [[\Psi_1]]}{X, X, \Phi_1 < [[\Psi_1, Y]]}} (<W) \qquad \frac{X, \Phi_1 < [[\Psi_1, Y]]}{C<} $$

and $\mathcal{D}^\dagger$ is:

$$ \frac{\vdots \quad \mathcal{D}_1}{\frac{X, X, \Phi_1 < [[\Psi_1]]}{X, \Phi_1 < [[\Psi_1]]}} (C<) \qquad \frac{X, \Phi_1 < [[\Psi_1, Y]]}{<W} $$

• $\mathcal{D}$ continues with ($<C$). There are two possibilities:

1. if $\mathcal{D}$ is of the form:
<figure><img src="image_2.png" /></figure>

then $\mathcal{D}^\dagger$ is:

<figure><img src="image_3.png" /></figure>

2. if $\mathcal{D}$ is of the form:

<figure><img src="image_4.png" /></figure>

then $\mathcal{D}^\dagger$ is:

<figure><img src="image_5.png" /></figure>

• $\mathcal{D}$ continues with (Amalg). Then $\mathcal{D}$ is of the form:

<figure><img src="image_6.png" /></figure>

and $\mathcal{D}^\dagger$ is:

<figure><img src="image_7.png" /></figure>
By successively replacing, within a derivation, each initial segment $D_0$ containing exactly one application of ($<W$) in penultimate position, by $D_0^\dagger$, one eventually obtains a derivation of the same sequent where no application of ($<W$) is followed by an application of any other rule. $\square$

**Lemma D.2** If $\Phi < X$ is derivable in GLK$^{ra}$, it has a derivation where no application of (Tran$<$) or (Bind$<$) follows an application of (Amalg).

*Proof* We consider any GLK$^{ra}$-derivation $\mathcal{D}$ where (Amalg) is applied exactly once in last position. We consider all possible rule applications for the continuation $D_0$ of $\mathcal{D}$ and show how to construct of derivation of the same sequent where no application of (Tran$<$) or (Bind) follows an application of (Amalg).

* $\mathcal{D}$ cannot continue with (Tran$<$).

* $\mathcal{D}$ continues with (Bind). $D_0$ is of the form:

$$ \frac{\frac{\Phi_1, X < [[\Psi_1]] \qquad \Phi_2 < [[\Psi_2]]}{\Phi_1, \Phi_2, X < [[\Psi_1, \Psi_2]]} \quad (\text{Amalg}) \qquad \Phi_3, \bar{X} < [[\Psi_3]]}{\Phi_1, \Phi_2, \Phi_3 < [[\Psi_1, \Psi_2, \Psi_3]]} \quad (\text{Bind}) $$

and $D_0^\dagger$ is:

$$ \frac{\frac{\Phi_1, X < [[\Psi_1]] \qquad \Phi_3, \bar{X} < [[\Psi_3]]}{\Phi_1, \Phi_3 < [[\Psi_1, \Psi_3]]} \quad (\text{Bind}) \qquad \Phi_2 < [[\Psi_2]]}{\Phi_1, \Phi_2, \Phi_3 < [[\Psi_1, \Psi_2, \Psi_3]]} \quad (\text{Amalg}) $$

* $\mathcal{D}$ continues with (C$<$), ($<C$) or (Amalg). Then $D_0^\dagger = D_0$.

By successively replacing, within a derivation $\mathcal{D}$ of $\Phi < X$, each initial segment $D_0$ containing exactly one application of (Amalg) in penultimate position, by $D_0^\dagger$, one eventually obtains a derivation of $\Phi < X$ where no application of (Tran$<$) or (Bind) follows an application of (Amalg). $\square$

**Theorem D.1** If $\Phi < X$ is derivable in GLK$^a$, it has an analytic GLK$^a$-derivation.

*Proof* Let $D_0$ be any GLK$^a$-derivation of $\Phi < X$. By Lemma D.1, $\Phi < X$ has a derivation $D_1$ where no application of ($<W$) is followed by an application of any other rule. In other words, $D_1$ has a largest ($<W$)-free initial segment $D_2$, which is a GLK$^{ra}$-derivation of the sequent $\Phi_1 < X_1$. By Lemma D.2, $\Phi_1 < X_1$ has a derivation $D_3$ where no application of (Tran$<$) or (Bind) follows an application of (Amalg).
So $\mathcal{D}_3$ has a largest (Amalg)-free initial segment $\mathcal{D}_4$ that is a GLK<sup>r</sup>-derivation of the sequent $\Phi_2 < X_2$. Now, let $\mathcal{D}_5$ be the result of replacing, within $\mathcal{D}_1$, the initial segment $\mathcal{D}_2$ with $\mathcal{D}_3$. Then $\mathcal{D}_5$ is an analytic GLK<sup>a</sup>-derivation of $\Phi < X$. $\square$

**Corollary D.1**

* If $\Phi < [[\Psi]]$ is derivable in GLK, it has an analytic GLK-derivation.

* If $\Phi < [[\Psi]]$ is derivable in GLK<sup>r</sup>, it has an analytic GLK<sup>r</sup>-derivation.

* If $\Phi < [[\Psi]]$ is derivable in GLK<sup>ra</sup>, it has an analytic GLK<sup>ra</sup>-derivation.

**Proposition D.1** If $< [[\Psi]]$ has an analytic GLK<sup>ra</sup>-derivation $\mathcal{D}$, all members of $\Psi$ are contributors in $\mathcal{D}$.

*Proof* By definition, all analytic GLK<sup>ra</sup>-derivations of $< [[\Psi]]$ have an empty Weakening Phase. Therefore, all members of $\Psi$ are contributors there. $\square$

**Lemma D.3** If $X_1$ and $X_2$ are two contributors in an analytic GLK<sup>r</sup>-derivation, they are co-contributors.

*Proof* By induction on the length of the Binding Phase of an analytic GLK<sup>r</sup>-derivation $\mathcal{D}$. By assumption, there are at least two contributors, so in the base case the Binding Phase is of the form:

$$ \frac{\Phi_1, Y < [[X_1]] \qquad \Phi_2, \bar{Y} < [[X_2]]}{\Phi_1, \Phi_2 < [[X_1, X_2]]} \quad (\text{Bind}) $$

Then $X_1$ and $X_2$ are immediate co-contributors and a fortiori co-contributors. The induction step is trivial in cases where the Binding Phase ends with ($<C$) or with ($C<$). Suppose $\mathcal{D}$ ends with an application of (Bind). It has the form:

$$ \frac{\Phi_1, Y < [[\Psi_1]] \qquad \Phi_2, \bar{Y} < [[\Psi_2]]}{\Phi_1, \Phi_2 < [[\Psi_1, \Psi_2]]} \quad (\text{Bind}) $$

If $X_1, X_2 \in \Psi_i$ with $i \in \{1, 2\}$, they are co-contributors by the induction hypothesis. If $X_1 \in \Psi_1$ and $X_2 \in \Psi_2$, then we focus on the closest ancestor of $Y$ that occurs in the Grounding Phase of $\mathcal{D}_1$. By construction, this ancestor occurs in a grounding sequent with an $s$-term on the right containing exactly one signed formula, call it $X_3$. Let $X_4$ be the unique signed formula in the $s$-term of the grounding sequent of the Grounding Phase of $\mathcal{D}_2$ where $\bar{Y}$ has its closest ancestor. By construction, $X_3$ and $X_4$ are immediate co-contributors, and hence co-contributors. By the induction
hypothesis, $X_1$ and $X_3$ are co-contributors, and $X_2$ and $X_4$ are co-contributors. By transitivity, $X_1$ and $X_2$ are co-contributors.

☐

**Theorem D.2** For all analytic GLK-derivation $\mathcal{D}$ of $< [[\Phi]]$, all the contributors in $\mathcal{D}$ are joint contributors.

*Proof* Let $X_1$ and $X_2$ be any two contributors in an analytic GLK-derivation $\mathcal{D}$. They are a fortiori two contributors in the GLK$^r$-derivation $\mathcal{D}'$ obtained by removing the Weakening Phase of $\mathcal{D}$. By Lemma D.3, they are co-contributors. Since the Weakening Phase of $\mathcal{D}$ adds no further contributors, we conclude that the contributors of $\mathcal{D}$ are joint contributors.

☐

**Corollary D.2** For all analytic GLK$^r$-derivation $\mathcal{D}$ of $< [[\Phi]]$, all members of $\Phi$ are joint contributors in $\mathcal{D}$.

**Lemma D.4** If $< [[\Phi]]$ has an analytic GLK$^a$-derivation where the members of $\Phi$ are joint contributors, $< [[\Phi]]$ has a GLK$^r$-derivation.

*Proof* Let $\mathcal{D}$ be an analytic GLK$^a$-derivation of $< [[\Phi]]$ where the members of $\Phi$ are joint contributors. Then its Weakening Phase is empty (otherwise, some member of $\Phi$ would not be a contributor). If its Amalgamation Phase is empty, $\mathcal{D}$ is a GLK$^r$-derivation. If its Amalgamation Phase is not empty, let $< [[\Psi_1]]$ and $< [[\Psi_2]]$ be any two inputs to the Amalgamation Phase inherited from the Binding Phase, and let $X_1 \in \Psi_1$ and $X_2 \in \Psi_2$. By construction $X_1$ and $X_2$ are not co-contributors. The only way for the members of $\Phi$ to be joint contributors is for a descendent of $X_1$ or of $X_2$ to be contracted out by ($<C$). It follows that all the inputs to the Amalgamation Phase share the same signed formulas. Then $< [[\Phi]]$ is derivable from some such input by ($<C$) alone, and thus has a GLK$^r$-derivation.

☐

**Theorem D.3** $\Gamma \succcurlyeq \Delta$ is relevant$_T$ iff for some analytic GLK$^a$-derivation $\mathcal{D}$ of $< [[\Gamma^-, \Delta^+]]$, all sequent elements are joint contributors in $\mathcal{D}$.

*Proof* (left-to-right) By Theorem C.2, Corollaries D.1, and D.2. (right-to-left) Let $\mathcal{D}$ be an analytic GLK$^a$-derivation of $< [[\Gamma^-, \Delta^+]]$ where all sequent elements are joint contributors. By Lemma D.4 and Theorem C.2, $\Gamma \succcurlyeq \Delta$ is relevant$_T$.

☐

**Theorem D.4** $\Gamma \succcurlyeq \Delta$ is relevant$_{DM}$ iff for all analytic GLK$^a$-derivations of $< [[\Gamma^-, \Delta^+]]$, all sequent elements are joint contributors.
*Proof* (left-to-right) Suppose that $\Gamma \supset \Delta$ is relevant<sub>DM</sub> and let $\mathcal{D}$ be an analytic GLK<sup>a</sup>-derivation of < [[$\Gamma^-, \Delta^+$]]]. Its Weakening Phase is empty, otherwise, some proper subsequent of $\Gamma \supset \Delta$ would be valid, by Proposition C.1 and Theorem C.1. If its Amalgamation Phase is empty, $\mathcal{D}$ is an analytic GLK<sup>r</sup>-derivation. By Corollary D.2, all sequent elements are joint contributors in $\mathcal{D}$. If its Amalgamation Phase is non-empty, let < [[$\Pi^-, \Sigma^+$]] be some input to the Amalgamation Phase inherited from the Binding Phase. Because $\Gamma \supset \Delta$ is relevant<sub>DM</sub>, $\Pi \supset \Sigma$ cannot be a subsequent of $\Gamma \supset \Sigma$. Since < [[$\Pi^-, \Sigma^+$]] is an input to the Amalgamation Phase, for all $X$, if $X \in \Pi^- + \Sigma^+$, then $X \in \Gamma^- + \Delta^+$. It follows that < [[$\Gamma^-, \Delta^+$]] is obtainable from < [[$\Pi^-, \Sigma^+$]] by applications of ($<C$) only and thus has an analytic GLK<sup>r</sup>-derivation, which is a fortiori an analytic GLK<sup>a</sup>-derivation where all sequent elements are joint contributors, by Corollary D.2. (right-to-left) By contraposition. Suppose that $\Gamma \supset \Delta$ has a valid proper subsequent $\Pi \supset \Sigma$. By Theorems C.1 and D.1, < [[$\Gamma^-, \Sigma^+$]] has an analytic GLK-derivation. It is easy to continue that derivation by enough applications of ($<W$) as necessary to reach < [[$\Gamma^+, \Delta^+$]]. That derivation is an analytic GLK<sup>a</sup>-derivation relative to which some sequent elements are not contributors. A fortiori, it is not the case that all sequent elements are joint contributors. $\square$

**Theorem D.5** $\Gamma \supset \Delta$ is relevant<sub>B</sub> iff for all analytic GLK<sup>a</sup>-derivation of < [[$\Gamma^-, \Delta^+$]] all sequent elements are contributors.

*Proof* (left-to-right) Suppose that $\Gamma \supset \Delta$ is relevant<sub>B</sub>. By Definition of relevance<sub>B</sub>, for all $A \in \Gamma + \Delta$, there is a perfectly valid subsequent $\Pi_A \supset \Sigma_A$ of $\Gamma \supset \Delta$. By Theorem D.4, for all $A \in \Gamma + \Delta$, and all analytic GLK<sup>a</sup>-derivation of < [[$\Pi_A^-, \Sigma_A^+$]], all sequent elements are joint contributors. By Lemma D.4 and Corollary D.1, [[$\Pi_A^-, \Sigma_A^+$]] has an analytic GLK<sup>r</sup>-derivations, for all $A \in \Gamma + \Delta$. Then by means of (Amalg) and ($<C$), one obtains an analytic GLK<sup>ra</sup>-derivation of < [[$\Gamma^-, \Delta^+$]] where all sequent elements are contributors, by Proposition D.1. (right-to-left) We prove the contra-positive. Suppose that for some $A \in \Gamma + \Delta$ all subsequent $\Pi \supset \Sigma$ such that $A \in \Pi + \Sigma$ is either invalid or imperfectly valid. If at least one such subsequent $\Pi \supset \Sigma$ is imperfectly valid, it has a valid proper subsequent. By Theorem D.1, one has an analytic GLK<sup>a</sup>-derivation of < [[$\Pi^-, \Sigma^+$]] which, by appropriate Weakening steps, is easily extended to a GLK<sup>a</sup>-derivation of [[$\Gamma^-, \Delta^+$]] where the sequent elements introduced by Weakening are not contributors. If all subsequents $\Pi \supset \Sigma$ of $\Gamma \supset \Delta$ such that $A \in \Pi + \Sigma$ are invalid, then $\Gamma \supset \Delta$ is invalid, and by Theorem C.1, not derivable in GLK, nor in GLK<sup>a</sup>, by Proposition C.1. A fortiori, it has no analytic GLK<sup>a</sup>-derivation of < [[$\Gamma^-, \Delta^+$]] where all sequent elements are contributors. $\square$

**Theorem D.6** $\Gamma \supset \Delta$ is relevant<sub>BT</sub> iff for some analytic GLK<sup>a</sup>-derivation of < [[$\Gamma^-, \Delta^+$]] all sequent elements are contributors.

*Proof* (left-to-right) Suppose that $\Gamma \supset \Delta$ is relevant<sub>BT</sub>. By Proposition B.5, for all $A \in \Gamma + \Delta$, there is a perfectly valid subsequent $\Pi_A \supset \Sigma_A$ of $\Gamma \supset \Delta$. By Theorem C.2 and Corollary D.1, < [[$\Pi_A^-, \Sigma_A^+$]] has an analytic GLK<sup>r</sup>-derivation for all such $A$. By means of (Amalg) and ($<C$) one constructs a analytic GLK<sup>ra</sup>-derivation of
< [[Γ⁻, Δ⁺]], where all sequent elements are contributors, by Proposition D.1. (right-to-left) Suppose we have an analytic GLKª-derivation D of < [[Γ⁻, Δ⁺]] where all sequent elements are contributors. Then D can be decomposed into a number of GLKª-derivations < [[Γ₁⁻, Δ₁⁺]], ..., < [[Γₙ⁻, Δₙ⁺]] which branch together by a series of applications of (Amalg), (C<) and (<C). By Theorem C.2, Γ₁ > Δ₁, ..., Γₙ > Δₙ are perfectly valid subsequents of Γ > Δ. Then, for all A ∈ Γ + Δ, there is a perfectly valid subsequent Π > Σ of Γ > Δ such that A ∈ Π + Σ. By Proposition B.5, Γ > Δ is relevant<sub>BT</sub>. □

Corollary D.3 Γ > Δ is relevant<sub>BT</sub> iff < [[Γ⁻, Δ⁺]] is derivable in GLK<sup>ra</sup>.

E Inferentialism

Our account has shown a bias toward semantic Bolzano-Tarski-style accounts
of entailment. It would be interesting if our account could also be given an
inferentialist interpretation.

Now, bilateralism is so central to our approach, that it will need to be
conserved in any inferentialist interpretation. Fortunately, there is a brand
of inferentialist bilateralism that takes acceptance and rejection as primitive
attitudes in terms of which the meaning of connectives and the notion of
entailment can be defined (Restall, 2005; Ripley, 2017). This is the route we
will follow here.

As a matter of fact, Litland (2023) has proposed a bilateralist inferen-
tialist account of logical grounding, formally very similar to ours, where a
notion of rejection is introduced. According to Litland’s inferentialism about
ground, grounding claims express explanatory arguments. In the bilateral ver-
sion of this inferentialism, one considers in addition *antigrounding* claims,
which express explanatory rejections. While an explanatory argument is an
answer to the question: “why *p* ?”, an explanatory rejection is an answer to
the question: “why not *p* ?” In other words, Litland’s bilateral move consists
in introducing an attitude of rejection (rather than falsity as a semantical
primitive notion), on par with acceptance when it comes to the outputs of
explanatory reasoning.

It is natural for us to propose a similar understanding of the system GK, except that we do not limit the attitude of rejection to the output of explanatory reasoning, but also allow for premises to be rejected, as well as accepted.<fnref n="30" />

In general terms, a grounding sequent of the form

$$
A_1^+, \dots, A_n^+, B_1^-, \dots, B_m^- < C^+
$$

<footnote n="30">³⁰We use “explanatory reasoning” here as an umbrella term covering both Litland’s explanatory arguments and explanatory rejections. From our standpoint, Litland’s *explanatory arguments* are explanatory reasonings from acceptances to acceptances, and explanatory rejections are explanatory reasonings from acceptances to rejections.</footnote>
says that the acceptance of $A_1, \dots, A_n$ and the rejection of $B_1, \dots, B_m$ together ground the acceptance of $C$. Symmetrically, a grounding sequent of the form

$$A_1^+, \dots, A_n^+, B_1^-, \dots, B_m^- < C^-$$

says that the acceptance of $A_1, \dots, A_n$ and the rejection of $B_1, \dots, B_m$ together ground the rejection of $C$. The calculus GLK is easily seen to be sound for such an interpretation.

When it comes to entailment, an influential proposal of Restall (2005) defines multiple-conclusion entailments in terms of acceptance and rejection as follows:

**Definition 24** $\Gamma > \Delta$ is valid if it is incoherent to accept all the premises and reject all the conclusions.

Now, our calculi GLK and GLK<sup>r</sup> do not straightforwardly fit this interpretation, but a simple modification enables them to accommodate it. Instead of representing sequents by terms of the form $[[A_1^-, \dots, A_n^-, B_1^+, \dots, B_m^+]]$ where the comma is read disjunctively, we could represent them by terms of the form:

(28) $\sim (A_1^+, \dots, A_n^+, B_1^-, \dots, B_m^-)$ where the plus and the minus still denote acceptance and rejection, respectively, the symbol $\sim$ stands for the modality of incoherence, and the commas are now interpreted conjunctively.

In the general case, grounding sequents of the form:

(29) $A_1^+, \dots, A_n^+, B_1^-, \dots, B_m^- < \sim (C_1^+, \dots, C_l^+, B_1^-, \dots, B_k^-)$

mean that the acceptance of $A_1, \dots, A_n$ and the rejection of $B_1, \dots, B_m$ together ground the incoherence of simultaneously accepting $C_1, \dots, C_l$ and rejecting $D_1, \dots, D_k$.

It is not difficult to adapt GLK to this new syntax, as the definition of GLK' shows. It is not difficult either to check that the results about GLK carry over to GLK' –one only needs to add a modality in front of sequent terms and invert the signs within them. So our ground-theoretic analysis of relevant entailment can also be adopted from such a bilateralist inferentialist point of view.
Pure logic:

$$ \frac{X < X}{\perp} \text{ (Irr<)} $$

$$ \frac{\Phi_1 < X \quad X, \Phi_2 < Y}{\Phi_1, \Phi_2 < Y} \text{ (Tran<)} $$

$$ \frac{X < \sim (\bar{X})}{\text{Ent'}} $$

$$ \frac{\Phi_1, A^+ < \sim (\Psi_1) \quad \Phi_2, A^- < \sim (\Psi_2)}{\Phi_1, \Phi_2 < \sim (\Psi_1, \Psi_2)} \text{ (Bind')} $$

$$ \frac{\Phi < \sim (\Psi)}{\Phi < \sim (X, \Psi)} \text{ (<W')} $$

$$ \frac{\Phi, X, X < (\Psi)}{\Phi, X < \sim (\Psi)} \text{ (C<')} $$

$$ \frac{\Phi < \sim (X, X, \Psi)}{\Phi < \sim (X, \Psi)} \text{ (<C')} $$

Impure logic:

Each axiom of GK is an axiom of GLK' of the same name.

<caption>**Table 6:** The system GLK'</caption>