Relevant entailment and logical ground

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 system 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 differences between alternative contribution-based accounts of relevance.

Appendices

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.

Preliminaries

1.1 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 $\langle X, \lambda \rangle$, where $\lambda : X \mapsto \mathbb {N}$ assigns a number of copies to each member of X.

We shall use double braces to describe multisets extensionally. The string “ $\{\hspace{-2.77771pt}\{a,a,b \}\hspace{-2.77771pt}\}$” describes the multiset ${\textbf {X}}=\langle X,\lambda _X\rangle$ such that $\lambda _X(a)=2$, $\lambda _X(b)=1$, and $\lambda (x)=0$ 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 ${\textbf {X}}=\langle X,\lambda _X \rangle$ and ${\textbf {Y}}=\langle Y, \lambda _Y \rangle$:

• $x \in {\textbf {X}}$ if $\lambda _X(x) \ge 1$ (membership)

• ${\textbf {X}} \subseteq {\textbf {Y}}$ if $X \subseteq Y$ and for all $x \in X$, $\lambda _X(x) \le \lambda _Y(x)$ (inclusion)

• ${\textbf {X}} + {\textbf {Y}}=\langle X \cup Y, \lambda _{X+Y} \rangle$, where for all $x \in X \cup Y$, $\lambda _{X+Y}(x)=\lambda _X(x)+\lambda _Y(x)$ (addition)

• ${\textbf {X}} - {\textbf {Y}}=\langle X \cup Y, \lambda _{X-Y} \rangle$, 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)

1.2 Syntax

Definition 17

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

$$\begin{aligned} A := \mathcal {V} \mid (\lnot A) \mid (A \wedge B) \mid (A \vee B) \mid (A \supset B) \end{aligned}$$

Since we shall not work with any other language, $\mathcal {L}$-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 “ $\lnot \lnot p \supset p$” instead of “ $((\lnot (\lnot p)) \supset p)$”.

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 “ ” is called a sequent.

We shall use commas for multiset-theoretic addition and have formulas stand for their own singleton multisets. Thus, “ ” stands for “ ”.

Definition 19

(Substitution map) A substitution map on $\mathcal {L}$ is a map $\sigma : \mathcal {V} \mapsto \mathcal {L}$ 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 $A \in \mathcal {V}$, then $A^\sigma =\sigma (A)$,

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

• 3. if $A= B \circ C$, then $A^\sigma =B^\sigma \circ C^\sigma$, for all $\circ \in \{ \wedge , \vee , \supset \}$.

We shall also use the following handy abbreviation, when $\Gamma =\{\hspace{-2.77771pt}\{A_1, \dots , A_n \}\hspace{-2.77771pt}\}$:

$$\begin{aligned} \Gamma ^\sigma :=\{\hspace{-2.77771pt}\{A_1^\sigma , \dots , A_n^\sigma \}\hspace{-2.77771pt}\}\end{aligned}$$

Definition 21

(Substitution instance)

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

• A sequent is a substitution instance of if it is a $\sigma$-substitution instance of , for some substitution map $\sigma$.

Classical relevance

We shall not rehearse here the standard definition of classical validity for classical propositional logic.²⁸

1.1 Classical logic

We shall take the Mset–Mset sequent calculus $\textsf{LK}$ (see Table 4) as our official proof-theoretic presentation of Classical Propositional Logic.

Table 4 The system $\textsf{LK}$

1.2 Perfect validity

Proposition B.1

A sequent is perfectly valid iff it is valid and all its premises and conclusions make a difference to the validity of .

Proof

(left-to-right) Immediate. (right-to-left) Let be any proper subsequent of 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, is invalid. By the monotonicity of classical validity, is also invalid. We conclude that every proper subsequent of is invalid and thus that is perfectly valid. $\square$

1.3 Perfectible validity

We shall make use of the system $\textsf{LK}^\text {r}$ 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.²⁹

Table 5 The system $\textsf{LK}^\text {r}$

Theorem B.1

is $\textsf{LK}^\text {r}$-derivable iff it is perfectibly valid.

Proof

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

There is a close connection between $\textsf{LK}^\text {r}$ and $\textsf{LK}$. It can be shown that one recovers the full power of $\textsf{LK}$ simply by adding the structural rules of Weakening (W ) and ( W) to $\textsf{LK}^\text {r}$. Let us call $\textsf{LK}^*$ the resulting sequent calculus.

Proposition B.2

is $\textsf{LK}^*$-derivable iff it is $\textsf{LK}$-derivable.

Proof

(left-to-right) It is enough to show that ( ), ( ), ( ) and ( ) are derivable in $\textsf{LK}$. 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 ( ), leaving the three remaining cases to the reader:

(right-to-left) In light of the well-known eliminability of (Cut) in $\textsf{LK}$ (Gentzen, 1969), it is enough to show that ( ), ( ), and ( ) are derivable in $\textsf{LK}^*$. In the first two cases, the rule is easily derived by applying the corresponding starred rule, followed by appropriate contraction steps, as illustrated by the ( ) case, leaving the symmetrical derivation of ( ) to the reader:

The derivability of ( ) is only slightly more complex:

$\square$

1.4 Relevance_B

Proposition B.3

If is relevant_B and is relevant_B, then is relevant_B.

Proof

See Proposition 3.2 in (Brauer, 2020, p.446). $\square$

Proposition B.4

The set of relevant_B sequents is the smallest set comprising the set of relevant_DM sequents and closed under (Amalgamation), (C ) and ( C).

Proof

( $\subseteq$) That all relevant_DM sequents are relevant_B is immediate from the definitions. The closure of relevance_B under (Amalgamation) follows from Proposition B.3. Contracting a relevant_B sequent preserves, for all $A \in \Gamma + \Delta$, the existence of a perfect subsequent $\Pi \Rightarrow \Sigma$ of $\Gamma + \Delta$ such that $A \in \Pi + \Delta$. ( $\supseteq$) Suppose that is relevant_B. For all $A \in \Gamma + \Delta$, let be a perfectly valid subsequent of such that $A \in \Pi _A + \Sigma _A$. Let $\Phi$ be the set of all sequents such that $A \in \Gamma + \Delta$. One then recovers by Amalgamating two by two the members of $\Phi$ and contracting the supernumerary copies of formulas. So is part of the smallest set comprising the set of relevant_DM sequents and closed under (Amalgamation), (C ) and ( C). $\square$

1.5 Relevance_BT

Proposition B.5

is relevant_BT iff for all $A \in \Gamma + \Delta$, there is a perfectibly valid subsequent of such that $A \in \Pi + \Sigma$.

Proof

(left-to-right) By assumption, is relevant_BT. Then has a relevant_B abstraction . Let $\sigma$ be the corresponding substitution map, and let $A \in \Gamma + \Delta$. Then $A=\sigma (B)$ for some $B \in \Pi + \Sigma$. By the definition of relevance_B, there is a perfectly valid subsequent of such that $B \in \Pi ' + \Sigma '$. Then from , after appropriate Contraction steps, one gets a perfectibly valid subsequent of and $\sigma (B)=A \in \Pi + \Sigma$. (right-to-left) For each $A \in \Gamma +\Delta$, let be any perfectibly valid subsequent of . By the definition of perfectibility, there is a perfectly valid sequent such that can be obtained from , possibly after some Contraction steps, for some substitution map $\sigma _{A}$. Let . We will use these perfectly valid sequents to construct a relevant_B abstraction of . To do so, we reletter the members of $\Phi$ 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 $\Psi$ share a propositional variable. By so doing, we have defined for each $A \in \Gamma + \Delta$ an injective substitution map $\tau _{A}$ from $\mathcal {V}$ to $\mathcal {V}$, such that . Thus the inverse map $\tau _{A}^{-1}$ exists and is also a substitution map, for all A. We define the substitution map $\sigma$ as follows:

Let $\Theta _{A}=\Pi ^{\tau _{A}}_{A}$ and $\Omega _{A}=\Sigma ^{\tau _{A}}_{A}$. Now, for all A, is obtained by the uniform substitution of variables to variables from the perfectly valid . Therefore, is also perfectly valid. Let be the result of Amalgamating all the such that $A \in \Gamma + \Delta$. By Proposition B.3, is relevant_B. By the construction of $\sigma$, suitable contractions of give us back . Thus is a relevant_B abstraction of , which makes the latter relevant_BT. $\square$

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 Fig. 1:

Fig. 1

Systems of logical grounding

The system $\textsf{GK}$, 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:

• $\textsf{GLK}^\text {r}$= $\textsf{GK}$ + (Bind) + (C<) + (<C)

• $\textsf{GLK}$= $\textsf{GLK}^\text {r}$ + (<W)

• $\textsf{GLK}^\text {ra}$= $\textsf{GLK}^\text {r}$ + (Amalg)

• $\textsf{GLK}^\text {a}$= $\textsf{GLK}$ + (Amalg)

Proposition C.1

$\Phi < X$ is $\textsf{GLK}$-derivable iff it is $\textsf{GLK}^\text {a}$-derivable.

Proof

(left-to-right) Immediate. (right-to-left) The Amalgamation rule is derivable in $\textsf{GLK}$ by (<W). $\square$

Definition 22

$X=A^x$ is an absolute subformula of $Y=B^y$ if A is a subformula of B.

Definition 23

A $\textsf{GLK}^\text {a}$-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 $\textsf{GLK}^\text {ra}$-derivations have the absolute-subformula property.

Proof

An easy induction on the length of $\textsf{GLK}^\text {ra}$-derivations. $\square$

The following abbreviations will prove handy. If $\Gamma = \{\hspace{-2.77771pt}\{A_1, \dots , A_n \}\hspace{-2.77771pt}\}$, then:

• $\Gamma ^+ = \{\hspace{-2.77771pt}\{A_1^+, \dots , A_n^+ \}\hspace{-2.77771pt}\}$

• $\Gamma ^- = \{\hspace{-2.77771pt}\{A_1^-, \dots , A_n^- \}\hspace{-2.77771pt}\}$.

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

• 1. .

• 2. .

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

Lemma C.1

If $\Psi < X$ is derivable in $\textsf{GLK}$, then $(\Psi < X)^\tau$ is valid.

Proof

By induction on the length of a $\textsf{GLK}$-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 $\textsf{LK}^*$ 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 $\textsf{LK}^*$-derivation is:

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

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

The proof also establishes the following:

Corollary C.1

If $\Phi < X$ is derivable in $\textsf{GLK}^\text {r}$, then $(\Phi < X)^\tau$ is derivable in $\textsf{LK}^\text {r}$.

Lemma C.2

If is derivable in $\textsf{LK}^*$, then ${< [[\Gamma ^-, \Delta ^+]]}$ is derivable in $\textsf{GLK}$.

Proof

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

To show that each rule of $\textsf{LK}^*$ preserves the property, it suffices to show how to simulate within $\textsf{GLK}$ each rule of $\textsf{LK}^*$. 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 $< [[ \Psi , A_i^- ]]$, from which we want to obtain a derivation $\mathcal {D}$ of $< [[ \Psi , A_1 \wedge A_2^- ]]$. There are two possibilities:

• 1. $\mathcal {D}_0$ is of the form:

  

  then $\mathcal {D}$ is:

  

  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:

  

  then $\mathcal {D}$ is:

  

  where $\mathcal {D}'_1$ is obtained by substituting $A_1 \wedge A_2^-$ to all descendents 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 is derivable in $\textsf{LK}^\text {r}$, then ${< [[ \Gamma ^-, \Delta ^+ ]]}$ is derivable in $\textsf{GLK}^\text {r}$.

Theorem C.1

$< [[\Gamma ^-, \Delta ^+]]$ is derivable in $\textsf{GLK}$ iff is valid.

Proof

By Lemmas C.1 and C.2, Proposition B.2, and the adequacy of $\textsf{LK}$ for classical validity.

Theorem C.2

$< [[\Gamma -, \Delta ^+]]$ is derivable in $\textsf{GLK}^\text {r}$ iff is perfectibly valid.

Proof

By Corollaries C.1 and C.2, and Theorem B.1.

Analytic derivations

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

We shall approach the problem of finding an analytic derivation for every $\textsf{GLK}^\text {a}$-derivable grounding sequent in a constructive way, by providing an effective method for transforming any given $\textsf{GLK}^\text {a}$-derivation into an analytic $\textsf{GLK}^\text {a}$-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 $\textsf{GLK}^\text {a}$, 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:

  

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

  

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

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

  

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

  

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

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

    

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

    

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

    

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

    

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

  

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

  

By successively replacing, within a derivation, each initial segment $\mathcal {D}_0$ containing exactly one application of (<W) in penultimate position, by $\mathcal {D}_0^\dagger$, 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 $\textsf{GLK}^\text {ra}$, it has a derivation where no application of (Tran $_<$) or (Bind $_<$) follows an application of (Amalg).

Proof

We consider any $\textsf{GLK}^\text {ra}$-derivation $\mathcal {D}$ where (Amalg) is applied exactly once in last position. We consider all possible rule applications for the continuation $\mathcal {D}_0$ 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). $\mathcal {D}_0$ is of the form:

  

  and $\mathcal {D}_0^\dagger$ is:

  

• $\mathcal {D}$ continues with (C<), (<C) or (Amalg). Then $\mathcal {D}_0^\dagger =\mathcal {D}_0$.

By successively replacing, within a derivation $\mathcal {D}$ of $\Phi < X$, each initial segment $\mathcal {D}_0$ containing exactly one application of (Amalg) in penultimate position, by $\mathcal {D}_0^\dagger$, 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 $\textsf{GLK}^\text {a}$, it has an analytic $\textsf{GLK}^\text {a}$-derivation.

Proof

Let $\mathcal {D}_0$ be any $\textsf{GLK}^\text {a}$-derivation of $\Phi < X$. By Lemma D.1, $\Phi < X$ has a derivation $\mathcal {D}_1$ where no application of (<W) is followed by an application of any other rule. In other words, $\mathcal {D}_1$ has a largest (<W)-free initial segment $\mathcal {D}_2$, which is a $\textsf{GLK}^\text {ra}$-derivation of the sequent $\Phi _1 < X_1$. By Lemma D.2, $\Phi _1 < X_1$ has a derivation $\mathcal {D}_3$ 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 $\textsf{GLK}^\text {r}$-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 $\textsf{GLK}^\text {a}$-derivation of $\Phi < X$. $\square$

Corollary D.1

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

• If $\Phi < [[\Psi ]]$ is derivable in $\textsf{GLK}^\text {r}$, it has an analytic $\textsf{GLK}^\text {r}$-derivation.

• If $\Phi < [[\Psi ]]$ is derivable in $\textsf{GLK}^\text {ra}$, it has an analytic $\textsf{GLK}^\text {ra}$-derivation.

Proposition D.1

If $< [[ \Psi ]]$ has an analytic $\textsf{GLK}^\text {ra}$-derivation $\mathcal {D}$, all members of $\Psi$ are contributors in $\mathcal {D}$.

Proof

By definition, all analytic $\textsf{GLK}^\text {ra}$-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 $\textsf{GLK}^\text {r}$-derivation, they are co-contributors.

Proof

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

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:

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 $\overline{Y}$ 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. $\square$

Theorem D.2

For all analytic $\textsf{GLK}$-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 $\textsf{GLK}$-derivation $\mathcal {D}$. They are a fortiori two contributors in the $\textsf{GLK}^\text {r}$-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. $\square$

Corollary D.2

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

Lemma D.4

If $< [[ \Phi ]]$ has an analytic $\textsf{GLK}^\text {a}$-derivation where the members of $\Phi$ are joint contributors, $< [[ \Phi ]]$ has a $\textsf{GLK}^\text {r}$-derivation.

Proof

Let $\mathcal {D}$ be an analytic $\textsf{GLK}^\text {a}$-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 $\textsf{GLK}^\text {r}$-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 $\textsf{GLK}^\text {r}$-derivation. $\square$

Theorem D.3

is relevant_T iff for some analytic $\textsf{GLK}^\text {a}$-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 $\textsf{GLK}^\text {a}$-derivation of $< [[ \Gamma ^-, \Delta ^+ ]]$ where all sequent elements are joint contributors. By Lemma D.4 and Theorem C.2, is relevant_T. $\square$

Theorem D.4

is relevant_DM iff for all analytic $\textsf{GLK}^\text {a}$-derivations of ${< [[ \Gamma ^-, \Delta ^+ ]]}$, all sequent elements are joint contributors.

Proof

(left-to-right) Suppose that is relevant_DM and let $\mathcal {D}$ be an analytic $\textsf{GLK}^\text {a}$-derivation $\mathcal {D}$ of $< [[ \Gamma ^-, \Delta ^+ ]]$. Its Weakening Phase is empty, otherwise, some proper subsequent of would be valid, by Proposition C.1 and Theorem C.1. If its Amalgamation Phase is empty, $\mathcal {D}$ is an analytic $\textsf{GLK}^\text {r}$-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 is relevant_DM, cannot be a subsequent of . 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 $\textsf{GLK}^\text {r}$-derivation, which is a fortiori an analytic $\textsf{GLK}^\text {a}$-derivation where all sequent elements are joint contributors, by Corollary D.2. (right-to-left) By contraposition. Suppose that has a valid proper subsequent . By Theorems C.1 and D.1, $< [[ \Pi ^-, \Sigma ^+ ]]$ has an analytic $\textsf{GLK}$-derivation. It is easy to continue that derivation by enough applications of (<W) as necessary to reach $< [[\Gamma ^+, \Delta ^-]]$. That derivation is an analytic $\textsf{GLK}^\text {a}$-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

is relevant_B iff for all analytic $\textsf{GLK}^\text {a}$-derivation of ${< [[ \Gamma ^-, \Delta ^+ ]]}$ all sequent elements are contributors.

Proof

(left-to-right) Suppose that is relevant_B. By Definition of relevance_B, for all $A \in \Gamma + \Delta$, there is a perfectly valid subsequent of . By Theorem D.4, for all $A \in \Gamma + \Delta$, and all analytic $\textsf{GLK}^\text {a}$-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 $\textsf{GLK}^\text {r}$-derivations, for all $A \in \Gamma + \Delta$. Then by means of (Amalg) and (<C), one obtains an analytic $\textsf{GLK}^\text {ra}$-derivation of $< [[\Gamma ^-, \Delta ^+]]$ where all sequent elements are contributors, by Proposition D.1. (right-to-left) We prove the contrapositive. Suppose that for some $A \in \Gamma + \Delta$ all subsequent such that $A \in \Pi + \Sigma$ is either invalid or imperfectly valid. If at least one such subsequent is imperfectly valid, it has a valid proper subsequent. By Theorem D.1, one has an analytic $\textsf{GLK}^\text {a}$-derivation of $<[[\Pi ^-, \Sigma ^+]]$ which, by appropriate Weakening steps, is easily extended to a $\textsf{GLK}^\text {a}$-derivation of $[[\Gamma ^-, \Delta ^+]]$ where the sequent elements introduced by Weakening are not contributors. If all subsequents of such that $A \in \Pi + \Sigma$ are invalid, then is invalid, and by Theorem C.1, not derivable in $\textsf{GLK}$, nor in $\textsf{GLK}^\text {a}$, by Proposition C.1. A fortiori, it has no analytic $\textsf{GLK}^\text {a}$-derivation of $< [[ \Gamma ^-, \Delta ^+ ]]$ where all sequent elements are contributors. $\square$

Theorem D.6

is relevant_BT iff for some analytic $\textsf{GLK}^\text {a}$-derivation of $< [[ \Gamma ^-, \Delta ^+ ]]$ all sequent elements are contributors.

Proof

(left-to-right) Suppose that is relevant_BT. By Proposition B.5, for all $A \in \Gamma + \Delta$, there is a perfectibly valid subsequent of . By Theorem C.2 and Corollary D.1, $<[[\Pi _A^-, \Sigma _A^+]]$ has an analytic $\textsf{GLK}^\text {r}$-derivation for all such A. By means of (Amalg) and (<C) one constructs a analytic $\textsf{GLK}^\text {ra}$-derivation of $< [[ \Gamma ^-, \Delta ^+ ]]$, where all sequent elements are contributors, by Proposition D.1. (right-to-left) Suppose we have an analytic $\textsf{GLK}^\text {a}$-derivation $\mathcal {D}$ of $< [[ \Gamma ^-, \Delta ^+ ]]$ where all sequent elements are contributors. Then $\mathcal {D}$ can be decomposed into a number of $\textsf{GLK}^\text {r}$-derivations $< [[ \Gamma _1^-, \Delta _1^+ ]]$, ..., $< [[ \Gamma _n^-, \Delta _n^+ ]]$ which branch together by a series of applications of (Amalg), (C<) and (<C). By Theorem C.2, , ..., are perfectibly valid subsequents of . Then, for all $A \in \Gamma + \Delta$, there is a perfectibly valid subsequent of such that $A \in \Pi + \Sigma$. By Proposition B.5, is relevant_BT. $\square$

Corollary D.3

is relevant_BT iff ${< [[\Gamma ^-, \Delta ^+]]}$ is derivable in $\textsf{GLK}^\text {ra}$.

Inferentialism

Our integrated account of logical ground and entailment 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 preserved 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 inferentialist 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 version 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 $\textsf{GK}$, 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.³⁰

In general terms, a grounding sequent of the form

$$\begin{aligned} A^+_1, \dots , A^+_n, B^-_1, \dots , B^-_m < C^+ \end{aligned}$$

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

$$\begin{aligned} A^+_1, \dots , A^+_n, B^-_1, \dots , B^-_m < C^- \end{aligned}$$

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 $\textsf{GK}$ 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

is valid if it is incoherent to accept all the premises and reject all the conclusions.

Now, our calculi $\textsf{GLK}$ and $\textsf{GLK}^\text {r}$ 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:

In the general case, grounding sequents of the form:

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 $\textsf{GLK}$ to this new syntax, as the definition of $\textsf{GLK}$’ shows (see Table 6). It is not difficult either to check that the results about $\textsf{GLK}$ carry over to $\textsf{GLK}$’ – 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.³¹

Table 6 The system $\textsf{GLK}$’