2 Tarskian consequence relations: from sets to multisets

We’ll recall the standard, Tarskian, definition of consequence relation, and consider reasons to alter certain key features of it. Let us fix a set $\mathit {Fm}$ of formulas. Unless said otherwise, we do not assume that formulas have any internal structure.Footnote ³

We’ll be concerned throughout with axiomatic presentations of logics, and we’ll wind up constructing them out of consecutions, but before doing so in formal detail, let’s just consider an informal example to note some properties of axiomatic derivations in a tree form. Consider the following proof in classical logic ( $\mathbf {CL}$ ) of the fact that $\varphi \rightarrow ({\psi \rightarrow \chi }),{\varphi \land \psi }\vdash _{\mathbf {CL}}\chi $ , using a tacit, but fairly standard axiomatic presentation of classical (or indeed intuitionistic) logic in the $\{\land ,\rightarrow \}$ -fragment:

Notice that the structure relies on using the deduction rule of modus ponens, and that the premise $\varphi \land \psi $ is used twice. In a standard axiomatic presentation (in the Fmla framework, to use Humberstone’s terminology [Reference Humberstone23]), axioms are taken to be formulas and deduction rules to operate thereon. We’ll employ a slightly different presentation by dealing with consecutions: these are objects of the form $\Gamma \rhd \varphi $ where $\Gamma $ is a finite set of formulas, $\varphi $ a formula and $\rhd $ is an entailment relation. Axioms, then, are consecutions of the form $\varnothing \rhd \varphi $ , and all the other consecutions are deduction rules: for instance, we would have the axiom $\varnothing \rhd \varphi \land \psi \rightarrow \varphi $ and the deduction rule $\varphi \land \psi \rightarrow \psi ,\varphi \land \psi \rhd \psi $ in a consecution-version of the above proof tree.

This is a standard definition, but in adapting it too swiftly to a non-classical setting, we accidentally introduce some problems; of particular focus here are problems of relevance and problems concerning multiple uses of premises. For the first, clearly any Tarskian consequence relation will allow for valid arguments in which some premises are irrelevant, and not needed: for instance, if we have $\varnothing \rhd \varphi \rightarrow \varphi $ among the axioms of $\mathcal {AS}$ , then we’ll have $\psi \vdash _{\mathcal {AS}}\varphi \rightarrow \varphi $ , despite the fact that $\psi ,\varphi $ may have nothing to do with one another. Stated in terms of the use criterion, there is, at best, only a rather attenuated sense in which $\psi $ can be said to be used in a derivation of an irrelevant $\varphi \rightarrow \varphi $ .

As mentioned in the introduction, relevant logics have usually been studied either as sets of formulas rather than as Tarskian consequence relations, or the Tarskian consequence relation has not been understood to really express the relation of relevant entailment (with the exception of [Reference Meyer and McRobbie28, Reference Meyer and McRobbie29]), which is rather expressed by an implication connective $\rightarrow $ . These amount to a similar point, which is that relevant logics have usually been studied in terms of internal consequence relations, rather than external ones: that is, they are studied in terms of that relation which obtains between a collection of formulas $\Gamma $ and a formula $\varphi $ when a formula $\gamma \rightarrow \varphi $ is valid, where $\gamma $ is a complex formula combining the elements of $\Gamma $ with some kind of conjunction, usually either the lattice conjunction $\land $ or an intensional conjunction $\circ $ , commonly called “fusion”. Both of these choices bring along certain difficulties, and if we use the latter, $\circ $ , then, since there are reasons of relevance for rejecting $\varphi \circ \varphi \rightarrow \varphi $ (see [Reference Anderson and Belnap1, sec. 29.5]), we cannot take the internal consequence relation to hold between sets of formulas and formulas. In particular, we may not always add new copies of formulas and expect to preserve validity. This suggests that an approach employing multisets, like that taken in [Reference Avron4, Reference Meyer and McRobbie28, Reference Meyer and McRobbie29], is better suited to the task at hand.Footnote ⁴

The standard relevant logics, like R and its neighbours, are somewhat complex—for instance, do not have finite characteristic matrix presentations—so let’s consider a logic with a simpler presentation which nicely exemplifies some of these difficulties: Abelian logic.

Example 2.5. We present a variant of Abelian logic in the language consisting of binary connectives $\wedge $ , $\vee $ , and $\circ $ , a unary connective $\neg $ , and truth constants ${\bar {0}}$ and ${\bar {1}}$ .Footnote ⁵ Its semantics is given by the algebra ${\boldsymbol {\mathit {Z}}} = \langle \mathbb {Z}, \min ,\max , +, -, 0, 1 \rangle $ , which is in fact a characteristic algebra for Abelian logic. We also consider a defined binary connective $\to $ , where $\varphi \to \psi $ is defined as $\neg (\varphi \circ \neg \psi )$ , and constants for the integers, namely, $\overline {n+1}$ defined as ${\bar {n}}\circ {\bar {1}}$ and $\overline {-n}$ as $\neg {\bar {n}}$ , for all $n\ge 1$ .

If we take the usual order $\leq $ on ${\boldsymbol {\mathit {Z}}}$ , then there are three natural things we might try to define a consequence relation; for any formulas ${\varphi _1,\ldots ,\varphi _n},\psi $ , let us define:Footnote ⁶

$$ \begin{align*} \begin{array}{llll} \displaystyle{\varphi_1,\ldots,\varphi_n} &\!\!\!\!\vdash_{\boldsymbol {\mathit{Z}}}^{p} \psi, &\qquad\displaystyle\mathrm{iff} \qquad \displaystyle{\boldsymbol {\mathit{Z}}}&\!\!\!\!\models 0\leq \min({\varphi_1,\ldots,\varphi_n}) \Longrightarrow 0\leq \psi,\\ \displaystyle{\varphi_1,\ldots,\varphi_n} &\!\!\!\!\displaystyle\vdash_{\boldsymbol {\mathit{Z}}}^{\leq} \psi, &\displaystyle\qquad\mathrm{iff} \qquad {\boldsymbol {\mathit{Z}}} &\!\!\!\!\displaystyle\models\min({\varphi_1,\ldots,\varphi_n}) \leq \psi,\\ {\varphi_1,\ldots,\varphi_n} &\!\!\!\!\displaystyle\vdash' \psi, &\displaystyle\qquad\mathrm{iff} \qquad {\boldsymbol {\mathit{Z}}}&\!\!\!\!\displaystyle\models \varphi_1 + \varphi_2 + \dots + \varphi_n \leq \psi. \end{array} \end{align*} $$

Let’s consider these in turn. The relation $\vdash ^{p}_{{\boldsymbol {\mathit {Z}}}}$ is the most natural external consequence relation, as we are concerned with preservation of designated values (namely, the values $\geq 0$ ). Indeed, this is a (finitary) Tarskian consequence relation (in particular, it’s obviously monotone). However, it does not satisfy the deduction theorem: note that $0\leq \min (1,2)$ implies $0\leq 1$ , and yet $0\leq 1$ and $0\nleq 2\rightarrow 1$ . A natural way to try to buy this back is to internalise just a bit more, as in $\vdash ^{\leq }_{{\boldsymbol {\mathit {Z}}}}$ . This is, again, a finitary Tarskian consequence relation (studied in [Reference Paoli, Spinks and Veroff34]), but it also fails to satisfy the deduction theorem (the same choice of values as before suffices to show this). In addition, this also fails to satisfy the deduction rule of modus ponens, as $\min ({-\!1}\ {\to}\ {-\!2},{-\!1})\nleq\ {-\!2}$ .

In light of this, the most obvious way to go more internal is to use $+$ rather than $\min $ (i.e., $\circ $ rather than $\land $ ) to bunch premises, resulting in $\vdash '$ . Unfortunately, this definition, which is apparently the intended one, fails to be well-defined for sets of premises. By the definition, we have that $1\leq 1$ iff $ \{\overline {1}\}\vdash '\overline {1}$ iff $ \{\overline {1},\overline {1}\}\vdash '\overline {1}$ iff $2\leq 1$ . By using sets of premises, we have clearly distorted the intended meaning, and produced an obviously bad result here: the definition we want calls out to be given in a multiset framework.

These issues, caused by disregarding the number of occurrences of a formula, motivate a move to multiset consequence.Footnote ⁷ Let us recall now some basic notions.

Definition 2.6. A multiset over a set A is a mapping $M\colon A\to \mathbb N$ . The natural number $M(a)$ , for $a\in A$ , is called the multiplicity of the element $a$ in the multiset M. The set $|M|=\{a\in A\mid M(a)>0\}$ is called the support (or root set) of M.

We say that a multiset is finite if it has a finite support.Footnote ⁸ We use the usual square bracket notation $M=[{a_1,\ldots ,a_n}]$ for finite multisets, where $M(a)$ is the number of occurrences of $a$ in the list ${a_1,\ldots ,a_n}$ , provided the domain $A\supseteq \{{a_1,\ldots , a_n}\}$ of M is clear from the context or does not need to be specified.

We denote the set of all multisets over A by ${\mathcal {P}^{\#}}(A)$ and the set of all finite multisets over A by ${\mathcal {P}_{\!\mathit {fin}}^{\#}}(A)$ .

A set $B\subseteq A$ is identified with the multiset $M_B\colon A\to \mathbb N$ such that $M_B(a)=1$ if $a\in B$ and $M_B(a)=0$ if $a\notin B$ (i.e., $M_B$ is the characteristic function of B on A). The empty multiset will be written .

A multiset $M\in {\mathcal {P}^{\#}}(A)$ is a submultiset of a multiset $N\in {\mathcal {P}^{\#}}(A)$ , written $M\le N$ , if $M(a)\le N(a)$ for all $a\in A$ . The pointwise operations $\cup ,\cap ,\uplus ,\mathbin {\dot -}$ on multisets over A are defined as follows, for all $a\in A$ :

$$ \begin{align*} (M_1\cap M_2)(a)&= \min\{M_1(a), M_2(a)\}, && {\mathit{Intersection}} \\ (M_1\cup M_2)(a)&= \max\{M_1(a), M_2(a)\}, && {\mathit{Union}}\\ (M_1\uplus M_2)(a)&=M_1(a)+M_2(a), && {\mathit{Sum}}\\ (M_1\mathbin{\dot-} M_2)(a)&= \max\{0, M_1(a)-M_2(a)\}. && {\mathit{Difference}} \end{align*} $$

Returning to the example above, we can now adapt the definition of $\vdash '$ , using multisets, to be what we really want, namely:

$$ \begin{align*} [{\varphi_1,\ldots,\varphi_n}] \vdash_{\boldsymbol {\mathit{Z}}} \psi, \qquad\text{iff} \qquad & {\boldsymbol {\mathit{Z}}}\models \varphi_1 + \varphi_2 + \dots + \varphi_n \leq \psi. \end{align*} $$

This definition does get us what we want—we have the deduction theorem and the validity of modus ponens, for instance, as well as a variety of other nice properties. The move to multisets, and the ensuing rejection even of the version of monotonicity which we found to fail when premises are combined with $\circ $ (namely that $[\varphi ,\varphi ]\nvdash \varphi $ ), is quite natural. As mentioned above, the rejection of even this weak form of monotonicity is also motivated for reasons of relevance. This, along with the example, motivates our move to multiset consequence as providing a good basis for relevant consequence: essentially, the reason is that when we reject monotonicity, we want to really reject monotonicity. Giving such an account will be the goal of the paper.

3 From relevant tree proofs to relevant consequence relations

Our goal, as mentioned, is to obtain the right abstract account of relevant consequence. Our approach will be to start from a concrete approach, defining what it takes for a tree proof to be relevant (in accordance with the use criterion), and working from there.

The kind of proof system we’ll be interested in are axiomatic, composed out of consecutions, at first as above, objects of the form $\Gamma \rhd \varphi $ , where $\Gamma $ is a multiset of formulas and $\varphi $ a formula, and then, later, as objects $\Gamma \rhd \Delta $ where both $\Gamma ,\Delta $ are multisets of formulas. In general, a consecution system will comprise a set of axioms and inference rules, as before. After presenting proofs in such a system, we’ll present an abstract definition, and relate the latter to the former by way of justification.

We start with a pair of definitions giving us the grist for our mill.

It is worth observing that in Definition 3.2 we are not requiring closure under substitution, contrary to the most commonly used notions of axiomatic systems in the literature.

Let us fix now the central notion of this section: tree-proofs in a given axiomatic system $\mathcal {AS}$ from multisets of premises (once we’ve done so, we’ll pick out the relevant ones). As in the case of proof from sets of premises, proofs will be certain finite trees labeled by formulas. There are two main differences:

• • As the rules now have multisets of premises, when applying them to obtain formulas labeling non-leaves we have to speak about the multisets of labels of the immediate predecessors.

• • As some formula could label multiple leaves, we have to make sure that it is among the premises sufficiently many times (unless it is an axiom).

While the former difference requires only minimal changes in the notion of tree proof, the latter one requires a bigger change: to stress its importance we shall formulate it as an extra (fourth) condition in the definition of the tree-proof, even though it makes the second condition redundant. Recall that our aim is to capture the notion of relevant proof, which, given the use criterion, means that a formula should occur among the premises exactly as many times as we need it for the proof. Let’s say a bit more about why we take this rather strong reading of the use criterion, requiring that we list premises as many times as they’re used, in addition to requiring that they are used. As noted in the introduction, in the standard relevant logics, we generally do not have that $(\varphi \circ \varphi )\rightarrow \varphi $ is valid. Indeed, adding this to R gives rise to some rather irrelevant looking consequences, such as $\neg (\varphi \to \varphi )\rightarrow (\psi \to \psi )$ .Footnote ¹⁰ This indicates that keeping track not just of what premises are used, but also how many times they’re used is important for relevance purposes.

Given our robust version of the use criterion, there remains a subtle problem of how to deal with axioms which could, in principle, occur among premises; our definition below gives, in our opinion, the appropriate condition, but note that other options are available, and some will be discussed in Remark 3.4. Roughly speaking, we shall require axioms to be used in the proof at least as many times as they appear among the premises (i.e., if you mention it, you have to use it).

Let us state the definitions and then discuss a bit further. The notion of a tree-proof from a multiset of premises featuring in the following definition is a straightforward adaptation of the usual one, concerning sets. Our definition is just an adaptation of that introduced in [Reference Cintula, Gil-Férez, Moraschini and Paoli12, def. 5.26], but versions of this have been presented in [Reference Paoli33, def. 2.31] and [Reference Troelstra43, def. 7.4]; here, we additionally give a condition when a tree-proof is relevant (according to the use criterion).

We use the following notation to simplify the restriction in (relevant) proofs mentioned above: Given a tree T labeled by formulas, by $\Lambda _T$ we denote the multiset of labels of its leaves. Using this notation we can express the fourth condition in any of the following equivalent ways:

• • If $\chi $ is not an axiom of $\mathcal {AS}$ , then $\Lambda _T(\chi ) \le \Gamma (\chi )$ .

• • Each element of $|\Lambda _T\mathbin {\dot -} \Gamma |$ is an axiom of $\mathcal {AS}$ .

• • There is a finite multiset of axioms $\Delta $ such that $\Lambda _T \le \Gamma \uplus \Delta $ .

The relevant condition can then be simply expressed as $\Gamma \le \Lambda _T$ and its conjunction with the fourth condition is then equivalent to:

• • There is a finite multiset of axioms $\Delta $ such that $\Lambda _T = \Gamma \uplus \Delta $ .

Remark 3.4. Condition (R) is a simple implementation of the Use Criterion, requiring just that in a relevant tree derivation, each premise is used. The way the premises must be used, in particular, is that they should be inputs to one of the rules of inference used to obtain the conclusion. In many cases, there is only one rule of inference, modus ponens, so in such systems, to quote Meyer [Reference Meyer27, p. 54], “use in a deduction is use in an application of modus ponens.”

The definition of a relevant proof is further complicated by the fact that some axioms of $\mathcal {AS}$ could appear in $\Gamma $ , and there is a question of whether we should require these to be used in the same way that non-axioms are. Consideration of this question naturally gives rise to two related alternative definitions we could use to replace the relevance condition. A tree-proof T in $\mathcal {AS}$ from a mutliset of premises $\Gamma $ is:

1. (wr) Weakly relevant if each element of $|\Gamma \mathbin {\dot -} \Lambda _T|$ is an axiom, i.e., for each $\chi $ which is not an axiom of $\mathcal {AS}$ we have $\Lambda _T(\chi ) = \Gamma (\chi )$ .

2. (sr) Strongly relevant if $\Gamma = \Lambda _T$ .

Using the suggestive notation $\vdash _{\!\mathcal {AS}}^{\mathit {wr}}$ and $\vdash _{\!\mathcal {AS}}^{\mathit {sr}}$ , we observe that, as the names suggest:

$$\begin{align*}{\vdash_{\!\mathcal{AS}}^{\mathit{sr}}} \subseteq {\vdash_{\!\mathcal{AS}}^{\mathit{r}}} \subseteq {\vdash_{\!\mathcal{AS}}^{\mathit{wr}}} \subseteq {\vdash_{\!\mathcal{AS.}}} \end{align*}$$

The difference is easily demonstrated: consider a set $\mathit {Fm}= \{x,y,z\}$ and an axiomatic system $\mathcal {AS}$ with the axioms $x$ and $y$ and no rules. Then the only tree-proof of $x$ in $\mathcal {AS}$ we can write is the one-node tree T whose only node is labeled by $x$ , i.e., $\Lambda _T = [x]$ . We can easily determine when T is a (weakly/strongly) (relevant) proof of $x$ from a multiset $\Gamma $ :

Therefore we have:

• • $[x,z]\vdash _{\!\mathcal {AS}} x$ , but not $[x,z]\vdash ^{\mathit {wr}}_{\!\mathcal {AS}} x$ .

• • $[x,y]\vdash ^{\mathit {wr}}_{\!\mathcal {AS}} x$ , but not $[x,y]\vdash ^{\mathit {r}}_{\!\mathcal {AS}} x$ .

• • , but not .

Note that if $\mathcal {AS}$ has no axioms, then:

$$ \begin{align*}{\vdash^{\mathit{r}}_{\!\mathcal{AS}}} = {\vdash^{\mathit{sr}}_{\!\mathcal{AS}}} = {\vdash_{\!\mathcal{AS}}^{\mathit{wr}}}. \end{align*} $$

Throughout, we’ll adopt the relevant condition, leaving aside consideration of weak and strong relevance, but we note the distinction here in order to draw attention to the fact that there is a choice point and to give some indication of why we have gone the way we did.

Remark 3.5. It’s worth noting that the relations $\vdash ^r_{\mathcal {AS}}$ and $\vdash _{\mathcal {AS}}$ may coincide, e.g., whenever $\mathcal {AS}$ contains all instances of the inference rules of Modus Ponens ( $\varphi ,\varphi \to \psi \rhd \psi $ ) and weakening ( $\varphi \rhd \psi \to \varphi $ ).

In such a case we can take any tree-proof T of $\varphi $ from $\Gamma \uplus [\chi ]$ , for any formula $\chi $ , and construct a tree-proof $T^\chi $ of $\varphi $ from $\Gamma \uplus [\chi ]$ such that $\Lambda _{T'} = \Lambda _{T}\uplus [\chi ]$ : indeed it suffices to add a new root labeled $\chi \to \varphi $ connected to the original one by weakening and then adding a new leaf labeled $\chi $ and new root labeled by $\varphi $ again and connecting them using Modus Ponens. In a picture, with weakening we can move from the left tree proof below to the right one, increasing the number of uses of $\chi $ by one.

Repeating this trick as many times as necessary yields a relevant tree-proof S of $\varphi $ from $\Gamma \uplus [\chi ]$ .

Our next goal is to find some simple description of (relevant) provability relations ${\vdash _{\!\mathcal {AS}}^{\mathit {(r)}}}$ . Below is our definition appropriate to the work we aim to do in this paper and, as discussed, it differs in key regards from other standard accounts which involve monotonicity and contraction as defining conditions (the latter just in virtue of using sets).

We use some usual notational conventions, stated next to their set versions for easy comparison:

With that remark out of the way, let us note that the following facts are immediate:

Let us note that any consequence relation can be seen as a set of consecutions. This allows us to state and prove several interesting facts. The first one is obvious.

Next we prove that the relation $\vdash ^{r}_{\!\mathcal {AS}}$ is indeed an example of a consequence relation, and furthermore that it is an especially simple example of such. This goes part of the way towards justifying our abstract definition in terms of the concrete examples which obey the Use Criterion.

Proof. First we show that $\vdash ^{r}_{\!\mathcal {AS}}$ (resp. $\vdash _{\!\mathcal {AS}}$ ) is a monotone consequence relation on $\mathit{Fm}$ . The Reflexivity for both $\vdash _{\!\mathcal {AS}}$ and $\vdash ^{r}_{\!\mathcal {AS}}$ is obvious as a tree with a single node labeled by $\varphi $ is a (relevant) tree-proof of $\varphi $ from the premises $[\varphi ]$ . The Monotonicity for $\vdash _{\!\mathcal {AS}}$ is also straightforward (clearly any tree-proof T from premises $\Gamma $ is a tree-proof from $\Gamma \uplus \Delta $ for any $\Delta $ ). Finally, we have to deal with Cut: suppose that we have a (relevant) tree-proof T of $\varphi $ from $\Gamma \uplus [\psi ]$ and a (relevant) tree-proof S of $\psi $ from $\Delta $ and we have to find a (relevant) proof R of $\varphi $ from $\Gamma \uplus \Delta $ . First note that in the relevant case $\psi $ has to label a leaf ${\boldsymbol {l}}$ of T (as $\Gamma \uplus [\psi ] \le \Lambda _T$ ) and in the monotonic one we can, without loss of generality, assume it at well (indeed otherwise already T would be the proof of $\varphi $ from $\Gamma \uplus \Delta $ ). Thus we can define R as the tree resulting from T by replacing ${\boldsymbol {l}}$ by the tree S and note that for each formula $\chi $ we have:

$$ \begin{align*}\Lambda_{R}(\chi) = \begin{cases} \Lambda_{T}(\chi) + \Lambda_{S}(\chi), & \text{if } \chi \neq \psi,\\ \Lambda_{T}(\chi) + \Lambda_{S}(\chi) - 1, & \text{if } \chi = \psi. \end{cases} \end{align*} $$

To show that R is indeed a tree-proof of $\varphi $ in $\mathcal {AS}$ from $\Gamma \uplus \Delta $ we consider a formula $\chi $ which is not an axiom and recall from the assumptions we know that:

$$ \begin{align*}\Lambda_{T}(\chi) \le (\Gamma\uplus[\psi])(\chi) \qquad\text{and}\qquad \Lambda_{S}(\chi) \le \Delta(\chi). \end{align*} $$

Thus for $\chi \neq \psi $ we get:

$$ \begin{align*}\Lambda_{R}(\chi) = \Lambda_{T}(\chi) + \Lambda_{S}(\chi) \le (\Gamma\uplus[\psi])(\chi) + \Delta(\chi) = \Gamma(\chi) + \Delta(\chi) = (\Gamma\uplus \Delta)(\chi), \end{align*} $$

and for $\chi = \psi $ we get:

$$ \begin{align*}\Lambda_{R}(\psi) &= \Lambda_{T}(\psi) + \Lambda_{S}(\psi) - 1 \le (\Gamma\uplus[\psi])(\psi) + \Delta(\psi) - 1 \\&= \Gamma(\psi) + 1 + \Delta(\psi) - 1 = (\Gamma\uplus \Delta)(\psi). \end{align*} $$

The proof that R is relevant if T and S are is analogous; we only observe that in this case we have inequalities converse to those above even for the axioms.

The fact that $\mathcal {AS} \subseteq {\vdash ^{(r)}_{\!\mathcal {AS}}}$ is obvious. In order to show the minimality condition, let us focus on the relevant case (the monotonic one is just its simpler variant). Consider any consequence relation $\vdash $ such that $\mathcal {AS}\subseteq {\vdash }$ ; we need to show that ${\vdash ^{r}_{\!\mathcal {AS}}}\subseteq \,\vdash $ . Assume that we have a relevant tree-proof T of $\varphi $ from $\Gamma $ ; for each node ${\boldsymbol {n}}$ , let $\varphi _{\boldsymbol {n}}$ denote the formula labeling it and let $\Gamma _{\boldsymbol {n}}$ denote the multiset of formulas labeling the leaves of the subtree of T containing the node ${\boldsymbol {n}}$ and all its predecessors.

We prove by induction over the tree that $\Gamma _{\boldsymbol {n}}\vdash \varphi _{\boldsymbol {n}}$ (in the monotone case we prove directly that $\Gamma \vdash \varphi _{\boldsymbol {n}})$ . For ${\boldsymbol {n}}$ being a leaf we have $\Gamma _{\boldsymbol {n}} = [\varphi _{\boldsymbol {n}}]$ and so the claim follows by the Reflexivity of $\vdash $ . Let $[{\boldsymbol {n}}_1, \dots , {\boldsymbol {n}}_k]$ be the multiset of immediate predecessors of ${\boldsymbol {n}}$ ; then we know that $\varphi _{{\boldsymbol {n}}_1}, \dots , \varphi _{{\boldsymbol {n}}_k}\rhd \varphi _{\boldsymbol {n}} \in \mathcal {AS}$ and thus also $\varphi _{{\boldsymbol {n}}_1}, \dots , \varphi _{{\boldsymbol {n}}_k}\vdash \varphi _{\boldsymbol {n}}$ (because $\mathcal {AS} \subseteq {\vdash }$ ). By the induction hypothesis we have $\Gamma _{{\boldsymbol {n}}_i}\vdash \varphi _{{\boldsymbol {n}}_i}$ for each $i \leq k$ and thus by the Relevant Cut we obtain $\Gamma _{{\boldsymbol {n}}_1}, \dots , \Gamma _{{\boldsymbol {n}}_k} \vdash \varphi _{\boldsymbol {n}}$ . Observe that $\Gamma _{\boldsymbol {n}} = \Gamma _{{\boldsymbol {n}}_1} \uplus \Gamma _{{\boldsymbol {n}}_2} \dots \uplus \Gamma _{{\boldsymbol {n}}_k}$ , which completes the proof of this part.

To complete the whole proof it suffices to observe that for the root ${\boldsymbol {r}}$ we have $\Gamma _{\boldsymbol {r}} = \Lambda _T$ thus $\Lambda _T \vdash \varphi $ . Because T is relevant proof we know that there is a finite multiset $\Delta $ of axioms of $\mathcal {AS}$ such that $\Lambda _T = \Gamma \uplus \Delta $ . From the fact that $\mathcal {AS} \subseteq {\vdash }$ we know that $\Delta $ consists of theorems of $\vdash $ , and therefore by Theorem Removal we know that $\Gamma \vdash \varphi $ .

So we can obtain the least (relevant) consequence relation for $\mathcal {AS}$ by taking that generated by its (relevant) tree proofs as defined above, and these were defined so as to guarantee observation of the Use Criterion. So while the abstract consequence relations we have defined simply remove monotonicity and a form of contraction from the Tarskian case, nonetheless this relatively minor adaptation is adequate to capture the behaviour of the concrete cases in which we’ve been most interested. Indeed, we can say a bit more to justify this connection.

Clearly any consequence relation $\vdash $ can be seen as an axiomatic system and so due to Proposition 3.12 it is its own relevant presentation (i.e., ${\vdash } = {\vdash ^r_{\vdash }}$ ); if furthermore $\vdash $ is monotone, then it is its own presentation ( ${\vdash } = {\vdash _{\vdash }}$ ). Therefore we can obtain a variant of Łoś–Suszko theorem.

It is worth noting that the previous theorem, perhaps surprisingly, implies that every monotone consequence relation has a relevant presentation. However, we have seen already in Remark 2.2, that we could have ${\vdash _{\!\mathcal {AS}}} = {\vdash ^{r}_{\!\mathcal {AS}}}$ which implies ${\vdash ^{r}_{\!\mathcal {AS}}}$ could be a monotone consequence relation: in general, relevance of proof is a property that holds regardless of one’s choice of axioms and inference rules. If one chooses irrelevant axioms and rules, then making the proofs ‘relevant’ in our sense won’t do you much good as far as avoiding relevance is concerned.Footnote ¹⁴ The next two propositions (with obvious proofs) explicate this phenomenon.

Proposition 3.15. Let $\mathcal {AS}$ be an axiomatic system. Then $\vdash ^{r}_{\!\mathcal {AS}}$ is a monotone consequence relation iff

$$ \begin{align*}{\vdash_{\!\mathcal{AS}}} = {\vdash^{r}_{\!\mathcal{AS.}}} \end{align*} $$

Proposition 3.16. Let $\vdash $ be a consequence relation. Then for the least monotone consequence relation $\vdash _{m}$ containing $\vdash $ , known as the monotonic companion of $\vdash $ , we have:

$$ \begin{align*}\Gamma \vdash_{m} \varphi \qquad\text{iff}\qquad \Delta \vdash \varphi \quad\text{for some } \Delta\leq \Gamma. \end{align*} $$

Furthermore, any relevant presentation of $\vdash $ is a presentation of $\vdash _m$ .

4 An example: $\mathbf {BCI}$ and some related systems

Let’s take some time to consider some concrete examples of relevant consequence relations over some familiar logics in the vicinity of the relevant logic family. We’ve seen a bit of Abelian logic as an example, adding some justification to using multisets, but there are a number of better known systems we can study using these tools. Let’s start with the implication fragment of the well-known substructural logic $\mathbf {BCI}$ :

1. (I) $\varphi \rightarrow \varphi ,$

2. (B) $(\varphi \rightarrow \psi )\rightarrow ((\chi \rightarrow \varphi )\rightarrow (\chi \rightarrow \psi )),$

3. (C) $(\varphi \rightarrow (\psi \rightarrow \chi ))\rightarrow (\psi \rightarrow (\varphi \rightarrow \chi )),$

4. (mp) $\varphi \rightarrow \psi ,\varphi \rhd \psi .$

This is a sublogic of very many systems, including the relevant logic $\mathbf {R}$ , about which we’ll have more to say shortly, as well as fuzzy logics, intuitionistic logic, and others. It’s easy to extend this system by an intensional conjunction (or fusion) connective $\circ $ , with the additional axioms:

1. (Res₁) $((\varphi \circ \psi )\rightarrow \chi )\rightarrow (\varphi \rightarrow (\psi \rightarrow \chi )),$

2. (Res₂) $(\varphi \rightarrow (\psi \rightarrow \chi ))\rightarrow ((\varphi \circ \psi )\rightarrow \chi ).$

In general, we’ll just refer to $\mathbf {BCI}$ , and let context determine which connectives are around (for now, it’s just $\rightarrow $ and $\circ $ ). First off, it’s an obvious, nice feature of the definition of relevant tree proof that we can obtain a proof of a deduction-like theorem for the internal consequence relation of $\mathbf {BCI}$ and some relevant logics expanding it (as we’ll see, there are some problems that come up for considering the standard relevant systems in their full vocabularies, but we can give this argument for at least some of the salient systems).

This proposition indicates that, in at least some cases, we’re getting what we want: that is, that our definition results in a match between the external consequence relation (of the kind we’ve defined) and the internal consequence relation, given by provable implication in the logic. $\mathbf {BCI}$ is a particularly nice logic, and this result extends to such extensions as the ‘intensional’ fragment of R (including implication and multiplicative conjunction and disjunction). However, there are limitations, related to our use of multisets rather than some more discerning kind of structure. First we’ll note that the above result fails in some logics weaker than $\mathbf {BCI}$ , and then go on to discuss issues with incorporating the lattice conjunction of R, which is usually (for reasons we’ll mention) formulated as a meta-rule.

For the first point, note about the inclusion of the permutation axiom (C) that it ensures that $(\psi \circ (\varphi \circ \chi ))\rightarrow ((\psi \circ \varphi )\circ \chi )$ is derivable in $\mathbf {BCI}$ . This, along with the prefixing axiom (B), ensures that $\circ $ is associative (hence showing that $\mathbf {BCI}$ is an extension of the associative Lambek calculus, with the left division as a notational variant of the right). Note, however, that $\circ $ may not be associative in the case of logics without (C), and that this has a substantial consequence for the deduction theorem just proved. Consider the relevant logic $\mathbf {T}_{\rightarrow ,\circ }$ , the implication/fusion fragment of $\mathbf {T}$ , which can be axiomatised by replacing (C) with the suffixing axiom (B $'$ ) and the contraction axiom (W):

1. (B $'$ )   $(\varphi \rightarrow \psi )\rightarrow ((\psi \rightarrow \chi )\rightarrow (\varphi \rightarrow \chi )).$

2. (W)  $(\varphi \rightarrow (\varphi \rightarrow \psi ))\rightarrow (\varphi \rightarrow \psi ).$

The failure of the associativity of $\circ $ gives rise, in $\mathbf {T}_{\rightarrow ,\circ }$ , to a remarkable failure of the above deduction-like theorem, given our definitions, as we can see with a simple matrix argument. We have one direction immediately, given the inclusion of (mp), but the converse direction fails.

Proof. The positive part is straightforward, so we’ll just present the matrix to show the negative part: the desired valuation has $\varphi \mapsto 2,\psi \mapsto 0,\chi \mapsto 1$ , where $\varphi $ is true on a valuation $v$ just when $v(\varphi )=3$ .Footnote ¹⁵

This fact also holds for $\mathbf {E}_{\rightarrow ,\circ }$ , which retains a (very) weak form of permutation but, like $\mathbf {T}_{\rightarrow ,\circ }$ , doesn’t validate $\varphi \circ (\psi \circ \chi )\rightarrow (\varphi \circ \psi )\circ \chi $ and, hence, is not fully associative. However, the famous relevant logic $\textbf {R}_{\rightarrow ,\circ }$ , which extends $\mathbf {BCI}$ by (W) does enjoy something like the above deduction-like theorem (and note, $\mathbf {BCI}$ is the implication-fusion fragment of the contraction-free relevant system RW).

Among the usual relevant logics studied in [Reference Anderson and Belnap1, Reference Routley, Meyer, Plumwood and Brady38], $\mathbf {R}$ is the upper limit of deductive strength, and most of the other systems do not have (C), so among the usual family of relevant logics, $\vdash ^{r}$ only ‘externalises’ the internal consequence relation (that expressed by the provability of a formula of the form $\varphi _1\circ \dots \circ \varphi _n\rightarrow \psi $ ) of a handful of (famous, well-studied) systems. For the others, it seems that even the use of multisets does not bring in enough structure, and that a move to sequences or even trees (as the data types of premises) is necessary. This is a natural avenue for future research, but we leave it aside, focusing on the simple case with multisets.

On to the second point, in order to add a lattice conjunction $\land $ to the relevant systems of [Reference Anderson and Belnap1, Reference Routley, Meyer, Plumwood and Brady38], it’s not adequate just to use axioms and inference rules, as we’ve done so far. In particular, these systems have it that if $\varphi $ and $\psi $ are both derivable from no premises, then $\varphi \land \psi $ is so derivable. Importantly, this fact is not reflected in an axiom, such as the usual $\varphi \rightarrow (\psi \rightarrow (\varphi \land \psi ))$ , as this delivers $\varphi \rightarrow (\psi \rightarrow \varphi )$ in that context, the avoidance of which is one of the main aims of the relevant enterprise (and the fact that $(\varphi \land \psi )\rightarrow (\varphi \land \psi )$ is derivable does us no good at all). Similarly, using an inference rule won’t help, as then, given Cut, if we had $\varphi ,\psi \rhd \varphi \land \psi $ , we’d have $\varphi \land \psi \vdash ^r\varphi $ and thus $\varphi ,\psi \vdash ^r\varphi $ , buying us back all the monotonicity we have worked to avoid. What we need to add will have the effect of what Smiley [Reference Tarski41] called a rule of proof: something of the form “if $\vdash \varphi _1,\dots ,\vdash \varphi _n$ then $\vdash \psi $ ”. This is the form of a simple kind of metarule, with zero premises, and perhaps the most famous example is the necessitation rule of the global consequence relations of modal logics. The usual rule of adjunction in relevant logics is given in the form “if $\vdash _{\textbf {R}}\varphi $ and $\vdash _{\textbf {R}}\psi $ , then $\vdash _{\textbf {R}}\varphi \land \psi $ ”, and as mentioned this is for a good reason. In some relevant logics much weaker than R, such as those studied in [Reference Routley, Meyer, Plumwood and Brady38], we need to add further rules of proof (such as rule versions of prefixing, suffixing, and contraposition).

So, the full relevant systems do not fall under the definition of “axiom system” we gave before without some tweaking—for instance, by taking all the theorems of some other presentation of $\mathbf {R}$ and making that the set of axioms and using the rule (mp) we could do it (though that smells a little of cheating, it is an option).Footnote ¹⁶ The point of this discussion of limitations is to suggest further avenues for improving on the proposal we give here, and motivates the claim that doing so is desirable to capture a wider range of relevant logics (with all their connectives). This task we leave for another occasion, focusing on the simpler setting of multiset-consequence.

5 Symmetric relevant consequence relations

With our adaptation of Tarskian consequence settled, a natural next step is toward multiple conclusion (or “Scott”) consequence relations. As mentioned, the approach we’ll take here is inspired by Bolzano, employing a ‘conjunctive’ reading of the conclusion.Footnote ¹⁷ This is simple, and in keeping with a common approach (for instance, in [Reference Font19, Reference Galatos and Tsinakis22]), so we’ll adopt it, leaving investigations into other options for future work.

The rough picture we’re trying to capture is that a collection of premises should have a collection of conclusions as consequences just in case each conclusion is a consequence of some of the premises (enforcing the relevance constraints we have taken on board in the single conclusion case, so that each premise is used in order to obtain some conclusion or other). This is useful as a heuristic but it doesn’t pin down any particular formal representation as yet. One choice point concerns when to take a multiset of premises to have the empty (multi)set of conclusions as a consequence. In the disjunctive reading, familiar from Gentzen systems, an empty conclusion expresses the falsum, but this seems inappropriate for the conjunctive reading. Should we take it that only the empty (multi)set of formulas has itself as a consequence? This approach has the advantage of simplicity, but it is quite out of step with the usual, monotone, case. In that case, if a set of premises has the empty set as a consequence, then it also has every theorem as a consequence—this follows immediately from the usual definition. Without monotonicity, however, this is not a consequence of the definition, unless we build it in explicitly. To a certain extent, the choice here is arbitrary, but so far we have worked with consequence relation notions that collapse into the Tarskian versions when we reimpose the Tarskian conditions: taking this on board as a desideratum suggests that the treatment of empty conclusions should be similar.

Let’s get to the definitions, starting from a relatively simple case. If $\vdash $ is a monotone and contractive consequence relation, we can define a symmetrization $\vdash ^s$ in a simple way, letting $\Gamma \uplus [{{\chi }_1,\ldots ,{\chi }_n}]$ be a finite multiset of formulas:

$$ \begin{align*}\Gamma \vdash^s [{{\chi}_1,\ldots,{\chi}_n}], \qquad\text{iff} \qquad \Gamma \vdash \chi_i \text{ for each }\ i\leq n. \end{align*} $$

Note that this definition has “right-side” contraction built in; indeed we obviously have:Footnote ¹⁸

• • If $\Gamma \vdash ^s \Delta ,\varphi ,\varphi $ , then $\Gamma \vdash ^s \Delta ,\varphi $ . r-Contraction

(Clearly, we do not have right-side monotony, just in virtue of the conjunctive reading of the conclusions. This holds even in the standard case.)

Removing contraction but keeping monotony makes things a bit more complicated, but still rather intuitive. The basic idea is to pay attention to what parts of $\Gamma $ are needed to derive each $\chi _i$ : so we ‘partition’ $\Gamma $ into the multiset union of a collection $\Gamma _1,\dots ,\Gamma _i$ of multisets, requiring that each $\chi _i$ is such that $\Gamma _i\vdash \chi _i$ holds for some $\Gamma _i$ . So we don’t build in contraction, as we’re paying attention to how often each premise among $\Gamma $ is used, even when multiple copies may be used to obtain different $\chi _i$ ’s among the conclusions. Stated formally:

$$ \begin{align*} \Gamma \vdash^s [{{\chi}_1,\ldots,{\chi}_n}], \qquad\text{iff} \qquad & \text{for each } i \leq n \text{ there is } \Gamma_i \le \Gamma \text{ such that } \Gamma_i \vdash \chi_i \text { and}\\& \Gamma_1\uplus\dots\uplus\Gamma_n \le \Gamma. \end{align*} $$

In full generality we will obviously want the relation of equality instead of submultisethood to obtain between the set of premises the multiset union of its subsets needed to prove each of the conclusions, note however that such a definition would entail that

holds iff

. As mentioned earlier, however, this is out of step with the usual, Tarskian, case. So we’ll build in the usual behaviour (that some premises $\Gamma $ have

as a consequence just in case $\Gamma $ already has every theorem as a consequence). These considerations give rise to the following definition, for $n\ge 1$ :

It is easy to see that by adding the monotony (and contraction), this general definition gives rise to the more standard one with which we started. With our intuitions fixed and formally expressed, let us formulate the essential properties of symmetric consequence relations.Footnote ¹⁹

Let us define the asymmetric variant $\vdash ^a$ of a symmetric consequence relation $\vdash $ as follows:

$$ \begin{align*}\Gamma \vdash^{a} \varphi, \qquad\text{iff}\qquad \Gamma \vdash [\varphi]. \end{align*} $$

With this, we can prove the following proposition, showing that the symmetric and asymmetric variants of consequence relations interact as one might hope.

Proof. (i): In the proof of the first part of (i), only transitivity poses a potential problem: let us assume that $\Gamma \vdash ^s [\delta _1,\ldots ,\delta _k]$ and $[\delta _1,\ldots ,\delta _k] \vdash ^s [{{\psi }_1,\ldots ,{\psi }_n}]$ and show that $\Gamma \vdash ^s [{{\psi }_1,\ldots ,{\psi }_n}]$ .

Let us first deal with the case where $k=0$ : if $n = 0$ as well then the claim is trivial. Otherwise from the second assumption we know that the $\psi _i$ s are theorems of $\vdash $ and so, due to the first assumption, $\Gamma \vdash \psi _1$ holds and thus fixing $\Gamma _1 = \Gamma $ and for each $i < n$ , we obtain the desired partition of $\Gamma $ , which completes the proof.

Next assume that $k> 0$ and $n =0$ : then from the second assumption we know that $\delta _1,\ldots ,\delta _k\vdash \chi $ holds for each theorem $\chi $ of $\vdash $ and, due to the first assumption, there are multisets $\Gamma _1,\ldots ,\Gamma _k$ such that $\Gamma _i \vdash \delta _i$ , and $\Gamma _1\uplus \dots \uplus \Gamma _k = \Gamma $ . Therefore due to Relevant Cut we know that $\Gamma \vdash \chi $ as required.

The case where $k> 0$ and $n> 0$ is only a bit more complex: we know that there are multisets $\Gamma _1,\ldots ,\Gamma _k$ and ${{\Delta }_1,\ldots ,{\Delta }_n}$ such that $\Gamma _1\uplus \dots \uplus \Gamma _k = \Gamma $ and $\Delta _1\uplus \dots \uplus \Delta _n = [\delta _1,\ldots ,\delta _k]$ and for each $i\leq k$ and $j\leq n$ we have $\Gamma _i \vdash \delta _i$ and $\Delta _j \vdash \psi _j$ . Without loss of generality, we can assume that there is a non-decreasing sequence $0 = s^1 \leq s^2 \leq \dots \leq s^k < k$ of integers and a sequence $n^1, \dots , n^{k}$ of non-negative integers such that $\Delta _j = [\delta _{s^j+1},\dots , \delta _{s^j + n^j}]$ (note that this includes the possibility that $n = 0$ , i.e., ). To complete the proof it suffices to set $\Gamma _j = \Gamma _{s^{j}+1} \uplus \dots \uplus \Gamma _{s^j + n^j}$ .

The monotone and/or contractive conditions are easily checked and the rest of the proof of the first claim is straightforward, so we have that:

$$ \begin{align*}\Gamma \vdash \varphi, \qquad\text{iff}\qquad \Gamma \vdash^s [\varphi] \qquad\text{iff}\qquad \Gamma \mathrel{(\vdash^s)^{a}} \varphi. \end{align*} $$

(ii): For the first part of the proof we will show, on the assumption that $\vdash $ is a symmetric consequence relation, that $\vdash ^a$ satisfies Cut and Reflexivity. The second part is immediate, since $\varphi \vdash \varphi $ holds by the hypothesis on $\vdash $ . So assume that $\Gamma , \psi \vdash ^a \varphi $ and $\Delta \vdash ^a \psi $ , in order to show that $\Gamma , \Delta \vdash ^a \varphi $ or, equivalently, that $\Gamma ,\Delta \vdash \varphi $ . We have that $\Gamma ,\psi \vdash \varphi $ and $\Delta \vdash \psi $ , and by Compatibility, it follows that $\Gamma ,\Delta \vdash \Gamma , \psi $ and, hence, one application of Transitivity gives that $\Gamma , \Delta \vdash \varphi $ . Checking Monotonicity and Contraction are also simple exercises left to the reader.

Finally, we verify that ${\vdash } \supseteq {(\vdash ^a)^s}$ . By definition, $\Gamma \mathrel {(\vdash ^a)^{s}} [{{\chi }_1,\ldots ,{\chi }_n}]$ only if $\text {for each } i \leq n \text { there is } \Gamma _i \le \Gamma \text { such that } \Gamma _i \vdash ^a \chi _i \text { and } \Gamma _1\uplus \dots \uplus \Gamma _n = \Gamma $ . The latter, in turn, implies that for each such $\Gamma _i$ , we have that $\Gamma _i \vdash \chi _i$ . Now, given $\Gamma _2 \vdash \chi _2$ , by Compatibility, we have that $\chi _1,\Gamma _2 \vdash [\chi _1, \chi _2]$ , and similarly since $\Gamma _1 \vdash [\chi _1]$ , we may obtain that $\Gamma _1, \Gamma _2 \vdash \chi _1, \Gamma _2$ and hence, by Transitivity, we get $\Gamma _1, \Gamma _2 \vdash [\chi _1, \chi _2]$ . Proceeding in this manner, it follows that $\Gamma _1\uplus \dots \uplus \Gamma _n = \Gamma \vdash [{{\chi }_1,\ldots ,{\chi }_n}]$ , which is what we wanted.

(iii): In the presence of monotonicity and contraction, our definition of the $\vdash ^s$ relation is the one at the beginning of the section. We can see that in this case, $\chi _i \vdash \chi _i$ by Reflexivity, and then $[{{\chi }_1,\ldots ,{\chi }_n}] \vdash [\chi _i]$ by Monotonicity. Hence, $\Gamma \vdash [{{\chi }_1,\ldots ,{\chi }_n}]$ implies that $\Gamma \vdash [\chi _i]$ by Transitivity, so we have that for each $i \leq n$ , $\Gamma \vdash ^a \chi _i$ and hence, $\Gamma \mathrel {(\vdash ^a)^{s}} [{{\chi }_1,\ldots ,{\chi }_n}]$ as desired.

Let us consider two examples of different ways of symmetrizing an asymmetric consequence relation, and ways these options can diverge. In so doing, we’ll show that the converse inclusion of point (ii) of the previous proposition can fail.

This shows the principle of the thing, but we can give a more natural example.

Example 5.4. Recall the consequence relation $\vdash _{\boldsymbol {\mathit {Z}}}$ , given in Example 3.8, and define an alternate symmetrization $\vdash $ as follows (recall that we treat the empty sum as $0$ ):

$$ \begin{align*}\varphi_1, \dots, \varphi_m \vdash [{\psi_1,\ldots,\psi_n}] \qquad\text{iff} \qquad {\boldsymbol {\mathit{Z}}} \models \varphi_1 + \varphi_2 + \dots + \varphi_m \leq \psi_1 + \psi_2 + \dots + \psi_n. \end{align*} $$

It is easy to see that ${\vdash ^a} = {\vdash _{\boldsymbol {\mathit {Z}}}}$ but ${\vdash } \neq {\vdash ^s_{\boldsymbol {\mathit {Z}}}}$ and so ${\vdash } \neq (\vdash ^a)^s$ . Indeed we have $\vdash [\overline {1},\overline {-1}]$ , but since $\nvdash _{\boldsymbol {\mathit {Z}}}\overline {-1}$ , it follows that $\nvdash ^s_{\boldsymbol {\mathit {Z}}}[\overline {1},\overline {-1}]$ .

Another example could be built using the logic $\mathbf{BCI}$ expanded by conjunction $\circ $ and its unit, the truth constant $t$ described by axioms $\varphi \circ t\to \varphi $ and $\varphi \to t\circ \varphi $ : let us define an alternate symmetrization $\vdash $ for it as (treating the empty conjunction as constant $t$ ):

$$ \begin{align*}\varphi_1, \dots, \varphi_m \vdash [{\psi_1,\ldots,\psi_n}], \qquad\text{iff} \qquad \vdash_{\mathbf{BCI}} \varphi_1 \circ \varphi_2 \circ \dots \circ \varphi_m \to \psi_1 \circ \psi_2 \circ \dots \circ \psi_n. \end{align*} $$

As in the previous case, it is easy to see that ${\vdash ^a} = {\vdash _{\mathbf {BCI}}}$ (thanks to Proposition 4.1) but ${\vdash } \neq {\vdash ^s_{\mathbf {BCI}}}$ and so ${\vdash } \neq (\vdash ^a)^s$ . Indeed we have $\varphi \circ \varphi \vdash [\varphi ,\varphi ]$ but if $\nvdash _{\mathbf {BCI}}\varphi $ we have $\varphi \circ \varphi \nvdash ^s_{\mathbf {BCI}}[\varphi ,\varphi ]$ .

Let us consider some additional properties of symmetric relations between finite multisets and their interplay with the defining conditions of symmetric consequence relations. The following are some natural candidates:

• • $\Gamma ,\Delta \vdash \Gamma $ . Generalized Reflexivity

• • . Theorem Reflexivity

• • If $\Gamma \vdash \Delta $ and $\Delta ,\Psi \vdash \Phi $ , then $\Gamma ,\Psi \vdash \Phi $ . Multi-Cut

• • If and $\Delta , \Psi \vdash \Phi $ , then $\Psi \vdash \Phi $ . Theorem Removal

The proofs of the following are straightforward, and left to the enthusiastic reader.

6 Derivations in the symmetric case

In the asymmetric case we started from the concrete, with the notion of a (relevant) tree proof, and moved to the abstract: in the symmetric case, we’re in the process of doing the opposite. So far we’ve fixed the abstract notion of symmetric consequence, building on the asymmetric one, but that leaves the natural question of how to understand derivations of multiple conclusions from multiple premises, in our sense. To that end, let us fix some definitions and infer some immediate results indicating that our abstract notion of symmetric consequence captures an intuitive concrete one (relying, as always, on our assumptions of finitude).

Definition 6.2 (Finitary derivation).

Let $\mathcal {AS}$ be a symmetric axiomatic system in $\mathit{Fm}$ . A derivation of $\Delta $ from $\Gamma $ in $\mathcal {AS}$ is a finite sequence $\Gamma = \Gamma _1,\dots , \Gamma _n \ge \Delta $ of finite multisets of formulas such that for every $1< i \leq n$ , there is a rule $\Psi \rhd \Psi '\in \mathcal {AS}$ , such that

$$ \begin{align*}\Psi \le \Gamma_{i}\qquad\qquad\text{and} \qquad\qquad \Gamma_i = (\Gamma_{i-1}\mathbin{\dot-} \Psi) \uplus \Psi'. \end{align*} $$

A derivation of $\Delta $ is relevant if $\Gamma _n = \Delta $ . We say that $\Delta $ is (relevantly) derivable from $\Gamma $ in $\mathcal {AS}$ , and write $\Gamma \vdash ^{(r)}_{\!\mathcal {AS}}\Delta $ , if there is a (relevant) derivation of $\Delta $ from $\Gamma $ in $\mathcal {AS}$ .

The following lemma supports the adequacy of the given definition.

Example 6.4. Recall $\mathbf {BCI}$ from Section 4. Let’s consider some provable and unprovable symmetric consecutions in this logic. As a very simple example, note that:

$$\begin{align*}\varphi\to\psi,\varphi\to\chi,\varphi,\varphi \nvdash[\varphi,\psi,\chi] \end{align*}$$

as while we can obtain asymmetric proofs for $\varphi \to \psi ,\varphi \vdash \psi $ and $\varphi \to \chi ,\varphi \vdash \chi $ , we cannot find any further formulas from which to infer the $\varphi $ occurring among the premises. However, this is the case if we take:

$$ \begin{align*}\varphi\to\psi,\varphi\to\chi,\varphi,\varphi,\varphi \vdash[\varphi,\psi,\chi].\end{align*} $$

Furthermore, the relevance constraints mean that, in general, $\varphi ,\psi \nvdash ^r[\varphi ]$ . Interestingly, while in the asymmetric case we have $\varphi \nvdash ^r\varphi \to \varphi $ , when we go symmetric we obtain something related, namely, $\varphi \vdash ^r[\varphi \to \varphi ,\varphi ]$ , for we can obtain, in the asymmetric setting, $\vdash ^r\varphi \to \varphi $ and $\varphi \vdash ^r\varphi $ . This is related to the fact that, if we add $\circ $ , along with a constant formula $t$ expressing (i.e., something like a left identity in the sense of [Reference Dunn and Hardegree17]) to the language, then $\varphi \to (\varphi \to \varphi )$ is not a valid implication, but we do have $\varphi \to ((\varphi \to \varphi )\circ \varphi )$ , as $t\to (\varphi \to \varphi )$ and $\varphi \to \varphi $ are both valid, and thus so is $\varphi \to (t\circ \varphi )\to ((\varphi \to \varphi )\circ \varphi )$ .

The next propositions spell out the relation between derivations and tree-proofs.

Using this proposition we know that the symmetric versions of Abelian logic and logic $\mathbf{BCI}$ introduced in Example 5.4 have no single-conclusion axiomatic systems.

7 Theories

Now that we have pictures of (relevant) symmetric consequence in abstract and concrete terms, the question of deductive closure, and the properties of theories is quite natural. The standard definition of this concept in the Tarskian setting is well known, but there are some complications which come up in our case. Let us work our way towards a natural definition, and show that it has some properties.

Recall that we work with a fixed set of formulas $\mathit {Fm}$ . In the classical Tarskian setting (defined over potentially infinite sets of formulas, cf. Remark 2.2), a $\vdash $ -theory is a deductively closed set of formulas (i.e., a set of formulas which contains all its consequences) and the set of all $\vdash $ -theories forms a closure system on the set of formulas which fully determines both the consequence relation and its symmetrization:

• • $\Gamma \vdash \varphi $ iff $\varphi $ belongs to all $\vdash $ -theories containing $\Gamma $ as a subset.

• • $\Gamma \vdash \Delta $ iff $\Delta $ is a subset of the $\vdash $ -theory generated by $\Gamma $ (i.e., the set $\{\varphi \mid \Gamma \vdash \varphi \}$ ).

The problem with this approach for our setting is that both conditions clearly entail the Monotonicity condition. Further problems are caused by restriction to finite sets and by working with multisets rather than sets. Our guiding idea will be to see theories as particular sets of finite multisets of formulas retaining as many nice properties of the Tarskian case as possible [Reference Cintula, Gil-Férez, Moraschini and Paoli12].

As we have shown in Section 6, not every symmetric consequence relation arises from an asymmetric one; therefore we define all the notions in this section for the symmetric case, which can be retold for asymmetric relations via their symmetrizations. Let us start by observing that due to Reflexivity and Transitivity, every symmetric consequence relation $\vdash $ is a preorder on the set of finite multisets of formulas and thus we can speak about $\vdash $ -upsets, i.e., sets T such that if $\Gamma \in T$ and $\Gamma \vdash \Delta $ , then $\Delta \in T$ . We say that a $\vdash $ -upset T is principal if there is a finite multiset of formulas $\Gamma $ such that $T = \{\Delta \mid \Gamma \vdash \Delta \}$ .

Let us now define the notion of $\vdash $ -theory which will correspond to the notion of finitely generated theory in the Tarskian setting; we will refrain from defining a general notion of $\vdash $ -theory as it would be alien to our needs here.

We will denote theories by uppercase Latin letters T, S, R, etc. Also if $\vdash $ is a consequence relation, we speak about $\vdash $ -theories instead of $\vdash ^s$ -theories and analogously for all upcoming notions in this section.

Obviously, $\mathrm {Th}_{\vdash }(\Gamma )$ is the least theory containing $\Gamma $ (this is an easy exercise left to the reader).Footnote ²⁰ Furthermore, our notion of theory intuitively captures the idea that theories are deductively closed; it’s just that the objects of deductive interest are themselves multisets of formulas, and not just formulas, unlike in the standard setting. Despite this ‘type-lifting’, though, as we’ll see in the next pair of propositions, our notion of theory has a desirable property from the Tarskian setting, and collapses down to it when we build in the properties distinguishing relevant from Tarskian consequence relations. Consider the two bulleted conditions above.

Proposition 7.2. Let $\vdash $ be a symmetric consequence relation. Then for each finite multisets of formulas $\Gamma $ and $\Delta $ we have:

$$ \begin{align*}\Gamma\vdash \Delta \qquad \text{iff}\qquad \Delta \in \bigcap \{T\in \mathrm{Th}(\vdash) \mid \Gamma\in T\} \qquad \text{iff}\qquad \mathrm{Th}_{\vdash}(\Delta) \subseteq \mathrm{Th}_{\vdash}(\Gamma). \end{align*} $$

Recall that if $\vdash $ is monotone and contractive, we can see it as a binary relation on finite sets of formulas (instead of finite multisets of formulas), so theories can be seen as sets of finite sets of formulas and we can formulate and prove the following claim (implicit already in [Reference Cintula and Paoli14]), roughly saying that the usual Tarskian theory is the union of our theory.

Proposition 7.3. Let $\vdash $ be a monotone and contractive symmetric consequence relation. Then for each $\vdash $ -theory T we have:

$$ \begin{align*}T = \{\Gamma \mid \Gamma\subseteq \textstyle\bigcup T\}. \end{align*} $$

Now that we have a collection of objects, we would love to find an algebraic structure on them: similarly to the Tarskian case, we can find a rather natural one. In particular, ${\mathcal {P}_{\!\mathit {fin}}^{\#}}(\mathit {Fm})$ can be seen as the domain of an Abelian monoid with a neutral element and addition $\uplus $ : we’ll denote this structure by ${\boldsymbol {\mathit {Fm}}}^\#$ . Recall that given an Abelian monoid ${\boldsymbol {\mathit {A}}} = \langle {A, +, 0}\rangle $ and an order $\leq $ on A we say that $\langle {{\boldsymbol {\mathit {A}}}, \leq }\rangle $ is an ordered Abelian monoid if $a\leq b$ implies $a+c \leq b+c$ (i.e., $+$ is monotone w.r.t. $\leq $ ). Let’s verify that we can find such a monoid structure on the class of theories.

Theorem 7.4. Let $\vdash $ be a symmetric consequence relation. Then the algebra ${\boldsymbol {\mathit {Th}}}_{\vdash } = \langle {\mathrm {Th}(\vdash ), +_{\vdash }, 0_{\vdash }}\rangle $ where and

$$ \begin{align*}\mathrm{Th}_{\vdash}(\Gamma) +_{\vdash} \mathrm{Th}_{\vdash}(\Delta) = \mathrm{Th}_{\vdash}(\Gamma \uplus \Delta) \end{align*} $$

is an Abelian monoid and the mapping $\mathrm {Th}_{\vdash }\colon {\boldsymbol {\mathit {Fm}}}^\# \to {\boldsymbol {\mathit {Th}}}_{\vdash }$ is a surjective homomorphism.

Furthermore $\langle {{\boldsymbol {\mathit {Th}}}_{\vdash }, \subseteq }\rangle $ is an ordered Abelian monoid and, finally, $\vdash $ is monotone iff $\mathrm {Th}_{\vdash }$ is a monotone mapping from $\langle {{\boldsymbol {\mathit {Fm}}}^\#, \le }\rangle $ into $\langle {{\boldsymbol {\mathit {Th}}}_{\vdash }, \subseteq }\rangle $ .

Similarly, it’s desirable that we be able to define a congruence on ${\boldsymbol {\mathit {Fm}}}^\#$ using $\vdash $ , which has other nice properties. This is straightforward: consider a symmetric consequence relation $\vdash $ and define a binary relation $\dashv \vdash $ of finite multisets of formulas as

$$ \begin{align*}\Gamma\dashv\vdash \Delta, \qquad\text{iff}\qquad \Gamma\vdash \Delta\ \text{ and }\ \Delta\vdash \Gamma. \end{align*} $$