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Published online by Cambridge University Press:  13 July 2022

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We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalisation of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on trees are sound and complete for the basic positive distributive substructural logic $\mathsf {B}^+$ , that collection frames on multisets are sound and complete for $\mathsf {RW}^+$ (the relevant logic $\mathsf {R}^+$ , without contraction, or equivalently, positive multiplicative and additive linear logic with distribution for the additive connectives), and that collection frames on sets are sound for the positive relevant logic $\mathsf {R}^+$ . The completeness of set frames for $\mathsf {R}^+$ is, currently, an open question.

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1 Ternary Relational Frames

The ternary relational semantics for relevant logics is a triumph. The groundbreaking results of Routley and Meyer [Reference Routley and Meyer45Reference Routley, Meyer and Leblanc47] have significantly clarified our understanding of relevant logics.Footnote 1 After 20 years of viewing relevant logics with Hilbert-style axiomatisations, natural deduction systems and algebraic semantics, we finally had a truth-conditional semantics which modelled relevant logics in the same way that Kripke semantics provide models for normal modal logics and intuitionistic and intermediate logics.Footnote 2

Propositions are modelled as sets of points, and connectives are interpreted as operations on such sets, some (namely the modal operators, intuitionistic conditional and negation, and in the case of relevant logics, relevant implication and the intensional conjunction, fusion) using accessibility relations on the class of points. In the case of the distinctively relevant conditional connective ‘ $\to $ ’, the two-place connective is naturally interpreted by a three-place accessibility relation, the eponymous ternary relation of the ternary relational semantics.

That a ternary relation should feature in a frame semantics for relevant logics should not have surprised anyone. The pieces had been in place for quite some time. Jónsson and Tarski’s papers, from the 1950s, on Boolean algebras with operators [Reference Jónsson and Tarski26, Reference Jónsson and Tarski27], showed how Boolean algebras with n-ary operators satisfying appropriate distributive laws can be concretely modelled as power set algebras where each n-place operator is interpreted using an ( $n+1$ )-place relation. Generalising these results from Boolean algebras to distributive lattices makes some of the details a little more complicated, but the picture is mostly unchanged. The details for how to make that generalisation of Jónsson and Tarski’s work to arbitrary distributive lattices with operators—including relevant logics—were worked out by Dunn in his papers on gaggle theory in the early 1990s [Reference Dunn14Reference Dunn, Max and Stelzner17].Footnote 3 The picture is extremely natural and well motivated. The ternary relational semantics for relevant and substructural logics is powerful, and it has resulted in significant advances in our understandings of these logics.

Nonetheless, it cannot be said that the ternary relational semantics has met with anything like the reception of the Kripke semantics for modal and constructive logics. Some of the difference is no doubt due to the size of the respective audiences. Substructural and relevant logic is a boutique interest when compared to the modal industrial complex of the late twentieth and early twenty-first centuries. However, it seems to us that this does not explain all of the differences in the scale and quality of the reception of the respective semantic frameworks. Some of the relative dissatisfaction with the ternary relational semantics centres on philosophy and the question of the intelligibility of the semantics [Reference Beall, Brady, Dunn, Hazen, Mares, Meyer, Priest, Restall, Ripley, Slaney and Sylvan2, Reference Copeland11]. We think those questions have been well dealt with in the literature, and that to a large degree the proof of this pudding is in the eating, rather than adding to the already long discussion of pudding interpretation. The ternary relational semantics is not problematic because it lacks interpretive power or philosophical intelligibility. The problem with the ternary relational semantics is that it is fiddly.

Consider Kripke semantics for modal logics. All you need to make a Kripke frame is a non-empty set of points, and a binary relation on those points. Nothing more. Propositions are modelled by sets of points. The Boolean operators correspond to the set functions of union, intersection and complementation, and the modal operators are simple universal or existential projections along the binary relation. This is simple, it is robust, and once you see it, you find this pattern everywhere. Structures for modal logics are ubiquitous.

Kripke semantics for intuitionistic logic is a little more complicated, but not by much. We must have a partial order on our set of points (or possibly a preorder) and propositions are sets of points closed upward along that order. Conjunction and disjunction are unchanged from the modal case, as intersection and union preserve the property of being upward closed. However, complementation, and the corresponding operation to model the material conditional, do not preserve the property of being closed, so they are replaced by operations that utilise the partial order and respect the upward closure condition. Again, this is all very straightforward. When you have an ordered collection of states, carrying information preserved along that order, constructive logic is a natural tool, and Kripke models for intuitionistic logic are correspondingly natural.

Now compare the general framework for substructural logics.Footnote 4 One natural presentation of the semantics takes this form: a frame is a 4-tuple $\langle P,R,\sqsubseteq ,N\rangle $ , where P is a non-empty set of points, R is a ternary relation on P, $\sqsubseteq $ is a binary relation on P, and N is a subset of P, where the following conditions are satisfied.

  • $\sqsubseteq $ is a partial order.

  • R is $\sqsubseteq $ -downward preserved in the first two positions, and $\sqsubseteq $ -upward preserved in the third. That is, if $Rxyz$ and $x^-\sqsubseteq x$ , $y^-\sqsubseteq y$ and $z\sqsubseteq z^+$ then $Rx^-y^-z^+$ .

  • $y\sqsubseteq z$ if and only if there is some x where $Nx$ and $Rxyz$ .

Notice that these models have three distinct moving parts: the ternary relation R, the partial order $\sqsubseteq $ , and the distinguished set N of points. Propositions are sets of points closed upward under the partial order $\sqsubseteq $ . R is used to interpret the conditional connective ‘ $\to $ ’ (and the intensional conjunction ‘ $\circ $ ’, if present), while the set N of so-called normal, or regular, points is the set of points at which logical truths are taken hold.Footnote 5 The need for N is a distinctive feature of relevant logics, as logical truths (like, say, $p\to p$ ) need not hold at all points. Since, for example, $q\to (p\to p)$ is not a theorem of $\mathsf {R}^+$ , so some models feature have counterexamples to the conditional. Those models have at least one point where q is supported but $p\to p$ is not. But $p\to p$ is still a logical truth according to $\mathsf {R}^+$ . Logical truths are guaranteed to hold at some points (namely, those in N), but not necessarily at all points. So, our models have three distinct moving parts: $\sqsubseteq $ for providing our closure conditions for propositions,Footnote 6 R for modelling ‘ $\to $ ’ and ‘ $\circ $ ’, and N for modelling the logical truths.

We challenge anyone to find this kind of formal semantics to be as straightforward to apply as the Kripke semantics for modal and constructive logics. While it is relatively easy to find preorders or binary relations on sets under every bush, it is rather harder to see where ternary relations, partial orders and special sets of normal points are to be found. Perhaps they are there somewhere, but they do not seem particularly easy to spot. It is not for nothing that modal and constructive logics have been applied in many domains where relevant and substructural logics have not.Footnote 7

It is true that the choice of primitives in the ternary frame semantics is somewhat arbitrary. We could take $\sqsubseteq $ to be defined in terms of N and R, but then the condition that it is a partial order (or a preorder) and that R is preserved along that order become even more complex and unnatural to state. In models for some of our logics (not all) we could impose the condition that $\sqsubseteq $ is the identity relation (and hence, all algebras of propositions arising out of such frames would be at least implicitly Boolean algebras, so this works only for logics conservatively extended with Boolean negation) [Reference Priest and Sylvan38]. It is possible, for some substructural logics, to trade in our set N for a single point g (and restrict our attention to so-called reduced models), cutting down further on the number of models generated, but the conceptual complexity remains [Reference Giambrone21, Reference Routley, Plumwood, Meyer and Brady48Reference Standefer and Bimbó51].

When you consider ternary relational models alongside point semantics for normal modal logics and constructive logics, the contrast is plain for all to see. Ternary relational models are significantly less elegant, and they have many different moving parts than Kripke models for modal and constructive logics. It is not for nothing that those of us working in the area have sought to simplify the semantics, but try as we might, significant complexity remains after such all such efforts [Reference Priest and Sylvan38, Reference Restall39].Footnote 8

In this paper we introduce a new class of models for positive relevant and substructural logics, which at the same time generalises and simplifies the ternary relational semantics. Collection frames generalise ternary relational frames in the sense that every ternary relational frame can be seen as a collection frame, but that there are also collection frames that do not arise as ternary relational frames. Collection frames simplify ternary relational frames in the sense that there are significantly fewer independent parts and conditions connecting different components of the semantics. While the resulting models are not quite as simple as Kripke semantics for modal logics—some complexity is inevitable, given that we are aiming to model an intensional two-place connective—the gain in simplicity over the traditional presentation of the ternary relational frame semantics for relevant logics is significant.

Simplifying the semantics is one motivation for our work. The second motivation for an alternate approach to frames for these logics arises out of noticing the following fact: When we work with particular substructural logics—such as $\mathsf {R}^+$ , $\mathsf {RW}^+$ , and $\mathsf {TW}^+$ —it is very natural to consider not only the ternary relation R but its generalisations to more places: $R^2a(bc)d$ is defined as $(\exists x)(Rbcx \land Raxd)$ , and $R^2(ab)cd$ is defined as $(\exists x)(Rabx\land Rxcd)$ . In $\mathsf {R}^+$ and $\mathsf {RW}^+$ , $R^2a(bc)d$ holds if and only if $R^2(ab)cd$ holds, so we can simplify our notation, and generalise further: for $n>0$ , we define $R^{n}$ to be the $(n+2)$ -ary relation on P, setting $R^1=R$ , and setting $R^{n+1}a_1a_2a_3\cdots a_{n+3}$ to hold if and only if $(\exists x)(Ra_1a_2x\land R^{n}xa_3\cdots a_{n+3})$ . This generalisation into an arbitrary n-ary relation, where $n\ge 3$ is extremely natural, and conditions on $R^2$ and still higher orders of R play a role in the specification of various substructural logics.Footnote 9

Our attempt to understand the phenomenon of higher order accessibility relations—and how they relate to each other—is the starting point for a new, simpler characterisation of frame semantics for substructural logics. In the next section we will start with one case, frames for the logics $\mathsf {RW}^+$ and $\mathsf {R}^+$ . In later sections we will then branch out to a wider class of substructural logics.

2 Multiset Frames

A guiding idea in ternary relational semantics for relevant logics is the notion of information application or combination. The ternary relation R relates the triple of points $x,y,z$ (that is, $Rxyz$ ) if and only if applying the information in x to the information in y results in information that is in z. In the logics $\mathsf {R}^+$ and $\mathsf {RW}^+$ , information application is commutative (applying x to y results in the same information as applying y to x), and associative (applying x to y and then applying the results to z results in the same things as applying x to a result of applying y to z). In models for $\mathsf {R}^+$ , combination is also idempotent, to the effect that the result of applying x to itself doesn’t take you outside x (so we have $Rxxx$ ). Associativity and commutativity of application (or combination) means that we could simplify our ternary relation R by thinking of it not so much as a ternary relation where all three slots act independently, but rather, at least in the case of these logics, as a relation between unordered pairs of points on the one hand, and points on the other. The fact rendered as $Rxyz$ in the ternary semantics could instead be represented as

$$\begin{align*}[x,y] R z \end{align*}$$

where we have the (unordered) pair of x and y on the one hand, and the z on the other. The fact that this is an unordered pair, and not a set is important, because when we consider $Rxxz$ what we have is

$$\begin{align*}[x,x] R z, \end{align*}$$

where x is applied to itself. But as far as order of application goes, $[x,y]Rz$ is the very same fact as $[y,x]Rz$ . When it comes to associativity, what we have in models for $\mathsf {RW}^+$ , traditionally presented, is the following complex fact:

$$\begin{align*}(\exists u)(Rxyu \land Ruzw) \textrm{ iff } (\exists v)(Ryzv \land Rxvw). \end{align*}$$

If we are willing to abuse notation a little more, what we have in this biconditional is two different ways of representing the one single fact

$$\begin{align*}[x,y,z] R w \end{align*}$$

to the effect that x, y and z together, combined in any order, are related to w. Collection frames arise from taking what was an abuse of notation literally. In collection frames, an accessibility relation relates collections of points to points.

This shifted perspective on R comes with advantages. Not only will this relation R do the job of the original ternary relation, in the case where the multiset has two elements, and not only can it represent $R^2$ and relations of higher arities with larger multisets. It also has the capacity to represent the binary relation $\sqsubseteq $ in the case where the collection being related is a singleton, and it also represents the predicate N, in the case where the collection being related is the empty multiset. The translation manual is straightforward:

$$\begin{align*}\begin{array}{lrcl} \mathrm{(F1)} & Nx & \textit{becomes} & [\;] Rx, \\ \mathrm{(F2)} & x\sqsubseteq y & \textit{becomes} & [x]Ry, \\ \mathrm{(F3)} & Rxyz & \textit{becomes} & [x,y]Rz. \end{array} \end{align*}$$

What was represented by three different fundamental concepts in traditional Routley–Meyer frames becomes three different aspects of one underlying relation. The conditions linking N, $\sqsubseteq $ and (ternary) R become corollaries of the fundamental structure of the one multiset relation R.

To make things explicit, a collection frame for $\mathsf {RW}^+$ has a non-empty set P of points and a single accessibility relation R on $M(P)\times P$ , where $M(P)$ is the class of finite multisets of elements of P. Since multisets are not in very wide use,Footnote 10 we would do well to be explicit about them and their properties.

Definition 1 (Finite multisets, ground)

A multiset is a collection in which order is irrelevant, but multiplicity of membership is relevant. There are various ways to formally define the notion. One way is this: a finite multiset of objects taken from some class P can be represented as a function $m:P\to \omega $ where $m(x)=0$ for all but finitely many values of x. If x is in P, then $m(x)$ is the number of times $x$ is a member of the multiset m. The multisets $m_1$ and $m_2$ from P are identical if they have the same members to the same multiplicities: that is, $m_1=m_2$ if and only if $m_1(x)=m_2(x)$ for each $x$ in P.

For any two multisets $m_1$ and $m_2$ , their union is the multiset with function $m_1+m_2$ . We also write ‘ $m_1\cup m_2$ ’ using the traditional notation for union. Note, however, that $m_1\cup m_1$ is now not (typically) the same multiset as $m_1$ .

We say that $m_1\le m_2$ (a generalisation of the subset relation to multisets) if $m_1(a)\le m_2(a)$ for all a in P.

We use the familiar bracket notation for multisets: for example, $[a,a,b]$ is the multiset where $m(a)=2$ and $m(b)=1$ and $m(x)=0$ for every other value of $x$ . So, $[a,b]\cup [a,c,c]=[a,a,b,c,c]$ .

As with sets, we will use the symbol ‘ $\in $ ’ for multiset membership. Here, ‘ $x\in m$ ’ will be taken to mean that $m(x)>0$ , that is, the object $x$ is in the multiset m a non-zero number of times.

For any multiset m on P, its ground $g(m)$ is the subset of P consisting of all objects x with non-zero multiplicity in m, that is, $g(m)=\{x\in P~|~m(x)>0\}$ .

Now we know enough about multisets for us to introduce the multiset semantics for $\mathsf {RW}^+$ and for $\mathsf {R}^+$ . As we have already indicated, a collection frame consists of a set P of points (with at least one member), and a relation R on $M(P)\times P$ , which relates multisets of points to points. Henceforth, we will call relations R on $M(P)\times P$ multiset relations.

The intended application of R in a multiset frame is straightforward: $X R y$ holds when, and only when, the information in the points X taken together also holds in y. There are aspects, in this reading, of the partial order from constructive logics, and just like that case, there must be at least some condition on this relation for such an interpretation to make sense. The relation R cannot be entirely arbitrary. In the case of the semantics for constructive logic, there are two parts to the constraint on the order relation. First, that it be reflexive, and second, that it is transitive.Footnote 11 In the case of multiset relations for frames for $\mathsf {RW}^+$ , the condition has much the same form: a transitivity component and a reflexivity component. The strictest and most natural form of reflexivity would be we require that the information in the singleton multiset of points $[x]$ is indeed carried by the x itself. This says very little about combining points, of course. For transitivity, we require that combination compose in a straightforward manner: if $XRy$ and $[y]\cup YRz$ then $(X\cup Y)Rz$ .Footnote 12 However, we require something stronger than just composition in this direction: we also require its converse. That is, if $(X\cup Y)Rz$ then we can find some ‘value’ y where $XRy$ and $([y]\cup Y)Rz$ . We call these two conditions compositionality because we think of R as a generalised combination relation, selecting for each collection of points the single points which are suitable to represent it. The compositionality condition says that this relation can be composed or decomposed piecewise. So, we have the following definition:

Definition 2 (Compositionality)

A multiset relation $R$ on $M(P)\times P$ is said to be compositional if and only if for all multisets X and Y and for all points z,

$$\begin{align*}(\exists y)(XRy\textrm{ and }([y] \cup Y)Rz)\textrm{ iff }(X\cup Y)Rz\textrm{.} \end{align*}$$

In addition, a compositional multiset relation is reflexive iff for all points x, we have

$$\begin{align*}[x]Rx. \end{align*}$$

We break the compositionality condition into two parts, the left to right direction we will call Transitivity, for obvious reasons. The right to left direction we will call Splitting.Footnote 13 These two parts of the condition play different roles in exploring the properties of this semantics, so we will highlight these roles by mentioning at each point whether Transitivity or Splitting is being appealed to.

Fig. 1 The two directions of compositionality.

The intuitions behind the two directions are represented in Figure 1. The intuition behind Transitivity is that if one can combine the information in X to obtain x and combine the information in Y together with x to obtain y, represented by the solid lines, then one could have just as well have used the information in the combination of X and Y to obtain y, represented by the broken lines. If we restrict our attention to the case where $X=[x]$ and $Y=[\;]$ then we see that Transitivity gives us the transitivity of the binary relation $\lambda x.\lambda y.[x]Ry$ on points.

The intuition behind Splitting is that if one can obtain y from some information Z, which can be split into components X and Y, then one could evaluate the X portion to obtain something, x, which can be combined with the information in Y to obtain y.Footnote 14 If we restrict our attention to the case where $X=[x]$ and $Y=[\;]$ , then Splitting gives us the density of the binary relation $\lambda x.\lambda y.[x]Ry$ . That is, if $[x]Rz$ then there is some y where $[x]Ry$ and $[y]Rz$ . Notice that the density of this relation holds automatically in the case where reflexivity holds, but this condition is strictly weaker than reflexivity.Footnote 15

Since this binary relation $\lambda x.\lambda y.[x]Ry$ is so important in our frames, we will reserve special notation for it. In ternary frames the usual notation is ‘ $\sqsubseteq $ ’. Since our frames will not require reflexivity (but we will allow it), let us write ‘ $\sqsubset $ ’ for this binary relation induced by the multiset relation R. We have seen proved the following lemma.

Lemma 3. If R is a compositional multiset relation then the induced binary relation $\sqsubset $ (given by setting $x\sqsubset y$ iff $[x]Ry$ ) is transitive and dense.

Before we continue spelling out the semantics, we would do well to pause to consider some examples of simple multiset relations, and their properties.

Example 4 (Compositional multiset relations on $\omega $ )

Here are some examples of compositional multiset relations on the set $\omega $ of natural numbers.

  • [ The Product] $XRy$ if and only if y is the product of all the members of X.Footnote 16 (This is genuinely and distinctively a multiset relation, which distinguishes repeated elements in the multiset. For this relation, $[2,2]R4$ holds, but $[2]R4$ does not.) This is compositional, which fact is left to the reader.

  • [ Some Product] $XRy$ if and only if y is some product of the members of X, using each instance in X at most once. (Unlike the product, this relation is not functional.

  • [ The Sum, and Some Sum] In the same way, the relation R given by setting $XRy$ iff $\Sigma X=y$ is compositional (given that we set $\Sigma [\;]=0$ ), as is the relation given by setting $XRy$ iff $\Sigma X'=y$ for some $X'\le X$ . As with the product relations, one is functional, and the other is not. Each of the relations discussed so far makes essential use of the multiset structure. The multiset $[2,2]$ is related to different numbers in each case, than the singleton multiset $[2]$ . In the next example, the multiplicity of members makes no difference at all.

  • [Maximum-or-zero-if-empty] In this case, $XRy$ if and only if y is the largest member of X, and is $0$ if X is empty. This satisfies the reflexivity condition, as well as Transitivity and Splitting.

  • [The Empty Relation] Another multiset relation, trivially compositional, is the empty multiset relation. It is straightforward to verify that this relation satisfies both the Transitivity and the Splitting conditions. Of course, this relation fails to be reflexive, unlike the other relations we have considered so far.

That is a range of compositional multiset relations on $\omega $ . Not every multiset relation, however, is compositional.

Example 5 (Non-compositional multiset relations on $\omega $ )

These relations fail to be compositional in different ways.

  • [Larger than the product of] $XRy$ holds if and only if $y>\Pi X$ . Clearly this is not reflexive. While Splitting holds, the Transitivity direction of compositionality fails.

  • [Largest two] $XRy$ if and only if y is one of the largest two elements of X. This relation fails transitivity.

  • [Membership] $XRy$ if and only if $y\in X$ . This relation enjoys Transitivity but not Splitting.

    Although membership is not a compositional multiset relation on $M(P)\times P$ , it is compositional if we restrict our attention to inhabited Footnote 17 multisets. (We will discuss this restricted form of compositionality below.)

  • [Between] $XRy$ iff y occurs between the smallest and the largest members of X, inclusive. So $[2,4]$ is related to $2$ and to $4$ and to $3$ but to no other number. This, like membership, is compositional on the inhabited multisets but not the full collection of multisets.

We will end this series of examples with two more compositional relations, this time, on the rational numbers ${\mathbb Q}$ and the reals, ${\mathbb R}$ , rather than on $\omega $ , so we have scope for examples of non-reflexive but dense order relations.

Example 6 (Non-reflexive multiset relations)

These examples of multiset relations make use of the density of the underlying order $<$ on ${\mathbb Q}$ and on ${\mathbb R}$ .

  • [larger than] $XRy$ if and only if $y>x$ for each $x\in X$ . So, $[\;] Ry$ for every y (in this case, the condition is vacuously satisfied). This relation satisfies Transitivity and Splitting but not reflexivity.

    In this case, the relation makes no distinction between multisets with the same ground. $[2,2]$ is related to all the numbers greater than $2$ , as is $[2]$ and $[2,2,2]$ .

  • [larger than the sum of] Here, $XRy$ if and only if $y$ is larger than the sum of all the members of X (counting their multiplicities, as in the case of the sum relation given previously). As before, we set $\Sigma [\;]=0$ . While this fails to be reflexive, it is compositional.

This flock of examples was longer than it strictly needed to be, if not for one thing. A complaint about the ternary relational semantics is that examples are hard to come by, hard to construct and above all, hard to picture. That there is such a list of naturally occurring examples of compositional multiset relations, both reflexive and irreflexive, and which exhibit significantly different behaviours, but are straightforward both to reason with and to understand, goes quite some way towards answering that complaint.

It is disappointing, however, that membership and betweenness failed to count as compositional relations. In fact, as we noted, those multiset relations are compositional if we restrict our attention to the class $M'(P)$ of inhabited multisets of points. We can make this notion precise in a definition.

Definition 7 (Compositional inhabited-multiset relations)

A relation R on $M'(P)\times P$ is said to be compositional if and only if for all multisets X and Y where $X\neq [\;]$ , and for all points z,

$$\begin{align*}(\exists y)(XRy\textrm{ and }([y]\cup Y)Rz)\textrm{ iff }(X\cup Y)Rz\textrm{.} \end{align*}$$

This is the appropriate definition of compositionality for a relation on inhabited multisets. You may wonder why, in this definition, X inhabited, but Y is allowed to be empty. Isn’t that outside the spirit of restricting our attention to inhabited multisets? This is a natural restriction of compositionality to this setting, because it is the smallest modification to the condition that ensures that the left relatum of any R-fact is nonempty. (Since we require that X be inhabited, for $XRy$ to make sense in this context, this is enough to guarantee that $X\cup Y$ is also inhabited, and $[y]\cup Y$ is inhabited by design.) A satisfying upshot of this result is the fact under this condition (allowing for Y to be empty), the proof of Lemma 15 works in the case of inhabited-multiset relations, too. That special case of transitivity, spelled out, is this: $XRy$ and $[y]Rz$ implies $XRz$ . We have also appealed to this condition in the proof Lemma 3. We will also see below, when we turn to more general structures, like lists and trees, that the general form of compositionality involves trading in a single item in a structure (here, a member of a multiset) for another structure. In the case of a multiset, any multiset with a member y can be written in the form $X\cup [y]$ . For this representation to work, in general, we need to allow the case where X is empty, even if our attention is fixed on inhabited multisets, for we may wish to trade in the y in a singleton multiset $[y]$ for some other multiset.

Example 8 (Compositional inhabited-multiset relations)

With this expanded definition, we can enlarge our class of models even further. We have already seen that membership and between give us compositional relations on inhabited multisets. So are these:

  • [maximum, and minimum] maximum-or-zero-if-empty is a compositional multiset relation on $\omega $ . Without the need to have a maximum for $[\;]$ , we can remove the “or-zero-if-empty” dodge, and restrict our attention to the largest member of the multiset. Or the smallest, if we choose, and the result is a compositional inhabited-multiset relation.

  • [ the sum, and some sum on subsets of ω] If we no longer have the requirement that the empty multiset $[\;]$ have a sum, then given any subset $S$ of $\omega $ , closed under addition (so if $x,y\in S$ , then so is $x+y$ ) we can define a compositional inhabited-multiset relations R and $R'$ on S, setting $XRy$ iff $y=\Sigma X$ , and $XR'y$ iff $y=\Sigma X'$ where $X'$ is an inhabited multiset where $X'\le X$ . For example, we can let $S=\{1,2,3,\ldots \}=\omega \backslash \{0\}$ to provide a very different kind of model, once $0$ is left out of the domain.

  • [ the product, and some product on subsets of ω] In exactly the same way, we can generate models defining R on subsets of $\omega $ closed under product, without having to include $1$ as the product of the empty multiset.

In what follows, we will consider both compositional multiset relations and, at times, compositional inhabited-multiset relations. For any compositional multiset relation, its restriction to inhabited multisets is, of course, also compositional. For the converse, we have the following lemma, which shows that there is a way to extend a compositional inhabited multiset relation R on $M'(P)\times P$ to a compositional multiset relation on $M(P\cup \{\infty \})\times (P\cup \{\infty \})$ , where we add a new ‘point at infinity’ to our point set.

Lemma 9. If R is a compositional inhabited-multiset relation on $M'(P)\times P$ , and $\infty \not \in P$ , then the multiset relation $R^{\times }$ on $M(P\cup \{\infty \})\times (P\cup \{\infty \})$ , defined as follows, is compositional.

$$\begin{align*}XR^{\times} z\textrm{ iff } \begin{cases} z=\infty, & \textrm{if }X\backslash \infty=[\;], \\ (X\backslash\infty)Rz, & \textrm{if }{X\backslash\infty}\neq[\;].\end{cases} \end{align*}$$

Furthermore, if R is reflexive, then so is $R^{\times }$ .

(In the definition of $R^{\times }$ we use the notation ‘ $X\backslash y$ ’ for the multiset formed by removing all instances of y from X. So, for example, $[a,b,b,c,c]\backslash c=[a,b,b]$ . We reserve ‘ $X\backslash Y$ ’ for the multiset formed by removing the number of occurrences in Y from X, so $[a,b,b,c,c]\backslash [c]=[a,b,b,c]$ .)

Proof. Let’s suppose that $(X\cup Y)R^{\times } z$ , in order to find some y where $YR^{\times } y$ and $(X\cup [y])R^{\times } z$ . By definition $(X\cup Y)R^{\times } z$ holds if and only if $z=\infty $ (if $(X\cup Y)\backslash \infty =[\;]$ ) or $((X\cup Y)\backslash \infty ))Rz$ (otherwise). Let’s take these cases in turn. If $(X\cup Y)\backslash \infty =[\;]$ then clearly $X\backslash [\;]$ and $Y\backslash [\;]$ , so in this case, both $YR^{\times }\infty $ and $(X\cup [\infty])R^{\times } \infty $ , as desired. So, now consider the second case: we have $((X\cup Y)\backslash \infty ))Rz$ and $(X\cup Y)\backslash \infty \neq [\;]$ . We aim to find some y where $YR^{\times } y$ and $(X\cup [y])R^{\times } z$ . If $Y\backslash \infty =[\;]$ , then we choose $\infty $ for y. We have, then, $YR^{\times } \infty $ and since $((X\cup Y)\backslash \infty ))Rz$ , we have $(X\backslash \infty )Rz$ , so we have $(X\cup \{\infty \})R^{\times } z$ as desired. On the other hand, if Y has some element other than $\infty $ , since $((X\cup Y)\backslash \infty ))Rz$ , we have $((X\backslash \infty )\cup (Y\backslash \infty ))Rz$ , and since R is compositional, there is some y where $(Y\backslash \infty )Ry$ and $((X\backslash \infty )\cup [y])Rz$ , which gives us $YR^{\times } y$ and $(X\cup [y])R^{\times } z$ as desired.

Now for the second half of the compositionality condition for $R^{\times }$ , suppose that there is some y where $YR^{\times } y$ and $(X\cup [y])R^{\times } z$ . We aim to show that $(X\cup Y)R^{\times } z$ . If $YR^{\times } y$ then either $y=\infty $ and Y contains at most $\infty $ , or otherwise $(Y\backslash \infty )Ry$ . In the first case, $(X\cup [y])R^{\times } z$ tells us that $(X\cup [\infty])R^{\times } z$ , which means either that $(X\backslash \infty )Rz$ , or X also contains at most $\infty $ and then $z=\infty $ . In the either of these cases, we have $(X\cup Y)R^{\times } z$ , as desired. So, let’s suppose $y\neq \infty $ . In that case we have $(Y\backslash \infty )Ry$ , and then, since $(X\cup [y]))R^{\times } z$ , we have $((X\cup [y])\backslash \infty )Rz$ , and by the compositionality of R, $((X\cup Y)\backslash {\infty })Rz$ , which gives $(X\cup Y)R^{\times } z$ , as desired.

Finally, $R^{\times }$ is reflexive follows immediately from the reflexivity of R and the fact that $[\infty]R^{\times }\infty $ .

With this result, it is possible for us to use examples like membership and betweenness as compositional multiset relations, with the full complement of logical resources, including the set of normal points, identified as those related to the empty multiset $[\;]$ .

Now we are in a position to define multiset frames and models. We will begin with the more standard ternary relational frames for $\mathsf {RW}^+$ .

Definition 10 (Ternary relational $\mathsf {RW}^+$ frames, models)

A ternary relational frame for $\mathsf {RW}^+$ is a quadruple $\langle P,R,\sqsubseteq , N\rangle $ obeying the following conditions.

  1. 1. $\sqsubseteq $ is a partial order.

  2. 2. If $x\sqsubseteq w$ , $y\sqsubseteq u$ , $v\sqsubseteq z$ , and $Rwuv$ , then $Rxyz$ .

  3. 3. $y\sqsubseteq z$ iff $\exists x\in N$ , $Rxyz$ .

  4. 4. If $x\in N$ and $x\sqsubseteq y$ , then $y\in N$ .

  5. 5. $Rxyz$ only if $Ryxz$ .

  6. 6. $Rwxyz$ only if $Rw(xy)z$ .

To get a ternary relational frame for $\mathsf {R}^+$ , one adds the condition that if $Rxyz$ , then $Rxyyz$ .

A ternary relational model is a quintuple $\langle P, R, \sqsubseteq , N, \Vdash \rangle $ where the first four components make up a frame and the final component is a binary relation between P and the set of atoms such that if $x\Vdash p$ and $x\sqsubseteq y$ , then $y\Vdash p$ . This is extended to the whole language according to the following clauses.

  • $x\Vdash A\land B$ iff $x\Vdash A$ and $x\Vdash B$ .

  • $x\Vdash A\lor B$ iff $x\Vdash A$ or $x\Vdash B$ .

  • $x\Vdash A\to B$ iff for each $y,z$ where $Rxyz$ , if $y\Vdash A$ then $z\Vdash B$ .

  • $x\Vdash A\circ B$ iff for some $y,z$ where $Rxyz$ , both $y\Vdash A$ and $z\Vdash B$ .

  • $x\Vdash \mathsf {t}$ iff $x\in N$ .

  • $x\Vdash \bot $ never.

Next, we define multiset frames.

Definition 11 (Multiset frame)

A multiset frame $\langle P,R\rangle $ is an inhabited set P of points together with a compositional multiset relation R on P.

This definition is, in one sense, starkly simpler than the traditional frame semantics for $\mathsf {RW}^+$ , in that the three elements N, $\sqsubseteq $ and the ternary relation R are subsumed into one fundamental relation, the compositional multiset relation. They are also more general, because we consider not only models in which $\sqsubset $ is reflexive (as it is in ternary relational frames), but the more general class of frames allowing for the underlying order relation $\sqsubset $ to be non-reflexive, or even irreflexive. In fact, we allow as a frame the case where R is the empty relation. So, this is a wider class of frames. The multiset frames subsume the traditional ternary relational frames for $\mathsf {RW}^+$ , following the conditions (F1), (F2), and (F3) from Section 2. The one relation in a multiset frame encodes the three different moving parts of a ternary frame. We have the following fact:

Lemma 12. Each ternary frame $\langle P,R,\sqsubseteq ,N\rangle $ for $\mathsf {RW}^+$ determines a reflexive multiset frame $\langle P,R'\rangle $ , defined by setting:

  • $[\;] R'x$ iff $x\in N$ ,

  • $[x]R'y$ iff $x\sqsubseteq y$ ,

  • $[x,y]R'z$ iff $Rxyz$ ,

  • If Y is a multiset of size two or more, $([x]\cup Y)R'z$ iff for some y, $YR'y$ and $[x,y]R'z$ .

Proof. We first need to show that the definition is $R'$ coherent: that the third clause, to the effect that $[x,y]R'z$ iff $Rxyz$ , that the last clause, according to which $([x]\cup Y)R'z$ iff for some y, $YR'y$ and $[x,y]R'z$ , could both hold. For the third clause, we need to be sure that $Rxyz$ holds iff $Ryxz$ holds, since $[x,y]=[y,x]$ , lest the clause give inconsistent guidance as about $[x,y]R'z$ . But in any ternary frame $\langle P,R,\sqsubseteq ,N\rangle $ for $\mathsf {RW}^+$ , we have $Rxyz$ iff $Ryxz$ , so this clause is coherent.

For the last clause, if $[x]\cup Y$ is the same multiset as $[x']\cup Y'$ , we need to show that

$$\begin{align*}(\exists y)(YR'y\land [x,y]R'z) \textrm{ if and only if } (\exists y')(Y'R'y'\land [x',y']R'z) \end{align*}$$

in order to ensure that this clause also gives consistent guidance concerning $R'$ . We prove this by induction on the size of $[x]\cup Y$ . When Y has size $2$ , this reduces to the case $(\exists y)([x_2,x_3]R'y\land [x_1,y]R'z$ iff $(\exists y')([x_1,x_3]R'y'\land [x_2,y']R'z)$ , but given the definition of $R'$ on two-element multisets in terms of the ternary R, this reduces to the biconditional $(\exists y)(Rx_2x_3y\land Rx_1yz)$ iff $(\exists y)(Rx_1x_3y'\land Rx_2y'z)$ , but this is the biconditional between $R^2x_1(x_2x_3)z$ and $R^2x_2(x_1x_3)z$ , which indeed holds in our $\mathsf {RW}^+$ frame.

Suppose the equivalence has been proved for all multisets of size n (where $n>2$ ) and we have a multiset $[x_1]\cup Y=[x_2]\cup Y'$ of size $n+1$ . Let Z be such that $Z\cup [x_{2}]=Y$ and $Z\cup [x_{1}]=Y'$ . Note that we may assume $x_{1}\neq x_{2}$ , as otherwise the case is trivial. We wish to show that

$$\begin{align*}(\exists y)(([x_{2}]\cup Z)R'y\land [x_1,y]R'z) \textrm{ iff } (\exists y')(([x_{1}]\cup Z)R'y'\land [x_2,y']R'z). \end{align*}$$

By the inductive hypothesis, $(\exists y)(([x_{2}]\cup Z)R'y\land [x_1,y]R'z)$ is equivalent to

$$\begin{align*}(\exists y)(\exists w)(ZR'w\land [w,x_{2}]R'y\land [x_1,y]R'z). \end{align*}$$

From the definition of $R'$ , the latter two conjuncts suffice for $Rx_{1}(x_{2}w)z$ , which is equivalent to $Rx_{2}(x_{1}w)z$ , as in the base case. Therefore,

$$\begin{align*}(\exists y')(\exists w)(ZR'w\land [w,x_{1}]R'y'\land [x_2,y']R'z), \end{align*}$$

which in turn is equivalent, by the inductive hypothesis, to

$$\begin{align*}(\exists y')(([x_{1}]\cup Z)R'y' \land [x_2,y']R'z). \end{align*}$$

So, we have shown by induction that the definition is coherent.

Now, it suffices to show that $R'$ , so defined, is reflexive and compositional. Reflexivity follows from the reflexivity of $\sqsubseteq $ , and Transitivity follows straightforwardly from the definition of $R'$ itself, albeit with many cases to check. It remains to show that Splitting holds.

We want to show that if $(X\cup Y)R'z$ , then there is some y where $XR'y$ and $([y]\cup Y)R'z$ . Given the definitions, we need to consider the cases where $X\cup Y$ has zero, one, two, or more elements. In the case where $X\cup Y$ is empty, then we have $[]R'z$ . So, then we have $[]R'z$ and $[z]R'z$ , as desired.

If $X\cup Y$ has size 1, then there are two subcases. Subcase: X is $[x]$ . By assumption we have $[x]R'z$ , so we then have $[x]R'x$ , by Reflexivity, and $[x]R'z$ , satisfying Splitting. Subcase: X is empty and Y is $[x]$ . Since $[x]R'z$ , $x\sqsubseteq z$ , so there is some $y\in N$ such that $Ryxz$ . We then have $[]R'y$ and $[y,x]R'z$ , satisfying Splitting.

Suppose $X\cup Y$ has size 2. Subcase: X is empty. We need a y such that $[]R'y$ and $[y,y_{1},y_{2}]R'z$ . Since $y_{1}\sqsubseteq y_{1}$ , there is a $u\in N$ such that $Ruy_{1}y_{1}$ . By assumption we have $Ry_{1}y_{2}z$ , so it follows that $Ruy_{1}y_{2}z$ , which is $[uy_{1}y_{2}]R'z$ , as desired. Subcase: X is [x] and Y is $[y_{1}]$ . In this subcase we have $[x,y_{1}]R'z$ . Since $[x]R'x$ , it follows that there is a y such that $[x]R'y$ and $[y,y_{1}]R'z$ , namely x. Subcase: X is $[x_{1},x_{2}]$ and Y is empty. By assumption we have $[x_{1},x_{2}]R'z$ and we need a y such that $[x_{1},x_{2}]R'y$ and $[y]R'z$ . Since $[z]R'z$ , we can simply take z as y.

Suppose $X\cup Y$ has size 3 or greater. Subcase: X is empty. The argument is similar to the subcase of the previous case where X is empty. Subcase: Y is empty. The argument is similar to the subcase of the previous case where Y is empty. Subcase: X and Y are inhabited, so $X= [x]\cup X'$ and $Y=[y_{1},\ldots ,y_{n}]$ and $([x]\cup X'\cup [y_{1},\ldots ,y_{n})R'z$ . From the definition of $R'$ , it follows that for some $z_{1}$ , $([x]\cup X'\cup [y_{2},\ldots ,y_{n}])R'z_{1}$ and $[y_{1}, z_{1}]R'z$ . Repeated use of the definition results in $z_{2},\ldots ,z_{n}$ such that $[y_{1}, z_{1}]R'z$ , $[y_{2}, z_{2}]R'z_{1}$ , …, $[y_{n}, z_{n}]R'z_{n-1}$ , and $([x]\cup X')R'z_{n}$ . Repeated use of Transitivity then yields $[y_{1},\ldots ,y_{n}, z_{n}]R'z$ , so we can let $z_{n}$ be the desired y.

All of the cases have been covered, so we conclude that $R'$ obeys Splitting.

So, the lemma is proved.

Now let us turn to consider what it is for a formula to hold at a point in a multiset frame. Given our understanding of the relation R, if $[x]Ry$ then the information in x also holds in y. So, if a formula holds at x, it is given by the multiset consisting of $[x]$ alone. But then, it should also hold at y, since the information given by $[x]$ is (perforce, according to R at least) also true at y, and there is nothing else in $[x]$ to take together with x. So, an appropriate heredity condition for truth-at-a-point in a multiset frame is given by the multiset relation R:

Definition 13 (Heredity)

A relation $\Vdash $ between points and formulas is hereditary along R for some class $\mathcal {F}$ of formulas if and only if whenever $[x]Ry$ (that is, when $x\sqsubset y$ ) and $x\Vdash A$ then $y\Vdash A$ , for each formula $A$ in $\mathcal {F}$ .

Given a hereditary relation $\Vdash $ for all atomic formulas on a multiset frame, we can extend it to a hereditary relation on all formulas in the language of $\mathsf {RW}^+$ as follows:

Definition 14 (Truth-at-a-point in a multiset model)

For any multiset frame $\langle P,R\rangle $ and a hereditary relation $\Vdash $ defined on atomic formulas in our language, we extend the relation $\Vdash $ to the whole vocabulary, defining $x\Vdash A$ recursively as follows:

  • $x\Vdash A\land B$ iff $x\Vdash A$ and $x\Vdash B$ .

  • $x\Vdash A\lor B$ iff $x\Vdash A$ or $x\Vdash B$ .

  • $x\Vdash A\to B$ iff for each $y,z$ where $[x,y]Rz$ , if $y\Vdash A$ then $z\Vdash B$ .

  • $x\Vdash A\circ B$ iff for some $y,z$ where $[y,z]Rx$ , both $y\Vdash A$ and $z\Vdash B$ .

  • $x\Vdash \mathsf {t}$ iff $[\;] Rx$ .

  • $x\Vdash \bot $ never.

Lemma 15. In any multiset frame $\langle P,R\rangle $ , the evaluation relation $\Vdash $ defined above, between points and arbitrary formulas is hereditary along R.

Proof. We aim to show that whenever $[x]Ry$ and $x\Vdash A$ then $y\Vdash A$ . This is an easy induction on the structure of the formula A. The result holds by fiat for atomic formulas, and the induction step is trivial for conjunctions and disjunctions.

For conditionals, suppose $[x]Ry$ and $x\Vdash A\to B$ . We wish to show that $y\Vdash A\to B$ too. Take $u,v$ where $[y,u]Rv$ . We wish to show that if $u\Vdash A$ then $v\Vdash B$ . By compositionality, since $[x]Ry$ and $[y,u]Rv$ , we have $[x,u]Rv$ . Since $x\Vdash A\to B$ , if $u\Vdash A$ then $v\Vdash B$ as desired.

Similarly, if $[x]Ry$ and $x\Vdash A\circ B$ , we wish to show that $y\Vdash A\circ B$ . So, we wish to find $u,v$ where $[u,v]Ry$ , $u\Vdash A$ and $v\Vdash B$ . Since $x\Vdash A\circ B$ , we have $u,v$ where $[u,v]Rx$ , $u\Vdash A$ and $v\Vdash B$ . By compositionality, $[u,v]Rx$ and $([x]\cup [\;])Ry$ gives us $([u,v]\cup [\;])Ry$ , i.e., $[u,v]Ry$ as desired.

Finally, if $[x]Ry$ and $x\Vdash \mathsf {t}$ , then we have $[\;] Rx$ . Notice that compositionality ensures that $[\;] Rx$ and $([x]\cup [\;] )Ry$ give $([\;]\cup [\;])Ry$ , i.e., from $[\;] Rx$ and $[x]Ry$ , we have $[\;] Ry$ , so if $\mathsf {t}$ holds at x and $[x]Ry$ , then $\mathsf {t}$ holds at y too.

So, evaluation relations on frames allow us to interpret formulas from the language of $\mathsf {RW}^+$ or $\mathsf {R}^+$ at points. Note that in the case for fusion, we needed to consider the multiset $[x]\cup [\;]$ , which is the special case highlighted in the definition of compositionality for inhabited-multiset relations. We call the combination of a frame $\langle P,R\rangle $ and an evaluation relation $\Vdash $ on that frame a model, and we abuse notation slightly to think of the triple $\langle P,R,\Vdash \rangle $ as a model.

Another way to represent how formulas are evaluated at points in frames is, for each formula A, to collect together the points that support A. We use the notation for the set $\{x\in P:x\Vdash A\}$ , the extension of the formula A in the model. The results of this section show that the set is upwardly closed along the relation $\sqsubset $ , and the evaluation conditions for atomic formulas are simply that for each atomic formula p, its extension is an upwardly closed set.

We pause to note that the evaluation conditions on ternary frames agree with those on multiset frames. In other words we have the following lemma:

Lemma 16 (Model equivalence)

If $\langle P,R,\sqsubseteq ,N,\Vdash \rangle $ is a ternary relational model for $\mathsf {RW}^+$ (or $\mathsf {R}^+$ ), then $\langle P,R',\Vdash \rangle $ is a multiset model defined on the multiset frame $\langle P,R'\rangle $ .

The proof is immediate, given that $[x,y]R'z$ iff $Rxyz$ , and $[\;] R'x$ iff $x\in N$ .

So, we have shown that reflexive multiset frames correspond tightly to ternary relational frames. We have also seen that compositional inhabited-multiset relations arise naturally as structures in the same general family as compositional multiset relation. A frame $\langle P,R\rangle $ which is furnished with an inhabited-multiset relation R can also be used to model our propositional vocabulary. Given an inhabited-multiset frame $\langle P,R\rangle $ and a hereditary evaluation relation $\Vdash $ on atomic formulas, we can extend $\Vdash $ to the propositional language except for the Ackermann constant $\mathsf {t}$ , in the manner given in Definition 14. The proof that $\Vdash $ so defined is heredity follows in exactly the same way. The only point at which the condition that R relate only inhabited multisets is violated in that proof is at the clause for $\mathsf {t}$ . The rest of the proof goes through as expected.

With multiset frames, we can model the relevant logic $\mathsf {RW}^+$ . To make this precise, we introduce the logic $\mathsf {RW}^+$ by way of a sequent calculus. The calculus utilises sequents of the form , where A is a formula and $\Gamma $ is a structure, generated by the following grammar:

$$\begin{align*}\Gamma \mathrel{:=} A~~|~~\epsilon~~|~~(\Gamma,\Gamma)~~|~~(\Gamma;\Gamma). \end{align*}$$

In other words, a structure is a formula A, the empty structure $\epsilon $ , the extensional combination $(\Gamma ,\Gamma ')$ of two structures, or the intensional combination $(\Gamma ;\Gamma ')$ of two structures. When presenting structures, we often omit the outer layer of parentheses (so $A,B$ is a structure, as is $A;(B,C)$ ), but we do not omit interior parentheses: $A,(B,C)$ differs from $(A,B),C$ in the order of combination, even though they will end up having the same logical force, due to the structural rules of the proof calculus.Footnote 18 We will also treat binary structural connectives as binding less tightly than any formula connectives, so $A\to B; C$ will be $(A\to B); C$ .

When specifying rules of inference, we use parentheses in another way: $\Gamma (A)$ is a structure with a particular subformula A singled out. Given $\Gamma (A)$ , the structure $\Gamma (\Gamma ')$ is found by substituting that instance of A by $\Gamma '$ . The same goes for other structures. So, $\Gamma (\Gamma ',\Gamma '')$ is a structure in which the structure $\Gamma ',\Gamma ''$ is found somewhere as a constituent, and the structure $\Gamma (\Gamma '',\Gamma ')$ is found by reversing the order of $\Gamma '$ and $\Gamma ''$ inside that structure. For future reference, we will call the part of the structure $\Gamma (A)$ around the instance A the context of A in $\Gamma (A)$ , and we will use the notation ‘ $\Gamma (-)$ ’ to refer to that context.

A derivation in this sequent calculus is a tree of sequents, of which every leaf is an axiom, where each transition is an inference rule. The fundamental rules in the sequent calculus are the axioms of Identity and the inference rule, Cut.Footnote 19

The next series of rules are structural rules, governing extensional and intensional structure combination respectively. Extensional combination allows for commutativity and associativity (at arbitrary depth inside a structure), as well as contraction and weakening, while intensional combination allows for only commutativity and associativity. In addition, $\epsilon $ acts as an identity for intensional combination.

The remaining rules are left and right rules for each connective. These are totally modular, in the sense that we can choose to include a connective or to leave it out. No rule for one connective requires the presence of any other connective in the vocabulary.

For $\mathsf {R}^+$ , we add one more rule: contraction for intensional combination.

With IW, we can derive new sequents, which could not be derived without it. For example, we can derive


The proof theory for logics like $\mathsf {RW}^+$ and $\mathsf {R}^+$ is well known, and so is the ternary relational semantics. Given our perspective on collection frames, it is worth taking the time to reconsider the relationship between proofs and models. Consider the proof given above, of the sequent

. What does this say about $\mathsf {R}^+$ models? It does not tell us that $(A\land (A\to B))\to B$ holds at every point in those models, only that it holds at normal points, those points x where $[\;] Rx$ . In other words, the sequent

should tell us that

For every point x, if $[\;] Rx$ then $x\Vdash (A\land (A\to B))\to B$ .

Scanning back to our derivation to its second line, we have

. This does not tell us that if $A\to B$ is true at a point and that A is true at that point, then B is true there too (if that were all the sequent said, the conditional would be irrelevant). The appropriate way to understand the ‘cash value’ of the derivation of this sequent according to our frames is that

For all x, y and z, if $x\Vdash A\to B$ and $y\Vdash A$ , if $[x,y]Rz$ then $z\Vdash B$ .

In the first of these cases, we have involved the R relation on empty multiset. In the second of these cases, we have used the R relation on a two-element multiset. The natural thing to consider when it comes to the sequent

, then, would be to understand the sequent as telling us this:

For all x and y, if $x\Vdash A\land (A\to B)$ and $[x]Ry$ then $y\Vdash B$ .

This is how we will understand validity of sequents on our frames. A single-premise single-conclusion sequent

is valid on a frame if and only if:

For all x and y, if $x\Vdash A$ and $[x]Ry$ then $y\Vdash B$ .

This agrees with the traditional understanding of validity of a sequent on a frame (that for each point x, if $x\Vdash A$ then $x\Vdash B$ too) when that frame is reflexive. (Take any reflexive frame. If a sequent has a counterexample according to the old definition, that provides a counterexample in the new definition too, by the reflexivity of the frame. Conversely, if we had points x and y where $[x]Ry$ and A holds at x but B fails at y, then by heredity on our frame, B must also fail at x, since $[x]Ry$ , and so we have a counterexample according to the traditional definition). This understanding of validity diverges only in cases where the frame is not reflexive.Footnote 20 Since non-reflexive frames are a proper generalisation of ternary relational frames, the question of how to interpret sequents on them is open. We have argued here that invoking R, and evaluating the lhs of our sequent at one point and the rhs at another is in keeping with how we have always interpreted zero-premise and multiple-premise sequents on ternary frames. It is also in keeping with the interpretation of conditionals in these frames. It would be surprising if the conditional-like notion of entailment in a relevant logic did not share in the features that the semantics ascribes to the conditional in that logic. So, we proceed with this new understanding of what it is for a sequent to be valid in a model.Footnote 21

So, when is a sequent valid in some model $\langle P,R,\Vdash \rangle $ ? We have considered sequents of the form , those of the form and those of the form . What about those involving the extensional combiner, the comma? When is the sequent valid in our frame? One candidate (generalising the case of the single formula on the left) is to say that whenever $x\Vdash A$ and $x\Vdash B$ then when $[x]Ry$ , we have $y\Vdash C$ . However, an equivalent way of formulating this claim will be more natural in our setting. Instead, we can say that is valid on a frame if and only if

For all x, y and z, if $x\Vdash A$ and $y\Vdash B$ , if $[x]Rz$ and $[y]Rz$ , then $z\Vdash C$ .

The parallel with the case for the semicolon is clear. We look for points where the lhs formulas are true, and we combine them, using R to locate where to check the rhs formula. Here we check C at all common descendants of x and of y, rather than those points found by combining x and y together. This choice allows us to give a particularly straightforward interpretation of the validity of sequents in our models. We start with the notion of the shadow cast by a structure in a model.

Definition 17 (The shadow cast by a structure)

For a structure $\Gamma $ its shadow $\{\!\{\Gamma \}\!\}$ in the model $\langle P,R,\Vdash \rangle $ is a set of points, defined recursively as follows:

  • $\{\!\{\epsilon \}\!\} = \{x\in P:[\;] Rx\}$ ,

  • ,

  • $\{\!\{\Gamma ,\Gamma '\}\!\}=\{x\in P:(\exists y\in \{\!\{\Gamma \}\!\})(\exists z\in \{\!\{\Gamma '\}\!\})([y]Rx\land [z]Rx)\}$ ,

  • $\{\!\{\Gamma ;\Gamma '\}\!\}=\{x\in P:(\exists y\in \{\!\{\Gamma \}\!\})(\exists z\in \{\!\{\Gamma '\}\!\})[y,z]Rx\}$ .

When a structure is a single formula A, then $\{\!\{A\}\!\}$ , the shadow it casts is not the formula’s extension, , but rather, it is the set of points upward from some point in the extension. If R is reflexive, then , so where R is reflexive, the distinction between shadows and extensions makes no significant difference. In any model, whether reflexive or not, .

It is worth pausing to understand the behaviour of shadows in a specific non-reflexive frame. Consider the multiset frame $\langle {\mathbb R},<\rangle $ with the multiset relation given by taking a multiset X of reals to relate to all and only those reals larger than each member of X. Here, the underlying order $\sqsubset $ is the order $<$ on ${\mathbb R}$ . So, the extension of a formula must be upwardly closed on ${\mathbb R}$ . So, an extension must have one of the forms $(-\infty ,\infty )$ , $[r,\infty )$ , or $(r,\infty )$ for some real r, or be empty. A shadow, on the other hand, cannot have the form $[r,\infty )$ . If , then $\{\!\{A\}\!\}=(r,\infty )$ , and if then $\{\!\{A\}\!\}=(r,\infty )$ too. The possible values of shadows are $(-\infty ,\infty )$ , and $(r,\infty )$ for each real r, and the empty set of reals.

It is also worth pausing to note that the notion of a shadow can be applied equally well in inhabited-multiset frames, provided that our structures do not contain the marker ‘ $\epsilon $ ’ for the empty structure. So, for the rest of this section, we will consider two kinds of models: those on multiset frames, and those on inhabited-multiset frames. The first kind will be models of the whole calculus, while inhabited-multiset frames can be used as models for the fragment of the proof calculus in which $\epsilon $ is absent: that is, the calculus without the rules $\epsilon \textit {I}$ , $\epsilon \textit {E}$ , $\mathsf {t}\textit {L}$ and $\mathsf {t}\textit {R}$ . We will call the calculi for $\mathsf {RW}^+$ and $\mathsf {R}^+$ without $\epsilon $ , $\mathsf {RW}^+_{-\epsilon }$ and $\mathsf {R}^+_{-\epsilon }$ respectively, to make explicit the absence of sequents with $\epsilon $ .

We have seen that the shadow $\{\!\{A\}\!\}$ of a formula A is related to its extension in a natural way. $x\in \{\!\{A\}\!\}$ iff there is some where $[y]Rx$ (that is, $y\sqsubset x$ ). This transition from extension to shadow is an operation on sets of points, and it is worth singling out with some notation.

Definition 18 ( $\sqsubset $ on sets of points)

$X_{\sqsubset }$ is defined as $\{x\in P:(\exists y\in X)y\sqsubset x\}$ .

So, this lemma is immediate:

Lemma 19 (From extensions to shadows)


This operation satisfies two useful conditions.

Lemma 20 ( $\sqsubset $ is monotone and idempotent)

For any sets X and Y, if $X\subseteq Y$ then $X_{\sqsubset } \subseteq Y_{\sqsubset }$ . Furthermore, $X_{\sqsubset }=X_{\sqsubset \sqsubset }$ .

Proof. For monotony, if $z\in X_{\sqsubset }$ then there is some $x\in X$ where $[x]Rz$ . Since $x\in Y$ , $z\in Y_{\sqsubset }$ too. For idempotence, we appeal to the density and transitivity of $\sqsubset $ . If $z\in X_{\sqsubset }$ then since there is some $x\in X$ where $[x]Rz$ then by density there is some y where $[x]Ry$ (so $y\in X_{\sqsubset }$ ) and $[y]Rz$ , ensuring that $x\in X_{\sqsubset \sqsubset }$ . Conversely, if $z\in X_{\sqsubset \sqsubset }$ then there is some $y\in X_{\sqsubset }$ where $[y]Rz$ and some $x\in X$ where $[x]Ry$ . By transitivity, $[x]Rz$ , ensuring that $z\in X_{\sqsubset }$ .

The shadow of a formula A is the set of points above that formula’s extension, . A shadow of a structure is not defined by taking the points above the extension of some formula, but nonetheless, it too is a fixed point for the operation $\sqsubset $ .

Lemma 21 (Shadows and order)

For each shadow $\{\!\{\Gamma \}\!\}$ , we have $\{\!\{\Gamma \}\!\}=\{\!\{\Gamma \}\!\}_{\sqsubset }$ .

To prove this, it is simplest to characterise the sets fixed under $\sqsubset $ in general terms. We first prove a more general lemma, for which Lemma 21 is a corollary. For this, we need one more definition:

Definition 22 (Closed upwards and open downwards)

A set X is closed upwards along $\sqsubset $ if whenever $x\in X$ and $x\sqsubset x'$ then $x'\in X$ too. A set X is open downwards along $\sqsubset $ if whenever $x\in X$ , there is some $x'\sqsubset x$ where $x'\in X$ too.

In the multiset frame $\langle {\mathbb R},<\rangle $ discussed above, the intervals $[r,\infty )$ are closed upwards but not open downwards, while the intervals $(r,\infty )$ are both closed upwards and open downwards along the order $<$ . The properties of being closed upwards and open downwards are related to the operation $\sqsubset $ as follows:

Lemma 23 (Open and closed sets)

If the relation $\sqsubset $ is transitive, then if X is closed upwards, then $X_{\sqsubset }\subseteq X$ . If $\sqsubset $ is dense, then if X is open downwards, then $X\subseteq X_{\sqsubset }$ .

The proof is a simple matter of unpacking the definitions:

Proof. Suppose $\sqsubset $ is transitive and that X is closed upwards. Take $x\in X_{\sqsubset }$ . So there is some $x'\in X$ where $x'\sqsubset x$ . Since X is closed upwards, we have $x\in X$ . Suppose $\sqsubset $ is dense and X is open downwards. Take $x\in X$ . Since X is open downwards we have some $x'\in X$ where $x'\sqsubset x$ . It follows that $x\in X_{\sqsubset }$ .

So, the sets X that are closed upwards and open downwards are fixed points for the operation $\sqsubset $ . Since on any collection frame, $\sqsubset $ is transitive and dense, the shadow $\{\!\{\Gamma \}\!\}$ of any structure $\Gamma $ is both closed upwards and open downwards, and is a fixed point for the operation $\sqsubset $ .

Now we can return to the proof of Lemma 21.

Proof. Consider each kind of shadow, as given in Definition 17. A quick inspection of each clause shows that if R satisfies Transitivity and Splitting, then the shadow is closed upward and open downward. For one example, for $\{\!\{\epsilon \}\!\}$ , if $x\in \{\!\{\epsilon \}\!\}$ , for upward closure, assume that $x\sqsubset x'$ . Since $[\;] Rx$ and $x\sqsubset x'$ we have $[\;] Rx'$ by transitivity, and $x'\in \{\!\{\epsilon \}\!\}$ . For downward openness, since $[\;] R x$ , by Splitting we have some $x'$ where $[\;] Rx'$ (so $x'\in \{\!\{\epsilon \}\!\}$ ) and $x'\sqsubset x$ , as desired.

For the intensional composition case, if $x\in \{\!\{\Gamma ;\Gamma '\}\!\}$ , for upward closure, assume that $x\sqsubset x'$ . Since we have $y\in \{\!\{\Gamma \}\!\}$ and $z\in \{\!\{\Gamma '\}\!\}$ where $[y,z]Rx$ , and since $x\sqsubset x'$ , by transitivity we have $[y,z]Rx'$ , and $x'\in \{\!\{\Gamma ;\Gamma '\}\!\}$ as desired. For downward openness, since $[y,z]Rx$ , by Splitting we have some $x'$ where $[y,z]Rx'$ and $[x']Rx$ (so $x'\in \{\!\{\Gamma ;\Gamma '\}\!\}$ ) and $x'\sqsubset x$ , as desired.

The other two cases follow in the same way, so we can declare this lemma proved.

With this behaviour of shadows proved, we can see that the definition of the shadow of an extensional structure can be simplified. Since $\{\!\{\Gamma ,\Gamma '\}\!\}=\{x\in P:(\exists y\in \{\!\{\Gamma \}\!\})[y]Rx\}\cap \{x\in P:(\exists y\in \{\!\{\Gamma '\}\!\})[z]Rx\}=\{\!\{\Gamma \}\!\}_{\sqsubset }\cap \{\!\{\Gamma '\}\!\}_{\sqsubset }$ , we have the following consequence:

Corollary 24. $\{\!\{\Gamma ,\Gamma '\}\!\}=\{\!\{\Gamma \}\!\}\cap \{\!\{\Gamma '\}\!\}$ .

With the definition of a structure’s shadow, the statement the condition for validity on a model is straightforward.

Definition 25 (Model validity)

A sequent is valid in the model $\langle P,R,\Vdash \rangle $ if and only if . That is, the shadow cast by the structure $\Gamma $ is restricted to the extension of the formula A.

So, we are in a position to state our soundness theorem:

Theorem 26 ( $\mathsf {RW}^+$ is sound for multiset frames)

Any $\mathsf {RW}^+$ derivable sequent holds in each model $\langle P,R,\Vdash \rangle $ on a multiset frame. Furthermore, any $\mathsf {RW}^+_{-\epsilon }$ derivable sequent holds in each model on an inhabited-multiset frame.

To prove the soundness theorem, it helps to establish the following facts about shadows and contexts.

Lemma 27 (Contexts preserve order, and are prime)

If , then for any context $\Gamma '(-)$ , we have $\{\!\{\Gamma '(\Gamma )\}\!\}\subseteq \{\!\{\Gamma '(A)\}\!\}$ . In this sense, contexts are order preserving over valid sequents. Furthermore, $\{\!\{\Gamma '(A\lor B)\}\!\}=\{\!\{\Gamma '(A)\}\!\}\cup \{\!\{\Gamma '(B)\}\!\}$ , so contexts are prime, and $\{\!\{\Gamma '(\bot )\}\!\}=\{\!\{\bot \}\!\}=\emptyset $ .

Proof. Both facts follow from an easy induction on the construction of the context $\Gamma '(-)$ . An atomic context $\Gamma '(-)$ the hole ‘ $-$ ’ itself. In this case, primeness is trivial, and order preservation follows from the monotony and idempotence (Lemma 20). If , then by monotony, , but , so $\{\!\{A\}\!\}_{\sqsubset }\subseteq \{\!\{B\}\!\}$ , and since idempotence gives $\{\!\{A\}\!\}=\{\!\{A\}\!\}_{\sqsubset }$ , we have $\{\!\{A\}\!\}\subseteq \{\!\{B\}\!\}$ as desired.

For the induction steps, $\Gamma '(-)$ either has the form $\Gamma ''(-),\Gamma '''$ , or $\Gamma ''',\Gamma ''(-)$ , or $\Gamma ''(-);\Gamma '''$ , or $\Gamma ''';\Gamma ''(-)$ , in which case preservation and primeness follow immediately from the properties holding for the simpler context $\Gamma ''(-)$ .

For example, if $\{\!\{\Gamma ''(\Gamma )\}\!\}\subseteq \{\!\{\Gamma ''(A)\}\!\}$ , then $\{\!\{\Gamma ';\Gamma ''(\Gamma )\}\!\}=\{x\in P:(\exists y\in \{\!\{\Gamma '\}\!\})(\exists z\in \{\!\{\Gamma ''(\Gamma )\}\!\})[y,z]Rx\}$ , but since $\{\!\{\Gamma ''(\Gamma )\}\!\}\subseteq \{\!\{\Gamma ''(A)\}\!\}$ , it follows that this set is a subset of $\{x\in P:(\exists y\in \{\!\{\Gamma '\}\!\})(\exists z\in \{\!\{\Gamma ''(A)\}\!\})[y,z]Rx\}$ , which is $\{\!\{\Gamma ';\Gamma ''(A)\}\!\}$ , as desired. Similarly, given that $\{\!\{\Gamma ''(A\lor B)\}\!\}=\{\!\{\Gamma ''(A)\cup \Gamma ''(B)\}\!\}$ , then $\{\!\{\Gamma ';\Gamma ''(A\lor B)\}\!\}=\{x\in P:(\exists y\in \{\!\{\Gamma '\}\!\})(\exists z\in \{\!\{\Gamma ''(A\lor B)\}\!\})[y,z]Rx\}$ , which is equal to $\{x\in P:(\exists y\in \{\!\{\Gamma '\}\!\})(\exists z\in \{\!\{\Gamma ''(A)\}\!\}\cup \{\!\{\Gamma ''(B)\}\!\})[y,z]Rx\}$ , which is $\{x\in P:(\exists y\in \{\!\{\Gamma '\}\!\})(\exists z\in \{\!\{\Gamma ''(A)\}\!\})[y,z]Rx\}\cup \{x\in P:(\exists y\in \{\!\{\Gamma '\}\!\}) (\exists z\in \{\!\{\Gamma ''(B)\}\!\})[y,z]Rx\}$ , which is in turn $\{\!\{\Gamma ';\Gamma ''(A)\}\!\}\cup \{\!\{\Gamma ';\Gamma ''(B)\}\!\}$ . Finally, given that $\{\!\{\Gamma ''(\bot )\}\!\}=\emptyset $ , clearly $\{\!\{\Gamma ';\Gamma ''(\bot )\}\!\}=\{x\in P:(\exists y\in \{\!\{\Gamma '\}\!\})(\exists z\in \{\!\{\Gamma ''(\bot )\}\!\})[y,z]Rx\}=\{x\in P:(\exists y\in \{\!\{\Gamma '\}\!\})(\exists z\in \emptyset )[y,z]Rx\}=\emptyset $ , as desired.

Now we can return to our proof of the soundness theorem. As is usual, it is a straightforward induction on the length of a derivation. The technique is standard, and there are no surprises, despite the idiosyncratic interpretation of sequents to allow for the non-reflexive frames.Footnote 22

Proof. We prove soundness by induction on the length of a derivation for the sequent . The axiomatic sequent holds in every multiset frame and in every inhabited-multiset frame since . The sequent holds in every multiset frame, since in these frames we have .

For the Cut rule, suppose we have and . We wish to show that . Here we appeal to the fact that the context $\Gamma '$ preserves order. Since , we have $\{\!\{\Gamma '(\Gamma )\}\!\}\subseteq \{\!\{\Gamma '(A)\}\!\}$ , and since , we have as desired.

That the extensional structural rules preserve validity on frames is an immediate consequence of the fact that the outer context $\Gamma (-)$ preserves order, and the extensional structure is modelled by intersection of shadows. For example, for the weakening rule EK, since $\{\!\{\Gamma ',\Gamma ''\}\!\}=\{\!\{\Gamma '\}\!\}\cap \{\!\{\Gamma ''\}\!\}\subseteq \{\!\{\Gamma '\}\!\}$ , and since $\Gamma (-)$ preserves order, we know that if then we also have . In the same way, associativity, commutativity and contraction are assured.

Most of the intensional structural rules follow in the same way from the properties of multisets. For example, the associativity rule IB follows appeals to the compositionality of R. $\{\!\{(\Gamma ';\Gamma '');\Gamma '''\}\!\}=\{x\in P:(\exists y\in \{\!\{(\Gamma ';\Gamma '')\}\!\})(\exists z\in \{\!\{\Gamma '''\}\!\})[y,z]Rx\}$ unpacking the definition of $\{\!\{\Gamma ';\Gamma ''\}\!\}$ this set is identical to $\{x\in P:(\exists y\in P)(\exists u\in \{\!\{\Gamma '\}\!\})(\exists v\in \{\!\{\Gamma ''\}\!\})([u,v]Ry\land [y,z]Rx)\}$ . Applying compositionality, we see that $(\exists y\in P)([u,v]Ry\land [y,z]Rx)$ is equivalent to $[u,v,z]Rx$ , using Transitivity in one direction and Splitting in the other. Thus, the set $\{\!\{(\Gamma ';\Gamma '');\Gamma '''\}\!\}$ simplifies (as expected) to $\{x\in P:(\exists u\in \{\!\{\Gamma '\}\!\})(\exists v\in \{\!\{\Gamma ''\}\!\})(\exists z\in \{\!\{\Gamma '''\}\!\})[u,v,z]Rx\}$ where the left-associated structure $(\Gamma ';\Gamma '');\Gamma '''$ unwraps into the unassociated multiset $[u,v,z]$ . A moment’s reflection shows that the right-associated structure $\Gamma ';(\Gamma '';\Gamma ''')$ unwraps to exactly the same set, so $\{\!\{(\Gamma ';\Gamma '');\Gamma '''\}\!\}=\{\!\{\Gamma ';(\Gamma '';\Gamma ''')\}\!\}$ , showing that the associativity structural rule IB is valid on frames. It is simpler to show that IC holds, since $\{\!\{\Gamma ';\Gamma ''\}\!\}=\{\!\{\Gamma '';\Gamma '\}\!\}$ straightforwardly, given that $[y,z]=[z,y]$ for each y and z.

The $\epsilon $ I and $\epsilon $ E rules hold in models on multiset frames (but not in models on inhabited-multiset frames). Here, we have $\{\!\{\epsilon ;\Gamma '\}\!\}=\{\!\{\Gamma '\}\!\}$ since $\{\!\{\epsilon \}\!\}=\{x\in P:[\;] Rx\}$ and so $\{\!\{\epsilon ;\Gamma '\}\!\}=\{x\in P:\exists y ([\;] Ry \land \exists z\in \{\!\{\Gamma '\}\!\}[y,z]Rx)\}$ . However, if $[\;] Ry$ and $[y,z]Rx$ then by transitivity, $[\;]\cup [z]Rx$ , i.e., $[z]Rx$ . And conversely, by Splitting, if $[z]Rx$ then $[\;]\cup [z]Rx$ and so, there is some y where $[\;] Ry$ and $[y,z]Rx$ . So, our set $\{x\in P:\exists y([\;] Ry \land \exists z\in \{\!\{\Gamma '\}\!\}[y,z]Rx)\}$ is the set $\{x\in P:\exists z\in \{\!\{\Gamma '\}\!\}[z]Rx\}$ , which is $\{\!\{\Gamma '\}\!\}$ itself, by Lemma 21.

It remains to verify the validity of each of the connective rules. The validity of the left rules for conjunction, disjunction, and fusion follow immediately from the truth conditions for these connectives and the fact that the context $\Gamma (-)$ preserves order. For example, for $\circ $ L, if we know that holds in the model, then since , and the context $\Gamma (-)$ preserves order, it follows that too. The reasoning for the left rules for conjunction is similar, and so is the left rule for $\mathsf {t}$ when our attention is restricted to multiset frames.

The reasoning for the left rule for disjunction follows immediately from the primeness of the context $\Gamma (-)$ . If and then indeed . The left rule for $\bot $ is trivial, given that $\{\!\{\bot \}\!\}=\emptyset $ .

For the last left rule, for the conditional, to show that and ensures that , we appeal to the fact that $\Gamma '(-)$ preserves order. Using this fact, it suffices to show that , for then we indeed have as desired. That follows from by the definition of shadows for intensional combination. If $x\in \{\!\{A\to B;\Gamma \}\!\}$ then there are y and z where