A Van Benthem Characterization Result for Distribution-Free Logics

Abstract

This article contributes to recent results in the model theory of distribution-free logics (which include a Goldblatt-Thomason theorem and a development of their Sahlqvist theory) by lifting van Benthem’s characterization result for modal logic to the more general setting of the logics of normal lattice expansions. Our proof approach makes use of a fully abstract translation of the language of the logics of interest into the language of sorted residuated modal logic, building on an analogous translation of substructural logics recently published by the author. The article is intended as a demonstration and application of a project of reduction of non-distributive logics to (sorted) residuated modal logics. The reduction makes the proof of a van Benthem characterization of non-distributive logics possible, by adapting, reusing and generalizing existing results.

1. Introduction

Contributions to the model theory of non-distributive logics, arising as the logics of normal lattice expansions, are quite recent. They include published results on their Sahlqvist theory [1,2], proof of a Goldblatt-Thomason theorem [3], as well as modal translation semantics [4,5].

This article focuses on the logics of normal (in general, distribution-free) lattice expansions (lattices with quasi-operators). The relational interpretation of the logics is in sorted frames (with sorts $1,\partial$) $\mathfrak{F}=(Z_1,Z_{\partial },I,\dots )$, $I\subseteq Z_1\times Z_{\partial }$, where several additional sorted relations may be involved, depending on the signature of the logic. Our main result (Theorem 5) is a characterization of the fragment of the sorted first-order language of the structures $\mathfrak{F}$ consisting of formulae that are equivalent to a translation of a sentence of the language of our propositional logic into the sorted first-order language.

The result generalizes the well known van Benthem characterization for modal logic. Our proof approach is a demonstration and application of a project of reduction of non-distributive logics to (sorted) residuated modal logics. The reduction itself is an application of recent representation results [6,7] for normal lattice expansions and a generalization of a canonical and fully abstract translation [5] of the language of substructural logics into the language of their companion sorted residuated modal logics.

Section 2 introduces normal lattice expansions, their minimal propositional logics and the associated relational semantics, in sorted residuated frames. Issues relating to proving completeness are also briefly sketched.

A translation into the first-order language of sorted residuated frames is only given in Section 4. This is because the translation is defined as a composition of a translation into the language of the sorted residuated companion modal logics of our propositional logics and a subsequent standard translation of the modal language into the first-order language. Section 3 presents the sorted residuated modal logics of the sorted frames that provide Kripke (relational) semantics to the logics of normal lattice expansions. A fully abstract (parameterized family of) translation(s) is presented and full abstraction is proven in Theorem 1.

In Section 4 the sorted first-order language is presented, notation is fixed, the reduction to the language of unsorted first-order logic is recalled, as detailed in [8], and a fully abstract standard translation of the modal language into the first-order language is given. The rest of the section deals with lifting to the sorted case the model theoretic machinery used to prove van Benthem’s characterization theorem for modal logic. The proof of the main fact on the existence of $\omega$-saturated extensions for the sorted case is detailed in Theorem 3, defining the extension as the ultrapower ${\prod }_U\mathfrak{F}$ of the structure $\mathfrak{F}$ over a countably incomplete ultrafilter U over some index set J, just as in the unsorted case.

The article concludes with Section 5 where sorted bisimulations, bisimilarity of structures and modal equivalence of models are defined. As in the unsorted case, the key fact to derive, detailed in Proposition 4, is that if two models are modally equivalent, then their $\omega$-saturated extensions constructed as explained above are also modally equivalent and, in addition, modal equivalence is in fact a bisimulation. Theorem 4 lifts the van Benthem result to the sorted residuated modal logic case and, after defining t-invariance, Theorem 5 concludes with the desired van Benthem characterization result for the logics of normal lattice expansions. Specifically, we prove that a sorted first-order formulae $\Phi \left(u\right)$ is equivalent to the translation $T\left(\phi \right)$ of a sentence in the propositional logic considered iff $\Phi \left(u\right)$ is invariant under sorted bisimulations and, in addition, it is invariant under the transformation t mapping $\Phi \left(u\right)$ to the formula $t(\Phi \left(u\right))={\forall }^{\partial }v(\mathbf{I}(u,v)⟶{\exists }^{1}z(\mathbf{I}(z,v)\wedge \Phi \left(z\right)))$.

2. Algebras, Languages and (Sorted) Frames

By a distribution type we mean an element $\delta$ of the set ${\{1,\partial \}}^{n+1}$, for some $n\ge 0$, typically to be written as $\delta =(i_1,\dots ,i_n;i_{n+1})$ and where ${\delta }_{(n+1)}=i_{n+1}\in \{1,\partial \}$ will be referred to as the output type of $\delta$. A similarity type $\tau$ is then defined as a finite sequence of distribution types, $\tau =\langle {\delta }_1,\dots ,{\delta }_k\rangle$.

If ${\mathcal{L}}_1,\dots ,{\mathcal{L}}_n,\mathcal{L}$ are bounded lattices, then a function $f:{\mathcal{L}}_1\times \cdots \times {\mathcal{L}}_n⟶\mathcal{L}$ is additive and normal, if for each i, f distributes over finite joins in ${\mathcal{L}}_i$ (including the empty join).

We write $\mathcal{L}$ for ${\mathcal{L}}^1$ and ${\mathcal{L}}^{\partial }$ for its opposite lattice (where order is reversed, often designated as ${\mathcal{L}}^{op}$).

Definition 1.

An n-ary map f on a lattice $\mathcal{L}$ is a normal lattice operator of distribution type $\delta =(i_1,\dots ,i_n;i_{n+1})$ if it is an additive and normal function $f:{\mathcal{L}}^{i_1}\times \cdots \times {\mathcal{L}}^{i_n}⟶{\mathcal{L}}^{i_{n+1}}$, where each $i_j$, for $j=1,\dots ,n,n+1$, is in the set $\{1,\partial \}$, i.e., ${\mathcal{L}}^{i_j}$ is either $\mathcal{L}$, or ${\mathcal{L}}^{\partial }$. We refer to $i_{n+1}$ as the output type of the operator.

Definition 2.

A normal lattice expansion is a structure $\mathcal{L}=(L,\le ,\wedge ,\vee ,0,1,{\left(f_i\right)}_{i\in k})$ where (a) $(L,\le ,\wedge ,\vee ,0,1)$ is a bounded lattice and (b) $k\ge 0$ is a natural number and for each $i\in k$, $f_i$ is a normal operator on $\mathcal{L}$ of some specified arity $\alpha \left(f_i\right)\in {\mathbb{N}}^{+}$ and distribution type $\delta \left(i\right)$. The similarity type of $\mathcal{L}$ is the k-tuple $\tau \left(\mathcal{L}\right)=\langle \delta \left(0\right),\dots ,\delta (k-1)\rangle$.

Example 1.

A bounded lattice with a box and a diamond operator $\mathcal{L}=(L,\le ,\wedge ,\vee ,0,1,\square ,◇)$ is a normal lattice expansion of similarity type τ, where $\tau =\langle (1;1),(\partial ,\partial )\rangle$ where $\delta (◇)=(1;1)$, i.e., $◇:\mathcal{L}⟶\mathcal{L}$ distributes over finite joins of $\mathcal{L}$, while $\delta (\square )=(\partial ;\partial )$, i.e., $\square :{\mathcal{L}}^{\partial }⟶{\mathcal{L}}^{\partial }$ distributes over “joins” of ${\mathcal{L}}^{\partial }$ (i.e. meets of $\mathcal{L}$), delivering “joins” of ${\mathcal{L}}^{\partial }$ (i.e. meets of $\mathcal{L}$).

Similarly for an implicative lattice, of similarity type ${\tau }^{\prime }=\langle (1,\partial ;\partial )\rangle$ and where $(1,\partial ;\partial )=\delta (\to )$ is the distribution type of the implication operator, regarded as a map $\to :\mathcal{L}\times {\mathcal{L}}^{\partial }⟶{\mathcal{L}}^{\partial }$ distributing over “joins” in each argument place, i.e., co-distributing over joins in the first place, turning them to meets, and distributing over meets (joins of ${\mathcal{L}}^{\partial }$) in the second place, delivering “joins” of ${\mathcal{L}}^{\partial }$, i.e., meets of $\mathcal{L}$.

An FL-algebra (Full Lambek algebra [9]) $\mathcal{L}=(L,\le ,\wedge ,\vee ,0,1,\leftarrow ,\circ ,\to )$ is a normal lattice expansion with similarity type ${\tau }^{″}=\langle (1,1;1),(1,\partial ;\partial ),(\partial ,1;\partial )\rangle$. In other words, it is a residuated lattice, with $\delta (\circ )=(1,1;1)$, $\delta (\to )=(1,\partial ;\partial )$ and $\delta (\leftarrow )=(\partial ,1;\partial )$.

Normal lattice operators can be grouped in just two families according to their output type (1 or ∂) and, for our purposes, nothing depends on the size of each group, or on the arity of the operators. Where $(L,\le ,\wedge ,\vee ,0,1)$ is a bounded lattice which may not be distributive, we shall therefore be only considering expansions $\mathcal{L}$ = (L, ≤, ∧, ∨, 0, 1, , ⊖), where , $\ominus$ are normal n-ary lattice operators with respective output types 1 and ∂. Let $\tau =\langle (i_1,\dots ,i_n;1),(i_1^{\prime },\dots ,i_n^{\prime };\partial ))$ be the similarity type of $\mathcal{L}$. The language ${\Lambda }_{\tau }$ of the lattice expansion is displayed below, where $\overline{\phi }=({\phi }_1,\dots ,{\phi }_n)$.

The minimal axiomatization of the logic adds to axioms and rules for Positive Lattice Logic (the logic of bounded lattices) the normality axioms (distribution axioms) determined by the distribution types of the operators.

Depending on the specific signature of logical operators, additional axioms can be added, giving rise to a wealth of logics. Examples include substructural logics, extensively presented and studied in [10], logics with a negation operator, such as orthologic, or logics including distribution in their axiomatization, such as the logic of De Morgan algebras and even classical (Boolean) logic, as recently studied in [11].

Normal lattice expansions are the algebraic models of (non-distributive) logics. Sorted relational semantic frameworks for substructural and, more generally, non-distributive logics have been considered by Suzuki [12,13], by Gehrke and co-workers [14,15,16], as well as by the present author [5,11,17,18]. In all three cases the semantics is supported by representation theorems for lattices [19,20], used in the construction of a canonical frame in a completeness proof. Single-sorted approaches have been also studied [21,22,23], based on a different representation [24,25].

Structures $\mathfrak{F}=(A,〧,B),〧 \subseteq A\times B$ have been introduced (named ‘polarities’) and studied by Birkhoff [26] and subsequently formed the basic structures of Formal Concept Analysis (FCA) [27] where they are called ‘formal contexts’. The dual structure ${\mathfrak{F}}^{+}$ of a formal context $\mathfrak{F}$ is its ‘formal concept lattice’, a complete lattice of ‘formal concepts’ $(C,D)$ where $C\subseteq A$ with C = ^〧(C^〧) (a Galois stable set) and $D$ = C^〧 (a Galois co-stable set), hence also $C$ = ^〧D, and where ( )^〧: (A) ⇆ (B): ^〧( ) is the Galois connection generated by the relation 〧,

$$\begin{array}{ccc}\hfill C^{〧}& =& \{d\in D|\forall c\in Cc〧d\}=\{d\in D|C〧d\}\hfill \\ \hfill {}^{〧}D& =& \{c\in C|\forall d\in Dc〧d\}=\{c\in C|c〧D\}.\hfill \end{array}$$

We let $\mathcal{G}\left(X\right),\mathcal{G}\left(Y\right)$ designate the (dually isomorphic) complete lattices of Galois stable and co-stable sets, respectively. In the general case, $\mathcal{G}\left(X\right),\mathcal{G}\left(Y\right)$ are non-distributive. A first-order frame condition for $\mathcal{G}\left(X\right)$ to be (completely) distributive was specified in [11] [Theorem 3.4]. Letting I be the complement of the Galois frame relation 〧, note that I generates a pair of residuated maps $\ell ,r$ such that the generated (by composition) closure operators $\ell r,r\ell$ coincide with those generated by the Galois connection.

It has been shown in [20] (following the FCA approach and building on Urquhart’s [24]) and in [19] (building on Goldblatt’s representation of ortholattices [28]) that every lattice can be represented as a sublattice of the formal concept lattice of a suitable formal context. This was generalized in [6,7] to the case of normal lattice expansions, using sorted frames with additional relations $\mathfrak{F}=(A,I,B,{\left(R_t\right)}_{t\in T})$. The representation in [6,7] is uniform and all cases reduce to the cases of relations of sorts $(1;i_1\cdots i_n)$ and $(\partial ;i_1^{\prime }\cdots i_n^{\prime })$, corresponding to normal lattice operators that take their values in the lattice $\mathcal{L}$, or in its opposite lattice ${\mathcal{L}}^{\partial }$. Hence we will be considering only sorted structures $(A,I,B,R,S)$ with relations $I\subseteq A\times B$, $R\subseteq A\times {\prod }_{j=1}^{j=n}{Z}^{i_j}$ and $S\subseteq B\times {\prod }_{j=1}^{j=n}{Z}^{i_j^{\prime }}$ (where $Z^{i_j}=A$ if $i_j=1$ and B otherwise), nothing depending on having more than one relation of each sort, or on having relations of different arities.

By the similarity type of a sorted frame $\mathfrak{F}=(A,I,B,{\left(R_k\right)}_{k\in K})$ we shall mean the tuple ${\langle \sigma \left(R_k\right)\rangle }_{k\in K}$ of sorts of the sorted frame relations $R_k$. Following the approach of [6], to an algebra of similarity type $\tau$, a frame of the same similarity type is considered for the relational interpretation of the language ${\Lambda }_{\tau }$.

Given a frame $\mathfrak{F}=(A,I,B,R,S)$, where the sort of $R,S$ corresponds to the distribution type of , $\ominus$, respectively, a model $\mathfrak{M}=(\mathfrak{F},V)$ is equipped with an interpretation that assigns a Galois-stable set $V\left(p_i\right)\in \mathcal{G}\left(X\right)$ to each propositional variable. A co-interpretation V^〧 is also defined, by setting $V^{〧}\left(p_i\right)={V\left(p_i\right)}^{〧}\in \mathcal{G}\left(Y\right)$. Interpretation and co-interpretation are extended to all sentences and we write $\left[\right[\phi \left]\right]\subseteq A,\left(\right|\phi \left|\right)\subseteq B$, respectively, so that $\left(\right|\phi \left|\right)={\left[\right[\phi \left]\right]}^{〧}$. Equivalently, we may say that each sentence $\phi$ is interpreted as a formal concept $(\left[\right[\phi \left]\right],\left(\right|\phi \left|\right))$ in the formal concept lattice of the frame. The satisfaction and co-satisfaction (refutation) relations $⊩\subseteq A\times \Lambda ,{⊩}^{\partial }\subseteq B\times \Lambda$ are defined as usual, $a⊩\phi$ iff $a\in \left[\right[\phi \left]\right]$, $b{⊩}^{\partial }\phi$ iff $b\in \left(\right|\phi \left|\right)$. Since interpretation and co-interpretation determine each other, it suffices to provide for each logical operator the clause for either ⊩, or ${⊩}^{\partial }$, as we do in

Table 1. (Co)Satisfaction Relations.

$a⊩p_i$               iff   $a\in V\left(p_i\right)$

$a⊩\top$               iff $a=a$

$b{⊩}^{\partial }\perp$             iff $b=b$

$a⊩\phi \wedge \psi$        iff $a⊩\phi$ and $a⊩\psi$

$b{⊩}^{\partial }\phi \vee \psi$        iff $b{⊩}^{\partial }\phi$ and $b{⊩}^{\partial }\psi$

b ⊩∂ $({\phi }_1,\dots ,{\phi }_n)$

iff $\forall u_1\cdots u_n({\bigwedge }_j^{i_j=1}(u_j⊩{\phi }_j)\wedge {\bigwedge }_r^{i_r=\partial }\left(u_r{⊩}^{\partial }{\phi }_r\right)⟶b{R}^{\prime }{u}_1\cdots u_n)$

$a⊩\ominus ({\phi }_1,\dots ,{\phi }_n)$

iff $\forall v_1\cdots v_n({\bigwedge }_j^{i_j=1}(v_j⊩{\phi }_j)\wedge {\bigwedge }_r^{i_r=\partial }\left(v_r{⊩}^{\partial }{\phi }_r\right)⟶a{S}^{\prime }{v}_1\cdots v_n)$

. The relations $R^{\prime },S^{\prime }$ in Table 1 are the Galois dual relations of $R,S$, defined by setting $R^{\prime }{u}_1\cdots u_n={(R{u}_1\cdots u_n)}^{〧}$ and, similarly, $S^{\prime }{v}_1\cdots v_n={}^{〧}(S{v}_1\cdots v_n)$, where we let $R{u}_1\cdots u_n$ designate the set $\left\{u\right|uR{u}_1\cdots u_n\}$ and similarly for $S{v}_1\cdots v_n$.

Example 2.

If $=\circ$ (the fusion (cotenability) operator) and $\ominus =\to$ (implication), of respective distribution types $(1,1;1),(1,\partial ;\partial )$, the semantic clauses run as follows

• $b{⊩}^{\partial }\phi \circ \psi \hspace{1em}iff\hspace{1em}\forall a,c\in A(c⊩\phi \wedge a⊩\psi ⟶b{R}^{\prime }ca)$
• $a⊩\phi \to \psi \hspace{1em}iff\hspace{1em}\forall c\in A\forall b\in B(c⊩\phi \wedge b{⊩}^{\partial }\psi ⟶a{S}^{\prime }cb)$

where suitable conditions on $R,S$ ensure residuation of the operators (cf [29] for details). Note that fusion ∘, which is a binary diamond operator, is dually interpreted as a binary box. This is in line with the principle of order-dual relational semantics proposed in [30] by the author.

For another example, consider the case of modal operators $\ominus =\square$, = ◇, of respective distribution types $(\partial ;\partial )$ and $(1;1)$. The semantic clauses run as follows, after some logical manipulation of the respective clauses in Table 1, with ◇ dually interpreted as necessity (cf [17,30] for details),

• $b{⊩}^{\partial }◇\phi \hspace{1em}iff\hspace{1em}\forall d\in B(b{R}^{″}d⟶d{⊩}^{\partial }\phi )$
• $a⊩\square \phi \hspace{1em} iff\hspace{1em}\forall c\in A(a{S}^{″}c⟶c⊩\phi )$

where we define $b{R}^{″}$ = ^〧$\left(b{R}^{\prime }\right)$ (and recall that $R^{\prime }$ is defined from $R\subseteq A\times A$ by setting $R^{\prime }c=\left(Rc\right)$^〧) and similarly $S^{″}$ is defined from $S\subseteq B\times B$ by first letting $S^{\prime }b=$ ^〧(Sb), then defining $a{S}^{″}$ = ^〧(aS′).

Axioms for the operators , $\ominus$ capture properties of the frame relations. These can be always regarded as properties of $R,S$, but it may be more natural to express them as properties of the Galois dual relations $R^{\prime },S^{\prime }$ or of some derived relation such as the relations $R^{″},S^{″}$ involved in the semantic clauses for the possibility and necessity operators. As an example, if = $\circ$ is the fusion operator, of distribution type $\delta (\circ )=(1,1;1)$, then the commutativity axiom for ∘ corresponds to the property $\forall a,c,c^{\prime }\in A(aRc{c}^{\prime }\leftrightarrow aR{c}^{\prime }c)$. The contraction axiom for ∘ corresponds to the property $\forall a\in AaRaa$, see [5] for details.

To a frame $\mathfrak{F}=(A,I,B,R,S)$, with $I\subseteq A\times B$ and $\sigma \left(R\right)=(1;i_1\cdots i_n)$ and $\sigma \left(S\right)=(\partial ;t_1\cdots t_n)$, both a (sorted) modal and a (sorted) first-order language can be associated, in the natural way, as we explain in the sequel. This raises the question of the characterization of the fragments of the modal, as well as of the first-order languages consisting of formulae equivalent to sentences of the propositional language ${\Lambda }_{\tau }$.

A translation of the language ${\Lambda }_{\tau }$ into the (necessarily sorted) first-order language needs to take into account the fact that propositional variables are interpreted as Galois stable sets C = ^〧(C^〧), hence merely introducing a unary predicate P_i for each propositional variable $p_i$ falls short of the goal. The obstacle can be sidestepped by observing that the relation I generates a pair of residuated operators ◇: (A) ⇆ (B): ■ such that the generated (by composition) closure operators coincide with those generated by the Galois connection generated by its complement 〧, i.e., ^〧(U^〧) = ■◇U and ^〧(V^〧) = □◆V, for $U\subseteq A,V\subseteq B$ and where $\square =-◇-$ and $◆=-■-$. This leads to considering the sorted modal logic of polarities with relations, a project initiated in [4,5].

3. Sorted Modal Logics of Polarities with Relations

3.1. Sorted Residuated Modal Logic

Fix any $\tau$-structure (a structure of similarity type $\tau$) $\mathfrak{F}=(A,B,I,R,S)$, where $I\subseteq A\times B$, and $R,S$ are $(n+1)$-ary relations of respective sorts $\sigma \left(R\right)=(1;i_1,\dots ,i_n)$ and $\sigma \left(S\right)=(\partial ;i_1^{\prime },\dots ,i_n^{\prime })$, i.e., $R\subseteq A\times {\prod }_{j=1}^{j=n}{Z}_{i_j}$ and $S\subseteq B\times {\prod }_{j=1}^{j=n}{Z}_{i_j^{\prime }}$, where $Z_1=A$ and $Z_{\partial }=B$. The relation I generates residuated operators ◇: (A) ⇆ (B):■

$$\begin{array}{ccc}\hfill ◇U& =& \{b\in B|\exists a\in A(aIb\wedge a\in U)\}\hfill \\ \hfill ■V& =& \{a\in A|\forall b\in B(aIb⟶b\in V)\}\hfill \end{array}$$

(1)

and each of $R,S$ generates a sorted image operator in the sense of [31,32]

(2)

(3)

where for each $j=1,\dots ,n$, $U_j\subseteq Z_{i_j}\in \{A,B\}$ and $V_j\subseteq Z_{i_j^{\prime }}\in \{A,B\}$.

Remark 1.

Each of , distributes over arbitrary unions of sets and it is therefore residuated in each argument place. Letting , ⊖ be the respective closure of the restriction of , to Galois sets (stable, or co-stable, according to the distribution type in question), it was shown in [6] that if (and similarly for ⊖) is residuated at some argument place, then its residual is simply the restriction of the residual of to Galois sets. Furthermore, it was also shown in [6] that if the generating relation R has the property that the section of its Galois dual relation $R^{\prime }$ corresponding to the argument place under consideration is a Galois set, then is indeed residuated at that argument place (hence it distributes over any joins at that argument place).

To the structure (frame) $\mathfrak{F}=(A,B,I,R,S)$ with sorts of $R,S$ as above, we may associate a residuated sorted modal logic with residuated modal operators $◇,■$ and sorted polyadic diamonds , of sorts determined by the sorts of the relations. The language $L=(L_1,L_{\partial })$ of sorted, residuated modal logic, given countable, nonempty and disjoint sets of propositional variables, is defined as follows

where $\overline{\theta }=({\theta }_1,\dots ,{\theta }_n)$, the sort of is σ() = $(i_1,\dots ,i_n;1)$ and if $\sigma \left(j\right)=1$, then ${\theta }_j\in L_1$, else ${\theta }_j\in L_{\partial }$. Similarly for , of sort ${\sigma }^{\prime }=(i_1^{\prime },\dots ,i_n^{\prime };\partial )$. Note that $\sigma (■)=(\partial ;1)$ and $\sigma (\square )=(1;\partial )$. The same symbols are used for negation and implication in the two sorts and we rely on context to disambiguate. Diamond operators $◇=\neg \square \neg$ and $◆=\neg ■\neg$ are defined as usual, except that the two occurrences of negation in each definition are of different sort. Sorted box operators , $\boxminus$ are defined accordingly from , , and negation. Disjunction and conjunction for each sort is defined in the classical way. We let $\top ,\perp \in L_1$ and $\mathtt{t},\mathtt{f}\in L_{\partial }$ designate the (definable) truth and falsity constants for each sort. The operators ${}^{\perp }\left(\right),{\left(\right)}^{\perp }$ are defined by ${}^{\perp }\beta =■(\neg \beta )$ and ${\alpha }^{\perp }=\square (\neg \alpha )$.

The weakest sorted normal modal logic is an extension of a KB type of system, where the B-axioms (one for each sort) are $\alpha \to ■◇\alpha$ and, for the second sort, $\beta \to \square ◆\beta$. In addition, it includes normality axioms for the polyadic sorted diamonds , . Note that in the K-axiom (one for each sort) implication on both sorts is involved, witness $\square (\alpha \to \eta )\to (\square \alpha \to \square \eta )$.

Example 3.

The sorted residuated companion modal logic of the logic of a normal lattice expansion is determined by the similarity type of the expansion. As an example, consider FL, the associative full Lambek calculus, with algebraic semantics in residuated lattices $\mathcal{L}=(L,\le ,\wedge ,\vee ,0,1,\leftarrow ,\circ ,\to )$, of similarity type $\tau =\langle (1,1;1),(1,\partial ;\partial ),(\partial ,1;\partial )\rangle$, and where $\delta (\circ )=(1,1;1)$, $\delta (\to )=(1,\partial ;\partial )$ and $\delta (\leftarrow )=(\partial ,1;\partial )$. Its companion modal logic includes three diamond operators , , of respective sort $(\partial ,1;\partial )$, $(1,1;1)$ and $(1,\partial ;\partial )$. An implication operator of the first sort is defined by setting α $\eta$ = ^⊥(α${\eta }^{\perp }$) (reminiscent of the classical definition of implication as $\phi \to \psi$$=\neg (\phi \wedge \neg \psi )$) and similarly for . The two languages are interpreted over the same class of frames, determined by the similarity type at hand. Frames $\mathfrak{F}=(A,I,B,L,F,R)$ include ternary relations of respective sorts $\sigma \left(L\right)=(\partial ;\partial 1)$, $\sigma \left(F\right)=(1;11)$ and $\sigma \left(R\right)=(\partial ;1\partial )$, in other words $L\subseteq B\times (B\times A)$, $F\subseteq A\times (A\times A)$, while $R\subseteq B\times (A\times B)$. Operators are generated on both arbitrary (as classical image operators) and (co)stable subsets (as closures of suitable compositions of the image operators with the Galois connection of the frame), as detailed in [7]. Appropriate frame conditions ensure that residuation obtains, as detailed in [5,29] (see also Remark 1). In this particular case, two of the relations can be dispensed with, as they are definable in terms of the third and the Galois connection (cf [5,29] for details).

Given a sorted frame $\mathfrak{F}=(A,I,B,R,S)$ as above, a model $\mathfrak{M}=(\mathfrak{F},V)$ on the frame $\mathfrak{F}$ is equipped with a sorted valuation function V such that $V\left(P_i\right)\subseteq A$ and $V\left(Q_i\right)\subseteq B$. The sorted modal language is interpreted in the expected way, as in

Table 2. Sorted Interpretation.

$a\vDash P_i$	iff $a\in V\left(P_i\right)$	bQi	iff $b\in V\left(Q_i\right)$
$a\vDash \neg \alpha$	iff a α	b ¬ β	iff b β
$a\vDash \alpha \wedge \eta$	iff $a\vDash \alpha$ and $a\vDash \eta$	b$\beta \wedge \delta$	iff b β and b δ
$a\vDash ■\beta$	iff $\forall b(aIb$ implies b β)	b$\square \alpha$	iff $\forall a(aIb$ implies $a\vDash \alpha )$
a ⊧ ($\overline{\theta }$)	iff $\exists w_1,\dots ,w_n(aR{w}_1\cdots w_n$ and ${\bigwedge }_{j=1}^n(w_j$ θj))
b($\overline{\theta }$)	iff $\exists w_1,\dots ,w_n(bS{w}_1\cdots w_n$ and ${\bigwedge }_{j=1}^n(w_j$ θj))

, where we use $\vDash \subseteq A\times L_1$ and ⊆ $B\times L_{\partial }$ for the two satisfaction relations. Put differently, the satisfaction relation is sorted = (⊧, ). Furthermore, we let ${\left[\right[\alpha \left]\right]}_{\mathfrak{M}}\subseteq A$ and ${\left(\right|\beta \left|\right)}_{\mathfrak{M}}\subseteq B$ be the generated interpretations of the sorted modal formulae of the first and second sort, respectively.

Let $\mathbb{F}$ be a class of sorted frames and $\mathbb{M}$ the class of models $\mathfrak{M}=(\mathfrak{F},V)$ over frames $\mathfrak{F}\in \mathbb{F}$. The following notions have a standard definition just as in the single sorted case. Consult [33] for details.

• $\alpha$ (or $\beta$) is locally true, or satisfiable in $\mathfrak{F}=(A,I,B,R,S)$
• $\alpha$ (or $\beta$) is globally true in $\mathfrak{F}$
• $\alpha$ (or $\beta$) is valid in the class$\mathbb{F}$ of frames
• A set $\Sigma$ of sentences (perhaps of both sorts) defines the class $\mathbb{F}$ of frames.

3.2. Translation Semantics for Logics of Normal Lattice Expansions

In [5], a translation of the language of substructural logics (in the language on the signature $\{\wedge ,\vee ,\top ,\perp ,\leftarrow ,\circ ,\to \}$) was introduced, proven to be full and faithful (fully abstract). To prove a characterization theorem, generalizing the van Benthem result for modal logic, we define a family of translations, parameterized on permutations of natural numbers, given an enumeration of modal sentences of the second sort.

Let ${\beta }_0,{\beta }_1,\dots$ be an enumeration of modal sentences of the second sort and $\pi :\omega \to \omega$ a permutation of natural numbers. The translation ${\mathrm{T}}_{\pi }^{•}$ and co-translation ${\mathrm{T}}_{\pi }^{\circ }$ of sentences of the language ${\Lambda }_{\tau }$ are defined as in

Table 3. Modal translation and co-translation.

${\mathrm{T}}_{\pi }^{•}\left(p_i\right)$	= $■{\beta }_{\pi \left(i\right)}$	${\mathrm{T}}_{\pi }^{\circ }\left(p_i\right)$	= $\square ◆\neg {\beta }_{\pi \left(i\right)}$
${\mathrm{T}}_{\pi }^{•}(\top )$	= ⊤	${\mathrm{T}}_{\pi }^{\circ }(\top )$	= $\square \perp$
${\mathrm{T}}_{\pi }^{•}(\perp )$	= $■\mathtt{f}$	${\mathrm{T}}_{\pi }^{\circ }(\perp )$	= $\mathtt{t}$
${\mathrm{T}}_{\pi }^{•}(\phi \wedge \psi )$	= ${\mathrm{T}}_{\pi }^{•}\left(\phi \right)$ ∧ ${\mathrm{T}}_{\pi }^{•}\left(\psi \right)$	$T_{\pi }^{\circ }(\phi \wedge \psi )$	= $\square (◆{\mathrm{T}}_{\pi }^{\circ }\left(\phi \right)\vee ◆{\mathrm{T}}_{\pi }^{\circ }\left(\psi \right))$
${\mathrm{T}}_{\pi }^{•}(\phi \vee \psi )$	= $■(◇{\mathrm{T}}_{\pi }^{•}\left(\phi \right)\vee ◇{\mathrm{T}}_{\pi }^{•}\left(\psi \right))$	${\mathrm{T}}_{\pi }^{\circ }(\phi \vee \psi )$	= ${\mathrm{T}}_{\pi }^{\circ }\left(\phi \right)\wedge {\mathrm{T}}_{\pi }^{\circ }(\psi )$
${\mathrm{T}}_{\pi }^{•}$$\left(({\phi }_1,\dots ,{\phi }_n)\right)$ = ■◇($\dots ,\underset{i_j=1}{\underbrace{{\mathrm{T}}_{\pi }^{•}\left({\phi }_j\right)}},\dots ,\underset{i_r=\partial }{\underbrace{{\mathrm{T}}_{\pi }^{\circ }\left({\phi }_r\right)}},\dots )$
		${\mathrm{T}}_{\pi }^{\circ }$($({\phi }_1,\dots ,{\phi }_n))$ = $\square \neg {\mathrm{T}}_{\pi }^{•}$($({\phi }_1,\dots ,{\phi }_n))$
${\mathrm{T}}_{\pi }^{•}(\ominus ({\phi }_1,\dots ,{\phi }_n))$ = $■\neg {\mathrm{T}}_{\pi }^{\circ }(\ominus ({\phi }_1,\dots ,{\phi }_n))$
		${\mathrm{T}}_{\pi }^{\circ }(\ominus ({\phi }_1,\dots ,{\phi }_n))$ = □◆$(\dots ,\underset{i_j=1}{\underbrace{{\mathrm{T}}_{\pi }^{•}\left({\phi }_j\right)}},\dots ,\underset{i_r=\partial }{\underbrace{{\mathrm{T}}_{\pi }^{\circ }({\phi }_r)}},\dots )$

.

Example 4.

Let be the language of implicative modal lattices $\mathbb{L}=(L,\le ,\wedge ,\vee ,0,1$, , →). As the distribution types of the operators are δ ()$=(1;1)$ and $\delta (\to )=(1,\partial ;\partial )$, the language of its companion sorted modal logic includes an operator such that for $\alpha \in {\mathcal{L}}_1$, also $\alpha \in {\mathcal{L}}_1$, as well as a sorted operator such that for $\alpha \in {\mathcal{L}}_1,\beta \in {\mathcal{L}}_{\partial }$ we have $(\alpha ,\beta )\in {\mathcal{L}}_{\partial }$.

Then, letting $T_{\pi }^{•}\left(p_0\right)=■{\beta }_{\pi \left(0\right)}=\alpha \in {\mathcal{L}}_1$, $p_0$ translates to $T_{\pi }^{•}$($p_0)=■◇$α.

Letting also $T_{\pi }^{\circ }\left(p_1\right)=\square ◆\neg {\beta }_{\pi \left(1\right)}=\xi \in {\mathcal{L}}_{\partial }$, $p_0\to p_1$ translates to the formula $T_{\pi }^{•}(p_0\to p_1)=■\neg \square ◆$ $(\alpha ,\xi )$ (which, by residuation, is equivalent to $■\neg$$(\alpha ,\xi )$).

Theorem 1.

Let $\mathfrak{F}=(X,I,Y,R,S)$ be a frame (a sorted structure) and $\mathfrak{M}=(\mathfrak{F},V)$ a model of the sorted modal language. For any enumeration ${\beta }_0,{\beta }_1,\dots$ of modal sentences of the second sort and any permutation $\pi :\omega ⟶\omega$ of the natural numbers define a model ${\mathfrak{N}}_{\pi }$ on $\mathfrak{F}$ for the language ${\Lambda }_{\tau }$ by setting $V_{\pi }\left(p_i\right)=■{\left(\right|{\beta }_{\pi \left(i\right)}\left|\right)}_{\mathfrak{M}}$. Then for any sentences $\phi ,\psi$ of ${\Lambda }_{\tau }$

1.

${\left[\right[\phi \left]\right]}_{\mathfrak{N}}={\left[\left[T_{\pi }^{•}\left(\phi \right)\right]\right]}_{\mathfrak{M}}={\left[[■\neg T_{\pi }^{\circ }\left(\phi \right)]\right]}_{\mathfrak{M}}={\left[[■◇T_{\pi }^{•}\left(\phi \right)]\right]}_{\mathfrak{M}}$

2.

${\left(\right|\phi \left|\right)}_{\mathfrak{N}}={\left(\right|T^{\circ }\left(\phi \right)\left|\right)}_{\mathfrak{M}}={\left(\right|\square \neg T^{•}\left(\phi \right)\left|\right)}_{\mathfrak{M}}={\left(\right|\square ◆T^{\circ }\left(\phi \right)\left|\right)}_{\mathfrak{M}}$

3.

$\phi ⊩\psi$ iff $T_{\pi }^{•}\left(\phi \right)\vDash T_{\pi }^{•}\left(\psi \right)$ iff $T_{\pi }^{\circ }\left(\psi \right)$ $T_{\pi }^{\circ }\left(\phi \right)$

Proof. 

The proof is a generalization of the proof given in [5]. In [5] we gave a translation of the language of substructural logics into the language of sorted modal logic and proved it to be fully abstract ([5], Theorem 4.1, Corollary 4.9). The difference with the current translation is that instead of fixing the translation of propositional variables we define a family of translations, parameterized by a permutation $\pi$ on the natural numbers. This is needed in the proof of a van Benthem type correspondence result for propositional logics without distribution (Theorem 5). The second difference is that we treat here arbitrary normal operators, rather than the operators $\leftarrow ,\circ ,\to$ of the language of a substructural logic.

Claim (3) is an immediate consequence of the first two, which we prove simultaneously by structural induction. Note that, for (2), the identities ${\left(\right|\phi \left|\right)}_{\mathfrak{N}}={\left(\right|\square \neg T^{•}\left(\phi \right)\left|\right)}_{\mathfrak{M}}={\left(\right|\square ◆T^{\circ }\left(\phi \right)\left|\right)}_{\mathfrak{M}}$ are easily seen to hold for any $\phi$, given the proof of claim (1), since

• $\left(\right|\phi \left|\right)\mathfrak{N}={\left[\right[\phi \left]\right]}_{\mathfrak{N}}^{〧}=\square (-{\left[\left[T^{•}\left(\phi \right)\right]\right]}_{\mathfrak{M}})={\left(\right|\square \neg T^{•}\left(\phi \right)\left|\right)}_{\mathfrak{M}}$
• $\left(\right|\phi \left|\right)\mathfrak{N}=\square (-{\left[\left[T^{•}\left(\phi \right)\right]\right]}_{\mathfrak{M}})=\square (-{\left[\left[■◇T^{•}\left(\phi \right)\right]\right]}_{\mathfrak{M}})={\left(\right|\square ◆\square \neg T^{•}\left(\phi \right)\left|\right)}_{\mathfrak{M}}$
• $\hspace{1em}\hspace{1em}\hspace{1em}={\left(\right|\square ◆T^{\circ }\left(\phi \right)\left|\right)}_{\mathfrak{M}}$

where $\square$ is the set map interpreting the syntactic operator □ (note the larger font size for the first).

For the induction proof, we separate cases.

(Case $p_i$)

${\left[\left[p_i\right]\right]}_{\mathfrak{N}}=V_{\pi }\left(p_i\right)=\left[[■{\beta }_{\pi \left(i\right)}]\right]=\left[\left[T^{•}\left(p_i\right)\right]\right]$, by definitions. The other two equalities of 1) are a result of residuation and of the fact that, by residuation again, every boxed formula is stable, i.e., $■\beta \equiv ■◇■\beta$. Similarly for (2), using definitions and residuation.

(Case $\top ,\perp ,\wedge ,\vee$)

See [4], or [5], Theorem 4.1.

(Case )

To prove this case, let be the sorted image operator generated by the relation R

let also be the operator on $\left(A\right)$ resulting by composition with the Galois connection and defined on $W_1,\dots ,W_n\subseteq A$ by

and let be obtained as the closure of the restriction of on stable subsets $C_s=■◇C_s\subseteq A$, for $s=1,\dots ,n$, i.e.,

A dual operator ^∂ on co-stable subsets $D_s=\square ◆D_s\subseteq B$ is defined by composition with the Galois connection

In particular, if $C_s={\left[\left[{\phi }_s\right]\right]}_{\mathfrak{N}}$ and $D_s={\left(\right|{\phi }_s\left|\right)}_{\mathfrak{N}}$ we obtain

Computing membership in the sets $(C_1,\dots ,C_n)$ and ^∂$(D_1,\dots ,D_n)$, see [7], Lemma 3.6, for the particular case $C_s={\left[\left[{\phi }_s\right]\right]}_{\mathfrak{N}}$ and $D_s={\left(\right|{\phi }_s\left|\right)}_{\mathfrak{N}}$ we obtain the interpretation of Table 1, i.e., [[${({\phi }_1,\dots ,{\phi }_n)]]}_{\mathfrak{N}}$ = $({\left[\left[{\phi }_1\right]\right]}_{\mathfrak{N}},\dots ,{\left[\left[{\phi }_n\right]\right]}_{\mathfrak{N}})$ and (|${({\phi }_1,\dots ,{\phi }_n)\left|\right)}_{\mathfrak{N}}$ = ^∂$({\left(\right|{\phi }_1\left|\right)}_{\mathfrak{N}},\dots ,{\left(\right|{\phi }_n\left|\right)}_{\mathfrak{N}})$. Comparing with the translation, which was defined in line with the representation results of [7] by

and using the induction hypothesis both claims 1) and 2) follow.

The case for ⊖ is similar to the case for . □

Corollary 1.

A modal formula $\alpha {\in }_1{L}_{\tau }$ is equivalent to a translation $T_{\pi }^{•}\left(\phi \right)$, for some permutation $\pi :\omega \to \omega$, of a formula φ in the language ${\Lambda }_{\tau }$ of normal lattice expansions of similarity type τ iff there is a modal formula $\beta {\in }_{\partial }{L}_{\tau }$ such that $\alpha \equiv ■\beta$ iff α is equivalent to $■◇\alpha$.

Similarly for a formula $\beta {\in }_{\partial }{L}_{\tau }$ and a translation $T_{\pi }^{\circ }\left(\phi \right)$, in which case $\beta \equiv T_{\pi }^{\circ }\left(\phi \right)$ iff $\beta \equiv \square \alpha$, for some $\alpha {\in }_1{L}_{\tau }$ iff $\beta \equiv \square ◆\beta$.

Proof. 

The direction left-to-right follows from Theorem 1. Conversely, every formula $■\beta$ is in the range of a translation $T_{\pi }^{•}$ for some permutation $\pi$. □

Call a modal formula $\alpha$ stable if it is equivalent to $■◇\alpha$, and analogously for co-stable. The stable fragment (analogously for the co-stable fragment) of the sorted modal logic is the fragment of modal formulae that are stable in the above sense. The translation $T^{•}$ maps a sentence of the non-distributive logic into the stable fragment of its companion modal logic. Analogously for the co-translation.

Note that in the statement and proof of Theorem 1 we did not need to place any restrictions on the frame relations $R,S$ and the proof remains valid when restricting to the class of frames where the relations $R,S$ satisfy the section stability requirement of Remark 1.

4. First-Order Languages and Structures

A sorted subset $S=(S_1,S_{\partial })\subseteq Z=(Z_1,Z_{\partial })$ is finite (more generally, of cardinallity $\kappa$) iff both $S_1,S_{\partial }$ are finite (resp. of cardinallity $\kappa$). The sorted membership relation $a{\in }_{1}Z,b{\in }_{\partial }Z$ means that $a\in Z_1=A$ and, respectively, $b\in Z_{\partial }=B$. For a pair $c=(a,b)$, with $a\in A,b\in B$, the statement $c\in Z$ has the obvious intended meaning. A sorted function $h:Z⟶Z^{\prime }$ is a pair of functions $h_1:Z_1⟶Z_1^{\prime },h_{\partial }:Z_{\partial }⟶Z_{\partial }^{\prime }$. A sorted $(n+1)$-ary relation is a subset $R\subseteq Z_{i_{n+1}}\times {\prod }_{j=1}^n{Z}_{i_j}$, where for each j, $i_j\in \{1,\partial \}$. The tuple $\sigma =(i_{n+1};i_1\cdots i_n)\in {\{1,\partial \}}^{n+1}$ is referred to as the sort of R and $i_{n+1}\in \{1,\partial \}$ as its output type. For an $(n+1)$-ary relation we typically use the notation $uR{v}_1\cdots v_n$ and sometimes, for notational transparency, $uR(v_1,\dots ,v_n)$.

Consider a structure $\mathfrak{F}=(A,B,R)$, where $Z_1=A,Z_{\partial }=B$ are the sort sets and R is a relation of some sort $\sigma =(i_{n+1};i_1\cdots i_n)$. Nothing of significance for our current purposes changes if we consider expansions $(A,B,{\left(R_s\right)}_{s\in S})$ with a tuple of relations $R_s$, with s in some index set S, each of some sort ${\sigma }_s$. In particular, what we say below applies to the structures of our current interest, of the form $(A,I,B,R,S)$.

The sorted first-order language with equality ${\mathcal{L}}_s^1(V_1,V_{\partial },\mathbf{R},{=}_1,{=}_{\partial })$ of a structure $\mathfrak{F}=(A,B,R)$, for some $(n+1)$-ary sorted relation, is built on a countable sorted set $(V_1,V_{\partial })$ of individual variables $v_0^1,v_1^1,\dots$ and $v_0^{\partial },v_1^{\partial },\dots$, respectively, and an $(n+1)$-ary sorted predicate R of some sort $\sigma =(i_{n+1};i_1\cdots i_n)$. As customary in a first-order context, we also refer to the structure $\mathfrak{F}$ as a model of the language ${\mathcal{L}}_s^1$. Well-formed (meaning also well-sorted) formulae are built from atomic formulae $v_r^1{=}_1{v}_t^1$, $v_n^{\partial }{=}_{\partial }{v}_m^{\partial }$ and $\mathbf{R}(v_{r_{n+1}}^{i_{n+1}},v_{r_1}^{i_1},\dots ,v_{r_n}^{i_n})$ using negation, conjunction and sorted quantification ${\forall }^1{v}_r^1\Phi$, ${\forall }^{\partial }{v}_t^{\partial }\Psi$. We typically simplify notation and write ${\forall }^{1}v\Phi$, ${\exists }^{\partial }v\Phi$ etc, with an understanding and assumption of well-sortedness. We assume the usual definition of other logical operators ($\vee ,\to ,{\exists }^1,{\exists }^{\partial }$) and of free and bound (occurrences) of a variable, as well as that of a closed formula (sentence), and we follow the usual convention about the meaning of displaying variables in a formula, as in $\Phi (v_0^1,v_1^{\partial })$.

Given a sorted valuation V of individual variables, $\mathfrak{F}{\vDash }_s\Phi \left[V\right]$ is defined as in the case of unsorted FOL. When $V\left(u_k^1\right)=a\in A=Z_1$, we may also display the assignment in writing $\mathfrak{F}{\vDash }_s\Phi \left(u_k^1\right)[u_k^1:=a]$, or just $\mathfrak{F}{\vDash }_s\Phi \left(u_k^1\right)\left[a\right]$, and similarly for more variables occurring free in $\Phi$. A formula $\Phi$ in n free variables is also referred to as an n-ary type. A valuation Vrealizes the type $\Phi$ in the structure $\mathfrak{F}$ iff V satisfies $\Phi$, $\mathfrak{F}{\vDash }_s\Phi \left[V\right]$. A structure $\mathfrak{F}$ realizes $\Phi$ iff some valuation V does (iff $\Phi$ is satisfiable in $\mathfrak{F}$), otherwise $\mathfrak{F}$ omits the type. Similarly for a set $\Sigma$ of n-ary types, which will itself, too, be referred to as an n-ary type.

An ${\mathcal{L}}_s^1$-theory T is a set of ${\mathcal{L}}_s^1$-sentences and a complete theory is a theory whose set of consequences $\{\Phi |T{\vDash }_s\Phi \}$ is maximal consistent. The (complete) ${\mathcal{L}}_s^1$-theory of a structure is designated by ${\mathtt{Th}}_s\left(\mathfrak{F}\right)$. If $C=(C_1,C_{\partial })\subseteq (A,B)$ is a sorted subset, the expansion ${\mathcal{L}}_s^1\left[C\right]$ of the language includes sorted constants $c^1\in C_1,c^{\partial }\in C_{\partial }$, for each member of $C_1,C_{\partial }$. We sometimes simplify notation writing $c_a,c_b$ for the constants naming the elements $a\in A,b\in B$. It is assumed, as usual, that a constant is interpreted as the element that it names. The extended structure interpreting the expanded signature of the language is designated by ${(\mathfrak{F},c)}_{c\in C}$, or just ${\mathfrak{F}}_C$.

For a similarity type $\tau$, ${\mathcal{L}}_{s,\tau }^1$ is the sorted first-order language that includes a predicate of sort $\sigma$, for each $\sigma$ in $\tau$, together with a distinguished binary predicate I of sort $(1;\partial )$.

To a structure $\mathfrak{F}=(A,B,R)$ we may also associate an unsorted (single-sorted) first-order language with equality${\mathcal{L}}^1(V^{\prime },{\mathbf{U}}_1,{\mathbf{U}}_{\partial },\mathbf{R},=)$ where the interpretation of ${\mathbf{U}}_1,{\mathbf{U}}_{\partial }$ is, respectively, $Z_1=A,Z_{\partial }=B$ and $V^{\prime }=V_1\cup V_{\partial }$. Assuming the sort of R is $\sigma =(i_{n+1};i_1\cdots i_n)$, the structure validates all sentences pertaining to sorting constraints, which are of the following form, with $i_r\in \{1,\partial \}$, for each r.

$$\begin{array}{c}\hfill \forall v_1\cdots \forall v_{n+1}(\mathbf{R}(v_{n+1},v_1,\dots ,v_n)⟶\underset{r=1}{\overset{n+1}{\bigwedge }}{\mathbf{U}}_{i_r}(v_r))\end{array}$$

(4)

$$\begin{array}{c}\hfill \forall v_1\forall v_2(v_1=v_2⟶(({\mathbf{U}}_1\left(v_1\right)\wedge {\mathbf{U}}_1\left(v_2\right))\vee ({\mathbf{U}}_{\partial }\left(v_1\right)\wedge {\mathbf{U}}_{\partial }\left(v_2\right)))\end{array}$$

(5)

In particular, (5) implies the sentence $\forall v({\mathbf{U}}_1\left(v\right)\vee {\mathbf{U}}_{\partial }\left(v\right))$. The (unsorted) ${\mathcal{L}}^1$-theory of $\mathfrak{F}$ will be designated by $\mathtt{Th}\left(\mathfrak{F}\right)$.

By sort-reduction (for details cf. [8], ch. 4), the language ${\mathcal{L}}_s^1$ can be translated into ${\mathcal{L}}^1$, by relativising quantifiers (where $i_r\in \{1,\partial \}$)

$$\mathrm{\Psi }={\forall }^{i_r}{u}_k^{i_r}\Phi ⤇{\mathrm{\Psi }}^{*}=\forall u_k^{i_r}({\mathbf{U}}_{i_r}\left(u_k^{i_r}\right)⟶{\Phi }^{*})$$

and replacing ${=}_1,{=}_{\partial }$ by a single equality predicate =. For later use we list the following result.

Theorem 2

(Enderton [8], ch. 4.3).

1.

(Sort-reduction) If ${\Phi }^{*}$ is the sort-reduct of Φ and V a valuation of variables, then $\mathfrak{F}{\vDash }_s\Phi \left[V\right]$ iff $\mathfrak{F}\vDash {\Phi }^{*}\left[V\right]$.

2.

(Compactness) If every finite subset of a set Σ of many-sorted sentences in ${\mathcal{L}}_s^1$ has a model, then Σ has a model. □

4.1. Standard Translation of Sorted Modal Logic

The standard translation of sorted modal logic into sorted FOL is exactly as in the single-sorted case, except for the relativization to two sorts, displayed in

Table 4. Standard Translation of the sorted modal language.

$\mathrm{S}{\mathrm{T}}_u\left({\mathbf{P}}_i\right)$	= ${\mathbf{P}}_i\left(u\right)$
$S{T}_u(\neg \alpha )$	= $\neg S{T}_u\left(\alpha \right)$
$S{T}_u(\alpha \wedge {\alpha }^{\prime })$	= $S{T}_u\left(\alpha \right)\wedge S{T}_u\left({\alpha }^{\prime }\right)$
$S{T}_u(■\beta )$	= $\forall v(\mathbf{I}(u,v)⟶S{T}_v\left(\beta \right))$
$S{T}_u$($\left(\overline{\theta }\right))$	= $\exists \overline{u}(\mathbf{R}(u,\overline{u})\wedge {\bigwedge }_{j=1,\cdots ,n}^{i_j=1}S{T}_{u_j}\left({\theta }_j\right)\wedge {\bigwedge }_{r=1,\cdots ,n}^{i_r=\partial }S{T}_{u_r}\left({\theta }_r\right))$
$S{T}_v\left(Q_i\right)$	= ${\mathbf{Q}}_i\left(v\right)$
$S{T}_v(\neg \beta )$	= $\neg S{T}_v\left(\beta \right)$
$S{T}_v(\beta \wedge {\beta }^{\prime })$	= $S{T}_v\left(\beta \right)\wedge S{T}_v\left({\beta }^{\prime }\right)$
$S{T}_v(\square \alpha )$	= $\forall u(\mathbf{I}(u,v)⟶S{T}_u\left(\alpha \right))$
$S{T}_v$($\left(\overline{{\theta }^{\prime }}\right)$	= $\exists \overline{v}(\mathbf{S}(v,\overline{v})\wedge {\bigwedge }_{j=1,\cdots ,m}^{i_j=1}S{T}_{v_j}\left({\theta }_j\right)\wedge {\bigwedge }_{r=1,\cdots ,m}^{i_r=\partial }S{T}_{v_r}\left({\theta }_r\right))$

, where $S{T}_u\left(\right),S{T}_v\left(\right)$ are defined by mutual recursion and $u,v$ are individual variables of sort $1,\partial$, respectively.

Proposition 1.

For any sorted modal formulae $\alpha ,\beta$ (of sort $1,\partial$, respectively), for any model $\mathfrak{F}=(A,I,B,R,S)$ of ${\mathcal{L}}_s^1$ and for any $a\in A,b\in B$, $\mathfrak{F},a\vDash \alpha$ iff $\mathfrak{F}\vDash S{T}_u\left(\alpha \right)[u:=a]$ and $\mathfrak{F}$, b β iff $\mathfrak{F}\vDash S{T}_v\left(\beta \right)[v:=b]$.

Proof. 

Straightforward. □

We next review and adapt to the sorted case the basics on ultraproducts and ultrapowers that will be needed in the sequel. Consult [34,35] for background and details.

4.2. Sorted Ultraproducts

Let ${\left({\mathfrak{F}}_j\right)}_{j\in J}={(A_j,B_j,R_j)}_{j\in J}$, with J some index set, be a family of structures with sorted relations $R_j$ of some fixed sort $\sigma$.

An ultrafilter over J is an ultrafilter (maximal filter) U of the powerset Boolean algebra $\left(J\right)$. Let ${\prod }_U{A}_j,{\prod }_U{B}_j$ be the ultraproducts of the families of sets ${\left(A_j\right)}_{j\in J},{\left(B_j\right)}_{j\in J}$ over the ultrafilter U. Members of ${\prod }_U{A}_j$ are equivalence classes $f_U$ of functions $f\in {\prod }_{j\in J}{A}_j$ (i.e., functions $f:J⟶{\bigcup }_j{A}_j$ such that for all $j\in J,f\left(j\right)\in A_j$) under the equivalence relation $f{\sim }_{U}g$ iff $\{j\in J|f\left(j\right)=g\left(j\right)\}\in U$.

Ultraproducts for polarities (sorted structures with a binary relation) have been recently also considered in [3,36,37], slightly generalizing the classical construction. We review the definition, adapting to the case of an arbitrary $(n+1)$-ary relation.

Definition 3

(Ultraproducts of Sorted Structures). Given a family ${\left({\mathfrak{F}}_j\right)}_{j\in J}$ of structures (models) with J some index set, theirultraproduct${\prod }_U{\mathfrak{F}}_j=({\prod }_U{A}_j,{\prod }_U{B}_j,R_U)$ is the sorted structure where

1.

${\prod }_U{A}_j,{\prod }_U{B}_j$ are the ultraproducts over U of the families of sets ${\left(A_j\right)}_{j\in J}$, ${\left(B_j\right)}_{j\in J}$.

2.

Where the sort of $R_j$ for all $j\in J$ is $\sigma =(i_{n+1};i_1,\dots ,i_n)$ and for each $r\in \{1,\dots ,n,n+1\}$ we have $h_{r,U}\in {\prod }_U{A}_j$, if $i_r=1$, and $h_{r,U}\in {\prod }_U{B}_j$ if $i_r=\partial$, the relation $R_U$, of sort σ is defined by setting

$$h_{n+1,U}{R}_U(h_{1,U},\dots ,h_{n,U})iff\{j\in J|h_{n+1}\left(j\right)R_j(h_1\left(j\right),\dots ,h_n\left(j\right))\}\in U$$

(6)

If for all $j\in J$, ${\mathfrak{F}}_j=\mathfrak{F}$, then the ultraproduct is referred to as the ultrapower ${\prod }_U\mathfrak{F}$ of $\mathfrak{F}$ over the ultrafilter U. □

Considering the structures ${\mathfrak{F}}_j$ as ${\mathfrak{L}}^1$-structures, by the fundamental theorem of ultraproducts (Łos’s theorem) we have

$$\begin{array}{c}\hfill {\prod }_U{\mathfrak{F}}_j\vDash \Phi [f_{1,U},\dots ,f_{t,U}]iff\{j\in J|{\mathfrak{F}}_j\vDash \Phi [f_1\left(j\right),\dots ,f_t\left(j\right)]\}\in U\end{array}$$

(7)

By sort reduction, Łos’s theorem holds when the ${\mathfrak{F}}_j$ are regarded as models of the sorted language (as ${\mathcal{L}}_s^1$-structures), as well. Indeed

• ${\prod }_U{\mathfrak{F}}_j{\vDash }_s\Phi [f_{1,U},\dots f_{t,U}]\mathrm{iff}{\prod }_U{\mathfrak{F}}_j\vDash {\Phi }^{*}[f_{1,U},\dots ,f_{t,U}]$
• $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{iff}\{j\in J|{\mathfrak{F}}_j\vDash {\Phi }^{*}[f_1\left(j\right),\dots ,f_t\left(j\right)]\}\in U$
• $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}iff\{j\in J|{\mathfrak{F}}_j{\vDash }_s\Phi [f_1\left(j\right),\dots ,f_t\left(j\right)]\}\in U$

We use the standard notation $\mathfrak{F}\equiv \mathfrak{G}$ for elementarily equivalent structures (satisfying the same set of sentences) and $\mathfrak{F}\prec \mathfrak{G}$ to designate the fact that $\mathfrak{G}$ is an elementary extension of $\mathfrak{F}$, meaning that $\mathfrak{F}\subset \mathfrak{G}$ ($\mathfrak{F}$ is a substructure of $\mathfrak{G}$) and for any n-ary type $\Phi (w_{i_1},\dots ,w_{i_n})$ of some sort $(i_1,\dots ,i_n)\in {\{1,\partial \}}^n$ and any valuation V for $\mathfrak{F}$ we have $\mathfrak{F}\vDash \Phi \left[V\right]$ iff $\mathfrak{G}\vDash \Phi \left[V\right]$. Finally, we recall that a map $h:\mathfrak{F}⪯\mathfrak{G}$ is an elementary embedding iff for any n-ary type $\Phi$ as above we have $\mathfrak{F}\vDash \Phi \left(\overline{w_{i_j}}\right)\left[V\right]$ iff $\mathfrak{G}\vDash \Phi \left(\overline{w_{i_j}}\right)[h\circ V]$.

The same argument as above, appealing to sort-reduction, applies to lift to the sorted case well-known consequences of Łos’s theorem (in particular, Corollary 4.1.13 of [34], restated for the sorted case below).

Corollary 2.

If $\mathfrak{F}$ is an ${\mathcal{L}}_s^1$-structure, J an index set and U an ultrafilter over J, then $\mathfrak{F}$ and the ultrapower ${\prod }_U\mathfrak{F}$ are elementarily equivalent, $\mathfrak{F}\equiv {\prod }_U\mathfrak{F}$. Furthermore, the embedding $e=(e_1:Z_1\to {\prod }_U{Z}_1,e_{\partial }:Z_{\partial }\to {\prod }_U{Z}_{\partial })$ sending elements $a\in Z_1=A$, $b\in Z_{\partial }=B$ to the respective equivalence classes $e\left(a\right)=e_1\left(a\right)=f_{a,U}$, $e\left(b\right)=e_{\partial }\left(b\right)=f_{b,U}$ of the constant functions $f_a\left(j\right)=a$, $f_b\left(j\right)=b$, for all $j\in J$, is an elementary embedding $e:\mathfrak{F}\prec {\prod }_U\mathfrak{F}$.

Sketch of Proof.

By appealing to sort-reduction (cf Theorem 2). In fact, a direct argument for the sorted case is literally the same as in the unsorted case, as seen by consulting for example the proof in [35] [Lemma 5.2.3]. □

Note, in particular, that for a unary type $\Phi \left(u^1\right)\in {\mathcal{L}}_s^1$ and any element say $a\in A$ (i.e., a valuation V such that $V\left(u^1\right)=a\in A$) we have (dropping the sorting superscript on the variable) the following

Corollary 3.

For $u\in V_1$, $\mathfrak{F}{\vDash }_s\Phi \left(u\right)[u:=a]$ iff ${\prod }_U\mathfrak{F}{\vDash }_s\Phi \left(u\right)[u:=f_{a,U}]$. The same holds for a type with a free variable $v\in V_{\partial }$.

Proof. 

• ${\prod }_U\mathfrak{F}{\vDash }_s\Phi \left(u\right)[u:=f_{a,U}]\hspace{1em}\mathrm{iff}\hspace{1em}{\prod }_U\mathfrak{F}{\vDash }_s{\Phi }^{*}\left(u\right)[u:=f_{a,U}]\hspace{1em}$(by sort-reduction)
• $\hspace{1em}\hspace{1em}\mathrm{iff}\hspace{1em}\left\{j\right|\mathfrak{F}\vDash {\Phi }^{*}\left(u\right)[u:=f_{a\left(j\right)}]\}\in U\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}$(by Łos’s theorem)
• $\hspace{1em}\hspace{1em}\mathrm{iff}\hspace{1em}\left\{j\right|\mathfrak{F}\vDash {\Phi }^{*}\left(u\right)[u:=a]\}\in U\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}$(∀j f_a(j) = a)
• $\hspace{1em}\hspace{1em}\mathrm{iff}\hspace{1em}\mathfrak{F}\vDash {\Phi }^{*}\left(u\right)[u:=a]$ (U is a filter, so {j|$\mathfrak{F}\vDash {\Phi }^{*}\left(u\right)[u:=a]$} ≠ ∅)
• $\hspace{1em}\hspace{1em}\mathrm{iff}\hspace{1em}\mathfrak{F}{\vDash }_s\Phi \left(u\right)[u:=a]$ (by sort-reduction)

and this proves the claim. □

Definition 4

(Ultrapowers of Models). If $\mathfrak{M}=(\mathfrak{F},V)$ is a model and U is an ultrafilter over an index set J, the ultrapower of $\mathfrak{M}$ is defined by ${\prod }_U\mathfrak{M}=({\prod }_U\mathfrak{F},V_U)$ where $V_U\left(u\right)=f_{a,U}$ iff $V\left(u\right)=a$.

4.3. Saturated Structures

Let $\mathfrak{F}=\langle A,B,R\rangle$ be an ${\mathcal{L}}_s^1$-structure. The structure $\mathfrak{F}$ is called ω-saturated iff for any finite subset $C\subseteq A\cup B$, every unary type $\Sigma$ of the expanded language ${\mathcal{L}}_s^1\left[C\right]$ that is consistent with the theory ${\mathtt{Th}}_s{(\mathfrak{F},c)}_{c\in C}$ is realized in ${(\mathfrak{F},c)}_{c\in C}$. If reference to sorting is disregarded, this is precisely the meaning of $\omega$-saturated structures for (unsorted) first-order languages. The definition generalizes to $\kappa$-saturated structures, for any cardinal $\kappa$, but we shall only have use of $\omega$-saturated structures in the sequel.

The following fact (stated in a more general form in [34] [Chapter 5]) will be useful in the sequel. It is this fact that needs to be used also in the proof of the characterization result in the case of classical modal logic but with polyadic, rather than unary only modalities.

Proposition 2.

A structure $\mathfrak{F}$ is ω-saturated iff for every $n\in \omega$ and every $C=\{u_1,\dots ,u_n\}\subseteq A\cup B$, the extended structure ${\mathfrak{F}}_C$ realizes every m-ary type $\Phi (x_1,\dots ,x_m)$ in the expanded language ${\mathcal{L}}_s^1\left[C\right]$ that is consistent with ${\mathtt{Th}}_s\left({\mathfrak{F}}_C\right)$.

Proof. 

The claim is a special case of claim (ii) in [34] [Proposition 5.1.1] which more generally refers to $\alpha$-saturated structures and extensions ${(\mathfrak{F},a_{\eta })}_{\eta \in \xi }$ obtained by any $\xi$-sequence of elements of the universe of the structure, for $\xi <\alpha$. □

$\omega$-saturated first-order (unsorted) structures can be constructed as unions of elementary chains, or as ultrapowers. Consult Bell and Slomson [35] [Theorems 11.1.7 and 11.2.1], or Chang and Keisler [34] [Chapter 5], for details. Any two elementarily equivalent $\kappa$-saturated structures (models) of the same cardinality are isomorphic ([34] [Theorem 5.1.13], [35] [Theorem 11.3.1]), so we only discuss ultrapowers. With some necessary adaptation, the original arguments for the unsorted case (for the existence of $\omega$-saturated extensions) can be reproduced for the sorted case. It is easier, however, to derive the result for the sorted case by reducing the problem to the unsorted case, using sort-reduction, as we do below.

Theorem 3.

Every ${\mathcal{L}}_s^1$-structure $\mathfrak{F}$ has an elementary ω-saturated extension $h:\mathfrak{F}⪯{\mathfrak{F}}^{♭}$.

Proof. 

By standard model-theoretic results ([34], Proposition 5.1.1, Theorem 6.1.1), for every first-order structure (${\mathcal{L}}^1$-structure) $\mathfrak{F}$ and any countably incomplete ultrafilter1 U over some index set J, its ultrapower ${\prod }_U\mathfrak{F}$ is an elementary $\omega$-saturated extension of $\mathfrak{F}$, $e:\mathfrak{F}\prec {\prod }_U\mathfrak{F}$, by the embedding of (the unsorted version of) Corollary 2 (see [34], Corollary 4.1.13).

Let $\mathfrak{F}=(A,B,R)$ be a sorted first-order structure, $C\subseteq A\cup B$ and $\Sigma \left(v\right)$, with $v\in V_1\cup V_{\partial }$, a unary type in the expanded language ${\mathcal{L}}_s^1\left[C\right]$ consistent with ${\mathtt{Th}}_s\left({\mathfrak{F}}_C\right)$. We claim that the sort reduct ${\Sigma }^{*}=\left\{{\Phi }^{*}\left(v\right)\right|\Phi \left(v\right)\in \Sigma \}$ is consistent with the (unsorted) theory $\mathtt{Th}\left({\mathfrak{F}}_C\right)$. Assuming for the moment that the claim is proved, by $\omega$-saturation of the ultrapower of the ${\mathcal{L}}^1$-structure $\mathfrak{F}$, ${\prod }_U{\mathfrak{F}}_C\vDash {\Sigma }^{*}\left[S\right]$ and then by sort reduction ${\prod }_U{\mathfrak{F}}_C{\vDash }_s\Sigma \left[S\right]$, i.e., the type $\Sigma$ in the sorted language ${\mathcal{L}}_s^1\left[C\right]$ is realized in ${\prod }_U{\mathfrak{F}}_C$ by some valuation S. Hence ${\prod }_U\mathfrak{F}$, regarded as an ${\mathcal{L}}_s^1$-structure, is an elementary $\omega$-saturated extension of $\mathfrak{F}$.

To prove the claim we made in the course of the above argument, recall that we assume that $\Sigma \left(v\right)$ is consistent with ${\mathtt{Th}}_s\left({\mathfrak{F}}_C\right)$, so that a structure $\mathfrak{N}$ and a valuation $V_N$ exist such that $V_N$ satisfies in $\mathfrak{N}$ every formula in $\Sigma \left(v\right)$ and sentence in ${\mathtt{Th}}_s\left({\mathfrak{F}}_C\right)$.

If ${\Sigma }^{*}$ is not consistent with the theory $\mathtt{Th}\left({\mathfrak{F}}_C\right)$, let $\Phi \in {\mathcal{L}}^1\left[C\right]$ be such that both $\Phi$ and $\neg \Phi$ are derivable from $\Sigma \left(v\right)\cup \mathtt{Th}\left({\mathfrak{F}}_C\right)$. By compactness, let ${\Phi }_1^{*}\left(v\right),\dots ,{\Phi }_n^{*}\left(v\right)\in {\Sigma }^{*}$ and ${\Theta }_1,\dots ,{\Theta }_k$ be sentences in $\mathtt{Th}\left({\mathfrak{F}}_C\right)$ such that ${\Phi }_1^{*}\left(v\right),\dots ,{\Phi }_n^{*}\left(v\right),{\Theta }_1,\dots ,{\Theta }_k⊢\Phi \wedge \neg \Phi$.

Since $\mathtt{Th}\left({\mathfrak{F}}_C\right)$ is a complete theory we may assume that $\Phi \in \mathtt{Th}\left({\mathfrak{F}}_C\right)$. It then follows that ${\Phi }_1^{*}\left(v\right),\dots ,{\Phi }_n^{*}\left(v\right),{\Theta }_1,\dots ,{\Theta }_k⊢\neg \Phi$ and then $\Phi ,{\Phi }_2^{*}\left(v\right),\dots ,{\Phi }_n^{*}\left(v\right),{\Theta }_1,\dots ,{\Theta }_k⊢\neg {\Phi }_1^{*}\left(v\right)$. Since $\mathfrak{N}$ with the valuation $V_N$ satisfies each of the formulas on the left of the turnstile it follows that $\mathfrak{N}\vDash \neg {\Phi }_1^{*}\left(v\right)\left[V_N\right]$. By sort-reduction ([8], Lemma 4.3A), $\mathfrak{N}{\vDash }_s\neg {\Phi }_1\left(v\right)\left[V_N\right]$, contradiction. □

5. Bisimulations and van Benthem Characterization

5.1. Bisimulations

Definition 5

(Sorted Bisimulation). Let $\mathfrak{F}=(A,I,B,R,S)$, ${\mathfrak{F}}^{\prime }=(A^{\prime },I^{\prime },B^{\prime },R^{\prime },S^{\prime })$ be frames (structures, or models of ${\mathcal{L}}_s^1$), where recall that the sorts of $R,S$ are $(1;i_1,\dots ,i_n)$ and $(\partial ;i_1^{\prime },\dots ,i_m^{\prime })$, respectively, and assume that $\precsim \subseteq Z\times Z^{\prime }$ (where $Z=A\cup B$ and $Z^{\prime }=A^{\prime }\cup B^{\prime }$) is a well-sorted relation (i.e. for $a\in A$, $b\in B$, the set $a^{\precsim }=\left\{a^{\prime }\right|a\precsim a^{\prime }\}$ is a subset of $A^{\prime }$ and similarly $b^{\precsim }\subseteq B^{\prime }$). Then the relation ≾ is asimulation, in symbols $\mathfrak{F}\precsim {\mathfrak{F}}^{\prime }$, iff

1.

If $a\precsim a^{\prime }$ then

-

if $aIb$, then $b\precsim b^{\prime }$ for some $b^{\prime }\in B^{\prime }$ such that $a^{\prime }{I}^{\prime }{b}^{\prime }$

-

if $aR{u}_1\cdots u_n$, then $a^{\prime }{R}^{\prime }{u}_1^{\prime }\cdots u_n^{\prime }$ for some $u_j^{\prime }$ such that $u_j\precsim u_j^{\prime }$

2.

If $b\precsim b^{\prime }$ then

-

if $aIb$, then $a\precsim a^{\prime }$ for some $a^{\prime }\in A^{\prime }$ such that $a^{\prime }{I}^{\prime }{b}^{\prime }$

-

if $bS{v}_1\cdots v_m$, then $b^{\prime }{S}^{\prime }{v}_1^{\prime }\cdots v_m^{\prime }$ for some $v_j^{\prime }$ such that $v_j\precsim v_j^{\prime }$

A relation ≾ is a simulation of models $\mathfrak{M}=(\mathfrak{F},V),{\mathfrak{M}}^{\prime }=({\mathfrak{F}}^{\prime },V^{\prime })$ iff

1.

for any propositional variable $P_i$ of the first sort, if $a\in V\left(P_i\right)$ and $a\precsim a^{\prime }$, then $a^{\prime }\in V^{\prime }\left(P_i\right)$

2.

for any propositional variable $Q_i$ of the second sort, if $b\in V\left(Q_i\right)$ and $b\precsim b^{\prime }$, then $b^{\prime }\in V^{\prime }\left(Q_i\right)$

3.

$\mathfrak{F}\precsim {\mathfrak{F}}^{\prime }$

A relation ≾ is a bisimulation if both ≾ and its inverse ${\precsim }^{-1}$ are simulations. We use the notation ∼ for bisimulations. If ∼ is a bisimulation for the frames, we write $\mathfrak{F}\sim {\mathfrak{F}}^{\prime }$. If ∼ is a bisimulation for the models, we write $\mathfrak{M}\sim {\mathfrak{M}}^{\prime }$ and we use $\mathfrak{M},w\sim {\mathfrak{M}}^{\prime },w^{\prime }$ when $w\sim w^{\prime }$ are points of either (but the same) sort. □

Remark 2.

Note that if we collapse the two sorts to one, $A=B$ and set I to be the identity relation, then we have a classical Kripke frame and, in addition, the notion of bisimulation of Definition 5 reduces to the classical one (cf. [33] [Section 2.2]) used in the proof of the van Benthem theorem for modal logic (cf. [33] [Theorem 2.65]).

Proposition 3.

Sorted modal formulas are invariant under bisimulation. In other words, if $\mathfrak{M},a\sim {\mathfrak{M}}^{\prime },a^{\prime }$ (resp. $\mathfrak{M},b\sim {\mathfrak{M}}^{\prime },b^{\prime }$), then $\mathfrak{M}\vDash \alpha \left(u\right)[u:=a]$ iff ${\mathfrak{M}}^{\prime }\vDash \alpha \left(u\right)[u:=a^{\prime }]$ (resp. $\mathfrak{M}$$\beta \left(v\right)[v:=b] \mathit{iff} {\mathfrak{M}}^{\prime }$$\beta \left(v\right)[v:=b^{\prime }]$).

Proof. 

By structural induction, observing that the argument for the base case is built into the definition of bisimulations, while for negations and implications it reduces to that for the subsentences and the induction hypothesis is used, while for modal operators the corresponding clauses in the definition of bisimulations allow directly the use of the inductive hypothesis on subsentences. □

For models $\mathfrak{M},{\mathfrak{M}}^{\prime }$, points $a\in A,a^{\prime }\in A^{\prime }$, respectively are modally equivalent iff for any $\alpha {\in }_1{L}_{\tau }$, $\mathfrak{M},a{\vDash }_s\alpha$ iff ${\mathfrak{M}}^{\prime },a^{\prime }{\vDash }_s\alpha$. In symbols $\mathfrak{M},a\stackrel{\square }{\leftrightsquigarrow }{\mathfrak{M}}^{\prime },a^{\prime }$. Similarly for points $b\in B,b^{\prime }\in B^{\prime }$.

Proposition 4.

Assume $\mathfrak{M},a\stackrel{\square }{\leftrightsquigarrow }{\mathfrak{M}}^{\prime },a^{\prime }$ and that U is a countably incomplete ultrafilter over some index set J. Then ${\prod }_U\mathfrak{M},f_{a,U}\stackrel{\square }{\leftrightsquigarrow }{\prod }_U{\mathfrak{M}}^{\prime },f_{a^{\prime },U}$ and the relation of modal equivalence on the ultrapowers is a bisimulation.

Proof. 

By Proposition 1 we have $\mathfrak{M},a\stackrel{\square }{\leftrightsquigarrow }{\prod }_U\mathfrak{M},f_{a,U}$ and so the first hypothesis implies that ${\prod }_U\mathfrak{M},f_{a,U}\stackrel{\square }{\leftrightsquigarrow }{\prod }_U{\mathfrak{M}}^{\prime },f_{a^{\prime },U}$.

From the second hypothesis and Theorem 3 it is obtained that ${\prod }_U\mathfrak{M}$, ${\prod }_U{\mathfrak{M}}^{\prime }$ are $\omega$-saturated and the claim is that this implies that modal equivalence is a bisimulation. The proof of this claim involves a notion of modally saturated models and is given in two steps. The first is to prove that in the class of modally saturated models the relation of modal equivalence of models is a bisimulation and the second is to prove that every $\omega$-saturated model is modally saturated.

The proofs of these claims are essentially the same as the corresponding proofs in the unsorted case (cf. [33], ch. 2, Proposition 2.54 and Theorem 2.65). For the reader’s convenience and for explicitness, we provide below the suitably adapted definition of modally saturated models (Definition 6) and sketch the proofs of the claims (Lemma 1, Lemma 2). □

Definition 6

(Modally Saturated Structures). Let $\mathfrak{F}=(A,I,B,R,S)$ be a frame and $\mathfrak{M}=(\mathfrak{F},V)$ a model on $\mathfrak{F}$. Let $\in \{◇,◆,$, }, with sort σ()$=(i_1,\dots ,i_n;i_{n+1})$ (where n depends on the arity of the particular diamond operator considered). Respectively, let $T\in \{I,I^{-1},R,S\}$ be the frame relation whose sort $\sigma \left(T\right)=(i_{n+1};i_1,\dots ,i_n)$ matches the sort of . Let w be any element of A, if $i_{n+1}=1$, and any element of B, otherwise. Let ${\Sigma }_1,\dots ,{\Sigma }_j,\dots ,{\Sigma }_n$ be any sequence of sets of sentences in the language of the sorted modal logic defined in Section 3 such that for $j=1$ to n, if $i_j=1$ then ${\Sigma }_j\subseteq {\mathcal{L}}_1$ and ${\Sigma }_j\subseteq {\mathcal{L}}_{\partial }$ otherwise.

If whenever (A) below holds, it is also the case that (B) below holds, then we say that the model $\mathfrak{M}$ ismodally saturated.

(A)

(Finite Satisfiability) For every sequence ${\left(F_j\right)}_{j=1}^n$ of finite subsets $F_j\subseteq {\Sigma }_j$, there are points $v_1,\dots ,v_n$ with $v_r\in A$ if $i_r=1$ and $v_r\in B$ otherwise such that $wT{v}_1\cdots v_n$ and $\mathfrak{M},v_r$$F_r$, for each $r=1,\dots ,n$.

(B)

There are points $v_1,\dots ,v_n$ with $v_r\in A$ if $i_r=1$ and $v_r\in B$ otherwise such that $wT{v}_1\cdots v_n$ and $\mathfrak{M},v_r$${\Sigma }_r$, for each $r=1,\dots ,n$.

Lemma 1.

In the class of modally saturated models, the relation of modal equivalence of models is a bisimulation.

Proof. 

Assume the hypothesis and suppose that $\mathfrak{M},w\stackrel{\square }{\leftrightsquigarrow }{\mathfrak{M}}^{\prime },w^{\prime }$. To show that $\stackrel{\square }{\leftrightsquigarrow }$ is a bisimulation (cf. Definition 5), the clause relating to requiring that $w,w^{\prime }$ satisfy the same propositional variables is trivial, since $w,w^{\prime }$ are modally equivalent.

Given that $w\stackrel{\square }{\leftrightsquigarrow }{w}^{\prime }$, suppose now that $wT{u}_1\cdots u_n$, for points $u_j$ of sort $i_j$ (recall that $\sigma \left(T\right)=(i_{n+1};i_1,\dots ,i_n)$). Let ${\left({\Sigma }_j\right)}_{j=1}^n$ be the sequence of sets of formulae such that for each $1\le j\le n$, ${\Sigma }_j$ is the set of formulas true at $u_j$ (which will be formulae in ${\mathcal{L}}_1$, if $i_j=1$, or formulae in ${\mathcal{L}}_{\partial }$, if $i_j=\partial$).

Since u_j${\Sigma }_j$, we get u_j$\bigwedge F_j$ for any finite subset $F_j\subseteq {\Sigma }_j$. Letting ${\varphi }_j=\bigwedge F_j$, it follows that w$({\varphi }_1,\dots ,{\varphi }_n)$. By modal equivalence of $w,w^{\prime }$, we also have w′$({\varphi }_1,\dots ,{\varphi }_n)$. By definition of the satisfaction relation for diamond operators, there exist points $u_j^{\prime }$, for $1\le j\le n$ such that $w^{\prime }{T}^{\prime }{u}_1^{\prime }\cdots u_n^{\prime }$ and $u_j^{\prime }$ϕ_j, for each $1\le j\le n$ (where recall again that = (⊧, ) is the sorted satisfaction relation, so that if $i_j=1$, then $u_j^{\prime }$ϕ_j is really $u_j^{\prime }\vDash {\varphi }_j$ and it is $u_j^{\prime }$ϕ_j in the case where $i_j=\partial$). This means exactly that condition (A) of finite satisfiability of Definition 6 holds and since the models considered are modally saturated we can conclude that there are points $v_1^{\prime },\dots ,v_n^{\prime }$ in ${\mathfrak{F}}^{\prime }$ such that $w^{\prime }{T}^{\prime }{v}_1^{\prime }\cdots v_n^{\prime }$ and $v_j^{\prime }$Σ_j, for each $1\le j\le n$.

To state then the obvious, assuming that $\mathfrak{M},w\stackrel{\square }{\leftrightsquigarrow }{\mathfrak{M}}^{\prime },w^{\prime }$ and that $wT{u}_1\cdots u_n$, we concluded that $w^{\prime }{T}^{\prime }{v}_1^{\prime }\cdots v_n^{\prime }$, for points $v_j^{\prime }$ in ${\mathfrak{F}}^{\prime }$ such that $u_j\stackrel{\square }{\leftrightsquigarrow }{v}_j^{\prime }$, for $1\le j\le n$.

Thus $\stackrel{\square }{\leftrightsquigarrow }$ is a bisimulation of the models. □

The property of the frame class in the statement of Lemma 1 is often referred to as the Hennessy-Milner property, since historically it was in the calculus now known as Hennessy-Milner logic that Hennessy and Milner identified it, in the context of their study of labelled transition systems and process equivalence.

Lemma 2.

Every ω-saturated model $\mathfrak{M}$ is modally saturated.

Proof. 

Assume that $\mathfrak{M}=(\mathfrak{F},V)$ is an $\omega$-saturated model over the frame $\mathfrak{F}=(A,I,B,R,S)$. Let $\in \{◇,◆,$, }, with sort σ() $=(i_1,\dots ,i_n;i_{n+1})$ and w be a state such that if $i_{n+1}=1$, then $w\in A$, else $w\in B$. Respectively, let $T\in \{I,I^{-1},R,S\}$ be the frame relation whose sort $\sigma \left(T\right)=(i_{n+1};i_1,\dots ,i_n)$ matches the sort of .

Let also ${\Sigma }_1,\dots ,{\Sigma }_n$ be a sequence of sets of formulae in the language of sorted residuated modal logic of Section 3 such that for $j=1$ to n, if $i_j=1$ then ${\Sigma }_j\subseteq {\mathcal{L}}_1$ and ${\Sigma }_j\subseteq {\mathcal{L}}_{\partial }$ otherwise.

Assume that the sequence ${\left({\Sigma }_j\right)}_{j=1}^n$ is finitely satisfiable, in the sense of condition (A) in Definition 6. Let ${\left(F_j\right)}_{j=1}^n$ be a sequence of finite subsets $F_j\subseteq {\Sigma }_j$.

Then, by finite satisfiability, there are points $u_1,\dots ,u_n$ such that $wT{u}_1\cdots u_n$ and $\mathfrak{F},u_j$ϕ_j, where we set ${\varphi }_j=\bigwedge F_j$ for each $1\le j\le n$.

Set $C=\{w,u_1,\dots ,u_n\}\subseteq A\cup B$, expand the sorted first-order language ${\mathcal{L}}_s^1$ of the frame with constants $\mathbf{w},{\mathbf{u}}_{\mathbf{1}},\dots ,{\mathbf{u}}_{\mathbf{n}}$, naming $w,u_1,\dots ,u_n$, which we denote by ${\mathcal{L}}_s^1\left[C\right]$, and let $\mathbf{T}$ be the predicate corresponding to the relation T.

For each $1\le j\le n$, let $S{T}_{x_j}\left({\Sigma }_j\right)=\left\{S{T}_{x_j}\left(\gamma \right)\right|\gamma \in {\Sigma }_j\}\subseteq {\mathcal{L}}_s^1$, where $x_j$ is an individual variable in $V_1$, or $x_j\in V_{\partial }$, accordingly as $i_j=1$, or $i_j=\partial$. Let $\Phi$ be the n-ary type defined by

$$\Phi (x_1,\dots ,x_n)=\left\{\mathbf{T}(\mathbf{w},x_1,\dots ,x_n)\right\}\cup \bigcup _{j=1}^{n}S{T}_{x_j}\left({\Sigma }_j\right)$$

By the assumption of finite satisfiability, $\Phi$ is consistent with the first-order theory ${\mathtt{Th}}_s\left({\mathfrak{F}}_C\right)$.

By the hypothesis of $\omega$-saturation of $\mathfrak{F}$ and Proposition 2, the type $\Phi$ is realized in ${\mathfrak{F}}_C$, i.e., there is an assignment V with $V\left(x_j\right)=w_j$ such that ${\mathfrak{F}}_C$Φ[V].

In particular, ${\mathfrak{F}}_C$$\mathbf{T}(\mathbf{w},x_1,\dots ,x_n)\left[V\right]$ and ${\mathfrak{F}}_C$$S{T}_{x_j}\left({\Sigma }_j\right)[x_j:=w_j]$.

Hence $\mathfrak{F}$$\mathbf{T}(x_0,x_1,\dots ,x_n)\left[V\right][x_0:=w]$ and $\mathfrak{F}$$S{T}_{x_j}\left({\Sigma }_j\right)[x_j:=w_j]$. By full abstraction of the standard translation of sorted residuated modal logic, Proposition 1, we obtain $\mathfrak{F},w_j$Σj, for each $1\le j\le n$.

It follows, by the definition of modal saturation, that $\mathfrak{M}$ is modally saturated. □

If $\Phi =\Phi \left(u\right)\in \mathcal{L}$ has only the displayed variable $u\in V_1$ free (i.e. it is a unary type) and it holds that $\Phi \vDash S{T}_u\left(\alpha \right)$, for some $\alpha$ (of sort 1) in the sorted modal language, then we say that ${\mathrm{ST}}_u\left(\alpha \right)$ is a modal 1-consequence of $\Phi$. Similarly, if $\Psi \left(v\right)\vDash {\mathrm{ST}}_v\left(\beta \right)$ then ${\mathrm{ST}}_v\left(\beta \right)$ is a modal ∂-consequence of $\Psi$. Let $m_1^1(\Phi )$, $m_{\partial }^1(\Psi )$ be the sets of 1- and ∂-consequences of $\Phi ,\Psi$, respectively.

Lemma 3.

Let $\Phi \left(u\right)$ be a sorted first-order formula in one free variable $u\in V_1$ and let $m_1^1(\Phi )$ be the set of its modal 1-consequences. If Φ is invariant under bisimulation, then $m_1^1(\Phi ){\vDash }_s\Phi$. Similarly, for a bisimulation invariant formula $\Psi \left(v\right)\in \mathcal{L}$, with $v\in V_{\partial }$, $m_{\partial }^1(\Psi ){\vDash }_s\Psi$.

Proof. 

The proof is again similar to that for the unsorted case, see for example the proof in Theorem 2.68 of [33]. We provide some details.

Let $\mathfrak{M}=\left(\right(A,I,B,R,S),V)$, $a\in A$, assume $\mathfrak{M}{\vDash }_s{m}_1^1(\Phi )[u:=a]$ and observe that $\Phi \cup m_1^1(\Phi )$ is consistent. If it were not, then by compactness of sorted FOL [8], we obtain that ${\vDash }_s\Phi \to \neg \bigwedge m_0^1(\Phi )$, for some finite $m_0^1(\Phi )\subseteq m_1^1(\Phi )$. Hence, $\neg \bigwedge m_0^1(\Phi )\in m_1^1(\Phi )$ which implies that $\mathfrak{M}{\vDash }_s\neg \bigwedge m_0^1(\Phi )$. This is in contradiction with the fact that $m_0^1(\Phi )\subseteq m_1^1(\Phi )$ and $\mathfrak{M}{\vDash }_{s}S{T}_u\left(\alpha \right)[u:=a]$ for all $S{T}_u\left(\alpha \right)\in m_1^1(\Phi )$.

By consistency of $\Phi \left(u\right)\cup m_1^1(\Phi )$, let ${\mathfrak{M}}^{\prime }=((A^{\prime },I^{\prime },B^{\prime },R^{\prime },S^{\prime }),V^{\prime })$ be a model and $a^{\prime }\in A^{\prime }$ such that ${\mathfrak{M}}^{\prime }{\vDash }_s\{\Phi \left(u\right)\}\cup m_1^1(\Phi )[u:=a^{\prime }]$. Then for any sentence $\alpha$ of the first sort in the sorted modal language, $\mathfrak{M},a{\vDash }_s\alpha$ iff ${\mathfrak{M}}^{\prime },a^{\prime }{\vDash }_s\alpha$. i.e., $a,a^{\prime }$ are modally equivalent. This is because if $\mathfrak{M},a{\vDash }_s\alpha$, then ${\mathrm{ST}}_u\left(\alpha \right)\in m_1^1(\Phi )$ and therefore by ${\mathfrak{M}}^{\prime }{\vDash }_s\{\Phi \left(u\right)\}\cup m_1^1(\Phi )[u:=a^{\prime }]$ and Proposition 1 it follows that ${\mathfrak{M}}^{\prime },a^{\prime }{\vDash }_s\alpha$. Conversely, if ${\mathfrak{M}}^{\prime },a^{\prime }{\vDash }_s\alpha$, then it must be that $\mathfrak{M},a{\vDash }_s\alpha$ for, if not, then $\mathfrak{M},a{\vDash }_s\neg \alpha$ and this implies ${\mathfrak{M}}^{\prime },a^{\prime }{\vDash }_s\neg \alpha$ which is a contradiction.

To obtain $\mathfrak{M}{\vDash }_s\Phi \left(u\right)[u:=a]$ from ${\mathfrak{M}}^{\prime }{\vDash }_s\Phi \left(u\right)[u:=a^{\prime }]$, let U be a countably incomplete ultrafilter over some index set J. By Proposition 4 and Corollaries 2 and 3 we obtain a sequence of implications:

• ${\mathfrak{M}}^{\prime }{\vDash }_s\Phi \left(u\right)[u:=a^{\prime }]⟹{\prod }_U{\mathfrak{M}}^{\prime }{\vDash }_s\Phi \left(u\right)[u:=f_{a,u}^{\prime }]$
• $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}⟹{\prod }_U\mathfrak{M}{\vDash }_s\Phi \left(u\right)[u:=f_{a,u}]$
• $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}⟹\mathfrak{M}{\vDash }_s\Phi \left(u\right)[u:=a]$

This establishes that $m_1^1(\Phi ){\vDash }_s\Phi$. The argument for a formula $\Psi \left(v\right)\in {\mathcal{L}}^1$, with $v\in V_{\partial }$ is similar. □

5.2. Van Benthem Characterization

Fix a similarity type $\tau$. Let ${\Lambda }_{\tau }$ be the language of normal lattice expansions of type $\tau$, $L_{\tau }={(L_1,L_{\partial })}_{\tau }$ be the sorted modal language of type $\tau$ and ${\mathcal{L}}_{s,\tau }^1$ the sorted first-order language of the same type $\tau$. All the necessary work to lift van Benthem’s characterization theorem to sorted modal logic has been presented and we state the result.

Theorem 4.

Let $\Phi \left(u\right)\in {\mathcal{L}}_{s,\tau }^1$ be a formula in one free variable in the sorted first-order language ${\mathcal{L}}_{s,\tau }^1$, with $u\in V_1$. Then Φ is equivalent to the translation ${\mathrm{ST}}_u\left(\alpha \right)$ of a modal formula $\alpha {\in }_1{L}_{\tau }$ iff Φ is bisimulation invariant. Similarly for a formula $\mathrm{\Psi }\left(v\right)$ with $v\in V_{\partial }$.

Proof. 

If $\Phi$ is equivalent to the translation ${\mathrm{ST}}_u\left(\alpha \right)$ of a modal formula $\alpha {\in }_1{L}_{\tau }$, then $\Phi$ is bisimulation invariant by Proposition 3.

For the converse, by Lemma 3 we obtain $m_1^1(\Phi ){\vDash }_s\Phi$. By compactness for sorted FOL (Theorem 2), let ${\mu }_1(\Phi )=\{S{T}_u\left({\alpha }_1\right),\dots ,S{T}_u\left({\alpha }_n\right)\}\subseteq m_1^1(\Phi )$ be a finite subset of $m_1^1(\Phi )$ such that ${\mu }_1(\Phi ){\vDash }_s\Phi$. Then ${\vDash }_s\Phi \leftrightarrow \bigwedge {\mu }_1(\Phi )$, hence ${\vDash }_s\Phi \leftrightarrow S{T}_u\left(\eta \right)$, where we set $\eta ={\alpha }_1\wedge \cdots \wedge {\alpha }_n$. □

It remains to adapt the result to the case of the logics of normal lattice expansions of similarity type $\tau$.

Definition 7.

$\Phi \left(u\right)$, with $u\in V_1$, is t-invariant if and only if it is equivalent to its transformation $t(\Phi \left(u\right))={\forall }^{\partial }v{\exists }^{1}z(\mathbf{I}(u,v)⟶\mathbf{I}(z,v)\wedge \Phi \left(z\right))$.

Theorem 5

(van Benthem Characterization). Fix a similarity type τ. Let $\Phi \left(u\right)\in {\mathcal{L}}_{s,\tau }^1$ be a formula with one free variable in the sorted first-order language ${\mathcal{L}}_{s,\tau }^1$, with $u\in V_1$. Then Φ is equivalent to the translation ${\mathit{ST}}_{\pi }^{•}\left(\phi \right)$, for some permutation $\pi :\omega \to \omega$, of a sentence in the language of lattice expansions of similarity type τ iff Φ is bisimulation and t-invariant.

Proof. 

The claim of the theorem follows immediately by combining the characterization result for sorted modal logic of similarity type $\tau$ (Theorem 4) and Corollary 1 (a consequence of Theorem 1). □

6. Final Comments

We conclude this article with two comments, one on the original van Benthem characterization theorem and another on a recent publication [38] bearing on a van Benthem characterization for non-distributive modal logic.

For the first comment, note that a classical Kripke frame $(W,R,\dots )$ is a special case of the type of sorted frames $\mathfrak{F}=(A,I,B,R,\dots )$ we have considered, arising when the two sorts coincide, $A=B=W$, taking the relation I to be the identity relation, in which case each of the set operators ◇, ■ is the identity on subsets. Hence, the set of stable sets is the whole of $\left(W\right)$ and the relation I need not be mentioned in the frame description. Additional frame relations are just classical accessibility relations on W. Having collapsed the two sorts to a single one, sorted bisimulations are ordinary bisimulations (see Remark 2). Note also that in the fist-order language of the frame the predicate $\mathbf{I}$ is just the equality predicate = and the t transformation of a formula $\Phi \left(u\right)$ in the single free variable u becomes $t(\Phi (u\left)\right)=\forall v(v=u⟶\exists z(z=v\wedge \Phi \left(z\right))$. Clearly then every formula $\Phi \left(u\right)$ is t-invariant and therefore Theorem 5 in this case is the classical van Benthem result, when the similarity type of the lattice expansion is that of a modal algebra.

For the second comment, while clearly related, this article and [38] (https://doi.org/10.48550/arXiv.2404.05574, 8 August 2024) differ in scope, method and results, as I explain below.

Despite the fact that van Benthem’s result was stated and proven for modal logic, characterization results are not specific to modal logic and can be stated and proven for any propositional logic for which a fully abstract translation of its language into a related first-order language can be given. This has determined the approach and scope of the present article which focused on establishing the van Benthem result for just any distribution-free propositional logic with normal operators. In [38], the authors have made the choice to address the problem specifically for distribution-free modal logic, which has been so far little studied [39,40].

Other than scope, the present article and [38] differ in method, too. A reductionist approach has been followed in the results presented here, which has been initiated in [4,5]. This approach aims at reducing a problem of significance for distribution-free logics to the same problem for classical, but sorted residuated modal logics (their sorted modal companions). In [38], the authors set out to prove that formulae in the sorted first-order logic are translations of sentences in the distribution-free modal logic (which is the system of their focus) iff they are invariant under a suitable notion of bisimulation ([38] [Definition 3.2]). They show that the Hennessy-Milner property fails ([38] [Example 3]), they proceed to refine their definition of bisimulation ( [38] [Definition 3.8]) and establish the Hennessy-Milner property [38] [Theorem 3.8]). The characterization theorem proven in [38] [Theorem 5.10] is weaker than the classical result (which explains the phrasing “Towards the van Benthem Characterization ...” of the title). The result in [38] [Theorem 5.10] introduces notions of being preserved, or reflected by simulations and it sets conditions for this to be the case. For preservation (and similarly for reflection), it states that a sorted first-order formula in a single variable $\phi \left(x\right)$ is preserved by simulations if and only if there exist sentences ${\psi }_1,\dots ,{\psi }_n$ in the language of the distribution-free modal logic such that $\phi \left(x\right)$ is equivalent to the disjunction of their translations ${\bigvee }_{i=1}^n{\mathrm{ST}}_x{\psi }_i$. What is, however, further needed for a van Benthem characterization result is the ability to restrict to a single sentence $\psi$, i.e., force $n=1$, and this step is not taken in [38].

The van Benthem characterization result proven in this article characterizes the fragment of sorted first-order logic, the sentences of which are translations of sentences of the propositional logic of interest, as the fragment of single-variable first order sentences that are both invariant under sorted bisimulations and t-invariant. The argument has two parts, one proceeding to lift the characterization result to sorted but classical modal logic with polyadic modalities, the other identifying (via a fully abstract translation) the language of the distribution-free logic of interest as a fragment of its companion sorted modal logic, namely the fragment of stable $\alpha \equiv ■◇\alpha$ modal sentences. This stability property leads to a requirement of t-invariance for first-order formulae that are translations of sentences in the distribution-free logic. The additional requirement of t-invariance collapses to a trivial one in the classical modal logic case, as detailed in the first comment of this section.

Both the current article and [38] set out to generalize and adapt the structure in the proof of the original van Benthem result. They also both use the same class of semantic structures, i.e., polarities with relations (or sorted residuated frames with additional relations, in our preferred terminology). As far as the present article is concerned, use of such structures is based on an older (1997) publication [19] of lattice representation (used in [41] to prove existence of canonical extensions), generalized in [7,18] and eventually in [6] to a full topological duality result for normal lattice expansions, subsequently used in applications, such as [11,42] and underlying this author’s current work (in progress) on distribution-free normal modal logics.

Finally, it deserves being pointed out that the present article was first publicly announced as an arxiv posting, https://doi.org/10.48550/arXiv.2001.00232, on 1 January 2020. After minor improvement, it was submitted to a journal nearly a year later, but after an extreme delay of two years and a half, waiting for reviewer reports (which never arrived), it was withdrawn, updated and submitted to the Logics MDPI journal. Though the unusual failure of the review process could have resulted in significant overlap with the arxiv posting [38], at https://doi.org/10.48550/arXiv.2404.05574, 8 August 2024 (included, as I have been informed, in the AiML 2024 (August 2024) conference proceedings, yet to appear), waste of research time has been fortunately avoided.