---
abstract: |
    Given a class $\clC$ of models, a binary relation $\clR$ between models, and a model-theoretic language $L$, we consider the modal logic and the modal algebra of the theory of $\clC$ in $L$ where the modal operator is interpreted via $\clR$. We discuss how modal theories of $\clC$ and $\clR$ depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside $L$. We calculate such theories for the submodel and the quotient relations. We prove a downward Löwenheim–Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.

    Keywords: *modal logic, modal algebra, robust modal theory, logic of submodels, logic of quotients, logic of forcing, provability logic, model-theoretic logic*
author:
- 'Denis I. Saveliev'
- 'Ilya B. Shapirovsky'
bibliography:
- 'modmod.bib'
title: 'On modal logics of model-theoretic relations[^1]'
---

Introduction {#introduction .unnumbered}
============

We consider modal systems in which the modal operator is interpreted via a binary relation on a class of models. Many instances of such systems can be found in the literature. During the last years, modal logics of various relations between models of set theory have been studied, see, e.g., [@Hamkins03; @HamkLowe; @BlockLowe2015; @ForcStruct2015; @innerMods2016]. A well established area in provability logic deals with modal axiomatizations of relations between models of arithmetic (and between arithmetic theories), see, e.g., [@Shavrukov1988; @Berarducci1990; @Ignatiev93; @VisserBigFirst; @Visser14; @Henk2015; @HamkinsArithmeticPotentialism2018]. In another extensively studied area modalities are interpreted by relations between Kripke and temporal models, see, e.g., [@Veltman96; @agostino_hollenberg_2000; @TemporalSubst2013] or the monograph [@vanBenthem2014]. In [@BarwiseVBenthem1999], the consequence along an abstract relation between models is studied, which is closely related to our consideration.

Let $f$ be a unary operation on sentences of a model-theoretic language $L$, and $T$ a set of sentences of $L$ (e.g., the set of theorems in a given calculus, or the set of sentences valid in a given class of models). Using the propositional modal language, one can consider the following “fragment” of $T$: variables are evaluated by sentences of $L$, and $f$ interprets the modal operator; the [*modal theory of $f$ on $T$*]{}, or just the [*$f$-fragment of $T$*]{}, is defined as the set of those modal formulas which are in $T$ under every valuation. A well-known example of this approach is a complete modal axiomatization of formal provability in Peano arithmetic given by Solovay [@Solovay1976]. Another important example is the theorem by Hamkins and Löwe axiomatizing the modal logic of forcing (introduced earlier by Hamkins in [@Hamkins03]) where the modal operator expresses satisfiability in forcing extensions [@HamkLowe]. Both these modal systems have good semantic and algorithmic properties; in particular, they have the finite model property, are finitely axiomatizable, and hence decidable.

These examples inspire the following observation. Let $\clC$ be an arbitrary class of models of the same signature, $T=\Th^L(\clC)$ the theory of $\clC$ in a model-theoretic language $L$, and $\clR$ a binary relation on $\clC$. Assuming that the satisfiability in $\clR$-images of models in $\clC$ can be expressed by an operation $f$ on sentences of $L$, i.e., for every sentence $\vf$ of $L$, and every $\stA\in\clC$,

> $\stA\mo f(\varphi)$ (“$\varphi$ is possible at $\stA$”) iff $\stB\mo\vf$ for some $\mathfrak B$ with $\stA\,\clR\,\stB$,

we can define the [*modal theory of $\clR$ in $L$*]{} as the $f$-fragment of $T$. In the general frame semantics, this modal theory is characterized by an enormous structure $
(\mathcal C,\mathcal R,\mathcal C_\vf:
\vf\text{ is a~sentence of~}L)
$ where $\clC_\vf$ is the class of models in $\clC$ validating $\vf$. We can also define the [*modal Lindenbaum algebra of $\Th^L(\clC)$ and $\clR$*]{}, i.e., the Boolean algebra of sentences of $L$ modulo the equivalence on $\clC$, endowed with the modal operator induced by $f$.[^2] In Section \[sec:defs\], we provide formal definitions and basic semantic tools for such modal theories. In particular, the algebra of $\Th^L(\clC)$ and $\clR$ can be represented as the modal algebra of a general frame consisting of complete theories $\{\Th^L(\stA):\stA\in\clC\}$. We use this in Section \[sec:sub\], where we calculate modal logics of the submodel relation $\supmod$; to express the satisfiability in submodels, we use second-order language.

In Section \[sec:upanddown\],[^3] we discuss the situation when the $\clR$-satisfiability is not expressible in a language $K$ (for example, this situation is typical when $K$ is first-order). In this case the modal algebra of $\Th^K(\clC)$ can be defined as the subalgebra of the modal algebra of $\Th^L(\clC)$ generated by the sentences of $K$, for any $L$ stronger than $K$ where the $\clR$-satisfiability on $\clC$ is expressible: the resulting modal algebra (and hence, its modal logic) does not depend on the way how we extend the language $K$. Under a natural assumption on $\clC$ and $\clR$, such an $L$ can always by constructed, and hence, [*the*]{} modal algebra of $\Th^K(\clC)$ and $\clR$ is well-defined for arbitrary $K$ (however, the resulting modal logic is not necessarily a “fragment” of the theory $\Th^K(\clC)$ anymore). Then we consider the finitary first-order language expanded with the modal operator for the extension relation $\submod$ and prove a version of the downward Löwenheim–Skolem theorem for this language.

In general, modal theories of $\clR$ depend on the model-theoretic language we consider. We say that a modal theory of $\clR$ is [*robust*]{} iff making the language stronger does not alter this theory (intuitively, the robust theory can be considered as the “true” modal logic of the model-theoretic relation $\mathcal R$). We discuss this notion in Section \[sec:rob\]. In Theorem \[thm: robust logicGOOD\], we show that under a certain assumption on $\clC$ and $\clR$, the robust logic is Kripke complete. Then we use this theorem to describe robust theories of the quotient and the submodel relations on certain natural classes.

A preliminary report on some results in Sections \[sec:defs\] and \[sec:sub\] can be found in [@SavelievShapirovsky2016].

Preliminaries
=============

To simplify reading, as a rule we denote the syntax of our object languages differently: we use inclined letters if they are related to model theory ($x,y,\ldots$ for individual variables, $\varphi,\psi,\ldots$ for formulas), and upright letters if they are related to modal logic ($\mathsf p,\mathsf q,\ldots$ for propositional variables, $\upvarphi,\uppsi,\ldots$ for formulas).

#### Model-theoretic languages

The languages we use for model theory are model-theoretic languages in sense of [@BarwiseFeferman] (where they are called “model-theoretic logics”). We are pointing out here only two of their features, which will be essential for the further investigations: if $L$ is a model-theoretic language under our consideration, then:

-   Satisfiability in $L$ is preserved under isomorphisms;

-   $L$ includes $L_{\omega,\omega}$, the standard first-order language with usual finitary connectives and quantifiers.

Also, we assume that $L$ is a set unless otherwise specified. For a model $\stA$, $\Th^L(\stA)$, or just $\Th(\stA)$, denotes its theory in $L$, i.e., the set of all sentences of $L$ holding in $\stA$; likewise for a class of models.

#### Modal logic {#subs:basicML}

By a [*modal logic*]{} we shall mean a normal propositional unimodal logic (see, e.g., [@BRV-ML] or [@CZ]). Modal formulas are built from a countable set $\PV$ of propositional variables $\mathsf{p},\mathsf{q},\ldots$, the Boolean connectives, and the modal connectives $\Di$, $\Box$. We assume that $\bot$, $\imp$, $\Di$ are basic, and others are abbreviations; in particular, $\Box$ abbreviates $\neg\,\Di\,\neg$. Let $\ML$ denote the set of modal formulas. A set of modal formulas is called a [*normal modal logic*]{} iff it contains all classical tautologies, two following modal axioms: $$\neg\,\Di\bot
\;\;\text{and}\;\;
\Di(\mathsf p\vee\mathsf q)\to\Di\mathsf p\vee\Di\mathsf q,$$ and is closed under three following rules of inference: $$\text{Modus Ponens}\;
\frac{\mvf,\;\mvf\to\mpsi}{\mpsi},
\;\;
\text{Substitution}\;
\frac{\mvf(\mathsf p)}{\mvf(\mpsi)},
\;\;
\text{Monotonicity}\;
\frac{\mvf\to\mpsi}{\Di\mvf\to\Di\mpsi}.$$

A [*general frame*]{} $\frF$ is a triple $(W,R,\clV)$ where $W$ is a nonempty set, $R\subseteq W\times W$, and $\clV$ is a subalgebra of the powerset algebra $\clP(W)$ closed under $R^{-1}$ (i.e., such that $R^{-1}(A)=\{x\in W:\exists y\in A\,\,x R y\}$ is in $\clV$ whenever $A$ is in $\clV$). By a [*Kripke frame*]{} $(W,R)$ we mean $(W,R,\clP(W))$. We shall often use the term [*frame*]{} for a general frame (and never for a Kripke frame). A [*modal algebra*]{} is a Boolean algebra endowed with a unary operation that distributes w.r.t. finite disjunctions. The [*modal algebra of a frame*]{} $(W,R,\clV)$ is $(\clV,R^{-1})$.

A [*Kripke model*]{} $\mathfrak M$ on a frame $\mathfrak F=(W,R,\clV)$ is a pair $(\mathfrak F,\theta)$ where $\theta:\PV\to\clV$ is a [*valuation*]{}. The [*truth relation*]{} $\mM,x\mo \mvf$ (“$\mvf$ is true at $x$ in $\mM$”) is defined in the standard way (see, e.g., [@CZ]); in particular, $\mM,x\mo\Di\mvf$ iff $\mM,y\mo\mvf$ for some $y$ with $xRy$. A formula $\mvf\in\MF$ is [*true in a model $\mM$*]{} iff it is true at every $x$ in $\mM$, and $\mvf$ is [*valid in a frame $\frF$*]{} iff it is true in every model on $\mathfrak F$. A modal formula $\mvf$ is [*valid in a modal algebra*]{} $\stA$ iff $\mvf=\top$ holds in $\stA$. In particular, it follows that if $\stA$ is the modal algebra of a frame $\frF$, then $\mvf$ is valid in $\stA$ iff $\mvf$ is valid in $\frF$. The set $\MLog(\frF)$ of all valid in $\frF$ formulas is called the ([*modal*]{}) [*logic of the frame $\frF$*]{}; likewise for algebras. It is well-known (see, e.g., [@BRV-ML] or [@CZ]) that a set of formulas is a consistent normal logic iff it is the logic of a general frame iff it is the logic of a non-trivial modal algebra.

In our paper we also consider modal logics of general frames $(\mathcal C,\mathcal R,\mathcal V)$ with a proper class $\mathcal C$; see Remarks \[rem:claclaclass1\] and \[rem:claclaclassGeneral\] in the next section.

${{{\mathrm{S4}}}}$ is the smallest logic containing the formulas $\Di\Di\mathsf p\imp\Di\mathsf p$ and $\mathsf p\imp\Di\mathsf p$, ${{{\mathrm{S4.2}}}}$ is ${{{\mathrm{S4}}}}+\Di\Box\mathsf p\;{\rightarrow}\;\Box\Di\mathsf p$, and ${{{\mathrm{S4.2.1}}}}$ is $
{{{\mathrm{S4}}}}+
\Box\Di\mathsf p\;{\leftrightarrow}\;\Di\Box\mathsf p
$, where $\Lambda+\mvf$ is the smallest logic that includes $\Lambda\cup\{\mvf\}$.

Modal theories of model-theoretic relations {#sec:defs}
===========================================

#### Definitions

Fix a signature $\Omega$ and a language $L=L(\Omega)$ based on $\Omega$. Let $L_s$ be the set of all sentences of $L$.

Consider a unary operation $f$ on sentences of $L$. A ([*propositional*]{}) [*valuation in $L$*]{} is a map from $\PV$ to $L_s$. A valuation $\vl$ extends to the set $\MF$ of modal formulas as follows: $$\begin{aligned}
{ \| \bot \|}
&=
\bot,
\\
{ \| \mvf\to\mpsi \|}
&=
{ \| \mvf \|}\to{ \| \mpsi \|},
\\
{ \| \Di\mvf \|}
&=
f({ \| \mvf \|}).\end{aligned}$$ Given a set of sentences $T\subseteq L_s$, we define the [*modal theory of $f$ on $T$*]{} $\mathrm{MTh}(T,f)$ as the set of modal formulas $\mvf$ such that for every valuation $\vl$ in $L$, ${ \| \mvf \|}$ is in $T$. Thus, modal formulas can be viewed as axiom schemas (for sentences) and we can think of the modal theory of $f$ on $T$ as a fragment of $T$.

Let $L$ be the usual first-order finitary language $L_{\omega,\omega}$.

1\. Let $\Omega$ be the signature of arithmetic and $f(\varphi)$ express the consistency of a sentence $\varphi$ in Peano arithmetic ${\mathrm{PA}}$. By the well-known Solovay’s results [@Solovay1976], the modal theory of $f$ on $\PA$ is the Gödel–Löb logic ${\mathrm{GL}}$, and the modal theory of $f$ on the true arithmetic ${\mathrm{TA}}$, the set of all sentences that are true in the standard model of arithmetic, is the (quasi-normal but not normal) Solovay logic ${\mathrm{S}}$. A survey on provability logics can be found in [@ArtemovBekl04Prov].

2\. Let $\Omega$ be the signature of set theory (i.e., $\Omega=\{\in\}$) and $f(\varphi)$ express that $\varphi$ holds in a generic extension. By Hamkins and Löwe [@HamkLowe], the modal theory of $f$ on ${\mathrm{ZFC}}$ is ${\mathrm{S4.2}}$.

In our work, we are interested in the case when $T$ is the theory of some class $\clC$ of models of $\Omega$. An easy observation shows that in this case the modal theory is closed under the rules of Modus Ponens and Substitution. Notice that it is not necessarily closed under Monotonicity, as the instance of the Solovay logic ${\mathrm{S}}$ shows.[^4] However, as we shall see, the modal theory is a normal logic whenever $f$ expresses the satisfiability in images of a binary relation between models.

Let $\clR$ be a binary relation on $\clC$ (or, perhaps, on a larger class $\clC'\supseteq\clC$). The [*$\clR$-satisfiability on $\clC$ is expressible in*]{} $L$ iff there exists $f:L_s\to L_s$ such that for every sentence $\vf\in L_s$ and every model $\stA\in\clC$, $$\label{eq:r-sat}
\stA\vDash f(\vf)
\;\Iff\;
\exists\,\stB\in\clC\;
(\stA\,\clR\,\stB\;\&\;\stB\vDash\vf).$$

Some examples of such expressibility and non-expressibility will be given below.

The following is straightforward:

If $f$ and $f'$ both express the $\mathcal R$-satisfiability on $\clC$, and $T=\Th(\clC)$, then $\mathrm{MTh}(T,f)=\mathrm{MTh}(T,f').$

Thus, in this case, $\mathrm{MTh}(T,f)$ does not depend on the choice of $f$.

\[def:exprMTH\] Let the $\mathcal R$-satisfiability be expressible in $L$ on a class $\mathcal C$ of models. The [*modal theory of $(\mathcal C,\mathcal R)$ in $L$*]{}, denoted by $\MTh^L(\mathcal C,\mathcal R)$, is the set of modal formulas $\mvf$ such that $\clC\mo{ \| \mvf \|}$ for every propositional valuation in $L$.

In this section, we just write $\mathrm{MTh}(\clC,\clR)$ assuming $L$ is fixed.

Given a propositional valuation $\vl$ in $L$, consider $\mM=(\clC,\clR,\theta)$, the enormous “Kripke model” with $\theta(\mathsf p)=\{\stA\in\clC:\stA\mo{ \| \mathsf p \|}\}$. By a straightforward induction on a modal formula $\mvf$, for every $\stA$ in $\clC$ $$\stA\mo{ \| \mvf \|}
\;\Iff\;
\mM,\stA\mo\mvf,$$ where $\mo$ on the left-hand and on the right-hand side of the equivalence denotes the satisfiability relation in model theory and the truth relation of a modal formula in a Kripke model, respectively. It follows that $$\clC\mo{ \| \mvf \|}
\;\Iff\;
\mM,\stA\mo\mvf\text{ for all }\stA\in\clC.$$ Let $\Mod(\psi)$ be the class of models of $\psi\in\L_s$, and $\clC_\psi=\Mod(\psi)\cap \clC$. Then validity of ${ \| \mvf \|}$ in $\clC$ for all propositional valuations in $L$ can be considered as validity of the modal formula $\mvf$ in an enormous “general frame of models” $(\clC,\clR,\clV)$ where $\clV$ “consists” of $\clC_\psi$ with $\psi\in L_s$.

\[rem:claclaclass1\] The collection $\mathcal V$ looks like a “class of classes”. In fact, however, $\mathcal V$ is a usual class defined by a formula of set theory with two parameters: if $\mathcal C$, $L_s$ are defined by formulas $\varPhi$, $\varPhi'$, respectively, then what we understand by $\mathcal V$ is the class of pairs $(\mathfrak A,\psi)$ defined by a formula $\varUpsilon$ that is constructed from $\varPhi$, $\varPhi'$ and expresses the satisfaction relation between models in $\mathcal C$ and sentences in $L_s$. Thus subclasses $\mathcal C_\psi$ of $\clC$ playing the role of “elements” of $\mathcal V$ are in fact defined by $\varUpsilon$ with a fixed second argument $\psi$. This allows us to formally extend the definition of validity in a (set) frame to the case of the “frame of models” $(\clC,\clR,\clV)$.

It follows that the modal logic $\MLog(\mathcal C,\mathcal R,\mathcal V)$, i.e., the set of all modal formulas that are valid in $(\mathcal C,\mathcal R,\mathcal V)$, coincides with the modal theory of $(\mathcal C,\mathcal R)$ in $L$. Namely, we have

\[thm:big-as-general\] If the $\mathcal R$-satisfiability on $\mathcal C$ is expressible in $L$, then $$\label{eq:big-as-general}
\mathrm{MTh}(\mathcal C,\mathcal R)=
\MLog(\mathcal C,\mathcal R,\mathcal V).$$

Consequently, $\MTh(\mathcal C,\mathcal R)$ is a normal logic.

\[prop:extendingexpr\] Assume that $f$ expresses the $\clR$-satisfiability on $\clC$ in $L$.

1.  If $\clD\subseteq \clC$ is $\clR$-upward closed (i.e., $\stA\in \clD\;\&\;\stA\,\clR\,\stB\;\&\;\stB\in\clC$ implies $\stB\in\clD$), then the $\clR$-satisfiability on $\clD$ is expressible in $L$ by $f$.

2.  If $\psi$ is a sentence of $L$, then the $\clR$-satisfiability on $\clC\cap\Mod(\psi)$ is expressible in $L$ by $g:\vf\mapsto f(\vf\wedge\psi)$.

Let $\clR^*(\stA)$ denote $\bigcup_{n<\omega} \clR^n(\stA)$, the least $\clR$-closed $\clD\subseteq\clC$ containing $\stA$. From (\[eq:big-as-general\]) and the generated subframe construction (see, e.g., [@CZ Section 8.5]), we obtain

\[cor:gener\] If the $\clR$-satisfiability on $\clC$ is expressible in $L$, then

$
\MTh(\clC,\clR)\,=\,
\bigcap\limits_{\mathfrak A\in\clC}
\MTh(\clR^*(\mathfrak A),\clR).
$

#### Frames of theories

Assume that the $\clR$-satisfiability is expressible on $\clC$ in $L$ by some $f:L_s\to L_s$.

We have observed in Theorem \[thm:big-as-general\] that $\MTh(\mathcal C,\mathcal R)$ can be viewed as the modal logic of a general frame of models. Now we provide other semantic characterizations of $\MTh(\mathcal C,\mathcal R)$.

Put $\Ths=\{\Th(\stA): \stA\in\clC\}$. Note that $\Ths$ is a set since $L$ is assumed to be a set. For theories $T_1,T_2 \in \Ths$, let $$\begin{aligned}
T_1\,\clR_{\Ths}\,T_2
&\tiff&
\EE\,\stA_1,\stA_2\in\clC\,\;
(\stA_1\mo T_1\;\&\;\stA_2\mo T_2\;\&\;\stA_1\clR\,\stA_2),
\\
T_1\,\Rmax\,T_2
&\tiff&
\AA\vf\in T_2\;\,f(\vf)\in T_1.
$$ Observe that $\Rmax$ does not depend on the choice of $f$, and $\clR_{\Ths}\subseteq\Rmax$.

For $\vf\in L_s$, put $\Ths_\vf=\{T\in\Ths:\vf\in T\}$. Clearly, $\Ths_{\vf\con\psi}=\Ths_{\vf}\cap\Ths_{\psi}$ and $\Ths_{\neg\,\vf}=\Ths\setminus\Ths_{\vf}$. Consider an arbitrary binary relation $\filtR$ such that $$\clR_{\Ths}\subseteq\filtR\subseteq\Rmax.$$ It follows from the definitions that $\filtR^{-1}(\Ths_\vf)=\Ths_{f(\vf)}$. Therefore, $$\clA=\{\Ths_{\vf}:\vf\in L_s\}$$ is a subalgebra of the powerset algebra $\clP(\Ths)$ closed under $\filtR^{-1}$, and $(\Ths,\filtR,\clA)$ is a general frame. We call $(\Ths,\clR_{\Ths},\clA)$ and $(\Ths,\Rmax,\clA)$ the [*minimal*]{} and the [*maximal frames of theories for $\clC,\clR$, and $L$*]{}.

The relation $\filtR$ can be viewed as a [*filtration of $\clR$*]{} (see, e.g., [@Goldblatt92 Section 4]). The frame $(\Ths,\Rmax,\clA)$ is known as the [*refinement*]{} of $(\clC,\clR,\mathcal V)$ (cf. [@CZ Chapter 8]).

For sentences $\vf,\psi$, put $\vf\approx\psi$ iff $\clC\mo\vf\lra \psi$. Let $L/{\approx}$ be the Lindenbaum algebra of the theory $\Th(\clC)$, i.e., the set $L_s$ of sentences of $L$ modulo $\approx$ with operations induced by Boolean connectives. We have $f(\vf)\approx f(\psi)$ whenever $\vf\approx\psi$, hence, $f$ induces the operation $f_\approx$ on $L/{\approx}$. It is easy to see that $(L/{\approx},f_\approx)$ is a modal algebra. It is called the [*modal (Lindenbaum) algebra of the language $L$ on $(\clC,\clR)$*]{}.

The above arguments yield

\[thm:minmax-ths\] If $\clR_{\Ths}\subseteq \filtR\subseteq\Rmax$, then the modal algebras $(L/{\approx},f_\approx)$ and $(\clA,\filtR^{-1})$ are isomorphic.

The isomorphism takes the $\approx$-class of a sentence $\vf$ to $\Ths_{\vf}$.

\[thm:maintoolnew\] Assume that the $\clR$-satisfiability on $\clC$ is expressible in $L$. Let $\clR_{\Ths}\subseteq \filtR \subseteq \Rmax$. Then $$\MTh(\clC,\clR)=
\MLog(\Ths,\filtR,\clA)=
\MLog(L/{\approx},f_\approx).$$

Every valuation $\vl$ in $L$ can be viewed as a valuation $\theta$ in the frame $\frF=(\Ths,\filtR,\clA)$, and vice versa. By induction on $\mvf$, for every $\stA$ in $\clC$ we have $
\stA\mo{ \| \mvf \|}\tiff
(\frF,\theta){,}\Th(\stA)\mo\mvf.
$ Thus $\MTh(\clC,\clR)=\MLog(\Ths,\filtR,\clA)$.

The second equality immediately follows from Theorem \[thm:minmax-ths\].

\[prop: finite frame\] If $\Ths$ is finite, then $\mathrm{MTh}(\mathcal C,\mathcal R)$ is the logic of the Kripke frame $(\Ths,\clR_{\Ths})$.

In this case $\clA=\clP(\Ths)$.

The family $\Ths$ of complete theories can be viewed as the quotient of $\clC$ by the $L$-equivalence $\equiv$ where $\stA\equiv \stB$ iff $\Th(\stA)=\Th(\stB)$. Theorem \[thm:maintoolnew\] can be generalized for the case of any equivalence $\sim$ on $\clC$ finer than $\equiv$; in particular, it holds for the isomorphism equivalence $\isom$ on models, or for the equivalence in a stronger model-theoretic language. Namely, we let: $$\begin{aligned}
\,[\stA]_\sim\clR_\sim[\stB]_\sim
&\,\Iff\,&
\exists\,\mathfrak A'\sim\mathfrak A\;
\exists\,\mathfrak B'\sim\mathfrak B\;\,
\mathfrak A'\,\mathcal R\,\mathfrak B',
\\
\,[\stA]_\sim\Rsimmax[\stB]_\sim
&\Iff&
\forall\,\varphi\in L_s\;\,
(\mathfrak B\vDash\varphi\,\Rightarrow\,
\mathfrak A\vDash f(\varphi));\end{aligned}$$ the algebra of valuations is defined as the collection $\clV_\sim$ “consisting” of classes $\clC_\psi/{\sim}$ for $\psi\in L_s$. (Again, since $[\mathfrak A]_\sim$ are in general proper classes, $\clV_\sim$ looks like a “class of classes of classes” but actually is nothing but a three-parameter formula; cf. Remarks \[rem:claclaclass1\] and \[rem:claclaclassGeneral\].) As in the proof of Theorem \[thm:maintoolnew\], for $\clR_\sim\subseteq\filtR\subseteq \Rsimmax$ one can obtain that $$\label{eq:quot}
\MTh(\clC,\clR)=\MLog(\clC/{\sim},\filtR,\clV_\sim).$$

To the best of our knowledge, Theorem \[thm:maintoolnew\] (or its analogue (\[eq:quot\])) has never been formulated explicitly before, although similar constructions related to frames of arithmetic theories were considered earlier (V.Yu. Shavrukov, an unpublished note, 2013; [@Henk2015 Remark 3]).

We conclude this section with a continuation of Remark \[rem:claclaclass1\].

\[rem:claclaclassGeneral\] It is possible to give a general definition of frames that are classes and their logics (in our metatheory which is assumed to be $\ZFC$ where “classes” are shorthands for formulas). We outline the idea and postpone details for a further paper. Below capital Greek letters $\varPhi,\varPsi,\ldots$ denote formulas of the metatheory.

Assume that $\mathcal C$, $\mathcal L$, $\mathcal R$ are (arbitrary) classes defined by formulas $\varPhi$, $\varPhi'$, $\varPsi$, respectively: $\mathcal C=\{x:\varPhi(x)\}$, $\mathcal L=\{y:\varPhi'(y)\}$, and $\mathcal R=\{(x,v):\varPsi(x,v)\}$. Let us say that a class $\mathcal V$ forms a *class modal algebra* (admissible for $\mathcal C$, $\mathcal L$, $\mathcal R$) iff it is defined by a formula $\varUpsilon$ that fulfills the conditions expressing that classes $\mathcal C_y=\{x:\varUpsilon(x,y)\}$ indexed by $y$ in $\mathcal L$ play the role of “elements” of $\mathcal V$. Namely, $\varUpsilon$ implies that all $\clC_y$ are subclasses of $\mathcal C$ and their collection is closed under Boolean operations and the modal operator given by $\mathcal R$; e.g., the latter is expressed as follows: $$\forall y\,\exists z\,\forall x\;
(\varUpsilon(x,z)\:\&\:\varPhi'(y)
\;\Leftrightarrow\;
\varPhi(x)\:\&\:\exists v\:
(\varPsi(x,v)\:\&\:\varUpsilon(v,y))).$$

Further, let us say that $\theta$ is a *valuation* of propositional variables in $\mathcal V$ iff $\theta$ is a (set) function with ${\mathrm{dom}}(\theta)=\PV$ and ${\mathrm{ran}}(\theta)\subseteq\mathcal L$, and that $\mathfrak F=(\mathcal C,\mathcal R,\mathcal V)$ is a *class general frame* and $\mathfrak M=(\mathcal C,\mathcal R,\mathcal V,\theta)$ a *class Kripke model* on $\mathfrak F$. To define the *truth* of modal formulas $\upvarphi$ at a point $x\in\mathcal C$ in the model $\mathfrak M$, denoted by $\mathfrak M,x\vDash\upvarphi,$ we first extend $\theta$ to a suitable $\bar\theta$ with ${\mathrm{dom}}(\bar\theta)=\MF$ and ${\mathrm{ran}}(\theta)\subseteq\mathcal L$. It can be shown that such an extension exists and is unique up to the equivalence $\sim$ on $\mathcal L$ defined by letting $y\sim z$ iff $
\forall x\,
(\varUpsilon(x,y)\Leftrightarrow\varUpsilon(x,z)).
$ Then we let $$\begin{aligned}
\mathfrak M,x\vDash\upvarphi
\;\Leftrightarrow\;
\varUpsilon(x,\bar\theta(\upvarphi)).\end{aligned}$$ This notion of truth has the expected properties; e.g., we have $$\begin{aligned}
\mathfrak M,x\vDash\Diamond\upvarphi
&\;\Leftrightarrow\;
\exists v\:(\varPsi(x,v)\:\&\:
\mathfrak M,v\vDash\upvarphi).\end{aligned}$$ A formula $\upvarphi\in\MF$ is *true* in $\mathfrak M$ iff $
\forall x\,(\varPhi(x)\Rightarrow
\mM,x
\vDash\upvarphi),
$ and *valid* in $\frF$ iff it is true in all models $\mathfrak M$ on $\frF$. The *modal logic $\MLog(\frF)$ of the class frame $\frF$* consists of those $\upvarphi\in\MF$ that are valid in $\frF$. It can be verified that this logic is a set defined by a $\ZFC$-formula constructed from the formulas $\varPhi,\varPhi',\varPsi,\varUpsilon$, and it is normal. By using formulas with additional parameters, one can imitate higher order class algebras and their modal logics.

Logics of submodels {#sec:sub}
===================

In this part, we apply Theorem \[thm:maintoolnew\] to calculate the modal theory of the [*submodel relation*]{} on the class of all models of a given signature.

#### Expressing the satisfiability {#subsec:relativ}

Given models $\mathfrak A$ and $\mathfrak B$ of a signature $\Omega$, let $\mathfrak A\supmod\mathfrak B$ mean “$\mathfrak A$ contains $\mathfrak B$ as a submodel”. As the initial step, we find a model-theoretic language for expressing the $\sqsupseteq$-satisfiability. Observe first that first-order languages are generally too weak for this.

\[prop:submod-non-exprr\] If $\Omega$ contains a predicate symbol of arity $\ge2$, then the $\sqsupseteq$-satisfiability is not expressible in $L_{\omega,\omega}$ and moreover, in the infinitary language $L_{\infty,\omega}$.

We can suppose w.l.g. that $\Omega$ contains a binary predicate symbol $<$ (otherwise mimic it by a predicate symbol of a bigger arity by fixing other arguments). Toward a contradiction, assume that some $f$ mapping the class of $L_{\infty,\omega}$-sentences into itself expresses the $\sqsupseteq$-satisfiability. Let $\varphi$ be an obvious $L_{\omega,\omega}$-sentence saying that there exists a $<$-minimal element, and let $\psi$ be the sentence $\neg\,f(\neg\,\varphi)$. Then $\psi$ says that each submodel has a $<$-minimal element (thus whenever $\Omega$ has no functional symbols then $\psi$ says that $<$ is well-founded). Let $\kappa$ be such that $\psi\in L_{\kappa,\omega}$. It follows from Karp’s theorem (see, e.g., [@rosenstein1982linear Theorem 14.29]) that there are models $\mathfrak A_0$ and $\mathfrak B_0$ of $\Omega_0=\{<\}$ such that $\mathfrak A_0$ is isomorphic to an ordinal while $\mathfrak B_0$ is not, and $\mathfrak A_0\equiv_{L_{\kappa,\omega}}\!\mathfrak B_0$. Add a $<$-last element to each of the models $\mathfrak A_0$ and $\mathfrak B_0$ and check that the resulting models $\mathfrak A_1$ and $\mathfrak B_1$ remain $L_{\kappa,\omega}$-equivalent (e.g., by using [@rosenstein1982linear Lemma 14.24]).

Expand $\mathfrak A_1$ and $\mathfrak B_1$ to models $\mathfrak A$ and $\mathfrak B$ of $\Omega$, respectively, by interpreting each predicate symbol other than $<$ by the empty set, each functional symbol of positive arity by the projection onto the first argument, and each constant symbol by the $<$-last element of the model. It is easy to see that in both $\mathfrak A$ and $\mathfrak B$ any formula of $\Omega$ is equivalent to a formula of $\Omega_0$; so we still have $\mathfrak A\equiv_{L_{\kappa,\omega}}\!\mathfrak B$. On the other hand, in both models every subset forms a submodel whenever it contains the $<$-last element of the whole model, whence it easily follows that $\mathfrak A\vDash\psi$ and $\mathfrak B\vDash\neg\,\psi$. A contradiction.

However, second-order language suffices to express the $\sqsupseteq$-satisfiability by the relativization argument (see, e.g., [@QuantHand14-2007 p. 242]). Given a second-order formula $\varphi$ and a unary predicate variable $U$ that does not occur in $\varphi$, let $\varphi^U$ be the relativization of $\varphi$ to $U$ defined in the standard way; in particular, if $P$ and $F$ are second-order predicate functional variables of arity $n$, then

-   $(\exists P\,\varphi)^U$ is $
    \exists P\,
    \bigl(\forall x_0\ldots\forall x_{n-1}\,
    \bigl(P(x_0,\ldots,x_{n-1})\to
    \bigwedge_{i<n}U(x_i)\bigr)\wedge\varphi^U\bigr),
    $

-   $(\exists F\,\varphi)^U$ is $
    \exists F\,
    \bigl(\forall x_0\ldots\forall x_{n-1}\,
    \bigl(\bigwedge_{i<n}U(x_i)\to
    U(F(x_0,\ldots,x_{n-1}))\bigr)
    \wedge\varphi^U\bigr).
    $

Let $\psi(U)$ be the formula expressing that $U$ is a submodel, i.e., saying that the interpretation of $U$ is non-empty and is closed under interpretations of functional symbols in $\Omega$. Then the map $\varphi\mapsto\exists U(\psi(U)\wedge\varphi^U)$ expresses the $\sqsupseteq$-satisfiability on the class of all models of $\Omega$. In view of Proposition \[prop:extendingexpr\], we obtain:

\[prop:submod-quot-exprr\] Let $\kappa=|\{F\in\Omega:F$ is a functional symbol$\}|.$

-   Let $\mathcal C$ be a class of models of $\Omega$ closed under submodels. Then the $\sqsupseteq$-satisfiability is expressible on $\mathcal C$ in $L^{2}_{\lambda,\omega}$ whenever $\lambda>\kappa$ and $\lambda\ge\omega$.

-   Let $T$ be a set of sentences of $L^{2}_{\mu,\omega}$ of $\Omega$. Then the $\sqsupseteq$-satisfiability is expressible on the class $Mod(T)$ in $L^{2}_{\lambda,\omega}$ whenever $\lambda>\max(\kappa,|T|)$ and $\lambda\geq\mu$.

These results on expressibility can be refined in several directions. In particular, the first statement of Proposition \[prop:submod-quot-exprr\] remains true for monadic language $L^{2}_{\lambda,\omega}$; the assumption $\lambda>\kappa$ is necessary; for details and further results, see [@Saveliev2019].

#### Axiomatization

Henceforth in this section we assume that $L$ expresses the $\supmod$-satisfiability on the class of models under consideration.

The next easy result is soundness for modal theories of the submodel relation.

\[thm:sqsup-soundness\] Let $\clC$ be a class of $\Omega$-models closed under submodels. Then $\MTh^L(\clC,\supmod)$ is a normal modal logic including $\mathrm{S4}$. If moreover, $\Omega$ contains a constant symbol, then $\MTh^L(\clC,\supmod)$ includes $\mathrm{S4.2.1}$.

Let $\stA\in\clC$, $\vl$ a valuation in $L$. Trivially, $\stA\mo{ \| \Di\Di \mathsf{p}\imp \Di \mathsf{p} \|}$ and $\stA\mo{ \| \mathsf{p}\imp \Di \mathsf{p} \|}$. If $\Omega$ contains a constant symbol, consider the submodel $\stB$ of $\stA$ generated by constants. It is straightforward that in this case ${\stA\mo { \| \Box\Di \mathsf{p} \|}} \tiff \stB\mo  { \| \Di \mathsf{p} \|} \tiff \stB\mo  { \| \Box  \mathsf{p} \|} \tiff
\stA \mo { \| \Di \Box \mathsf{p} \|}.$

We are going to prove completeness. Let $Q_n$ be the lexicographic product of $(n^{<n},\subseteq)$ (an $n$-ramified tree of height $n$) and $(n,n\!\times\!n)$ (a cluster of size $n$). Thus for $s,t\in n^{<n}$ and $i,j\in n$, in $Q_n$ we have $$(s,i)\le(t,j)
\;\Iff\;
s\subseteq t,$$ so $Q_n$ is a pre-tree which is $n$-ramified, has height $n$ and clusters of size $n$ at each point. Let also $Q_n'$ be the ordered sum of $Q_n$ and a reflexive singleton, thus $Q'_n$ adds to $Q_n$ an extra top element. The following fact is standard (see, e.g., [@CZ p. 563]).

\[prop:preetreeS4\] Let $\mvf$ be a modal formula. If $\mvf\notin{{{\mathrm{S4}}}}$, then $\mvf$ is not valid in $Q_n$ for some $n>0$. If $\mvf\notin{{{\mathrm{S4.2.1}}}}$, then $\mvf$ is not valid in $Q'_n$ for some $n>0$.

Let $\equiv$ be the $L$-equivalence. For a model $\mathfrak A$, let $\Sub(\stA)$ be the set of all its submodels, $\Sub(\stA)_\equiv$ abbreviate $\Sub(\stA)/{\equiv}$.

\[thm:models-An\] Let $\Omega$ have a functional symbol of arity $\geq 2$. For every positive $n<\omega$, there exists a model $\mathfrak A_n$ of $\Omega$ such that $$\bigl(\Sub(\mathfrak A_n)_\equiv,\sqsupseteq_\equiv\bigr)
\text{ is isomorphic to }
\left\{
\begin{array}{ll}
Q_n & \text{ if $\Omega$ has no constant symbols},
\\
Q'_n& \text{ otherwise}.
\end{array}
\right.$$

First, suppose that $\Omega$ has no constant symbols.

Without loss of generality we may assume that $\Omega$ has a binary functional symbol; we write $\,\cdot\;$ for it.

Fix $n\geq 1$. Let $X_n=n^{<n}\times \omega$. We define the model $\mathfrak{A}_n=(X_n,\cdot\,,\dots)$ of $\Omega$ as follows. Let $E$ be any injective map from $n^{<n}$ into $\omega$. For $s,t\in n^{<n}$ and $i,j\in\omega$, we put $$\begin{aligned}
(s,i)\cdot(t,j)=
\left\{
\begin{array}{ll}
(s,i+1)
&\text{if }s=t,\,i=j,
\\
(s^{\conc}({i\bmod n}),j)
&\text{if }s=t,\,j<i,\,|s|<n-1,
\\
(s,j+E(s))
&\text{if }s=t,\,j=i+1,\,i\equiv_n\!0,
\\
(\inf\{s,t\},\inf\{i,j\})
&\textrm{otherwise},
\end{array}
\right.\end{aligned}$$ where ${}^{\conc}$ denotes the concatenation, $|s|$ the length of $s$, $\bmod$ the remainder, and $\equiv_n$ the congruence modulo $n$. For other operations $F$ in $\mathfrak{A}_n$ we put $F((s,i),\ldots)=(s,i)$ and take the relations in $\mathfrak{A}_n$ to be empty.

For $(s,i)\in X_n$ let $
X_n(s,i)=
\{(t,j)\in X_n:(s,i)\preceq(t,j)\}
$, where we let $(s,i)\preceq(t,j)$ iff $s\subseteq t$ and $i\le j$.

\[lem:Aui\] An $X\subseteq X_n$ is the universe of a submodel of $\mathfrak{A}_n$ iff $X=X_n(s,i)$ for some $(s,i)\in X_n$.

Straightforward from the definition of $\mathfrak{A}_n$.

Let $\mathfrak{A}_n(s,i)$ be the submodel of $\mathfrak{A}_n$ with the universe $X_n(s,i)$.

\[lem:isom\] Let $(s,i),(s,j)\in X_n$. If $i\equiv_n\!j$, then the models $\mathfrak{A}_n(s,i)$ and $\mathfrak{A}_n(s,j)$ are isomorphic.

The map $(t,l)\mapsto(t,l+n)$ is an isomorphism between $\mathfrak{A}_n(s,i)$ and $\mathfrak{A}_n(s,i+n)$.

We define $p^0(x)$ as $x$, and $p^{k+1}(x)$ as $(p^{k}(x))\cdot (p^{k}(x))$. Then for $k<\omega$ $$\mathfrak{A}_n\vDash(t,j)=p^k(s,i)
\quad \text{iff}\quad s=t \text{ and } j=i+k.$$ For $s\in n^{<n}$, let $\varphi_s(x)$ be the following one-parameter formula: $$x\cdot p(x)=p^{E(s)+1}(x).$$

\[lem:x1\] Let $\mathfrak{A}$ be a submodel of $\mathfrak{A}_n$ and $(t,i)$ an element of $\mathfrak{A}$. Then $\mathfrak{A}\vDash\varphi_s(t,i)$ iff $s=t$ and $i\equiv_n\!0$.

We have $p(t,i)=(t,i+1)$. By the definition, $(t,i)\cdot(t,i+1)=(t,i+E(t)+1)$ if $i\equiv_n\!0$, and $(t,i)\cdot(t,i+1)=(t,i)$ otherwise.

\[lem:x2\] Let $\mathfrak{A}$ be a submodel of $\mathfrak{A}_n$. For every $s\in n^{<n}$, we have: [$\mathfrak{A}\vDash\exists x\,\varphi_s(x)$]{} iff $(s,i)$ is in $\mathfrak{A}$ for some $i\in\omega$.

The ‘only if’ part is immediate from Lemma \[lem:x1\]. For the ‘if’ part we use Lemmas \[lem:x1\] and \[lem:Aui\].

For $S\subseteq n^{<n}$, let $\chi_S$ be the sentence $
\bigwedge_{s\in S}\exists x\,\varphi_s(x)
\;\wedge\,
\bigwedge_{s\notin S}\neg\,\exists x\,\varphi_s(x),
$ and let $\chi_{\ge s}$ be $\chi_S$ for $S=\{t\in n^{<n}:s\subseteq t\}$.

\[lem:x3\] Let $\mathfrak{A}$ be a submodel of $\mathfrak{A}_n$. Then $\mathfrak{A}\vDash\chi_{\ge s}$ iff $\mathfrak{A}=\mathfrak{A}_n(s,i)$ for some $i\in\omega$.

Follows from Lemmas \[lem:x2\] and \[lem:Aui\].

Let $\psi(x)$ be the formula $\neg\,\exists y\,(x=p(y)).$ Then $$\mathfrak A_n(s,i)\vDash\psi(t,j) \text{ iff } j=i.$$

For $s\in n^{<n}$ and $k<n$, let $\chi_{s,k}$ be the following sentence: $$\exists x\,
\bigl(\varphi_s(p^{n-k}(x))\wedge\psi(x)\bigr)
\wedge
\chi_{\ge s}.$$

\[lem:character\] For every submodel $\mathfrak{A}$ of $\mathfrak{A}_n$ and every $k<n$, we have $\mathfrak{A}\vDash\chi_{s,k}$ iff $\mathfrak{A}=\mathfrak{A}_n(s,i)$ for some $i$ such that $i\equiv_n k$.

Follows from Lemmas \[lem:x3\] and \[lem:x1\].

\[lem:comp-important\] Let $\mathfrak{A},\mathfrak{B}$ be submodels of $\mathfrak{A}_n$. The following are equivalent:

1.  $\mathfrak{A}$ and $\mathfrak{B}$ are isomorphic,

2.  $\mathfrak{A}$ and $\mathfrak{B}$ are $L$-equivalent,

3.  $\mathfrak{A}$ and $\mathfrak{B}$ are elementarily equivalent, i.e., $L_{\omega,\omega}$-equivalent,

4.  $\mathfrak{A}=\mathfrak{A}_n(s,i)$ and $\mathfrak{B}=\mathfrak{A}_n(s,j)$ for some $s\in n^{<n}$ and $i,j<\omega$ such that $i\equiv_n j$.

The implications [(i)$\,\Imp\,$(ii)]{} and [(ii)$\,\Imp\,$(iii)]{} are immediate from our basic assumptions on model-theoretic languages. The crucial step [(iii)$\,\Imp\,$(iv)]{} follows from Lemmas \[lem:Aui\] and \[lem:character\]. Finally, [(iv)$\,\Imp\,$(i)]{} holds by Lemma \[lem:isom\].

From Lemma \[lem:Aui\] we conclude that $(\Sub(\mathfrak{A}_n),\sqsupseteq)$ is isomorphic to $(X_n,\preceq)$. Now it follows that $(\Sub(\mathfrak{A}_n)_\equiv,\sqsupseteq_\equiv)$ is isomorphic to $Q_n$, as required.

For the case when $\Omega$ has constant symbols, we add a new element $c$ to $X_n$ and define the model $\mathfrak{A}'_n$ on the set $X_n\cup\{c\}$. We extend the above defined operation $\cdot$ by letting $c\cdot x=x\cdot c=c$ for all $x$; all constant symbols in $\Omega$ are interpreted by $c$. The same arguments as above prove that $(\Sub(\mathfrak{A}'_n)_\equiv,\sqsupseteq_\equiv)$ is isomorphic to $Q'_n$.

This completes the proof of Theorem \[thm:models-An\].

Now the completeness result follows:

\[thm:sqsup-completeness\] Let $\Omega$ contain a functional symbol of arity $\ge 2$, and let $\mathcal C$ be the class of all models of $\Omega$. Then $$\mathrm{MTh}^L(\mathcal C,\sqsupseteq)=
\left\{
\begin{array}{ll}
\mathrm{S4}
&\text{ if $\Omega$~has no constant symbols},
\\
\mathrm{S4.2.1}
&\text{ otherwise}.
\end{array}
\right.$$

We have soundness by Theorem \[thm:sqsup-soundness\]. On the other hand, if $\stA\in \clC$, then $\mathrm{MTh}^L(\mathcal C,\sqsupseteq)$ is contained in the logic of the Kripke frame $(\Sub(\stA)_\equiv,\sqsupseteq_\equiv)$ by Corollaries \[cor:gener\] and \[prop: finite frame\]. Now completeness follows from Theorem \[thm:models-An\].

Notice that the binary operation used in the proof of Theorem \[thm:models-An\] is not associative. Modal axiomatizations of $\sqsupseteq$ in second-order language on semigroups, monoids, and groups are open questions.

Recall that Theorem \[thm:sqsup-completeness\] was formulated under the assumption that $L$ expresses the $\sqsupseteq$-satisfiability. In the next section we discuss how to define modal theories without such requirements. The logics calculated in Theorem \[thm:sqsup-completeness\] do not depend on $L$, while in general, modal theories depend on a chosen model-theoretic language; we shall discuss this in Section \[sec:rob\].

Inexpressible modalities {#sec:upanddown}
========================

In this section, we discuss the situation when the $\clR$-satisfiability is not expressible in a model-theoretic language.

#### Definition

The following question was raised by one of the reviewers on an earlier version of the paper: what is the modal theory of $(\clC,\clR)$ if the $\clR$-satisfiability on $\clC$ is not expressible in a given language? In particular, what is the modal theory of the relation $\supmod$ in the first-order case? Another natural question concerns the definition of modal theories in the case of $\submod$.

Assume that $L$ is a language stronger than $K$, and the $\clR$-satisfiability on $\clC$ is expressible in $L$. In this case, $\MTh^{K}(\clC,\clR)$ can be naturally defined as the logic of the subalgebra of the modal algebra of $L$ on $(\clC,\clR)$ generated by the sentences of $K$. Let us provide details.

First, we define an interim notion $\MTh^{\KL}(\clC,\clR)$. By Theorem \[thm:maintoolnew\], $\MTh^{L}(\clC,\clR)$ is the logic of the Lindenbaum modal algebra $\clA(L)=(L/{\approx},f^L_\approx)$. Let $\clA(\KL)$ be the subalgebra of $\clA(L)$ generated by the sentences of $K$ (more formally, $\clA(\KL)$ is generated by the set $\{[\vf]^L_\approx: \vf\in K_s\}$, where $[\vf]^L_\approx=\{\psi\in L_s: \clC\mo \vf\lra\psi\}$). Define $\MTh^{\KL}(\clC,\clR)$ as the modal logic of $\clA(\KL)$. (Formally, we have defined $\MTh^{\KL}(\clC,\clR)$ for the case $K\subseteq L$, but the same construction works for the case when $K$ is weaker than $L$ in $\clC$, i.e., if every $K$-definable subclass of $\clC$ is definable in $L$.)

It is immediate that $\MTh^{\KL}(\clC,\clR)$ is a normal logic, which includes $\MTh^L(\clC,\clR)$. Another simple (but important) observation is that $\MTh^{\KL}(\clC,\clR)$ does not depend on the choice of $L$. Indeed, if $M$ is another language stronger than $K$, which expresses the $\clR$-satisfiability on $\clC$, then the algebras $\clA(\KL)$ and $\clA({M[K]})$ are isomorphic: their elements can be thought as classes of models obtained from definable in $K$ subclasses of $\clC$ via Boolean operations and $\clR^{-1}$. Thus we have

\[prop:unspeak\] If languages $L$ and $M$ express the $\clR$-satisfiability on $\clC$ and are stronger than $K$, then:

1.  the algebras $\clA(\KL)$ and $\clA({M[K]})$ are isomorphic, and so

2.  $\MTh^{\KL}(\clC,\clR)=\MTh^{{M[K]}}(\clC,\clR)$.

\[def:secondMain\] If the $\clR$-satisfiability on $\clC$ is expressible in some language $L$ stronger than $K$, put $\MTh^K(\clC,\clR)=\MTh^{\KL}(\clC,\clR)$.

Trivially, $\MTh^{{L[L]}}(\clC,\clR)=\MTh^L(\clC,\clR)$ whenever $L$ expresses the $\clR$-satisfiability on $\clC$. Thus this definition generalizes Definition \[def:exprMTH\]. Note that, however, the modal logic $\MTh^K(\clC,\clR)$ is not necessarily a “fragment” of the theory $\Th^K(\clC)$ anymore.

Once we do not require the $\clR$-satisfiability to be expressible in the language, we can give an “external” definition of modal theory and modal algebra for $\clC$, $\clR$, and $K$. Let $K_\Di$ be the set of modal [*$K$-sentences*]{}, which are built from sentences of $K$ using Boolean connectives and $\Di$; namely, they are expressions of form $\mvf(\psi_1,\ldots,\psi_n)$ where $\mvf(\mathsf p_1,\ldots,\mathsf p_n)$ is a modal formula and $\psi_i$ are sentences of $K$. Such languages are regularly used in the context of modal logics of relations between models of arithmetic or set theory.[^5] Given $\clC$ and $\clR$, the $K_\Di$-satisfaction relation is defined in the straightforward way; in particular, $\stA\mo\Di\mvf(\psi_1,\ldots,\psi_k)$ iff $\stB\mo\mvf(\psi_1,\ldots,\psi_k)$ for some $\stB\in\clC$ with $\stA\,\clR\,\stB$. Assume that $\clR$ and $\clC$ satisfy the following natural condition: for all $\stA,\stA',\stB$ in $\clC$, $$\label{eq:isom-bisim}
\stA\isom\stA'\;\&\;\stA\:\clR\:\stB
\;\Rightarrow\;
\EE\stB'\in\clC\;
(\stB'\isom\stB\;\&\;\stA'\,\clR\,\stB')$$ (this condition says that the isomorphism equivalence $\isom$ is a [*bisimulation*]{} w.r.t. $\clR$ on $\clC$). In this case the $K_\Di$-satisfaction relation is preserved under isomorphisms, and so $K_\Di$ satisfies our basic assumptions on model-theoretic languages.

The operation on sentences of $K_{\Di}$ that takes $\vf$ to $\Di\vf$ expresses the $\clR$-satisfiability on $\clC$ in $K_{\Di}$. Hence, we can apply the constructions described in Section \[sec:defs\] to $K_\Di$. In particular, $\vf\approx \psi$ implies $\Di\vf\approx\Di\psi$ (recall that $\vf\approx\psi$ means $\clC\mo\vf\lra\psi$), so the connective $\Di$ induces the operation $\Di_\approx$ on $K_\Di/{\approx}$, and $\clA(K_\Di)=(K_\Di/{\approx},\Di_\approx)$ is a modal algebra. It is immediate that $\clA(K_\Di)=\clA(K_\Di[K])$, and hence $$\MTh^{K}(\clC,\clR)=\MTh^{K_\Di}(\clC,\clR)$$ by Definition \[def:secondMain\]. Therefore, assuming that $\clC$ and $\clR$ satisfy (\[eq:isom-bisim\]), we obtain the following generalization of our previous definitions for arbitrary $K$:

\[def:external-sat\] Let $\clC$ and $\clR$ satisfy (\[eq:isom-bisim\]). The [*modal (Lindenbaum) algebra of a language $K$ on $(\clC,\clR)$*]{} is the modal algebra $({K_\Di/{\approx}},\Di_\approx)$. The [*modal theory $\MTh^K(\clC,\clR)$ of $(\clC,\clR)$ in $K$*]{} is the modal logic of this algebra.

Let us emphasize that natural classes of models with relations between them always satisfy (\[eq:isom-bisim\]); in particular, so are the instances discussed in our paper. One might say that $\clR$ is a [*model theoretic relation on*]{} $\clC$ iff property (\[eq:isom-bisim\]) holds. E.g., the submodel relation on the class of models of a given signature is model-theoretic; hence, Theorem \[thm:sqsup-completeness\] remains true for arbitrary model-theoretic language.

#### Downward Löwenheim–Skolem theorem

Our next result provides a version of the downward Löwenheim–Skolem theorem for first-order language enriched with the modal operator for the extension relation between models.

For a cardinal $\kappa$, put $\clC_{\leq \kappa}=\{\stA\in\clC:|\stA|\leq\kappa\}$.

\[thm:submotapprox\] Let $K$ be $L_{\omega,\omega}$ based on a signature $\Omega$, let $K_\Di$ be $K$ enriched with the modal operator for the extension relation on models $\sqsubseteq$, and let $\clC$ be an elementary (in $K$) class of models. For every $\kappa\geq\omega+|\Omega|$, the following statements hold:

1.  Let $X$ be a set of elements of $\stA\in\clC$, and let $\lambda$ be a cardinal such that $|X|+\kappa\leq\lambda\leq|\stA|$. Then $\stA$ has a submodel $\stB$ of cardinality $\lambda$ such that $\Th^{K_\Di}(\stA)=\Th^{K_\Di}(\stB)$ and $\stB$ contains $X$.

2.  $
    \{\Th^{K_\Di}(\stA):\stA\in\clC\}=
    \{\Th^{K_\Di}(\stA):\stA\in\clC_{\leq\kappa}\}.
    $

3.  The modal algebras of $K$ on $(\clC,\submod)$ and on $(\clC_{\leq\kappa},\submod)$ coincide.

4.  $
    \MTh^K(\clC,\submod)=
    \MTh^K(\clC_{\leq \kappa},\submod).
    $

First, we provide a general observation about any $\clC$ and $\clR$ satisfying the condition (\[eq:isom-bisim\]), and arbitrary $K$. Let $\equiv$ be the $K$-equivalence on $\clC$, and let $\equiv_\Di$ be the $K_\Di$-equivalence on $\clC$. Hence, $\stA\equiv\stB$ means that $\Th^K(\stA)=\Th^K(\stB)$, and $\stA\equiv_\Di\stB$ that $\Th^{K_\Di}(\stA)=\Th^{K_\Di}(\stB)$. We recall that $\Th^{K_\Di}(\stA)$ depends not only on $\stA$ but also on $\clC$ and $\clR$.

\[prop:equive-str\] Assume that $\clC$ and $\clR$ satisfy  (\[eq:isom-bisim\]). Let $\equiv$ be a bisimulation w.r.t. $\clR$ on $\clC$, i.e., for all $\stA,\stA',\stB\in\clC$ we have $$\label{eq:equive-bisim}
\stA\equiv\stA'\;\&\;\stA\:\clR\:\stB
\;\Rightarrow\;
\EE\stB'\in\clC\;
(\stB'\equiv\stB\;\&\;\stA'\,\clR\,\stB').$$ Then $\equiv$ and $\equiv_\Di$ coincide on $\clC$.

By the standard bisimulation argument: an easy induction on the construction of $\mpsi(\mpv_1,\ldots,\mpv_n)$ shows that, whenever $\stA\equiv\stA'$ and $\vf_1,\ldots,\vf_n\in K_s$ then we have $
\stA\mo\mpsi(\vf_1,\ldots,\vf_n)
\,\Iff\,
\stA'\mo\mpsi(\vf_1,\ldots,\vf_n).
$

\[prop:equive-bisim\] Let $K=L_{\omega,\omega}$. If $\clC$ is an elementary class, then $\equiv$ is a bisimulation w.r.t. $\sqsubseteq$ on $\clC$.

Pick $\stA,\stA',\stB$ in $\clC$. If $\stA\equiv\stA'$, then there exists an ultrafilter $D$ such that the ultrapowers $\stA_D$ and $\stA'_D$ are isomorphic, due to the Keisler–Shelah isomorphism theorem (see, e.g., [@chang1990model Theorem 6.1.15]). If $\stA\submod\stB$, then $\stA_D$ is embeddable in $\stB_D$, the ultrapower of $\stB$. Since $\stA'$ is embeddable in $\stA'_D$, we obtain that $\stA'$ is embeddable in $\stB_D$, and thus $\stA'\submod\stB'$ for some $\stB'\isom\stB_D$. But then we have $\stB'\in\clC$ and $\stB'\equiv\stB$, which completes the proof.

By Propositions \[prop:equive-str\] and \[prop:equive-bisim\], $\equiv$ and $\equiv_\Di$ coincide on $\clC$, i.e., for all $\stA',\stA$ in $\clC$ we have: $$\label{eq:cool1}
\text{$\stA'$ and $\stA$ are $K$-equivalent}
\;\Iff\;
\textrm{$\stA'$ and $\stA$ are $K_\Di$-equivalent}.$$ Now the first statement of the theorem follows from the downward Löwenheim–Skolem theorem for the first-order case: $\stA$ has an elementary submodel $\stB$ of cardinality $\lambda$ such that $\stB$ contains $X$ (see, e.g., [@chang1990model Theorem 3.1.6]); hence $\stA\equiv_\Di\stB$ by (\[eq:cool1\]).

Let $(\Ths,\submod_\Ths,\clA)$ and $(\Ths^\kpp,\submod^\kpp_\Ths,\clA^\kpp)$ be the minimal frames of theories for $\clC$, $\submod$, $K_\Di$ and for $\clC_{\leq \kappa}$, $\submod$, $K_\Di$, respectively. Let us show that these two structures are equal.

We have $\Ths=\Ths^\kpp$, the second statement of the theorem, as an immediate corollary of the first one.

Suppose that $T_1\submod_\Ths T_2$, that is, $T_1=\Th^{K_\Di}(\stA)$ and $T_2=\Th^{K_\Di}(\stB)$ for some $\stA,\stB\in\clC$ with $\stA\submod\stB$. Then $\stA$ has a submodel $\stA'$ of cardinality $\leq\kappa$ with $\stA'\equiv_\Di\stA$ (indeed, if the cardinality of $\stA$ is less than $\kappa$ then we put $\stA'=\stA$, otherwise we use the first statement of the theorem). Likewise, $\stB$ has a submodel $\stB'$ of cardinality $\leq\kappa$ such that $\stB'\equiv_\Di\stB$ and $\stB'$ contains the universe of $\stA'$. Clearly, $\stA'$ is submodel of $\stB'$. It follows that $T_1\submod^\kpp_\Ths T_2$. Hence $\submod^\kpp$ includes $\submod$. The converse inclusion is obvious, so $\submod^\kpp$ equals $\submod$.

Let $\vf\in K_\Di$. Since $\Ths=\Ths^\kpp$, it is immediate that $\{T\in\Ths:\vf\in T\}=\{T\in\Ths^\kpp:\vf\in T\}$. Hence, $\clA^\kpp=\clA$.

Thus $
(\Ths,\submod_\Ths,\clA)=
(\Ths^\kpp,\submod^\kpp_\Ths,\clA^\kpp)
$. Now the third and consequently, the forth, statements are immediate from Definition \[def:external-sat\] and Theorem \[thm:minmax-ths\].

This result is based on the interplay between the downward Löwenheim–Skolem property of first-order language and the fact that $\equiv$ is a bisimulation w.r.t. $\sqsubseteq$. The property of $\equiv$ being a bisimulation w.r.t. a given $\mathcal R$ seems interesting enough per se. Contrary to the case of the relation $\submod$ (Proposition \[prop:equive-bisim\]), we have:

Let $\Omega$ contain a predicate symbol of arity $\ge2$. Then the (usual) elementary equivalence is not a bisimulation w.r.t. $\sqsupseteq$ on the class of models of $\Omega$.

Assume w.l.g. $\Omega$ contains a single binary predicate symbol ${\le}$, and let $\equiv$ denote $\equiv_{L_{\omega,\omega}}$. Let $\mathfrak A,\mathfrak A',\mathfrak B$ be the linearly ordered sets $\mathbb Q\cdot\mathbb Z$, $\mathbb Z$, $\mathbb Q$, respectively (where $\,\cdot\,$ denotes the lexicographical multiplication). We have $\mathfrak A\equiv\mathfrak A'$ (this fact can be established by using an Eurenfeucht–Fraïssé game, see, e.g., [@marker2002modelIntro Proposition 2.4.10]) and $\stB$ is embeddable in $\stA$. However, every $\mathfrak B'$ satisfying $\mathfrak B'\equiv\mathfrak B$ is a dense linearly ordered set without end-points. Clearly, no such $\mathfrak B'$ can at the same time be embeddable in $\mathfrak A'$.

Let us say that $\clR$ is [*image-closed under ultraproducts*]{} iff for every $\mathfrak A$, $(\mathfrak B_i)_{i\in I}$, and ultrafilter $D$ over $I$, if $\mathfrak A\,\clR\,\mathfrak B_i$ for all $i\in I$ then $\mathfrak A\,\clR\prod_D\mathfrak B_i$ (e.g., $\mathfrak A\,\mathcal R\,\mathfrak B$ may mean that, up to isomorphism, $\mathfrak B$ is an extension of $\mathfrak A$).

Let $\clC$ and $\clR$ satisfy $(\ref{eq:isom-bisim})$, let $\clR$ be image-closed under ultraproducts on $\clC$, and let $K=L_{\omega,\omega}$. If $\equiv$ and $\equiv_\Di$ coincide on $\clC$, then $\equiv$ is a bisimulation w.r.t. $\clR$ on $\clC$.

Let $\mathfrak A,\mathfrak A',\mathfrak B$ in $\clC$ be such that $\mathfrak A\equiv\mathfrak A'$ and $\mathfrak A\,\clR\,\mathfrak B$. Observe that for all $\varphi\in K_s$ we have: $$\begin{aligned}
\mathfrak B\vDash\varphi
&\;\Rightarrow\;
\mathfrak A\vDash\Di\varphi
\\
&\;\Rightarrow\;
\mathfrak A'\vDash\Di\varphi
\;\;\;(\text{since }\mathfrak A\equiv_\Di\mathfrak A'\,)
\\
&\;\Rightarrow\;
\mathfrak A'\,\clR\,\mathfrak B_\varphi
\;\&\;
\mathfrak B_\varphi\vDash\varphi
\text{ for some }\mathfrak B_\vf\in\clC.\end{aligned}$$ Let $T=Th^K(\mathfrak B)$. For any $\varGamma\in\mathcal P_{\omega}(T)$ choose a model $\mathfrak B_{\varGamma}$ in $\clC$ as in the observation above, i.e., such that $\mathfrak A'\,\clR\,\mathfrak B_{\varGamma}$ and $\mathfrak B_{\varGamma}\vDash\bigwedge\varGamma$. Pick any ultrafilter $D$ extending the centered family of sets $
S_\varphi=
\{\varGamma\in\mathcal P_{\omega}(T):
\varphi\in\varGamma\}
$ for all $\varphi\in T$ (i.e., a [*fine*]{} ultrafilter over $\mathcal P_{\omega}(T)$). Then the ultraproduct $\mathfrak B'=\prod_D\mathfrak B_\varGamma$ satisfies all $\varphi\in T$, thus $\mathfrak B'\vDash T$. Moreover, since $\clR$ is image-closed under ultraproducts, we have $\mathfrak A'\,\clR\,\mathfrak B'$. This completes the proof.

It suffices to assume that $\clR$ is image-closed under ultraproducts by ultrafilters over $|K|$ only. The proposition remains true for $K=L_{\kappa,\lambda}$ with any strongly compact $\kappa$.

Robustness and Kripke completeness {#sec:rob}
==================================

The logics of submodels calculated in Section \[sec:sub\] do not depend on choosing particular language (Theorem \[thm:sqsup-completeness\]). However, in general, modal theories depend on $L$.

\[ex:non-rob1\] Let $L$ be the first-order language $L_{\omega,\omega}$, $T$ the theory of dense linear orders without end-points. Trivially, the $\supmod$-satisfiability is expressible on $\clC=\Mod(T)$ in $L$: put $f(\vf)=\vf$. Then $\MTh^L(\clC,\supmod)$ contains the “trivial” formula $\mathsf{p} \leftrightarrow \Di \mathsf{p}$. Obviously, this formula is falsified if $L$ is second-order.

Henceforth we assume that $\clC$ and $\clR$ satisfy (\[eq:isom-bisim\]).

\[prop:language-monot\] Let $L$ and $K$ be two languages. Then $L\subseteq K$ implies $
\mathrm{MTh}^{L}(\mathcal C,\mathcal R)
\supseteq
\mathrm{MTh}^{K}(\mathcal C,\mathcal R).
$

Follows from Definition \[def:external-sat\], since on $(\clC,\clR)$, the modal algebra of $L$ is a subalgebra of the algebra of $K$.

One can think that $L$ describes the properties of $(\mathcal C,\mathcal R)$ in a robust way whenever the modal theory does not change under strengthening the language. Thus, we shall say that $\MTh^L(\mathcal C,\mathcal R)$ is [*robust*]{} iff for every language $K\supseteq L$ we have $$\mathrm{MTh}^{K}(\mathcal C,\mathcal R)=
\mathrm{MTh}^{L}(\mathcal C,\mathcal R).$$ Intuitively, the robust theory can be considered as a “true” modal logic of the model-theoretic relation $\mathcal R$ on $\mathcal C$.

By Definition \[def:external-sat\] and Theorem \[thm:maintoolnew\], modal theories are logics of general frames. The following construction shows that, under certain assumptions, robust theories are logics of Kripke frames (i.e., they are [*Kripke complete*]{}).

A relation $\clS$ is said to be [*image-set*]{} iff for every $\stA$ the image $\clS(\stA)=\{\stB:\stA\,\clS\,\stB\}$ is a set; $\clS$ is [*image-set on*]{} $\clC$ iff its restriction to $\clC$ is image-set. Recall that $\mathfrak A\isom\mathfrak B$ means that $\mathfrak A$ and $\mathfrak B$ are isomorphic, and that $\,[\stA]_\isom\clR_\isom[\stB]_\isom$ iff $
\exists\,\mathfrak A'\isom\mathfrak A\,
\exists\,\mathfrak B'\isom\mathfrak B\;
(\mathfrak A'\,\mathcal R\,\mathfrak B').
$ Let $\mathcal C_\isom$ abbreviate $\mathcal C/{\isom}$. Also, recall that $\clR^*(\stA)$ denotes $\bigcup_{n<\omega}\clR^n(\stA)$, the least $\clR$-closed $\clD\subseteq\clC$ containing $\stA$.

\[prop:sup-isom\] If $\clR_{\isom}$ is image-set on $\clC_\isom$, then

$\MTh^L(\clC,\clR) \supseteq
\,\bigcap\limits_{\mathfrak A\in\clC}
\MLog(\clR^*(\mathfrak A)_{\isom},\clR_\isom).$

Let $\stA\in\clC$. The inclusion $
\MTh^{L}(\clR^*(\stA),\clR)\supseteq
\MLog(\clR^*(\mathfrak A)_{\isom},\clR_\isom)
$ follows from (\[eq:quot\]) at the end of Section \[sec:defs\]. Now we apply Corollary \[cor:gener\].

\[thm: robust logicGOOD\] If $\clR_{\isom}$ is image-set on $\clC_\isom$, then

$\MTh^L(\clC,\clR)$ is robust iff $
\MTh^L(\clC,\clR)=
\bigcap\limits_{\mathfrak A\in\clC}
\MLog(\clR^*(\mathfrak A)_{\isom},\clR_\isom).
$

$(\Rightarrow)$ Put $\Lambda=\MTh^L(\clC,\clR)$. In view of Proposition \[prop:sup-isom\], we only have to show that $\Lambda$ is valid in the Kripke frame $(\clR^*(\mathfrak A)_{\isom},\clR_\isom)$ for every $\stA\in\clC$.

Fix $\stA\in\clC$ and put $Y=\clR^*(\stA)$. For a language $M$, let $({\Ths^M}\!,\clR_{\Ths^M},{\clA^M})$ be the minimal frame of its theories in the class $Y$.

Since $\clR_{\isom}$ is image-set on $\clC_\isom$, the quotient $Y_{\isom}$ is a set. Hence we can choose a language $K$ stronger than $L$ and such that the $K$-equivalence and the isomorphism relation coincide on $Y$. Observe that for every $M$ stronger than $K$, the Kripke frames $({\Ths^M}\!,\clR_{\Ths^M})$, $({\Ths^K}\!,\clR_{\Ths^K})$, and $(Y_{\isom},\clR_\isom)$ are isomorphic. We choose such $M$ that $$\begin{aligned}
&\AA\, T\in{\Ths^K}\:
\EE\,\vf_T\in M_s\:
\AA\,\stA\in Y\;
\bigl(\stA\mo\vf_T\,\Iff\,\Th^K(\stA)=T\bigr),
&\!\text{and moreover}
\\
&\AA\,\clS\subseteq{\Ths^K}\:
\EE\,\vf_\clS\in M_s\:
\AA\,\stA\in Y\;
\bigl(\stA\mo\vf_\clS\,\Iff\,\Th^K(\stA)\in\clS\bigr).&\end{aligned}$$ It follows that ${\clA^M}$ is $\mathcal P(\Ths^M)$, the powerset of $\Ths^M$, and so $(\Ths^M\!,\clR_{\Ths^M},{\clA^M})$ is a Kripke frame. By Theorem \[thm:maintoolnew\], $\MTh^M(Y,\clR)$ is the logic of this frame. Therefore, $\MTh^M(Y,\clR)$ is the logic of the Kripke frame $(Y_{\isom},\clR_\isom)$. Since $\Lambda$ is robust, we have $\Lambda = \MTh^M(\clC,\clR)$. By Corollary \[cor:gener\], $\Lambda \subseteq \MTh^M(Y,\clR)$.

$(\Leftarrow)$ Immediate from Propositions \[prop:language-monot\] and \[prop:sup-isom\].

Hence, when $\clR_{\isom}$ is image-set on $\clC_\isom$, this theorem describes a unique modal logic, the [*robust theory of $(\clC,\clR)$*]{}.

More generally, if $\mathcal K$ is a class of model-theoretic languages, a modal theory $\MTh^L(\mathcal C,\mathcal R)$ is *robust in* $\mathcal K$ iff it coincides with $\MTh^K(\mathcal C,\mathcal R)$ for every language $K\supseteq L$ in $\mathcal K$. It is easy to see that there exists at most one robust theory of $(\mathcal C,\mathcal R)$ whenever $\mathcal K$ is directed, and exactly one such theory whenever $\mathcal K$ is countably directed, in the sense that for every $\{K_n\in\mathcal K:n\in\omega\}$ there is $K\in\mathcal K$ which is stronger than any $K_n$.

Notice that even if $\clC$ is a set, Theorem \[thm: robust logicGOOD\] does not mean that the robust theory of $(\clC,\clR)$ is the logic of the Kripke frame $(\clC,\clR)$.

Let $\stA$ be the algebra $(2^{<\omega},\inf,{\mathrm l},{\mathrm r})$ with the binary operation $\inf(x,y)=x\cap y$ and the unary operations ${\mathrm l}(x)=x^\conc 0$, ${\mathrm r}(x)=x^\conc 1$, where ${}^{\conc}$ denotes the concatenation. It is easy to see that $(\Sub(\stA),\supmod)$, the structure of submodels of $\stA$, is isomorphic to the binary tree $\frT_2=(2^{<\omega},\subseteq)$. The modal logic of the Kripke frame $\frT_2$ is known to be ${{{\mathrm{S4}}}}$ (see [@GoldMink1980]). However, for every $L$, the modal theory $\MTh^L(\Sub(\stA),\supmod)$ is the trivial logic given by the axiom $\mathsf{p}\leftrightarrow\Di\mathsf{p}$ because every submodel of $\stA$ is isomorphic to $\stA$.

#### More completeness results {#par:morecompl}

We apply Theorem \[thm: robust logicGOOD\] to describe robust theories of the quotient and the submodel relations on certain natural classes.

The following fact most probably was known since 1970s.

\[prop:MedvedevTop\] The logic of the Kripke frame $(\mathcal P(\omega),\subseteq)$ is $\mathrm{S4.2.1}$.

It is not difficult to construct a family $\clV\subset\mathcal P(\omega)$ such that $\clV$ is downward-closed (i.e., $U_1\subseteq U_2\in\clV$ implies $U_1\in \clV$) and there exists a p-morphism of $(\clV,\subseteq)$ onto the binary tree $\frT_2=(2^{<\omega},\subseteq)$. (E.g., let $f$ be a bijection $\omega\to 2^{<\omega}$, and let $\clV$ consist of $U\subset \omega$ such that $f(U)$ is a finite chain in $\frT_2$; the required p-morphism takes $U$ to the greatest element of this chain.) This p-morphism obviously extends to the p-morphism of $(\clP(\omega),\subseteq)$ onto $\frT_2'$, the ordered sum of $\frT_2$ and a reflexive singleton. Since $\MLog(\frT_2')=\mathrm{S4.2.1}$ (see [@ShehtMink83]), it follows that ${\MLog(\mathcal P(\omega),\subseteq)}$ is included in $\mathrm{S4.2.1}$. The converse inclusion is clear.

\[thm:submRob\] Let $\Omega$ contain a functional symbol and a constant symbol, and let $\mathcal C$ be the class of all models of $\Omega$. Then the robust theory of $(\mathcal C,\sqsupseteq)$ is $\mathrm{S4.2.1}$.

Soundness is straightforward. Let us check the converse inclusion.

Let a signature $\Omega_0$ consist of a single unary functional symbol $F$; models of this signature are called [*unars*]{}. For any $n\in\omega{\setminus}\{0\}$, let $\mathfrak A_n$ be a cycle of length $n$, i.e., a one-generated unar of cardinality $n$ that satisfies $F^n(x)=x$. As easy to see, each $\mathfrak A_n$ has no proper submodels.

Let also $\mathfrak A$ be the disjoint sum of all these $\mathfrak A_n$. Clearly, $\mathfrak A$ is a countably generated unar, and any its submodel is given by a nonempty set $S\subseteq\omega$, and the map taking $S$ to the disjoint sum of $\mathfrak A_n$ for all $n\in S$ is an isomorphism between $(\mathcal P(\omega){\setminus}\{\emptyset\},\supseteq)$ and the frame $(Sub(\mathfrak A),\sqsupseteq)$ of submodels of $\stA$. Moreover, all submodels of $\mathfrak A$ are pairwise non-isomorphic, and hence, the Kripke frame $(Sub(\mathfrak A)_{\isom},\sqsupseteq_{\isom})$ is also isomorphic to $(\mathcal P(\omega){\setminus}\{\emptyset\},\supseteq)$.

Without loss of generality we may assume that $F$ is the only functional symbol in $\Omega$ (we can always mimic a unary functional symbol by any functional symbol of arity $\geq 1$). Expanding $\Omega_0$ to $\Omega$, let $\mathfrak A'$ be the model of $\Omega$ obtained from $\mathfrak A$ by adding a single extra point $a$ which interprets all constant symbols and assuming $F(a)=a$. Then the Kripke frame $(Sub(\mathfrak A')_{\isom},\sqsupseteq_{\isom})$ is isomorphic to $(\mathcal P(\omega),\supseteq)$.

By Theorem \[thm: robust logicGOOD\], the robust theory of $(\mathcal C,\sqsupseteq)$ is included in the logic $\MLog(\mathcal P(\omega),\supseteq)$, which is $\mathrm{S4.2.1}$ by Proposition \[prop:MedvedevTop\].

It follows from the above proof that for the class $\mathcal C$ of unars without constant symbols, the robust theory of $(\mathcal C,\sqsupseteq)$ is included in the intersection of the logics of [*Skvortsov frames*]{} $(\clP(\kappa){\setminus}\{\emptyset\},\supseteq)$ for all $\kappa$. However, it is unclear how the “soundness” part of the proof can be obtained. This question is connected to a long-standing open problem about properties of the modal Medvedev logic, see, e.g., [@ShehtMedv1990].

\[rem:compareTheorems\] The proof of Theorem \[thm:submRob\] is much simpler than the proof of Theorem \[thm:sqsup-completeness\], and covers more signatures in the case with constants. However, it is unclear whether the completeness result holds for second- or first-order language. In other words, we do not know whether the resulting theories are robust in the case of second- or first-order language.

Let $\mathfrak B\le\mathfrak A$ mean that $\mathfrak B$ is a quotient of $\mathfrak A$ (for the definition of quotients of arbitrary models, see [@Malcev73 Section 2.4]), and $
Quot(\mathfrak A)=\{\mathfrak B:
\mathfrak B\le\mathfrak A\}.
$ Quotients are, up to isomorphism, images under strong homomorphisms, hence the logic of quotients is the same that the logic of strong homomorphic images.

\[thm:quot\] Let $\Omega$ have a functional symbol of arity $\ge1$, and $\mathcal C$ the class of all models of $\Omega$. Then the robust theory of $(\mathcal C,\geq)$ is $\mathrm{S4.2.1}$.

Soundness is straightforward. In particular, every model $\stA$ has a unique single-point quotient $\stB$, and for any valuation $\vl$ in $L$ we have: ${\stA\mo{ \| \Box\Di \mathsf{p} \|}}$ iff $\stB\mo{ \| \Di \mathsf{p} \|}$ iff $\stB\mo{ \| \Box \mathsf{p} \|}$ iff $\stA \mo { \| \Di \Box \mathsf{p} \|}.$

By Theorem \[thm: robust logicGOOD\], the robust theory of $(\mathcal C,\ge)$ is included in the logic $(Quot(\mathfrak A)_{\isom},\ge_{\isom})$ for any model $\stA$ of $\Omega$. We shall construct $\mathfrak A$ such that $\MLog(Quot(\mathfrak A)_\isom,\ge_\isom)\subseteq \mathrm{S4.2.1}$. It suffices to handle the case when $\Omega$ consists of a single unary functional symbol $F$. As in Theorem \[thm:submRob\], let $\mathfrak A_n$ be a one-generated unar forming the cycle of length $n$. It is easy to see that all quotients of $\mathfrak A_n$ are (up to isomorphism) exactly $\mathfrak A_m$ for $n$ divisible by $m$; in particular, if $p$ is prime, $Quot(\mathfrak A_p)$ is $\{\mathfrak A_1,\mathfrak A_p\}$.

Let $\mathfrak A$ be the disjoint sum of the unars $\mathfrak A_p$ for $p=1$ or $p$ prime. Let $P$ denote the set of primes. Up to isomorphism, every quotient $\stB$ of $\stA$ is the disjoint sum of unars $\{\stA_p:p\in S\}$ and also $n$ copies of $\stA_1$ for some $S\subseteq P$ and $n\leq 1+|P{\setminus}S|$; we put $h(\stB)=h([\stB]_\isom])=S$ and $g(\stB)=n$. We claim that $h$ is a p-morphism of $(Quot(\mathfrak A)_\isom,\ge_\isom)$ onto $(\clP(P),\supseteq)$. Surjectivity and monotonicity are straightforward. Assume that $h([\stB]_\isom)=S$ and pick any $S'\subseteq S$. Let $\stB'$ be the disjoint sum of unars $\{\stA_p:p\in S'\}$ and $g(\stB)$ copies of $\stA_1$. Then $\stB'$ is isomorphic to a quotient of $\stB$, and $h([\stB']_\isom)=S'$, as required.

It follows that $\MLog(Quot(\mathfrak A)_\isom,\ge_\isom)$ is included in $\MLog(\clP(P),\supseteq)$. By Proposition \[prop:MedvedevTop\], the latter logic is ${{{\mathrm{S4.2.1}}}}$, which completes the proof.

This theorem remains true for (not necessarily strong) homomorphic images as well. That ${{{\mathrm{S4.2.1}}}}$ includes the modal theory of homomorphisms follows from the proof above (the signature of $\stA$ does not have predicate symbols). And ${{{\mathrm{S4.2.1}}}}$ is sound since $\mathcal C$ has a single-point model that gives a top element in $\mathcal C/{\isom}$.

\[ren:quot\] Similarly to the case of the submodel relation, the quotient relation is expressible in an appropriate second-order language. Given $\Omega$ and $\lambda>|\{F\in\Omega:F$ is a functional symbol$\}|$, for every $\varphi$ of $L^{2}_{\lambda,\omega}$ in $\Omega$ we pick a fresh binary predicate variable $U$ and define $\varphi_U$ by induction on $\varphi$: if $t_0,\ldots,t_{n-1}$ are terms in $\Omega$ and $P(t_0,\ldots,t_{n-1})$ is an atomic formula, then

-   $P(t_0,\ldots,t_{n-1})_U$ is $\exists x_0\ldots\exists x_{n-1}
    \bigl(\bigwedge_{i<n}U(x_i,t_i)
    \wedge P(x_0,\ldots,x_{n-1})\bigr)
    $

where $x_i$ are fresh first-order variables; if $F$ is a functional variable, then

-   $(\exists F\,\varphi)_U$ is $
    \exists F\,(\text{$U$~is a~congruence for~$F$}
    \wedge\varphi_U);
    $

we let the operation distribute w.r.t. Boolean connectives and quantifiers over first-order and predicate variables (e.g., $(\exists P\,\varphi\bigr)_U$ is $\exists P\,\varphi_U$ for every predicate variable $P$). If $\chi(U)$ says that the interpretation of $U$ is a congruence for interpretations of all functional symbols in $\Omega$, then the map $\varphi\mapsto\exists U(\chi(U)\wedge\varphi_U)$ expresses the $\geq$-satisfiability on $\Omega$-models.

However, in second-order language, we do not know whether the modal theory of quotients is ${{{\mathrm{S4.2.1}}}}$.

So far we have calculated modal theories only for classes consisting of all models of a given signature. In the following examples, we describe robust modal theories of classes consisting of models of a non-trivial theory; we only outline ideas and postpone complete proofs for a further paper.

Let $\mathcal D$ be the class of dense linearly ordered sets without end-points. It was observed in Example \[ex:non-rob1\] that the modal theory $\MTh^{L_{\omega,\omega}}(\mathcal D,\sqsupseteq)$ is trivial: this is the logic of a reflexive singleton axiomatized by the formula $\mathsf{p}\leftrightarrow\Diamond\mathsf{p}$. On the other hand, the robust theory of $(\mathcal D,\sqsupseteq)$ is $\mathrm{S4.2.1}$.

By the classical Cantor theorem, every two countable orders in $\mathcal D$ are isomorphic (see, e.g., [@rosenstein1982linear Theorem 2.8]). Therefore, $(\mathcal D_{\isom},\sqsupseteq_{\isom})$ has a top element (the order type of rationals), so its logic includes $\mathrm{S4.2.1}$ (in spite of the fact that we do not have constant symbols in the signature). To prove the converse inclusion, in view of Corollary \[cor:gener\], it suffices to show that S4.2.1 includes the logic of a generated subframe of the frame $(\mathcal D_{\isom},\sqsupseteq_{\isom})$.

Given $S\subseteq\mathbb R$, let $D_S=\{[X]_{\isom}:X\subseteq S$ is dense in $\mathbb R\}$. (Note that $[X]_{\isom}$ is the order type of $X$.) First we observe that, whenever $S\subseteq\mathbb R$ is dense, then $(D_S,\sqsupseteq_{\isom})$ forms an upper cone of $(\mathcal D_{\isom},\sqsupseteq_{\isom})$. Hence it suffices to find $S$ such that the logic of $(D_S,\sqsupseteq_{\isom})$ coincides with S4.2.1. For this, we use the following result by Sierpiński: there are two disjoint sets $E,F$ of reals both dense in $\mathbb R$, having cardinality $|E|=|F|=2^{\aleph_0}$, and such that $f(E)\not\subseteq E\cup F$ for any non-identity order embedding $f:\mathbb R\to\mathbb R$ (see, e.g., [@rosenstein1982linear Chapter 9, §2]). Let $S=E\cup F$, and for any dense subset $X$ of $S$, let $\pi([X]_\isom)=A$ if $X=E\cup A$ for some $A\subseteq F$, and $\pi([X]_\isom)=\emptyset$ otherwise. It can be verified that $\pi$ is a well-defined map and moreover, a p-morphism of $(D_S,\sqsupseteq_{\isom})$ onto $(\mathcal P(F),\supseteq)$. It induces a p-morphism of $(D_S,\sqsupseteq_{\isom})$ onto $(\mathcal P(\omega),\supseteq)$. Applying Proposition \[prop:MedvedevTop\], we conclude that $\MLog(D_S,\sqsupseteq_{\isom})$ is $\mathrm{S4.2.1}$, as required.

Let $\mathcal L$ and $\mathcal O$ be the classes of linearly ordered sets and partially ordered sets, respectively. The robust theories of $(\mathcal L,\sqsupseteq)$ and $(\mathcal O,\sqsupseteq)$ also coincide with $\mathrm{S4.2.1}$. To see this, we note that there exists a p-morphism of $(\mathcal L,\sqsupseteq)$ onto $(\mathcal D,\sqsupseteq)$ (condensing scattered segments, see, e.g., [@rosenstein1982linear Chapter 4]) and that $(\mathcal L,\sqsupseteq)$ is an upper cone of $(\mathcal O,\sqsupseteq)$. This induces the same relationships between $(\mathcal D_\isom\sqsupseteq_\isom)$, $(\mathcal L_\isom,\sqsupseteq_\isom)$, and $(\mathcal O_\isom,\sqsupseteq_\isom)$.

Let $\clC$ be the class of modal algebras. Similarly to the proof of Theorem \[thm:submRob\], one can show that the robust theory of $(\clC,\quot)$ is ${{{\mathrm{S4.2.1}}}}$. Namely, let $\frF$ be the disjoint sum of Kripke frames $\frF_n=(n,n\times n)$, and $\stA$ be the modal algebra of $\frF$. One can construct a p-morphism from $(Quot(\stA)_\isom,\quot_\isom)$ onto $(\clP(\omega),\supseteq)$. Thus, ${\mathrm{S4.2.1}}$ is the robust theory of the quotient relation on the class $\clC$ of modal algebras. We have the same axiomatization in the case when $\clC$ is the class of ${\mathrm{S4}}$-algebras or ${\mathrm{S5}}$-algebras, because $\stA$ is an ${{{\mathrm{S5}}}}$-algebra. (More generally, this holds if $\clC$ contains $\stA$ and is closed under quotients.)

The axiomatization in the case when $\clC$ is the class of Boolean algebras is an open question.

\[ex:freegr\] Let $\mathcal C$ be the class of free groups. Let $\mathbb F_\kappa$ be a $\kappa$-generated free group. All $\mathbb F_\kappa$ with $2\le\kappa\le\aleph_0$ are pairwise embeddable and non-isomorphic, while on all other $\mathbb F_\kappa$ their ordering by embedding coincides with their ordering by rank (the least cardinality of generators). Therefore, the Kripke frame $(\mathcal C_\isom,\sqsupseteq_\isom)$ has the top (corresponding to $\mathbb F_1$), a countable cluster immediate below the top (corresponding to $\mathbb F_\kappa$’s with $2\le\kappa\le\aleph_0$), and a structure isomorphic to $(Ord,\ge)$ (corresponding to $\mathbb F_\kappa$’s with $\kappa\ge\aleph_1$). By Theorem \[thm: robust logicGOOD\], it follows that the robust theory of $(\mathcal C,\sqsupseteq)$ is the modal logic of ordered sums $(\alpha,\geq)+S+S_0$ for $\alpha\in Ord$, a countable cluster $S$, and a singleton $S_0$. By the standard filtration technique, this logic has the finite model property (more precisely, we may assume that $\alpha$ and $S$ are finite) and decidable.

The class $\clC$ is closed under $\sqsupseteq$ as any subgroup of a free group is free by the classical Nielsen–Schreier theorem. Hence, $\sqsupseteq$-satisfiability on $\mathcal C$ is expressible in $L^{2}_{\omega,\omega}$ by Proposition \[prop:submod-quot-exprr\]. We do not know whether the modal theory of $\sqsupseteq$ on $\mathcal C$ in $L^{2}_{\omega,\omega}$ is robust.

We remark that in the first-order case the theory is not robust. Let $\equiv$ be the usual elementary equivalence. Observe that $\mathbb F_1\isom\mathbb Z\not\equiv\mathbb F_2$ (obvious), and $\mathbb F_\kappa\equiv\mathbb F_2$ for all $\kappa\ge2$ (by recently proved famous Tarski’s conjecture; see, e.g., [@ElementaryGroups Chapter 9]). It follows that $(\mathcal C_\equiv,\sqsupseteq_\equiv)$ is isomorphic to the ordinal $2$ (with the top corresponding to $\mathbb F_1$ and the bottom to other free groups). Moreover, the logic $\MTh^{L_{\omega,\omega}}(\clC,\sqsupseteq)$ is the logic of the ordinal 2. This follows from Corollary \[prop: finite frame\] in view of the following observation made by one the reviewers on an earlier version of the paper: the function $f:L_s\to L_s$ defined by letting $$f(\vf)=\left\{
\begin{array}{ll}
\top  & \text{ if }
\mathbb F_1\mo \vf,\\
\vf  & \text{ if }
\mathbb F_1\not\mo\vf\text{ but }\mathbb F_2\mo\vf,\\
\bot  & \text{ otherwise}
\end{array}
\right.$$ expresses the $\sqsupseteq$-satisfiability on $\clC$. This observation can be generalized as follows: if the elementary equivalence $\equiv$ is a bisimulation w.r.t. $\clR$ on $\clC$ and $\clC_\equiv$ is finite, then the $\clR$-satisfiability on $\clC$ is first-order expressible.

Even in the case of all models of a given signature, the axiomatization of the robust theory can be a very difficult problem. Consider the theories of the relation $\supmod$ on the class $\clC$ of all models of a signature $\Omega$ consisting of unary predicates and perhaps some constants.

The easiest is the degenerated case when $\Omega$ has no symbols other than constants. The theory of submodels on this class is ${\mathrm{Grz.3}}$, the Grzegorczyk logic with the linearity axiom. As well-known, this is the logic of conversely well-ordered sets, or of the set $\omega$ with the converse order, or else of all finite linearly ordered sets.

Furthermore, these theories include the logic ${\mathrm{Grz}}$ iff $\Omega$ contains finitely many predicates. We announce that in the case of $n<\omega$ predicates, the robust modal theory coincides with the modal logic of the direct product of $2^n$ copies of $\omega$ with the converse order if the signature has constants, and with the logic of this structure without the top element otherwise. These logics have the finite model property, decrease as $n$ grows, and their intersection coincides with $\MLog(\clP_{\omega}(\omega){\setminus}\{\emptyset\},\supseteq)$, the modal Medvedev logic. Despite this clear semantical description, no complete axiomatizations for these logics are known.

#### Acknowledgements.

We are grateful to the reviewers for their multiple suggestions and questions on earlier versions of the paper. We are also grateful to Lev Beklemishev, Philip Kremer, Fedor Pakhomov, Vladimir Shavrukov, Valentin Shehtman, Albert Visser for their attention to our work, useful comments, and discussions.

The work on this paper was supported by the Russian Science Foundation under grant 16-11-10252 and carried out at Steklov Mathematical Institute of Russian Academy of Sciences.

[^1]: The work on this paper was supported by the Russian Science Foundation under grant 16-11-10252 and carried out at Steklov Mathematical Institute of Russian Academy of Sciences.

[^2]: Modal algebras of theories are broadly used in provability logic, where they are called [*Magari*]{} (or [*diagonizable*]{}) [*algebras*]{}; they are known to keep a lot of information about theories containing arithmetic (see, e.g., [@ShavrukovPhD93]).

[^3]: A significant part of Sections \[sec:upanddown\] and \[sec:rob\] was motivated by reviewers’ questions on an earlier version of this paper.

[^4]: Let us also mention the logic of [*pure provability*]{} introduced by Buss in [@buss1990], which is not closed under the rule of substitution.

[^5]: In [@HamkinsArithmeticPotentialism2018], the language $K_\Di$ is called [*partial potentialist*]{}.
