Expressive Power and Intensional Operators

Abstract

In Entities and Indices, M. J. Cresswell argued that a first-order modal language can reach the expressive power of natural-language modal discourse only if we give to the formal language a semantics with indices containing infinite possible worlds and we add to it an infinite collection of operators ${{\varvec{actually}}}_n$ and $Ref _n$ which store and retrieve worlds. In the fourth chapter of the book, Cresswell gave a proof that the resulting intensional language, which he called ${\mathscr {L}}^*$, is as expressive as an extensional variant of it, called ${\mathscr {L}}$, which has full quantification over worlds. In both linguistics and philosophy, Cresswell’s book has been viewed as offering a compelling argument for preferring extensional systems in the study of natural language. In this paper, after providing a model-theoretic definition of the relation being as expressive as that can be applied to Cresswell’s languages ${\mathscr {L}}$ and ${\mathscr {L}}^*$, we show that the intensional language ${\mathscr {L}}^*$ is not as expressive as the extensional language ${\mathscr {L}}$. This result, we claim, undermines Cresswell’s argument to the effect that English modal discourse has the power of explicit quantification over worlds. Additionally, we show that ${\mathscr {L}}^*$ does become as expressive as ${\mathscr {L}}$ when we add Cresswell’s operator of universal modality $\square$ to ${\mathscr {L}}^*$, which provides an extra amount of expressive power. Recently, I. Yanovich has advocated a view that is similar to ours in important respects. At the end of the paper we offer a short discussion of his formalism.

1 Introduction

The present paper offers a detailed comparison of certain families of formal languages that are described in M. J. Cresswell’s book Entities and Indices.

In the first part of the book, Cresswell argues for the following theses:

• Modal Expressive Power (MEP): Modal discourse in natural language requires a semantics with the expressive power of full quantification over possible worlds.

• Temporal Expressive Power (TEP): Temporal discourse in natural language requires a semantics with the expressive power of full quantification over times.

Cresswell’s discussion of (MEP) and (TEP) has been quite influential in both linguistics and philosophy. The received view in these fields is that the facts about expressive power which Cresswell brought to light provide a rationale for preferring extensional variable-based approaches over intensional operator-based approaches in the study of natural languages.¹ To illustrate this point, let us consider how linguist A. Kratzer invokes Cresswell’s book in a discussion of the semantics of situations.

“Until the early seventies anaphoric reference to times and worlds in natural languages was believed to be constrained in precisely the way predicted by the evaluation index approach. The belief was challenged by work on temporal anaphora [...], however. Cresswell (1990) presented parallel arguments for modal anaphora, and showed more generally that natural languages have the full expressive power of object language quantification over worlds and times. Quantification over worlds or times is thus no different from quantification over individuals, and should be accounted for in the same way.

Exact analogues of Cresswell’s examples can be constructed to show that natural languages have the full expressive power of object language quantification over situations.

[...]

It is not hard to see that we can complicate such examples indefinitely, and that there would be no end to the number of evaluation indices needed. But that suggests that natural languages have the full power of object language quantification over situations. Quantification over situations is no different from quantification over individuals, then, as far as expressive power is concerned. Since natural languages have syntactically represented individual variables and it would be surprising if they used two different equally powerful quantification mechanisms, it seems to be at least a good bet that there are syntactically represented situation variables in natural languages [...] But then the situations quantified over or referred to [...] do not necessarily correspond to “unarticulated constituents”. They are syntactically represented, even though they might happen to be unpronounced.” Kratzer (2021, Section 5)

The argument that Kratzer offers in the passage is rather complex and, for simplicity, we omitted the specific examples that she examines.² We only want to highlight two points: (i) Kratzer concludes that there are situation-denoting variables in the syntax of natural languages and (ii) her argument for this conclusion relies crucially on Cresswell’s analysis of modal and temporal discourse in Cresswell (1990).

As we already mentioned, Kratzer is not the only theorist who has drawn substantive conclusions about the grammar of natural languages from the arguments given in Cresswell (1990, Part I). Philosophers of language and other linguists have done so too. In order to assess this trend in natural language semantics, a close inspection of Cresswell’s arguments for (MEP) and (TEP) is necessary.

His argument for (MEP) combines certain observations about English modal sentences with a formal proof involving two families of formal languages. Here is a structural representation of the argument³:

• Premise 1 Modal discourse in natural language requires the semantics of a modal language equipped (at the very least) with the operators ${{\varvec{actually}}}_n$ and $Ref _n$.

• Premise 2 Any modal language equipped with the operators ${{\varvec{actually}}}_n$ and $Ref _n$ is as expressive as a corresponding two-sorted language with full quantification over worlds.

• Conclusion (MEP): Modal discourse in natural language requires a semantics with the expressive power of full quantification over worlds.

Cresswell provides a parallel argument for (TEP) involving temporal analogues of Premise 1 and Premise 2. However, we will not consider Cresswell’s remarks about temporal discourse here. We will leave temporality aside and focus on modality.⁴

In a recent paper, I. Yanovich (2015) advocates the view that Cresswell’s argument for (MEP) is unsound. He claims that “Cresswell never proved the result that the subsequent linguistic and philosophical literature took him to have proven” (Yanovich, 2015, p. 67). Yanovich points out that Cresswell’s proof relies on the assumption that the kind of intensional modal language with actually$_{n}$/$Ref_{n}$ operators given in Cresswell (1990) (see Sect.  3.4) has an operator $\square$ of universal modality. According to Yanovich, this observation is crucial because the presence of $\square$ significantly increases the expressive power of the intensional language and its absence makes the intensional language less powerful than a system with explicit quantification over worlds. As we shall see later, Yanovich is right with respect to this point (Proposition 4.4.1 and Corollary 4.5.2). However, neither Cresswell nor Yanovich provide a formal definition of the relation being as expressive as, a task which seems crucial if one wants to prove or disprove (MEP). Yanovich considers a formalism slightly different from Cresswell’s. For this reason, a translation between the two formalisms is necessary in order to apply the results of Yanovich’s proofs to the formal languages of Cresswell (1990). One important advantage of the present paper is that we follow Cresswell’s formalism quite closely. This will enable us to examine his argument for (MEP) and show its drawbacks within his own formalism.

Our aim in this paper is to show, contrary to widespread opinion, that Cresswell’s argument for (MEP) is unpersuasive. The bulk of the paper will be devoted to disproving the claim made in Premise 2 by looking in detail at the formal languages which Cresswell compares in Cresswell (1990, Chapter 4). For the sake of exposition, in Sect. 1 we will briefly take a look at Cresswell’s case for Premise 1 (Cresswell scholars can safely skip Sect. 1 and move directly to Sect. 2). Since Premise 2 relies on a comparison of certain formal languages in terms of their expressive power, in Sect. 2 we provide a formal definition of the relation being as expressive as which fits Cresswell’s formal apparatus and gives us a basis for comparing the expressivity of Cresswell’s formal languages in a rigorous manner. Armed with this definition, we characterize Cresswell’s languages in Sect. 3 and disprove Premise 2 in Sect. 4 (Proposition 4.4.1). Here, what is essential in order to prove our main claim (Proposition 4.4.1) is that some of the intensional languages that Cresswell discusses have a local notion of truth with respect to worlds (see Lemmas 4.2.4 and 4.4.2), that is, the truth of a sentence does not depend on all the worlds, but only on a finite subset of them. It is this feature which makes some of these languages less expressive than their extensional counterparts. In Sect. 5.1, after giving a summary of our arguments, we asses an alternative reading of Cresswell’s argument for (MEP). In our final Sect. 5.2 we make a few remarks about Yanovich’s work in light of our discussion.

2 English Modal Discourse and Actually $_{n}$/$Ref_{n}$-Operators

Cresswell’s strategy to establish Premise 1 consists in arguing that an intensional treatment of English modal discourse requires a semantics with infinite sequences of worlds and a collection of operators actually$_{n}$/$Ref_{n}$ that shift those sequences. In this section we take a look at his discussion of this point in Chapter 3 of Cresswell (1990).

At the beginning of the chapter, Cresswell considers the following sentence:

(S1):

It might have been (the case) that everyone (who is) actually rich was poor.

According to the standard possible-world analysis of modals, the English modal “might” is associated with an accessibility relation between possible worlds. Let R be a binary predicate which denotes that relation and let $w_{0}$ be a constant that denotes the actual world. With these symbols, the truth-condition of (S1) can be specified as follows:

(S1)’s truth-condition:

There is some world $w_{1}$ such that $w_{0}Rw_{1}$—i.e. $w_{1}$ is accessible from the actual world—and, for every x, if x is rich in $w_{0}$, then x is poor in $w_{1}$.

In a system of two-sorted first-order logic equipped with variables that refer to individuals and variables that refer to worlds it is easy to provide a formula that expresses this truth-condition:

(S2):

$\exists w_{1}$($w_{0}Rw_1$ $\wedge$ $\forall x$($x \textit{ is rich in } w_{0}$ $\supset$ $x \textit{ is poor in } w_{1}$)).

The formula (S2) is a symbolization of (S1) that relies on explicit quantification over worlds.

Now, is it possible to symbolize (S1) in a modal language in which there are no world-variables? Let M be a modal operator that means ‘It might be the case that’. As Cresswell correctly points out, the formulas (S3) and (S4) of a standard system of first-order modal logic do not capture the truth-condition of (S1):

(S3):

M $\forall x$($x \textit{ is rich}$ $\supset$ $x \textit{ is poor}$)

(S4):

$\forall x$($x \textit{ is rich}$ $\supset$ M $x \textit{ is poor}$).

We can get the desired truth-condition if we introduce an actuality operator and adopt a semantics in which formulas are true with respect to pairs of worlds. In this sort of double-world system, (S1) can be symbolized as (S5):

(S5):

M $\forall x$(${{{\varvec{actually}}}}_1$ $x \textit{ is rich}$ $\supset$ $x \textit{ is poor}$).

In order to show how the system works, below we offer provisional semantic clauses for the operators M and ${{{\varvec{actually}}}}_1$, and then we give a derivation of (S1)’s truth-condition from (S5).

• A formula of the form $M \varphi$ is true at a pair of worlds $(w_{1}, w_{2})$ if and only if there is some world $w_{3}$ such that $w_{1}Rw_{3}$ and $\varphi$ is true at the pair of worlds $(w_{3}, w_{2})$.

• A formula of the form ${{{\varvec{actually}}}}_1 \varphi$ is true at a pair of worlds $(w_{1}, w_{2})$ if and only if $\varphi$ is true at the pair of worlds $(w_{2}, w_{2})$.

As we will see in Sect. 3, Cresswell adopts a formal framework in which formulas are interpreted by assigning them propositions (sets of semantical indices). For this reason, the semantic clauses that one finds in Cresswell (1990) do not look exactly as the two clauses that we gave above. Since formal precision is not a goal of this section, our discussion of Premise 1 will remain relatively informal for the sake of exposition.

To derive the truth-condition of a formula in the double-world system, we assess the formula with respect to the pair $(w_{0}, w_{0})$. Let us do this for formula (S5):

• The formula $M \forall {\varvec{x}}$(${{{\varvec{actually}}}}_1$ ${\varvec{x}}\,\,{{\textit{\textbf{is rich}}}}$ $\supset$ ${\varvec{x}}\,\,{{\varvec{is poor}}}$) is true at the pair $(w_{0}, w_{0})$

• $\Leftrightarrow$ there is some world $w_1$ such that $w_0Rw_1$ and the subformula $\forall {\varvec{x}}$(${{{\varvec{actually}}}}_1$ ${\varvec{x}}\,\,{{\varvec{is rich}}}$ $\supset$ ${\varvec{x}} {{\varvec{ is poor}}}$) is true at $(w_1,w_0)$. [by the semantic clause of M]

• $\Leftrightarrow$ there is some world $w_1$ such that $w_0Rw_1$ and, for every x, the subformula ${{{\varvec{actually}}}}_1$ ${\varvec{x}} {{\varvec{ is rich}}}$ $\supset$ ${\varvec{x}} {{\varvec{ is poor}}}$ is true of x at $(w_1,w_0)$.

• $\Leftrightarrow$ there is some world $w_1$ such that $w_0Rw_1$ and, for every x, if the subformula ${{{\varvec{actually}}}}_1$ ${\varvec{x}} {{\varvec{ is rich}}}$ is true of x at $( w_1,w_0)$, then the subformula ${\varvec{x}} {{\varvec{ is poor}}}$ is true of x at $(w_1,w_0)$.

• $\Leftrightarrow$ there is some world $w_1$ such that $w_0Rw_1$ and, for every x, if the subformula ${\varvec{x}} {{\varvec{ is rich}}}$ is true of x at $(w_0,w_0)$, then the subformula ${\varvec{x}} {{\varvec{ is poor}}}$ is true of x at $(w_1,w_0)$. [by the semantic clause of ${{{\varvec{actually}}}}_1$]

• $\Leftrightarrow$ there is some world $w_1$ such that $w_0Rw_1$ and, for every x, if x is rich in $w_0$, then x is poor in $w_1$.

The last step of the derivation is based on the convention that the first member of a given world-pair plays the role of the world of evaluation of the relevant atomic subformula. In the first step of the derivation, the operator M shifts the world-pair $(w_0,w_0)$ to a world-pair $(w_1,w_0)$, where $w_1$ is a world accessible from the actual world. The pair $(w_1,w_0)$ keeps a copy of the actual world. In the fourth step, the operator ${{{\varvec{actually}}}}_1$ shifts the world-pair $(w_1,w_0)$ to the world-pair $(w_0,w_0)$. In other words, ${{{\varvec{actually}}}}_1$ retrieves the stored copy of the actual world and turns it into the world of evaluation of the subformula ${\varvec{x}} {{\varvec{ is rich}}}$. In this way, the subformula ${\varvec{x}} {{\varvec{ is rich}}}$ is evaluated at $w_0$ while the subformula ${\varvec{x}} {{\varvec{ is poor}}}$ is evaluated at $w_1$, which is what the truth-condition of (S1) requires.

Cresswell argues that the addition of the operator ${{{\varvec{actually}}}}_1$ is just the beginning of a slippery-slope. The key example that he offers in order to make this point is (S6):

(S6):

If you had written some novels and people had rumored that you hadn’t written any of the novels you actually had written, you would be upset.⁵

To symbolize (S6), we need a storing operator $Ref _1$. Just as one can define an operator ${{{\varvec{actually}}}}_1$ that retrieves a given world and turns it into the current world of evaluation, one can also define an operator $Ref _1$ that stores the current world of evaluation. Here is its semantic clause:

• A formula of the form $Ref _1 \varphi$ is true at a pair of worlds $(w_{1}, w_{2})$ if and only if $\varphi$ is true at the pair of worlds $(w_{1}, w_{1})$.

Cresswell does not provide a specific symbolization of (S6). Nonetheless, he explains how it must look like. (S6) requires a symbolization along the lines of (S7):

(S7):

($\exists x$ (${{\varvec{novel }}} x$ $\wedge$ ${{\varvec{you write }}} x$) $\wedge$ $Ref _1$ people rumors that $\forall y$ (${{\varvec{novel }}} y$ $\wedge$ ${{\varvec{actually}}}_1$ ${{\varvec{you write }}} y$) $\supset$ $\lnot {{\varvec{ you write }}} y$)) you are upset.

where is the counterfactual conditional operator (see Cresswell, 1990, p. 10).

He then points out that sentence (S8) requires a modal language whose formulas are true or false relative to triples of worlds, and which is equipped with storing and retrieving operators $Ref _1$, $Ref _2$, ${{\varvec{actually}}}_1$, and ${{\varvec{actually}}}_2$ (where $Ref _2$ and ${{\varvec{actually}}}_2$ are analogously defined).

(S8):

If you had written some novels that were better than any of the novels you have actually written, and people had rumored that you hadn’t written the novels you had written, but had written some novels that were worse than any of the novels you actually have written, you would be upset.

The sentence (S8) requires to adopt a triple-indexed semantics and, moreover, to introduce operators ${{\varvec{actually}}}_2$ and $Ref _2$ that store and retrieve a third possible world. In this system, (S8) may be symbolized as (S9):

(S9):

$Ref _1$(($\exists x$ (${{\varvec{novel }}} x$ $\wedge$ ${{\varvec{you write }}} x$) $\wedge$ $\forall y$(${{\varvec{actually}}}_1$ ${{\varvec{you write }}} y$ $\supset$ $x {{\varvec{ better than }}} y$) $\wedge$ $Ref _2$ people rumors that $\forall z$ (${{\varvec{novel }}} z$ $\wedge$ ${{\varvec{actually}}}_2$ ${{\varvec{you write }}} z$) $\supset$ $\lnot {{\varvec{you write }}} z$ $\wedge$ $\exists x'$ ((${{\varvec{novel }}} x'$ $\wedge$ $\forall y'$(${{\varvec{novel }}} y'$ $\wedge$ ${{\varvec{actually}}}_1$ ${{\varvec{you write }}} y'$ $\supset$ $x' {{\varvec{ worse than }}} y'$))))) you are upset).

Since one can continue generating more complex variants of these examples, Cresswell concludes that modal English discourse is as powerful as a multiply-indexed intensional language equipped with infinite storing and retrieving operators ${{\varvec{actually}}}_n$ and $Ref _n$ that act upon infinite sequences of worlds. In Sect. 3.4, we describe in detail the semantics of an intensional language of this kind. For the time being, we would like to stress that the modal formulas that we have considered in this chapter—formulas like (S5), (S7), and (S9)—only contain modal operators of two kinds. On the one hand, they contain ordinary modal operators such as M and . On the other hand, they contain ${{\varvec{actually}}}_n$ and $Ref _n$ operators. In particular, they do not use any universal modal operator (such as the operator of logical necessity $\square$ introduced in Sect. 3.3). This observation will be crucial later on (see Sect. 5.1 below).

So far we have considered Cresswell’s argument for Premise 1. Regarding Premise 2, Cresswell establishes a comparison between the expressive power of different languages. However, he does not provide formal definitions of the relation being as expressive as for the formal languages that he discusses. In the next section we propose a general strategy to characterize the notion of expressive power for formal languages of different kinds, trying to remain as close as possible to Cresswell’s own tacit treatment of that relation.

3 Formal Languages and Expressive Power

What does it take for a formal language to be as expressive as another formal language? More often than not, this relation is taken for granted in the logic literature and formal definitions of it are seldom given. This is the case, in particular, in Cresswell’s Entities and Indices. The relation in question plays a central role through the whole book without being ever formally defined. Our goal in this section is to provide a definition of the relation is as expressive as that can be applied to the formal languages discussed in Cresswell (1990, Chapters 1–4) and that accords with Cresswell’s implicit understanding of this relation. To this end, we will adopt a model-theoretic approach very much in the spirit of Epstein (1990), García-Matos and Väänänen (2005), and Kocurek (2018).

Let us start by fixing some notation and terminology which will be helpful for our purposes.

3.1 Formal Languages

We will not provide a formal definition of what formal languages are. However, in the spirit of García-Matos and Väänänen (2005) and Kocurek (2018), we will assume throughout this article that every formal language ${\mathscr {L}}$ determines an ordered triple $(S_{\mathscr {L}}, F_{\mathscr {L}}, \models _{\mathscr {L}})$, where $S_{\mathscr {L}}$ and $F_{\mathscr {L}}$ are classes and $\models _{\mathscr {L}}$ is a relation between $S_{\mathscr {L}}$ and $F_{\mathscr {L}}$. Members of the class $S_{\mathscr {L}}$ will be called ${\mathscr {L}}$-structures, members of the class $F_{\mathscr {L}}$ will be called ${\mathscr {L}}$-formulas, and the relation $\models _{\mathscr {L}}$ will be called the modelling relation of ${\mathscr {L}}$. Intuitively, given an ${\mathscr {L}}$-structure M and an ${\mathscr {L}}$-formula $\varphi$, the assertion that $M\models _{\mathscr {L}}\varphi$ seeks to capture the idea that the formula $\varphi$ holds in M or is true in M.⁶ Let us consider a few simple examples of propositional formal languages.

Example 2.1.1

Let $P=\{p,q\}$ be a set of propositional letters. Consider a language ${\mathscr {L}}_1$ with the following characteristics:

The class $F_{{\mathscr {L}}_1}$ of ${\mathscr {L}}_1$-formulas of is recursively defined by

• if $\varphi \in P$, then $\varphi \in F_{{\mathscr {L}}_1}$;

• if $\varphi ,\psi \in F_{{\mathscr {L}}_1}$, then $(\varphi \wedge \psi )\in F_{{\mathscr {L}}_1}$.

The class $S_{{\mathscr {L}}_1}$ of ${\mathscr {L}}_1$-structures is the set of functions $M:P\rightarrow \{1,0\}$, where 1 and 0 respectively stand for the truth-values truth and falsity.

Finally, the modeling relation of ${\mathscr {L}}_1$—in symbols, $\models _{{\mathscr {L}}_1}$—is a relation on $S_{{\mathscr {L}}_1} \times F_{{\mathscr {L}}_1}$ defined by induction on ${\mathscr {L}}_1$-formulas: for any $M \in S_{{\mathscr {L}}_1}$ and $\varphi \in F_{{\mathscr {L}}_1}$,

• if $\varphi \in P$, then $M \models _{{\mathscr {L}}_1}\varphi$ if and only if $M(\varphi ) = 1$;

• if $\varphi$ is of the form $(\psi _1\wedge \psi _2)$, where $\psi _1,\psi _2\in F_{{\mathscr {L}}_1}$, then $M\models _{{\mathscr {L}}_1}\varphi$ if and only if $M \models _{{\mathscr {L}}_1} \psi _1$ and $M \models _{{\mathscr {L}}_1} \psi _2$.

Example 2.1.2

Let P be as in Example 2.1.1. Consider now a language ${\mathscr {L}}_2$ with the following characteristics:

The class of ${\mathscr {L}}_2$-formulas is recursively defined as follows:

• if $\varphi \in P$, then $\varphi \in F_{{\mathscr {L}}_2};$

• if $\varphi ,\psi \in F_{{\mathscr {L}}_2}$, then $(\varphi \wedge \psi )\in F_{{\mathscr {L}}_2}$;

• if $\varphi \in F_{{\mathscr {L}}_2}$, then $\lnot \varphi \in F_{{\mathscr {L}}_2}$.

The class of ${\mathscr {L}}_2$-structures is again the set of functions from P to $\{1,0\}$. So $S_{{\mathscr {L}}_2} = S_{{\mathscr {L}}_1}$. The modeling relation of ${\mathscr {L}}_2$ is defined by induction on ${\mathscr {L}}_2$-formulas as follows: for any $M \in S_{{\mathscr {L}}_2}$ and $\varphi \in F_{{\mathscr {L}}_2}$,

• if $\varphi \in P$, then $M\models _{{\mathscr {L}}_2}\varphi$ if and only if $M (\varphi ) = 1$;

• if $\varphi$ is of the form $(\psi _1\wedge \psi _2)$, where $\psi _1,\psi _2\in F_{{\mathscr {L}}_2}$, then $M\models _{{\mathscr {L}}_2}\varphi$ if and only if $M \models _{{\mathscr {L}}_2} \psi _1$ and $M \models _{{\mathscr {L}}_2} \psi _2$;

• if $\varphi$ is of the form $\lnot \psi$, where $\psi \in F_{{\mathscr {L}}_2}$, then $M\models _{{\mathscr {L}}_2}\varphi$ if and only if $M \not \models _{{\mathscr {L}}_2} \psi$.

Example 2.1.3

Let $P' =\{r,s\}$, where r, s are propositional letters. Let ${\mathscr {L}}_3$ be a language whose associated triple $(S_{{\mathscr {L}}_{3}}, F_{{\mathscr {L}}_{3}}, \models _{{\mathscr {L}}_{3}})$ is defined as the triple $(S_{{\mathscr {L}}_{1}}, F_{{\mathscr {L}}_{1}}, \models _{{\mathscr {L}}_{1}})$ of Example 2.1.1, but with P replaced by $P'$.

In Sect. 3 we will discuss formal languages with greater complexity.

3.2 Expressive Power

How do we compare the expressive power of the languages presented in the previous examples? We expect our readers to intuitively agree with the following statements:

$$\begin{aligned}&{\mathscr {L}}_2\text { is as expressive as }{\mathscr {L}}_1; \end{aligned}$$

(2.2.1)

$$\begin{aligned}&{\mathscr {L}}_3\text { is as expressive as }{\mathscr {L}}_1; \end{aligned}$$

(2.2.2)

$$\begin{aligned}&{\mathscr {L}}_1\text { is as expressive as }{\mathscr {L}}_3. \end{aligned}$$

(2.2.3)

When two languages ${\mathscr {L}}$ and ${\mathscr {L}}'$ share the same collection of structures (that is, $S_{{\mathscr {L}}}=S_{{\mathscr {L}}'}$), the most common definition of the relation ${\mathscr {L}}$ has less or equal expressive power ${\mathscr {L}}'$ is the following.

Definition 2.2.4

Suppose $S_{{\mathscr {L}}}=S_{{\mathscr {L}}'}$. Then we say that ${\mathscr {L}}'$ is as expressive as ${\mathscr {L}}$ (or, equivalently, that ${\mathscr {L}}$ has less or equal expressive power than ${\mathscr {L}}'$) if for every ${\mathscr {L}}$-formula $\varphi$ there is an ${\mathscr {L}}'$-formula $\varphi '$ such that for every $M\in S_{{\mathscr {L}}}$

$$\begin{aligned} M\models _{{\mathscr {L}}} \varphi \text { if and only if } M\models _{{\mathscr {L}}'} \varphi '. \end{aligned}$$

This first attempt of definition supports our intuition behind the statement (2.2.1). Clearly, both languages share the same structures. Furthermore, every ${\mathscr {L}}_1$-formula is an ${\mathscr {L}}_2$-formula and thus, given an ${\mathscr {L}}_1$-formula $\varphi$, one can take the ${\mathscr {L}}_2$-formula $\varphi '$ to be precisely the same formula $\varphi$. The truth of Definition 2.2.4 in this case follows almost tautologically.

However, the assumption that $S_{{\mathscr {L}}}=S_{{\mathscr {L}}'}$ in the previous definition seems too restrictive and does not support the intuition behind the statement (2.2.2).⁷ Indeed, although being basically the same, ${\mathscr {L}}_1$ and ${\mathscr {L}}_3$ do not share formally the same collection of structures (since the sets of propositional letters are different). This hints at the idea that the definition of the relation ${\mathscr {L}}$ has less or equal expressive power than ${\mathscr {L}}'$ comprises in many cases a certain choice of relating the sets $S_{{\mathscr {L}}}$ and $S_{{\mathscr {L}}'}$. Thus, we will work with the following relative version of Definition 2.2.4.

Definition 2.2.5

Let $\pi :S_{{\mathscr {L}}'}\rightarrow S_{{\mathscr {L}}}$ be a function. We say that ${\mathscr {L}}'$ is as expressive as ${\mathscr {L}}$ relative to $\pi$ (or, equivalently, that ${\mathscr {L}}$ has less or equal expressive power than ${\mathscr {L}}'$ relative to $\pi$), in symbols ${\mathscr {L}}\le _\pi {\mathscr {L}}'$, if and only if for every ${\mathscr {L}}$-formula $\varphi$ there is an ${\mathscr {L}}'$-formula $\varphi '$ such that for every $M\in S_{{\mathscr {L}}'}$⁸

$$\begin{aligned} \pi (M)\models _{{\mathscr {L}}} \varphi \text { if and only if } M\models _{{\mathscr {L}}'} \varphi '. \end{aligned}$$

When $S_{{\mathscr {L}}}=S_{{\mathscr {L}}'}$ and we take $\pi$ to be the identity function $\textrm{id}:S_{{\mathscr {L}}'}\rightarrow S_{{\mathscr {L}}}$, the relation ${\mathscr {L}}\leqslant _\textrm{id}{\mathscr {L}}'$ coincides with the definition given in Definition 2.2.4.

Going back to the examples, if one takes for $\pi :S_{{\mathscr {L}}_3}\rightarrow S_{{\mathscr {L}}_1}$ the function sending an ${\mathscr {L}}_3$-structure $M:\{r,s\}\rightarrow \{0,1\}$ to the ${\mathscr {L}}_1$-structure $\pi (M):\{p,q\}\rightarrow \{0,1\}$ defined by

$$\begin{aligned} \pi (M)(x){:}{=}{\left\{ \begin{array}{ll} M(q) &{}\quad \text {if }x=s\\ M(p) &{} \quad \text {if }x=r, \end{array}\right. } \end{aligned}$$

one can easily show that ${\mathscr {L}}_1\leqslant _\pi {\mathscr {L}}_3$. In addition, $\pi$ is a bijection, and we also have that ${\mathscr {L}}_{3} \leqslant _{\pi ^-1} {\mathscr {L}}_1$. This gives some ground to our intuition behind Statements 2.2.2 and 2.2.3.

Remark 2.2.6

It is worth noticing that if we impose $\pi$ to be surjective in Definition 2.2.5 (as in García-Matos and Väänänen (2005)), then we obtain a preorder in the following sense: first, it holds for every language ${\mathscr {L}}$ that ${\mathscr {L}}\leqslant _{\textrm{id}} {\mathscr {L}}$; second, if ${\mathscr {L}}_1\leqslant _{\pi } {\mathscr {L}}_2$ and ${\mathscr {L}}_2\leqslant _{\rho } {\mathscr {L}}_3$, then ${\mathscr {L}}_1\leqslant _{\rho \circ \pi } {\mathscr {L}}_3$.

Remark 2.2.7

Since Definition 2.2.5 is relative to a given function $\pi$, different choices for $\pi$ may give rise to different notions of being as expressive as for two given languages. How do we know which choice of $\pi$ is the right choice? Instead of trying to answer this question, we will simply rely on shared intuitions of which functions capture the intended relation for the languages we are considering. For example, whenever two languages share the same collection of structures, we will naturally compare their expressive power using the identity function in Definition 2.2.5. Examples 2.1.1 and 2.1.3 show that there can be more than one natural choice for $\pi$: indeed, any bijection between P and $P'$ is a natural choice.

In the case of Cresswell’s languages, the choice of the function $\pi$ is given by Cresswell himself (see Remark 3.4.1) and we will follow his choice.

4 Cresswell’s Intensional Languages

In this section we present some formal languages which Cresswell discusses in Cresswell (1990). Although we will characterize these languages in detail, some familiarity with Cresswell’s discussion will be assumed.

4.1 ${{\mathcal {L}}}$-Languages

In the first chapter of Cresswell (1990), Cresswell introduces a family of first-order intensional languages.⁹ We will call this family ${{\mathcal {L}}}$. A language ${\mathscr {L}}$ of the family ${{\mathcal {L}}}$ is characterized as follows:

The vocabulary of ${\mathscr {L}}$ consists of symbols of two types: primitive symbols and improper symbols. The primitive symbols are given by a countably infinite set of variables $Var ^{\mathscr {L}}$ and, for each $n \in {{\mathbb {N}}}$, a set of predicates $Pred _n^{\mathscr {L}}$ and a set of sentential functors (sentential operators) $Fun _n^{\mathscr {L}}$. The improper symbols of ${\mathscr {L}}$ are $\forall$ and parentheses.

Any well-formed formula of ${\mathscr {L}}$ (hereafter ${\mathscr {L}}$-formula) is a sequence of symbols from ${\mathscr {L}}$’s vocabulary.¹⁰ The class of ${\mathscr {L}}$-formulas is the smallest set containing the sequences of symbols that are generated by the following rules: for each $n\in {{\mathbb {N}}}$,

(R1):

if $P\in Pred _n^{\mathscr {L}}$ and $x_1,\ldots , x_n \in Var ^{\mathscr {L}}$ are variables, then $(P x_1\cdots x_n)$ is an ${\mathscr {L}}$-formula (an atomic ${\mathscr {L}}$-formula),

(R2):

if $\delta \in Fun _n^{\mathscr {L}}$ and $\varphi _1, \ldots , \varphi _n$ are ${\mathscr {L}}$-formulas, then $(\delta \varphi _1\cdots \varphi _n)$ is an ${\mathscr {L}}$-formula,

(R3):

if $x\in Var ^{\mathscr {L}}$ and $\varphi$ is an ${\mathscr {L}}$-formula, then $\forall x \varphi$ is an ${\mathscr {L}}$-formula.¹¹

An interpretation of ${\mathscr {L}}$ is a triple (W, D, V), where W and D are non-empty sets and V is a function with domain $\bigcup _{n\in {{\mathbb {N}}}} Pred _n^{\mathscr {L}}\cup Fun _n^{\mathscr {L}}$ such that

$$\begin{aligned}&\text { for }P\in Pred _n^{\mathscr {L}}, V(P)\text { is a function } V(P):D^n\rightarrow {\mathcal {P}}(W), \\&\text { for }\delta \in Fun _n^{\mathscr {L}}, V(\delta )\text { is a function } V(\delta ):{\mathcal {P}}(W)^n \rightarrow {\mathcal {P}}(W), \end{aligned}$$

where ${\mathcal {P}}$ is the power set function.

The sets D and W are intuitively regarded, respectively, as a domain of discourse containing individuals and a set of semantical indices. A semantical index, according to Cresswell, is any object with respect to which a sentence (or a formula) may be assessed as true or false. One can view W as a set of possible worlds, a set of moments of time, or a set of world/moment pairs, to name just a few options. Each subset of W represents a proposition.¹² The job of V is to assign denotations to the predicates and functors of ${\mathscr {L}}$. Intuitively, each n-place predicate denotes some intensional relation mapping n-tuples of individuals to propositions, and each n-place functor denotes some operation on propositions which maps any given n-tuple of propositions to a proposition.

Let ${\mathcal {I}} = (W,D,V)$ be an interpretation of ${\mathscr {L}}$. A variable assignment—with respect to ${\mathcal {I}}$—is any function from $Var ^{\mathscr {L}}$ to D. If v is a variable assignment and $x\in Var ^{\mathscr {L}}$, a variable assignment $\mu$ is called an x-alternative of v if, for every $y\in Var ^{\mathscr {L}}$ except possibly x, $\mu (y)=v(y)$.

Given an interpretation ${\mathcal {I}}$ of ${\mathscr {L}}$ and a variable assignment v, the function $V_v$ from the class of ${\mathscr {L}}$-formulas to ${\mathcal {P}}(W)$ is defined inductively on ${\mathscr {L}}$-formulas as follows:

For any $n \in {{\mathbb {N}}}$,

• (1) if $\varphi$ is an atomic ${\mathscr {L}}$-formula of the form $(P x_1\cdots x_n)$, where $P\in Pred _n^{\mathscr {L}}$ and $x_1,\ldots , x_n \in Var ^{\mathscr {L}}$ are n variables, then

  $$\begin{aligned} V_v(\varphi )=V(P)(v(x_1), \ldots , v(x_n)); \end{aligned}$$

• (2) if $\varphi$ is of the form $(\delta \varphi _1\cdots \varphi _n)$, where $\delta \in Fun _n^{\mathscr {L}}$ and $\varphi _1\ldots \varphi _n$ are n ${\mathscr {L}}$-formulas, then

  $$\begin{aligned} V_v(\varphi )=V(\delta )(V_v(\varphi _1), \ldots , V_v(\varphi _n)); \end{aligned}$$

• (3) if $\varphi$ is of the form $\forall x \psi$, where $x \in Var ^{{\mathscr {L}}}$ and $\psi$ is an ${\mathscr {L}}$-formula, then, for every $w\in W$,¹³

  $$\begin{aligned} w\in V_v(\varphi ) \text { if and only if } w\in V_\mu (\psi ), \text { for every }x\text {-alternative }\mu \text { of }v. \end{aligned}$$

Intuitively, $V_v$ assigns to every ${\mathscr {L}}$-formula $\varphi$ the proposition (subset of W) that $\varphi$ denotes relative to the interpretation (W, D, V) and the variable assignment v.

The languages of the family ${{\mathcal {L}}}$ can be characterized as formal languages in the sense of Sect. 2 as follows. If ${\mathscr {L}}$ is a language of this family, we identify $F_{\mathscr {L}}$ with the above-defined class of ${\mathscr {L}}$-formulas and we define the structures of ${\mathscr {L}}$ as quintuples (W, D, V, v, w), where (W, D, V) is an interpretation of ${\mathscr {L}}$, v is a variable assignment and w is an element of W.¹⁴ Given an ${\mathscr {L}}$-structure ${\mathcal {A}}= (W,D,V,v,w)$ and an ${\mathscr {L}}$-formula $\varphi$, the modelling relation $\models _{{\mathscr {L}}}$ of ${\mathscr {L}}$ is defined by

$$\begin{aligned} {{\mathcal {A}}}\models _{{\mathscr {L}}} \varphi \text { if and only if } w\in V_v(\varphi ). \end{aligned}$$

4.2 ${{\mathcal {L}}}_2$-Languages

After introducing the family ${{\mathcal {L}}}$, Cresswell describes other families that are obtained by extending the languages of ${{\mathcal {L}}}$ in certain ways. This subsection presents a family of languages that we will call ${{\mathcal {L}}}_2$.

The main difference between the family ${{\mathcal {L}}}$ and the family ${{\mathcal {L}}}_2$ is that languages in the second family deal with two different kinds of variables. Formally, given a language ${\mathscr {L}}$ in ${{\mathcal {L}}}_2$, the set $Var ^{{\mathscr {L}}}$ is the disjoint union of a countably infinite set of world-variables $Var _W^{{\mathscr {L}}}$ and a countably infinite set of domain-variables $Var _D^{{\mathscr {L}}}$. In addition, among the primitive symbols of ${\mathscr {L}}$ we have for each $n \in {{\mathbb {N}}}$ and each $k \in {{\mathbb {N}}}$, a collection of predicates $Pred _{(n,k)}^{{\mathscr {L}}}$, and a collection of sentential functors $Fun _n^{\mathscr {L}}$. The formulas of ${\mathscr {L}}$ are defined by the following formation rules:

For each $n, k \in {{\mathbb {N}}}$,

(R4):

if $P\in Pred _{(n,k)}^{{\mathscr {L}}}$, $x_1,\ldots , x_n\in Var _D^{{\mathscr {L}}}$ and $u_1,\ldots , u_k\in Var _w^{{\mathscr {L}}}$, then $(P x_1\cdots x_n u_1 \cdots u_k)$ is an ${\mathscr {L}}$-formula,

(R5):

if $\delta \in Fun _n^{{\mathscr {L}}}$ and $\varphi _1, \ldots , \varphi _n$ are ${\mathscr {L}}$-formulas, then $(\delta \varphi _1\cdots \varphi _n)$ is an ${\mathscr {L}}$-formula,

(R6):

if $z\in Var ^{{\mathscr {L}}}$ and $\varphi$ is an ${\mathscr {L}}$-formula, then $\forall z \varphi$ is an ${\mathscr {L}}$-formula.

An interpretation of ${\mathscr {L}}$ is a tuple (W, D, V), where:

• W and D are non-empty sets

• for every $P\in Pred _{(n,k)}^{{\mathscr {L}}}$, V(P) is a function $V(P):D^n\times W^k\rightarrow {{\mathcal {P}}}(W)$.

Let ${\mathcal {I}} = (W,D,V)$ be an interpretation of ${\mathscr {L}}$. A variable assignment—with respect to ${\mathcal {I}}$—is any function $v: Var ^{{\mathscr {L}}} \rightarrow D\cup W$ such that $v(x)\in D$ for $x\in Var _D^{{\mathscr {L}}}$ and $v(u)\in W$ for $u\in Var _W^{{\mathscr {L}}}$. As before, if v is a variable assignment and $x\in Var ^{{\mathscr {L}}}$, a variable assignment $\mu$ is called an x-alternative of v if, for every $y\in Var ^{{\mathscr {L}}}$ except possibly x, $\mu (y)=v(y)$.

Similarly as for languages in ${{\mathcal {L}}}$, given an interpretation ${\mathcal {I}}$ of ${\mathscr {L}}$ and a variable assignment v, a function $V_v$ from the class of ${\mathscr {L}}$-formulas to ${\mathcal {P}}(W)$ is defined inductively on ${\mathscr {L}}$-formulas as follows:

For any $n \in {{\mathbb {N}}}$,

• (1) if $\varphi$ is an atomic ${\mathscr {L}}$-formula of the form $(P x_1\cdots x_nu_1\cdots u_k)$, where $P\in Pred _{n,k}^{{\mathscr {L}}}$, $x_1,\ldots , x_n \in Var _D^{{\mathscr {L}}}$ and $u_1,\ldots , u_k \in Var _W^{{\mathscr {L}}}$, then

  $$\begin{aligned} V_v(\varphi )=V(P)(v(x_1), \ldots , v(x_n), v(u_1), \ldots , v(u_k) ); \end{aligned}$$

• (2) if $\varphi$ is of the form $(\delta \varphi _1\cdots \varphi _n)$, where $\delta \in Fun _n^{{\mathscr {L}}}$ and $\varphi _1\ldots \varphi _n$ are n ${\mathscr {L}}$-formulas, then

  $$\begin{aligned} V_v(\varphi )=V(\delta )(V_v(\varphi _1), \ldots , V_v(\varphi _n)); \end{aligned}$$

• (3) if $\varphi$ is of the form $\forall x \psi$, where $x \in Var ^{{\mathscr {L}}}$ and $\psi$ is an ${\mathscr {L}}$-formula, then, for every $w\in W$,

  $$\begin{aligned} w\in V_v(\varphi ) \text { if and only if } w\in V_\mu (\psi ), \text { for every }x\text {-alternative }\mu \text { of }v. \end{aligned}$$

As for the languages in ${{\mathcal {L}}}$, $V_v$ assigns to every ${\mathscr {L}}$-formula $\varphi$ the proposition that $\varphi$ denotes relative to the interpretation (W, D, V) and the variable assignment v.

The languages of the family ${{\mathcal {L}}}_2$ can be characterized as formal languages in the sense of Sect. 2 as follows. If ${\mathscr {L}}$ is a language of this family, we identify $F_{{\mathscr {L}}}$ with the above-defined class of ${\mathscr {L}}$-formulas and we define the structures of ${\mathscr {L}}$ as quintuples (W, D, V, v, w), where (W, D, V) is an interpretation of ${\mathscr {L}}$, v is a variable assignment and w is an element of W. Given an ${\mathscr {L}}$-structure $\mathcal {A}=(W,D,V,v,w)$ and an ${\mathscr {L}}$-formula $\varphi$, the modelling relation $\models _{{\mathscr {L}}}$ of ${\mathscr {L}}$ is defined by

$$\begin{aligned} {{\mathcal {A}}}\models _{{\mathscr {L}}} \varphi \text { if and only if } w\in V_v(\varphi ). \end{aligned}$$

4.3 The Language ${\mathscr {L}}_\square ^\dagger$

Let ${\mathscr {L}}$ be a language in ${{\mathcal {L}}}_2$. We will associate to ${\mathscr {L}}$ a language ${\mathscr {L}}_\square ^\dagger$. The sets of variables, predicates and operators are the same as in ${\mathscr {L}}$. In formal terms,

$$\begin{aligned} Var ^{{\mathscr {L}}_\square ^\dagger }= Var ^{{\mathscr {L}}} \hspace{1cm} Pred _{n,k}^{{\mathscr {L}}_\square ^\dagger }= Pred _{n,k}^{{\mathscr {L}}} \hspace{1cm} Fun _{n}^{{\mathscr {L}}_\square ^\dagger }= Fun _{n}^{{\mathscr {L}}} \end{aligned}$$

for all $n, k\in {{\mathbb {N}}}$. However, ${\mathscr {L}}_\square ^\dagger$ has a different set of improper symbols. In addition to $\forall$ and parentheses, ${\mathscr {L}}_\square ^\dagger$ has as improper symbols the operator of logical necessity $\square$ (see Cresswell, 1990, p. 8) and a collection of operators $\textrm{Ref}u$ and [u] for any $u \in Var _W^{{\mathscr {L}}_\square ^\dagger }$. Moreover, the universal quantifier $\forall$ of ${\mathscr {L}}_\square ^\dagger$ will only be allowed to bind domain variables (i.e. variables in $Var _D^{{\mathscr {L}}_\square ^\dagger }$). The formulas of ${\mathscr {L}}_\square ^\dagger$ are thus defined by the following formation rules:

For each $n,k \in {{\mathbb {N}}}$,

(R7):

if $P\in Pred _{(n,k)}^{{\mathscr {L}}_{\square }^{\dagger }}$, $x_1,\ldots , x_n\in Var _D^{{\mathscr {L}}}$ and $u_1,\ldots , u_k\in Var _w^{{\mathscr {L}}}$, then $(P x_1\cdots x_n u_1 \cdots u_k)$ is an ${\mathscr {L}}_{\square }^{\dagger }$-formula,

(R8):

if $\delta \in Fun _n^{{\mathscr {L}}_{\square }^{\dagger }}$ and $\varphi _1, \ldots , \varphi _n$ are ${\mathscr {L}}_\square ^ \dagger$-formulas then $(\delta \varphi _1\cdots \varphi _n)$ is an ${\mathscr {L}}_{\square }^{\dagger }$-formula,

(R9):

if $\varphi$ is an ${\mathscr {L}}_\square ^\dagger$-formula and $x\in Var _D^{{\mathscr {L}}_\square ^\dagger }$, then so is $\forall x \varphi$,

(R10):

if $\varphi$ is an ${\mathscr {L}}_\square ^\dagger$-formula, then so is $\square \varphi$,

(R11):

if $\varphi$ is an ${\mathscr {L}}_\square ^\dagger$-formula and $u\in Var _W^{{\mathscr {L}}_\square ^\dagger }$, then $(\textrm{Ref}u)\varphi$ is an ${\mathscr {L}}_\square ^\dagger$-formula,

(R12):

if $\varphi$ is an ${\mathscr {L}}_\square ^\dagger$-formula and $u\in Var _W^{{\mathscr {L}}_\square ^\dagger }$, then $[u]\varphi$ is an ${\mathscr {L}}_\square ^\dagger$-formula.

An interpretation ${{\mathcal {I}}}$ of ${\mathscr {L}}_{\square }^{\dagger }$ is simply an interpretation of ${\mathscr {L}}$. Given an interpretation ${{\mathcal {I}}}$ of ${\mathscr {L}}_{\square }^{\dagger }$, a variable assignment with respect to ${{\mathcal {I}}}$ is defined as for ${\mathscr {L}}$ (see Sect. 3.2). Given a variable assignment v, a variable $u\in Var _W^{{\mathscr {L}}_{\square }^{\dagger }}$ and an index $w\in W$, we will call (v, w/u) the variable assignment given by

$$\begin{aligned} (v,w/u)(x)= {\left\{ \begin{array}{ll} v(x) &{} \quad x\ne u,\\ w &{} \quad x=u. \end{array}\right. } \end{aligned}$$

The reader may think of (v, w/u) as the assignment that is equal to v except that it assigns the index w to the variable u.

Given an interpretation ${{\mathcal {I}}}=(W,D,V)$ of ${\mathscr {L}}_{\square }^{\dagger }$, the definition of the function $V_v$ from ${\mathscr {L}}_{\square }^{\dagger }$-formulas to ${{\mathcal {P}}}(W)$ is defined by induction of ${\mathscr {L}}_{\square }^{\dagger }$-formulas as follows:

• (1) if $\varphi$ is an atomic ${\mathscr {L}}$-formula of the form $(P x_1\cdots x_nu_1\cdots u_k)$, where $P\in Pred _{n,k}^{{\mathscr {L}}_{\square }^{\dagger }}$, $x_1,\ldots , x_n \in Var _D^{{\mathscr {L}}_{\square }^{\dagger }}$ and $u_1,\ldots , u_k \in Var _W^{{\mathscr {L}}_{\square }^{\dagger }}$, then

  $$\begin{aligned} V_v(\varphi )=V(P)(v(x_1), \ldots , v(x_n), v(u_1), \ldots , v(u_k) ); \end{aligned}$$

• (2) if $\varphi$ is of the form $(\delta \varphi _1\cdots \varphi _n)$, where $\delta \in Fun _n^{{\mathscr {L}}_{\square }^{\dagger }}$ and $\varphi _1\ldots \varphi _n$ are n ${\mathscr {L}}_{\square }^{\dagger }$-formulas, then

  $$\begin{aligned} V_v(\varphi )=V(\delta )(V_v(\varphi _1), \ldots , V_v(\varphi _n)); \end{aligned}$$

• (3) if $\varphi$ is of the form $\forall x \psi$, where $x \in Var _D^{{\mathscr {L}}_{\square }^{\dagger }}$ and $\psi$ is an ${\mathscr {L}}_{\square }^{\dagger }$-formula, then

  $$\begin{aligned} w\in V_v(\varphi ) \text { if and only if } w\in V_\mu (\psi ), \text { for every }x\text {-alternative }\mu \text { of }v. \end{aligned}$$

• (4) if $\varphi$ is of the form $\square \psi$, where $\psi$ is an ${\mathscr {L}}_{\square }^{\dagger }$-formula, then $V_v(\varphi )=W$ if $V_w(\psi )=W$, and $V_v(\varphi )=\emptyset$ otherwise;

• (5) if $\varphi$ is of the form $(\textrm{Ref}u)\psi$, where $u\in Var _W^{{\mathscr {L}}_{\square }^{\dagger }}$ and $\psi$ is an ${\mathscr {L}}_{\square }^{\dagger }$-formula, then $w \in V_v(\varphi )$ if and only if $w \in V_{(v,w/u)}(\psi )$;

• (6) if $\varphi$ is of the form $[u]\psi$, where $u\in Var _W^{{\mathscr {L}}_{\square }^{\dagger }}$ and $\psi$ is an ${\mathscr {L}}_{\square }^{\dagger }$-formula, then $w \in V_v(\varphi )$ if and only if $v(u)\in V_{v}(\psi )$.

As in the previous cases, we can treat a language ${\mathscr {L}}_\square ^\dagger$ within the formalism of Sect. 2. The structures of ${\mathscr {L}}_{\square }^{\dagger }$ are tuples ${{\mathcal {A}}}= (W,D,V,v,w)$, where (W, D, V) is an interpretation of ${\mathscr {L}}_{\square }^{\dagger }$, v is a variable assignment and $w\in W$. Given an ${\mathscr {L}}_{\square }^{\dagger }$-structure $\mathcal {A}= (W,D,V,v,w)$ and an ${\mathscr {L}}_{\square }^{\dagger }$-formula $\varphi$, the modelling relation $\models _{{\mathscr {L}}_\square ^\dagger }$ is defined by

$$\begin{aligned} {{\mathcal {A}}}\models _{{\mathscr {L}}_\square ^\dagger } \varphi \text { if and only if } w\in V_v(\varphi ). \end{aligned}$$

We let ${\mathscr {L}}^\dagger$ denote the language ${\mathscr {L}}_\square ^\dagger$ without the logical symbol $\square$. The formulas, structures and the modelling relation of ${\mathscr {L}}^\dagger$ are defined as for ${\mathscr {L}}_{\square }^{\dagger }$ but omitting the corresponding conditions for $\square$.

4.4 The Language ${\mathscr {L}}^*$

Let ${\mathscr {L}}$ be a language in ${{\mathcal {L}}}_2$ and let $\{u_i \mid i\in {{\mathbb {N}}}\}$ be an enumeration of $Var _W^{{\mathscr {L}}}$. The language ${\mathscr {L}}^*$ associated to ${\mathscr {L}}$ is defined as follows.

For each predicate $P\in Pred _{(n,k)}^{{\mathscr {L}}}$ and each functor $\delta \in Fun _n^{{\mathscr {L}}}$, there is a corresponding predicate $P^*\in Pred _n^{{\mathscr {L}}^*}$ and a corresponding functor $\delta ^*\in Fun_n^{{\mathscr {L}}^*}$. The formulas of ${\mathscr {L}}^*$ only have domain-variables and we set $Var ^{{\mathscr {L}}^*}= Var _D^{{\mathscr {L}}}$. Its improper symbols are $\forall$, parentheses, and operators $Ref _n$ and ${{\varvec{actually}}}_n$ for each $n\in {{\mathbb {N}}}$. The formation rules of ${\mathscr {L}}^*$ are as follows:

For each $n\in {{\mathbb {N}}}$,

(R13):

if $P\in Pred _n^{{\mathscr {L}}^*}$ and $x_1,\ldots , x_n \in Var ^{{\mathscr {L}}^*}$, then $(P x_1\cdots x_n)$ is an ${\mathscr {L}}^*$-formula (an atomic ${\mathscr {L}}^*$-formula),

(R14):

if $\delta \in Fun _n^{{\mathscr {L}}^*}$ and $\varphi _1, \ldots , \varphi _n$ are ${\mathscr {L}}^*$-formulas, then $(\delta \varphi _1\cdots \varphi _n)$ is an ${\mathscr {L}}^*$-formula;

(R15):

If $x\in Var ^{{\mathscr {L}}^*}$ and $\varphi$ is an ${\mathscr {L}}^*$-formula, then $\forall x \varphi$ is an ${\mathscr {L}}^*$-formula,

(R16):

if $\varphi$ is an ${\mathscr {L}}^*$-formula, then so are $( Ref _n)\varphi$ and $({{\varvec{actually}}}_n)\varphi$.

Each structure of ${\mathscr {L}}^*$ is obtained uniquely from an ${\mathscr {L}}$-structure as follows. Given a set of indices W, the set $W^\omega$ corresponds to the set of infinite sequences of elements of W. We represent an element $\sigma \in W^\omega$ as a function $\sigma :{{\mathbb {N}}}\rightarrow W$. Following Cresswell, given $\sigma \in W^\omega$ and $w\in W$, we use the notation $w\sigma$ to denote the element of $W^\omega$ defined by $w\sigma (0)=w$ and $w\sigma (i)=\sigma (i)$ for all $i\geqslant 1$. Given an ${\mathscr {L}}$-structure ${{\mathcal {A}}}=(W,D,V,v,w)$, the corresponding structure of ${{\mathcal {A}}}$ in ${\mathscr {L}}^*$ is the tuple ${{\mathcal {A}}}^*=(W^*, D^*, V^*,v^*,w^*)$, where

• $W^*=W^\omega$,

• $D^*=D$,

• $V^*$ is a function sending $P^*\in Pred_{n}^{{\mathscr {L}}^*}\mapsto V^*(P^*)$, where $V^*(P^*)$ is a function $V^*(P^*):D^n\rightarrow {\mathcal {P}}(W^*)$, and $\delta ^* \in Fun _n^{{\mathscr {L}}^*}\mapsto V^*(\delta ^*)$, where $V^*(\delta ^*)$ is a function $V^*(\delta ^*):{\mathcal {P}}(W^*)^n \rightarrow {\mathcal {P}}(W^*)$, respectively defined by

  $$\begin{aligned} V^*(P^*)(a_1,\ldots ,a_n) = \{\sigma \in W^* \mid \sigma (0)\in V(P)(a_1,\ldots ,a_n, \sigma (1),\ldots ,\sigma (k))\}. \end{aligned}$$

  where $(a_1,\ldots a_n)\in D^n$ and

  $$\begin{aligned} V^*(\delta ^*)(A_1,\ldots ,A_n) = \{\sigma \in W^* \mid \sigma (0)\in V(\delta )(B_1,\ldots ,B_n) \}, \end{aligned}$$

  where $(A_1,\ldots , A_n)\in {\mathcal {P}}(W^*)^n$ and $B_i{:}{=}\{w\in W\mid w\sigma \in A_i\}$ for each $1\leqslant i\leqslant n$.

• $v^*: Var ^{{\mathscr {L}}^*}\rightarrow D$ is the restriction of v to $Var _D^{{\mathscr {L}}}$;

• $w^*=w\sigma _v$, where $\sigma _v$ corresponds to the unique sequence in $W^*$ defined by $\sigma (i)=v(u_i)$ for all $i\geqslant 0$.

Given an element $\sigma \in W^*$ and natural numbers k, n, we let $\sigma [k/n]$ denote the element in $W^*$ given by

$$\begin{aligned} \sigma [k/n](m)= {\left\{ \begin{array}{ll} \sigma (k) &{}\quad m=n,\\ \sigma (m) &{} \quad m\ne n. \end{array}\right. } \end{aligned}$$

Given an ${\mathscr {L}}$-structure ${{\mathcal {A}}}=(W,D,V,v,w)$ and its corresponding ${\mathscr {L}}^*$-structure ${{\mathcal {A}}}^*=(W^*,D,V^*,v^*,w^*)$, the definition of the function $V_{v^*}^*$ from ${\mathscr {L}}^*$-formulas to ${{\mathcal {P}}}(W^*)$ is defined inductively on the complexity of ${\mathscr {L}}^*$-formulas: for any $n\in {{\mathbb {N}}}$,

• (1) if $\varphi$ is of the form $(P^* x_1\cdots x_n)$ for $P^*\in Pred _n^{{\mathscr {L}}^*}$, then

  $$\begin{aligned} V^*_{v^*}(\varphi )=V^*(P^*)(v^*(x_1), \ldots , v^*(x_n)), \end{aligned}$$

• (2) if $\varphi$ is of the form $(\delta ^* \varphi _1\cdots \varphi _n)$ for $\delta ^*\in Fun _n^{{\mathscr {L}}}$, then

  $$\begin{aligned} V^*_{v^*}(\varphi )=V^*(\delta ^*)(V^*_{v^*}(\varphi _1), \ldots ,V^*_{v^*}(\varphi _n)), \end{aligned}$$

• (3) if $\varphi$ is of the form $\forall x \psi$, then

  $$\begin{aligned} \sigma \in V^*_{v^*}(\varphi ) \text { if and only if } \sigma \in V^*_{\mu ^*}(\psi ) \text { for every }x\text {-alternative }\mu \text { of }v, \end{aligned}$$

• (4) if $\varphi$ is of the form $({{\varvec{actually}}}_n)\psi$, then

  $$\begin{aligned} \sigma \in V_{v^*}^*(({{\varvec{actually}}}_n)\psi )\text { if and only if }\sigma [n/0] \in V_{v^*}^*(\psi ), \end{aligned}$$

• (5) if $\varphi$ is of the form $( Ref _n)\psi$, then

  $$\begin{aligned} \sigma \in V_{v^*}^*(( Ref _n)\psi )\text { if and only if }\sigma [0/n] \in V_{v^*}^*(\psi ). \end{aligned}$$

Remark 3.4.1

Note that the function sending an ${\mathscr {L}}$-structure ${{\mathcal {A}}}$ to the ${\mathscr {L}}^*$-structure ${{\mathcal {A}}}^*$ is a bijection. This bijection is the function from ${\mathscr {L}}$-structures to ${\mathscr {L}}^*$-structures that Cresswell (1990, p. 45) chose as the basis for his proof that ${\mathscr {L}}^*$ is as powerful as ${\mathscr {L}}$. We will stick to it in comparing the expressivity of ${\mathscr {L}}$ and ${\mathscr {L}}^*$ (recall Remark 2.2.7).

As in all previous cases, given an ${\mathscr {L}}^*$-structure ${{\mathcal {A}}}^*=(W^*,D^*,V^*,v^*,w^*)$, the modelling relation $\models _{{\mathscr {L}}^*}$ of ${\mathscr {L}}^*$ is defined by

$$\begin{aligned} {{\mathcal {A}}}^*\models _{{\mathscr {L}}^*} \varphi \text { if and only if } w^*\in V^*_{v^*}(\varphi ). \end{aligned}$$

4.5 On the Choice of Logical Constants

According to Cresswell, formal languages can, at least in principle, be studied in abstraction from the choice of logical constants. He makes this point clear in the opening chapter of Cresswell (1973):

“Most treatments of propositional languages are interested in studying them as logics. This means that they pick out one or two symbols as ‘constants’ or ‘logical words’ and only consider interpretations which fit in with the intended meaning of these words. We are interested in propositional languages as languages and do not want our semantical framework to impose any particular meaning on any symbol.” Cresswell (1973, p. 20).

Here Cresswell is suggesting that, in the abstract study of languages, symbols do not have fixed meanings. However, as soon as one restricts the interpretations of a language in a way that the meanings of some of its symbols are fixed, one is treating those symbols as logical constants and a logic is obtained.

When we introduced the language families ${{\mathcal {L}}}$ and ${{\mathcal {L}}}_2$ in Sects. 3.1 and 3.2, we assumed that the symbol $\forall$ is the only logical constant of the languages of those families. The primitive symbols of such languages do not have fixed meanings. Their denotations vary from structure to structure (see fn. 14 above). By contrast, the symbol $\forall$ does not have different denotations in different structures. There is a semantic clause that fixes its intended meaning (see clause (3) in both Sects. 3.1 and 3.2), thereby ensuring that $\forall$ is interpreted as a universal quantifier in every structure—rather than being interpreted as one kind of quantifier in some structures and as another kind of quantifier in other structures.

One important consequence of this aspect of our characterization of the families ${{\mathcal {L}}}$ and ${{\mathcal {L}}}_2$ is that the operator of logical necessity $\square$ is not a logical constant of the languages of those families. We believe this is in line with the way Cresswell presents his formal framework in the first chapter of Cresswell (1990).

On page 7 Cresswell explicitly says that $\forall$ is the only logical constant of a language ${\mathscr {L}}$ of the family ${{\mathcal {L}}}$:

“${\mathscr {L}}$ contains no logical constants except for $\forall$ $[\ldots ]$ So there will be no wff true in all interpretations, and thus no intensional logic.” Cresswell (1990, p. 7).

Later on he makes a point similar to the point he had made in Cresswell (1973, Chapter 1) concerning logical constants, namely that a logic can be obtained from ${\mathscr {L}}$ by conferring on some of its symbols the status of logical constants:

“If not is a one-place sentential functor (i.e. a symbol in $Fun_{1}$) the interpretation closest to English would presumably give:

(10) $w \in V({{\varvec{not}}})(a)$ iff $w \notin a$, for any $a \subseteq W$.

[...]

I shall sometimes use $\sim$ in place of not, with the meaning of (10).

[...]

I shall use & for conjunction, $\vee$ for (inclusive) disjunction and $\bot$ for the standard false proposition:

(12) V( &)$(a,b) = a \cap b$

(13) $V(\vee )(a,b) = a \cup b$

(14) $V(\bot ) = \varnothing$

If we take all these as logical constants then we do indeed get a logic. It is a form of classical predicate logic. The existential quantifier $\exists x$ can be defined as ${\thicksim }\forall x{\sim }$.

We can also add modal operators. I shall distinguish logical necessity and possibility from relative necessity and possibility, and use $\square$ and $\Diamond$ for the former and L and M for the latter. Logical necessity is truth in all worlds and logical possibility is truth in at least one world:

(15) $V(\square )(a) = W$ if $a = W$, otherwise $V(\square )(a) = \varnothing$

(16) $V(\Diamond )(a) = W$ if $a \ne \varnothing$, otherwise $V(\Diamond )(a) = \varnothing$

The modal logic obtained from (15) and (16) will be S5 with variable functors.” Cresswell (1990, pp. 7–9).

Sections 3.1–3.4 reflect our understanding of these passages. The quantifier $\forall$ is the unique logical constant of any language ${\mathscr {L}}$ of the family ${{\mathcal {L}}}$ or the family ${{\mathcal {L}}}_2$. If the symbol $\square$ is one of the functors of ${\mathscr {L}}$, there are different ${\mathscr {L}}$-structures in which it denotes different operations, that is to say, it is not interpreted as an operator of logical necessity in every structure. Nevertheless, given a language ${\mathscr {L}}$ of the family ${{\mathcal {L}}}$ or the family ${{\mathcal {L}}}_2$, one can define new languages that resemble ${\mathscr {L}}$ in some relevant respects but have richer inventories of logical constants. For example, the logical constants of a language ${\mathscr {L}}_{\square }^{\dagger }$ (see Sect. 3.3) are $\forall$, the operator of logical necessity $\square$, and the operators $\textrm{Ref}u$, and [u] (for any world-variable u). A language ${\mathscr {L}}^\dagger$ (see Sect. 3.3) has the same logical constants as ${\mathscr {L}}_{\square }^{\dagger }$ except for the operator $\square$. Similarly, the logical constants of a language ${\mathscr {L}}^*$ (see Sect. 3.4) are $\forall$ and the operators $Ref _n$ and ${{\varvec{actually}}}_n$ (for any $n\in {{\mathbb {N}}}$). In Sect. 4.5 we will consider an extension of ${\mathscr {L}}^*$ in which $\square$ is added as an extra logical constant.

Clearly, identifying the stocks of logical constants of two languages is a crucial step when one wants to compare their expressive power. This is the reason why we carefully distinguished, in the previous sections, between languages with and without $\square$ as a logical constant. We will return to this point in Sect. 5.1. It is worth stressing that Cresswell’s proofs in Cresswell (1990, Chapter 4) presuppose that the meanings of the above mentioned operators ($\square$, $\textrm{Ref}u$, [u], $Ref _n$, and ${{\varvec{actually}}}_n$) are fixed, which is why we treated them as logical constants in Sects. 3.3 and 3.4.

5 Comparing Cresswell’s Intensional Languages

In Cresswell (1990), Cresswell states various relative results about the expressive power of the languages described in Sects. 3.1–3.4. In particular, he claims that the relation of being as expressive as holds between some of the above described languages. Although he never formally defines the relation being as expressive as, the proofs he presents in his text suggest that what he has in mind is something along the lines of Definition 2.2.5. However, recall that this definition is relative to a given function $\pi$. With this in mind, we will revisit some of Cresswell’s results in Cresswell (1990) and decide whether what he claims is indeed what he proves. Before starting, let us make some simple remarks about some relations between the intensional languages introduced in Sect. 3.

Since every ${\mathscr {L}}^\dagger$-formula is an ${\mathscr {L}}_{\square }^{\dagger }$-formula, it follows directly that

$$\begin{aligned} {\mathscr {L}}^\dagger \leqslant _{\textrm{id}} {\mathscr {L}}_\square ^\dagger . \end{aligned}$$

Similarly, given a language ${\mathscr {L}}$ in ${{\mathcal {L}}}$ and a language ${\mathscr {L}}'$ in ${{\mathcal {L}}}_2$ such that $Pred _n^{{\mathscr {L}}}\subseteq Pred _{n,0}^{{\mathscr {L}}'}$ and $Fun _n^{{\mathscr {L}}}\subseteq Fun _n^{{\mathscr {L}}'}$ for all $n\in {{\mathbb {N}}}$, it is also easy to show that

$$\begin{aligned} {\mathscr {L}}\leqslant _{\pi } {\mathscr {L}}', \end{aligned}$$

where $\pi$ is the restriction function, that is, the function that sends an ${\mathscr {L}}'$-structure ${{\mathcal {A}}}=(W,D,V,v,w)$ to the ${\mathscr {L}}$-structure $\pi ({{\mathcal {A}}})=(W,D,V,v',w)$ where $v'$ is the restriction of v to $Var _D^{{\mathscr {L}}'}$. Clearly, $\pi$ is surjective.

5.1 Comparing ${\mathscr {L}}$ and ${\mathscr {L}}_\square ^\dagger$

Let ${\mathscr {L}}$ be a language in ${{\mathcal {L}}}_2$ as defined in Sect. 3.2 and ${\mathscr {L}}_\square ^\dagger$ be its corresponding language as defined in Sect. 3.3, that is,

$$\begin{aligned} Pred _{n,k}^{\mathscr {L}}= Pred _{n,k}^{{\mathscr {L}}_{\square }^{\dagger }} \text { and } Fun _n^{\mathscr {L}}= Fun _n^{{\mathscr {L}}_{\square }^{\dagger }}, \end{aligned}$$

for all $n, k \in {{\mathbb {N}}}$. Cresswell showed in Cresswell (1990) that the language ${\mathscr {L}}_\square ^\dagger$ is as expressive as ${\mathscr {L}}$. We will formalize his argument in light of our terminology. First of all, we need to establish the function $\pi$ to compare the expressive power of ${\mathscr {L}}$ and ${\mathscr {L}}_\square ^\dagger$. Since in this case $S_{{\mathscr {L}}}=S_{{\mathscr {L}}_\square ^ \dagger }$, the identify function is the most natural choice for $\pi$. The following proposition is just a reformulation of what Cresswell shows in Cresswell (1990, Chapter 4, lines (7)–(10)). To simplify notation, we will sometimes omit the language subscript of the modelling relation $\models$ when no confusion arises.

Proposition 4.1.1

It holds that ${\mathscr {L}}\le _{\textrm{id}} {\mathscr {L}}_{\square }^{\dagger }$.

Proof

Let us show how to associate to each ${\mathscr {L}}$-formula $\varphi$ an ${\mathscr {L}}_{\square }^{\dagger }$-formula $\varphi '$ such that for every structure ${{\mathcal {A}}}\in S_{{\mathscr {L}}}$ (recall $S_{\mathscr {L}}=S_{{\mathscr {L}}_{\square }^{\dagger }}$)

$$\begin{aligned} {{\mathcal {A}}}\models _{{\mathscr {L}}} \varphi \text { if and only if } {{\mathcal {A}}}\models _{{\mathscr {L}}_{\square }^{\dagger }} \varphi '. \end{aligned}$$

(E)

We define the formula $\varphi '$ by induction on the complexity of $\varphi$ as follows:

• (i) if $\varphi$ is an atomic ${\mathscr {L}}$-formula, then $\varphi '=\varphi$;

• (ii) if $\psi _1', \ldots , \psi _n'$ have been defined, and $\varphi$ is $(\delta \psi _1\cdots \psi _n)$ for $\delta \in Fun _n^{\mathscr {L}}$, then $\varphi '$ is $(\delta \psi _1' \cdots \psi _n')$

• (iii) if $\psi '$ has been defined, and $\varphi$ is $\forall x\psi$ with $x\in Var ^{{\mathscr {L}}}$, then

  $$\begin{aligned} \varphi '{:}{=}{\left\{ \begin{array}{ll} \varphi &{} \quad \text { if } x\in Var _D^{{\mathscr {L}}} \\ (\textrm{Ref}y) \square (\textrm{Ref}x)[y] \psi ' &{}\quad \text { if } x\in Var _W^{{\mathscr {L}}} \text { (where }y\text { is a new variable in } Var _W^{{\mathscr {L}}_{\square }^{\dagger }}). \end{array}\right. } \end{aligned}$$

Let ${{\mathcal {A}}}$ be an ${\mathscr {L}}_\square ^\dagger$-structure. We prove that (E) holds by induction on $\varphi$:

• (i) For $\varphi$ an atomic formula, the result is obvious.

• (ii) Suppose the result holds for if $\psi _1, \ldots , \psi _n$ and let $\varphi$ be $(\delta \psi _1\cdots \psi _n)$. The result follows readily.

• (iii) Suppose the result holds for $\psi$ and that $\varphi$ is $\forall x\psi$. If $x\in Var _D^{\mathscr {L}}$ the result is again obvious, so suppose that $x\in Var _W^{\mathscr {L}}$. Then, given an ${\mathscr {L}}$-structure ${{\mathcal {A}}}=(W,D,V,v,w)$ we have

  

$\square$

5.2 Necessity of $\square$

Let ${\mathscr {L}}$ be a language in ${{\mathcal {L}}}_2$ and ${\mathscr {L}}^\dagger$ be its corresponding language as defined in Sect. 3.3. In Cresswell (1990), Cresswell claims that ${\mathscr {L}}\leqslant _{\textrm{id}} {\mathscr {L}}^\dagger$. We will show in this section that this is not the case, that is, that $\square$ is somehow necessary in the proof of Proposition 4.1.1. However, to be precise, we need to ensure that the languages ${\mathscr {L}}$ and ${\mathscr {L}}^\dagger$ contain at least some predicate involving world variables. Indeed, unsurprisingly, the following proposition shows that Cresswell’s claim does hold when ${\mathscr {L}}$ and ${\mathscr {L}}^\dagger$ contain no predicates involving world variables.

Proposition 4.2.1

Suppose that $Pred _{n,k}^{\mathscr {L}}$ is empty for all $k>0$. Then

$$\begin{aligned} {\mathscr {L}}\leqslant _{\textrm{id}} {\mathscr {L}}^\dagger . \end{aligned}$$

Proof

In order to show this, we will use the following claim, which can be proved by induction on formulas and is left to the reader as an exercise

Claim 4.2.2

Let $\psi$ be an ${\mathscr {L}}$-formula (resp. ${\mathscr {L}}^\dagger$-formula) and $x\in Var _W^{\mathscr {L}}$ (resp. in $x\in Var _W^{{\mathscr {L}}^\dagger }$). Then for every ${\mathscr {L}}$-structure ${{\mathcal {A}}}$,

$$\begin{aligned} {{\mathcal {A}}}\models \forall x\psi \Leftrightarrow {{\mathcal {A}}}\models \psi . \end{aligned}$$

To prove the statement of the proposition, let us show how to associate to each ${\mathscr {L}}$-formula $\varphi$ an ${\mathscr {L}}^\dagger$-formula $\varphi '$ such that for every structure ${{\mathcal {A}}}\in S_{{\mathscr {L}}}$

$$\begin{aligned} {{\mathcal {A}}}\models _{{\mathscr {L}}} \varphi \text { if and only if } {{\mathcal {A}}}\models _{{\mathscr {L}}^\dagger } \varphi '. \end{aligned}$$

(E2)

We define the formula $\varphi '$ by induction on the complexity of $\varphi$ as follows:

• (i) if $\varphi$ is an atomic ${\mathscr {L}}$-formula, then $\varphi '=\varphi$;

• (ii) if $\psi '$ has been defined, and $\varphi$ is $\forall x\psi$ with $x\in Var ^{{\mathscr {L}}}$, then

  $$\begin{aligned} \varphi '{:}{=}{\left\{ \begin{array}{ll} \varphi &{}\quad \text { if } x\in Var _D^{{\mathscr {L}}} \\ \psi ' &{}\quad \text { if } x\in Var _W^{{\mathscr {L}}}. \end{array}\right. } \end{aligned}$$

Let ${{\mathcal {A}}}=(W,D,V,v,w)$ be an ${\mathscr {L}}^\dagger$-structure. We prove (E2) holds by induction on $\varphi$:

• (i) For $\varphi$ an atomic formula, the result is obvious.

• (ii) Suppose the result holds for $\psi$ and that $\varphi$ is $\forall x\psi$. If $x\in Var _D^{\mathscr {L}}$ the result is again obvious, so suppose that $x\in Var _W^{\mathscr {L}}$. Then $\varphi '=\psi '$ and

  $$\begin{aligned} {{\mathcal {A}}}\models \varphi '&\Leftrightarrow {{\mathcal {A}}}\models \psi '&\\&\Leftrightarrow {{\mathcal {A}}}\models \psi&\text { (by induction)}\\&\Leftrightarrow {{\mathcal {A}}}\models \forall x\psi&\text { (by Claim}~4.2.2). \end{aligned}$$

$\square$

In contrast to Proposition 4.2.1, we will now show that $\square$ is necessary in the proof of Proposition 4.1.1 for languages ${\mathscr {L}}$ for which there are $n\in {{\mathbb {N}}}$ and $k>0$ such that $Pred _{n,k}^{\mathscr {L}}\ne \emptyset$.

Proposition 4.2.3

Assume ${\mathscr {L}}$ is a language such that for some $n\in {{\mathbb {N}}}$ and some $k>0$, there is $P\in Pred _{n,k}^{\mathscr {L}}$. Then ${\mathscr {L}}^\dagger$ is not as expressive as ${\mathscr {L}}$ with respect to the identity function, in symbols,

$$\begin{aligned} {\mathscr {L}}\not \leqslant _{\textrm{id}} {\mathscr {L}}^\dagger . \end{aligned}$$

The idea of the proof is to show that the modelling relation in ${\mathscr {L}}^\dagger$ is local with respect to world variables. Informally, this local character means that, even though the modelling relation of a fixed ${\mathscr {L}}^\dagger$-formula $\varphi$ may depend on all variables in $Var _D^{{\mathscr {L}}^\dagger }$, it only depends on a finite subset of the variables in $Var _W^{{\mathscr {L}}^\dagger }$, namely those that occur in $\varphi$. Formally, this is the content of following lemma.

Lemma 4.2.4

Let $\varphi$ be an ${\mathscr {L}}^\dagger$-formula. Suppose all world variables occurring in $\varphi$ are among $U=\{u_1,\ldots , u_r\}$. Let ${{\mathcal {A}}}=(W,D,V,v,w)$ and ${{\mathcal {A}}}'=(W,D,V',v',w)$ be two ${\mathscr {L}}^\dagger$-structures such that:

• (1) $v, v'$ agree on $Var _D^{{\mathscr {L}}^\dagger }\cup U$, i.e., $v(y)=v'(y)$ for any variable $y\in Var _D^{{\mathscr {L}}^\dagger } \cup U$;

• (2) for $P\in Pred _{n,k}^{{\mathscr {L}}^\dagger }$ with $k\leqslant r$ and $x=(x_1,\ldots ,x_n)$ a tuple of domain variables,

  $$\begin{aligned} V_v((Px u_{i_1} \cdots u_{i_k})) = V_{v'}'((P x u_{i_1} \cdots u_{i_k})); \end{aligned}$$

• (3) for every sentential functor $\delta$ in ${\mathscr {L}}^\dagger$, $V(\delta )=V'(\delta )$.

Then

$$\begin{aligned} {{\mathcal {A}}}\models \varphi \Leftrightarrow {{\mathcal {A}}}'\models \varphi . \end{aligned}$$

Proof

We proceed by induction on ${\mathscr {L}}^\dagger$-formulas. If $\varphi$ is atomic the result follows directly from assumptions (1) and (2). We split in cases for the inductive step:

Case 1: Suppose the result holds for $\psi _1,\ldots ,\psi _m$ and let $\varphi$ be the formula $(\delta \psi _1\cdots \psi _m)$. The result follows by induction and assumption (3).

Case 2: Suppose the result holds for $\psi$ and let $\varphi$ be the formula $(\textrm{Ref}u_j)\psi$. Since $v,v'$ agree on $Var _D^{{\mathscr {L}}^\dagger }\cup U$, the assignments $(v,w/u_j)$ and $(v',w/u_j)$ also agree on $Var _D^{{\mathscr {L}}^\dagger }\cup U$. Thus

$$\begin{aligned} (W,D,V,v,w)\models (\textrm{Ref}u_j)\psi&\Leftrightarrow w \in V_v((\textrm{Ref}u_j)\psi ) \\&\Leftrightarrow w \in V_{(v,w/u_j)}(\psi )\\&\Leftrightarrow (W,D,V,(v,w/u_j),w)\models \psi \\&\Leftrightarrow (W,D,V',(v',w/u_j),w)\models \psi \text { (by induction)}\\&\Leftrightarrow w \in V_{(v',w/u_j)}'(\psi ) \\&\Leftrightarrow w \in V_{v'}'((\textrm{Ref}u_j)\psi )\\&\Leftrightarrow (W,D,V',v',w)\models (\textrm{Ref}u_j)\psi . \end{aligned}$$

Case 3: Suppose the result holds for $\psi$ and let $\varphi$ be the formula $[u_j]\psi$. Then

$$\begin{aligned} (W,D,V,v,w)\models [u_j]\psi&\Leftrightarrow w \in V_v([u_j]\alpha ) \\&\Leftrightarrow v(u_j)\in V_{v}(\psi )\\&\Leftrightarrow (W,D,V,v,v(u_j))\models \psi \\&\Leftrightarrow (W,D,V',v',v'(u_j))\models \psi \text { (by induction)}\\&\Leftrightarrow v'(u_j) \in V_{v'}'(\psi ) \\&\Leftrightarrow w \in V_{v'}'([u_j]\psi )\\&\Leftrightarrow (W,D,V',v',w)\models [u_j]\psi . \end{aligned}$$

Case 4: Suppose the result holds for $\psi$ and let $\varphi$ be the formula $\forall x \psi$. Since v and $v'$ agree on $Var _D^{{\mathscr {L}}^\dagger }\cup U$, we have that for every $a\in D$, (v, a/x) and $(v',a/x)$ agree on $Var _D^{{\mathscr {L}}^\dagger }\cup U$. Then

$$\begin{aligned} (W,D,V,v,w)\models \forall x \psi&\Leftrightarrow w \in V_{(v,a/x)}(\psi ) \text { for all }a\in D \\&\Leftrightarrow (W,D,V,(v,a/x),w)\models \psi \text { for all }a\in D\\&\Leftrightarrow (W,D,V',(v',a/x),w)\models \psi \text { for all }a\in D\text { (by induction)}\\&\Leftrightarrow w \in V_{(v',a/x)}'(\psi ) \text { for all }a\in D \\&\Leftrightarrow (W,D,V',v',w)\models \forall x\psi . \end{aligned}$$

$\square$

Given a tuple of variables $x=(x_1,\ldots ,x_n)$ we use $\forall x \varphi$ as an abbreviation for $\forall x_1\cdots \forall x_n \varphi$.

Proof of Proposition 4.2.3

Let $P\in Pred _{n,k}^{\mathscr {L}}$ be a predicate with $n\in {{\mathbb {N}}}$ and $k>0$. Let $x=(x_1,\ldots ,x_n)$ be a tuple of domain variables and $u=(u_1,\ldots ,u_k)$ be a tuple of world variables. Suppose for a contradiction that ${\mathscr {L}}\leqslant _{\textrm{id}} {\mathscr {L}}^\dagger$. Then there is an ${\mathscr {L}}^\dagger$-formula $\varphi$ such that for all ${\mathscr {L}}^\dagger$-structures ${{\mathcal {A}}}$ we have

$$\begin{aligned} {{\mathcal {A}}}\models \forall x \forall u (Pxu) \text { if and only if } {{\mathcal {A}}}\models \varphi . \end{aligned}$$

(C)

Without loss of generality, suppose $U=\{u_1,\ldots , u_k, u_{k+1},\ldots , u_m\}$ contains all world variables occurring in $\varphi$. Let D be a non-empty set and let $a_0\in D$. Let W be a set of worlds $W{:}{=}\{w_1,\ldots ,w_{m+1}\}$. We will consider two structures ${{\mathcal {A}}}=(W,D,V,v,w_1)$ and ${{\mathcal {A}}}'=(W,D,V',v',w_1)$ where $V, V'$ and v are defined as follows. We define V and $V'$ to be the same function except for the predicate P. For P, we let $V(P):D^n\times W^k\rightarrow {{\mathcal {P}}}(W)$ be defined for all $a\in D^n$ by

$$\begin{aligned} v(P)(a, w_{i_1},\ldots ,w_{i_k}){:}{=}{\left\{ \begin{array}{ll} W &{} \quad \text { if }i_j\leqslant m\text { for all }1\leqslant j\leqslant k \\ \emptyset &{} \quad \text { otherwise}. \end{array}\right. } \end{aligned}$$

and $V'(P):D^n\times W^k\rightarrow {{\mathcal {P}}}(W)$ be defined as $V'(P)(a,w)=W$ for all $(a,w)\in D^n\times W^k$. Slightly less formally, if the world $w_{m+1}$ appears in one of the k world-coordinates of P, its interpretation with respect to V is the empty set, while the interpretation of P with respect to $V'$ is W for every possible tuple in $D^n\times W^k$. Finally, define the assignment v which sends every domain variable to $a_0$ and for world variables it is defined as

$$\begin{aligned} v(u_i)= {\left\{ \begin{array}{ll} w_i &{}\quad \text { if }1\leqslant i\leqslant m\\ w_{m+1} &{} \quad \text { otherwise}. \end{array}\right. } \end{aligned}$$

Note that $V_v((P x u_{i_1}\cdots u_{i_k}))=V_v'((Px u_{i_1} \cdots u_{i_k}))$ whenever $u_{i_1},\ldots ,u_{i_k}\in U$. Therefore, by Lemma 4.2.4 we have that

$$\begin{aligned} {{\mathcal {A}}}\models \varphi \Leftrightarrow {{\mathcal {A}}}'\models \varphi . \end{aligned}$$

However, by construction

$$\begin{aligned} {{\mathcal {A}}}\not \models \forall x \forall u (Pxu) \text { and } {{\mathcal {A}}}'\models \forall x \forall u (Pxu), \end{aligned}$$

thereby contradicting (C). $\square$

5.3 Comparing ${\mathscr {L}}^\dagger$ and ${\mathscr {L}}^*$

Let ${\mathscr {L}}$ be a language in ${{\mathcal {L}}}_2$ and let ${\mathscr {L}}^\dagger$ and ${\mathscr {L}}^*$ be the corresponding languages as defined in Sects. 3.3 and 3.4, respectively. In Cresswell (1990, Chapter 4), Cresswell shows that the language ${\mathscr {L}}^*$ is as expressive as the language ${\mathscr {L}}^\dagger$. In our terminology this amounts to proving ${\mathscr {L}}^\dagger \leqslant _{\pi } {\mathscr {L}}^*$ for a chosen function $\pi$ from ${\mathscr {L}}^*$-structures to ${\mathscr {L}}^\dagger$-structures. The function $\pi$ chosen by Cresswell is simply the inverse function of the function sending an ${\mathscr {L}}^\dagger$-structure ${{\mathcal {A}}}$ to the ${\mathscr {L}}^*$-structure ${{\mathcal {A}}}^*$ as defined in Sect. 3.4. For the sake of completeness, we will briefly sketch Cresswell’s proof.

Proposition 4.3.1

It holds that ${\mathscr {L}}^\dagger \leqslant _\pi {\mathscr {L}}^*$.

Proof

Let us show how to associate to each ${\mathscr {L}}^\dagger$-formula $\varphi$ an ${\mathscr {L}}^*$-formula $\varphi ^*$ such that for every ${\mathscr {L}}^\dagger$-structure ${{\mathcal {A}}}$

$$\begin{aligned} {{\mathcal {A}}}\models _{{\mathscr {L}}^\dagger } \varphi \text { if and only if } {{\mathcal {A}}}^*\models _{{\mathscr {L}}^*} \varphi ^*. \end{aligned}$$

(E3)

Following Cresswell’s proof, the formula $\varphi ^*$ is defined by induction on the complexity of $\varphi$ as follows:

• (i) for $\varphi$ an atomic ${\mathscr {L}}^\dagger$-formula of the form $(Px_1\cdots x_n u_1\cdots u_k)$ where $P\in Pred _{n,k}^{{\mathscr {L}}^\dagger }$, then $\varphi ^*=(P^*x_1\cdots x_n)$;

• (ii) for $\varphi$ of the form $(\delta \psi _1\cdots \psi _n)$ where $\delta \in Fun _{n}^{{\mathscr {L}}^\dagger }$ and $\psi _1,\ldots , \psi _n$ are ${\mathscr {L}}^\dagger$-formulas, then $\varphi ^*=(\delta ^* \psi _1^*\cdots \psi _n^*)$;

• (iii) for $\varphi$ of the form $\forall x\psi$ with $x\in Var ^{{\mathscr {L}}^\dagger }$, then $\varphi ^*=\forall x\psi ^*$.

• (iv) for $\varphi$ of the form $(\textrm{Ref}x)\psi$ with $x\in Var _W^{{\mathscr {L}}^\dagger }$ such that $x=u_n$, then $\varphi ^*= Ref _n\psi ^*$.

• (v) for $\varphi$ of the form $[x]\psi$ with $x\in Var _W^{{\mathscr {L}}^\dagger }$ such that $x=u_n$, then $\varphi ^*={{\varvec{actually}}}_n\psi ^*$.

Let ${{\mathcal {A}}}=(W,D,V,v,w)$ be an ${\mathscr {L}}^\dagger$-structure. We prove that (E3) holds by induction on $\varphi$:

• (i) For $\varphi$ an atomic formula, the result follows straightforwardly from the definition of $V^*_{v^*}(P^*)$.

• (ii) Suppose the result holds for $\psi _1,\ldots , \psi _n$ and that $\varphi$ is $(\delta \psi _1\cdots \psi _n)$. Again, the result follows easily from induction and the definition of $\delta ^*$.

• (iii) Suppose the result holds for $\psi$ and that $\varphi$ is of the form $\forall x\psi$. Then

  $$\begin{aligned} {{\mathcal {A}}}^*\models \varphi ^*&\Leftrightarrow {{\mathcal {A}}}\models \forall x\psi ^*&\\&\Leftrightarrow w^*\in V^*_{\mu ^*}(\psi ) \text { for every }x\text {-alternative }\mu \text { of }v\\&\Leftrightarrow (W^*, D^*, V^*, \mu ^*, w^*)\models \psi \text { for every }x\text {-alternative }\mu \text { of }v\\&\Leftrightarrow (W, D, V, \mu , w)\models \psi \text { for every }x\text {-alternative }\mu \text { of }v\\&\Leftrightarrow {{\mathcal {A}}}\models \forall x\psi&\\ \end{aligned}$$

• (iv) Suppose the result holds for $\psi$ and that $\varphi$ is of the form $\textrm{Ref}u_n\psi$. Then

  $$\begin{aligned} {{\mathcal {A}}}^*\models \varphi ^*&\Leftrightarrow {{\mathcal {A}}}\models ( Ref _n)\psi ^*&\\&\Leftrightarrow w^*\in V^*_{v^*}(( Ref _n)\psi ^*) \\&\Leftrightarrow w^*[0/n]\in V^*_{v^*}(\psi ^*) \\&\Leftrightarrow (W^*, D^*, V^*, v^*, w^*[0/n])\models \psi ^* \\&\Leftrightarrow (W, D, V, (v, w/u_n), w)\models \psi \\&\Leftrightarrow (W, D, V, v, w)\models (\textrm{Ref}u_n)\psi \end{aligned}$$

• (v) Suppose the result holds for $\psi$ and that $\varphi$ is of the form $[u_n]\psi$. Then

  $$\begin{aligned} {{\mathcal {A}}}^*\models \varphi ^*&\Leftrightarrow {{\mathcal {A}}}\models ({{\varvec{actually}}}_n)\psi ^*&\\&\Leftrightarrow w^*\in V^*_{v^*}(({{\varvec{actually}}}_n)\psi ^*) \\&\Leftrightarrow w^*[n/0]\in V^*_{v^*}(\psi ^*) \\&\Leftrightarrow (W^*, D^*, V^*, v^*, w^*[n/0])\models \psi ^* \\&\Leftrightarrow (W, D, V, v, v(n))\models \psi \\&\Leftrightarrow (W, D, V, v, w)\models [u_n]\psi \end{aligned}$$

$\square$

5.4 Comparing ${\mathscr {L}}$ and ${\mathscr {L}}^*$

Cresswell’s final goal in Cresswell (1990, Chapter 4) is to show that, given a language ${\mathscr {L}}$ in ${{\mathcal {L}}}_2$ and given its corresponding language ${\mathscr {L}}^*$ as defined in Sect. 3.4, we have that ${\mathscr {L}}\leqslant _{\pi } {\mathscr {L}}^*$, where $\pi$ corresponds to the function sending an ${\mathscr {L}}^*$-structure ${{\mathcal {A}}}^*$ to its corresponding ${\mathscr {L}}$-structure ${{\mathcal {A}}}$ (recall $S_{{\mathscr {L}}}=S_{{\mathscr {L}}^\dagger }$). The argument he presents seems to be a transitivity argument, based essentially in the following implication:

$$\begin{aligned} ({\mathscr {L}}\leqslant _\textrm{id}{\mathscr {L}}^\dagger \text { and } {\mathscr {L}}^\dagger \leqslant _\pi {\mathscr {L}}^*) \Rightarrow {\mathscr {L}}\leqslant _\pi {\mathscr {L}}^*. \end{aligned}$$

According to Definition 2.2.5 and Remark 2.2.6, the above implication is sound. However, as stated in Proposition 4.2.3, the first inequality of the conjunction does not hold in general. Although the falsity of the conjunction does not imply that ${\mathscr {L}}^*$ is not as expressive as ${\mathscr {L}}$, we will show in this section that this is indeed the case, that is, ${\mathscr {L}}\not \leqslant _\pi {\mathscr {L}}^*$ (at least in the case where ${\mathscr {L}}$ does contain predicates that apply to worlds, see Proposition 4.4.1 for the exact statement). The proof is a variant of the argument presented in Proposition 4.2.3.

Proposition 4.4.1

Assume ${\mathscr {L}}$ is a language such that for some $n\in {{\mathbb {N}}}$ and some $k>0$, there is $P\in Pred _{n,k}^{\mathscr {L}}$. Then ${\mathscr {L}}$ has not less or equal expressive power than ${\mathscr {L}}^*$ with respect to $\pi$ (in symbols, ${\mathscr {L}}\not \leqslant _{\pi } {\mathscr {L}}^*$).

We will need an analogue of Lemma 4.2.4 for ${\mathscr {L}}^*$. Here the modelling relation in ${\mathscr {L}}^*$ has a local character too, but its locality stems from the operators $( Ref _d)$ and $({{\varvec{actually}}}_d)$. In this case, having a local character means that, even though the modelling relation of a fixed ${\mathscr {L}}^*$-formula $\varphi$ may depend on all variables in $Var _D^{{\mathscr {L}}^*}$, it only depends on a finite subset of the variables in $Var _W^{{\mathscr {L}}}$ (where ${\mathscr {L}}$ is its corresponding language in ${{\mathcal {L}}}_2$) and a finite number of instances of the operators $( Ref _d)$ and $({{\varvec{actually}}}_d)$. Formally, this is the content of the following lemma.

Lemma 4.4.2

Let $\varphi$ be an ${\mathscr {L}}^*$-formula. Suppose N is a positive integer such that for every integer d for which either $( Ref _d)$ or $({{\varvec{actually}}}_d)$ occurs in $\varphi$, or $P\in Pred _{n,d}^{\mathscr {L}}$ for which $P^*$ occurs in $\varphi$, we have that $d<N$. For $i=1,2$, let ${{\mathcal {A}}}_i=(W,D,V_i,v_i,w_i)$ be an ${\mathscr {L}}$-structure and let ${{\mathcal {A}}}_i^*$ be its corresponding ${\mathscr {L}}^*$-structure. Suppose that

• (1) $v_1, v_2$ agree on $Var _D^{{\mathscr {L}}}\cup \{u_0,\ldots , u_N\}$;

• (2) for $P\in Pred _{n,d}^{{\mathscr {L}}}$, $V_1(P) = V_2(P)$ (with $d<N$)

• (3) for every sentential functor $\delta$ in ${\mathscr {L}}$, $V_1(\delta )=V_2(\delta )$.

Then

$$\begin{aligned} {{\mathcal {A}}}_1^*\models \varphi \Leftrightarrow {{\mathcal {A}}}_2^*\models \varphi . \end{aligned}$$

Proof

We show by induction on ${\mathscr {L}}^*$-formulas that the result holds for all possible pairs ${{\mathcal {A}}}_1, {{\mathcal {A}}}_2$ as in the statement. If $\varphi$ is atomic, then $\varphi$ is of the form $P^*(x_1,\ldots ,x_n)$ for $P\in Pred _{n,k}^{{\mathscr {L}}^*}$ with $k<N$ and $x_1,\ldots ,x_n\in Var ^{{\mathscr {L}}^*}$. In this case we have

$$\begin{aligned}&{{\mathcal {A}}}_1^* \models (P^* x_1\cdots x_n) \\&\quad \Leftrightarrow w\sigma _{v_1}\in (V_1^*)(P^*)(v_1^*(x_1),\ldots ,v_1^*(x_n)) \\&\quad \Leftrightarrow (w\sigma _{v_1})(0)\in (V_1)_{v_1}(P)(v_1(x_1),\ldots ,v_1(x_n), v_1(u_1),\ldots ,v_1(u_k)) \\&\quad \Leftrightarrow w\in (V_1)_{v_1}(P)(v_1(x_1),\ldots ,v_1(x_n), v_1(u_1),\ldots ,v_1(u_k)) \\&\quad \Leftrightarrow w\in (V_2)_{v_2}(P)(v_2(x_1),\ldots ,v_2(x_n), v_2(u_1),\ldots ,v_2(u_k)) \\&\quad \Leftrightarrow (w\sigma _{v_2})(0)\in (V_2)_{v_2}(P)(v_2(x_1),\ldots ,v_2(x_n), v_2(u_1),\ldots ,v_2(u_k)) \\&\quad \Leftrightarrow w\sigma _{v_2}\in (V_2^*)_{v_2^*}((P^* x_1\cdots x_n)) \\&\quad \Leftrightarrow {{\mathcal {A}}}_2^* \models (P^* x_1\cdots x_n). \end{aligned}$$

It remains to show the inductive step. We split in cases.

Case 1: Let $\varphi$ be the ${\mathscr {L}}^*$-formula $(\delta ^*\psi _1\cdots \psi _n)$ for $\delta \in Fun _n^{{\mathscr {L}}}$, and suppose the result holds for $\psi _1,\ldots ,\psi _n$. This implies in particular that for each $\psi _i$

$$\begin{aligned} \{w\in W \mid w\sigma _{v_1}\in (V_1^*)_{v_1^*}(\psi _i)\} = \{w\in W \mid w\sigma _{v_2}\in (V_2^*)_{v_2^*}(\psi _i)\}. \end{aligned}$$

For each $\psi _i$ and each $j=1,2$, let $C_{ij}$ denote the set $\{w\in W \mid w\sigma _{v_j}\in (V_i^*)_{v_i^*} (\psi _i)\}$. We thus have that

$$\begin{aligned} {{\mathcal {A}}}_1^* \models (\delta ^*\psi _1\cdots \psi _n)&\Leftrightarrow w\sigma _{v_1}\in (V_1^*)(\delta ^*)(V_1^*)_{v_1^*}(\psi _1),\ldots ,(V_1^*)_{v_1^*}(\psi _n)) \\&\Leftrightarrow (w\sigma _{v_1})(0)\in V_1(\delta )(C_{11},\ldots , C_{n1})\\&\Leftrightarrow w\in V_1(\delta )(C_{11},\ldots , C_{n1})\\&\Leftrightarrow w\in V_2(\delta )(C_{12},\ldots , C_{n2})\\&\Leftrightarrow (w\sigma _{v_2})(0)\in V_2(\delta )(C_{12},\ldots , C_{n2})\\&\Leftrightarrow w\sigma _{v_2}\in (V_2^*)(\delta ^*)(V_2^*)_{v_2^*}(\psi _1),\ldots ,(V_2^*)_{v_2^*}(\psi _n)) \\&\Leftrightarrow {{\mathcal {A}}}_2^* \models (\delta ^*\psi _1\cdots \psi _n). \end{aligned}$$

Case 2: Suppose $\varphi$ is the ${\mathscr {L}}^*$-formula $( Ref _n)\psi$ with $n<N$ and suppose the result holds for $\psi$. We have

$$\begin{aligned} {{\mathcal {A}}}_1^* \models ( Ref _n)\psi&\Leftrightarrow w\sigma _{v_1}\in (V_1^*)_{v_1^*}(( Ref _n)\psi ) \\&\Leftrightarrow w\sigma _{v_1}[0/n]\in (V_1^*)_{v_1^*}(\psi ) \\&\Leftrightarrow w\sigma _{\mu _1}\in (V_1^*)_{v_1^*}(\psi )&\text { with }\mu _1=(v_1,w/u_{n-1}) \\&\Leftrightarrow w\sigma _{\mu _2}\in (V_2^*)_{v_2^*}(\psi )&\text { with }\mu _2=(v_2,w/u_{n-1}) \\&\Leftrightarrow w\sigma _{v_2}[0/n]\in (V_2^*)_{v_2^*}(\psi ) \\&\Leftrightarrow {{\mathcal {A}}}_2^* \models ( Ref _n)\psi . \end{aligned}$$

Case 3: Suppose $\varphi$ is the ${\mathscr {L}}^*$-formula $({{\varvec{actually}}}_n)\psi$ with $n<N$ and that the result holds for $\psi$. Set $z\in W$ as

$$\begin{aligned} z{:}{=}{\left\{ \begin{array}{ll} w &{}\quad \text { if }n=0\\ v_1(u_{n-1}) (=v_2(u_{n-1})) &{} \text { if n>0} \end{array}\right. } \end{aligned}$$

We have in this case

$$\begin{aligned} {{\mathcal {A}}}_1^* \models ({{\varvec{actually}}}_n)\psi&\Leftrightarrow w\sigma _{v_1}\in (V_1^*)_{v_1^*}(({{\varvec{actually}}}_n)\psi ) \\&\Leftrightarrow w\sigma _{v_1}[n/0]\in (V_1^*)_{v_1^*}(\psi ) \\&\Leftrightarrow zw\sigma _{v_1}\in (V_1^*)_{v_1^*}(\psi ) \\&\Leftrightarrow zw\sigma _{v_2}\in (V_2^*)_{v_2^*}(\psi ) \\&\Leftrightarrow w\sigma _{v_2}[n/0]\in (V_2^*)_{v_2^*}(\psi ) \\&\Leftrightarrow {{\mathcal {A}}}_2^* \models ({{\varvec{actually}}}_n)\psi . \end{aligned}$$

$\square$

Proof of Proposition 4.4.1

Let $P\in Pred _{n,k}^{\mathscr {L}}$ be a predicate with $n\in {{\mathbb {N}}}$ and $k>0$. Let $x=(x_1,\ldots ,x_n)$ be a tuple of domain variables and $u=(u_1,\ldots ,u_k)$ be a tuple of world variables. Suppose for a contradiction that ${\mathscr {L}}\leqslant _{\pi } {\mathscr {L}}^*$. Then there is an ${\mathscr {L}}^*$-formula $\varphi$ such that for all ${\mathscr {L}}^*$-structures ${{\mathcal {A}}}^*$ we have

$$\begin{aligned} {{\mathcal {A}}}\models \forall x \forall u (Pxu)\text { if and only if } {{\mathcal {A}}}^*\models \varphi . \end{aligned}$$

(C')

Let N be a positive integer such that for all occurrences of ($Ref _d$) and $({{\varvec{actually}}}_d)$ in $\varphi$ we have $d<N$, and in addition if $Q\in Pred _{n,k}^{\mathscr {L}}$ is such that $Q^*$ occurs in $\varphi$, then $k<N$. Let $U=\{u_0,\ldots , u_{N}\}$ be the set of the first $N+1$ world-variables. Let D be a non-empty set and let $a_0\in D$. Let W be a set of worlds $W=\{w_1,\ldots ,w_{N+1}\}$. We will consider two ${{\mathcal {L}}}_2$-structures ${{\mathcal {A}}}=(W,D,V,v,w_1)$ and ${{\mathcal {A}}}'=(W,D,V',v,w_1)$ where $V, V'$ and v are defined as follows. We define V and $V'$ to be the same function except for the predicate P. For P, we let $V(P):D^n\times W^k\rightarrow {{\mathcal {P}}}(W)$ be defined for all $a\in D^n$ by

$$\begin{aligned} v(P)(a, w_{i_1},\ldots ,w_{i_k}):= {\left\{ \begin{array}{ll} W &{} \quad \text { if }i_j< N\text { for all }1\leqslant j\leqslant k \\ \emptyset &{} \quad \text { otherwise}. \end{array}\right. } \end{aligned}$$

and $V'(P):D^n\times W^k\rightarrow {{\mathcal {P}}}(W)$ be defined as $V'(P)(a,w)=W$ for all $(a,w)\in D^n\times W^k$. Finally, define the assignment v which sends every domain variable to $a_0$ and for world variables it is defined as

$$\begin{aligned} v(u_i)= {\left\{ \begin{array}{ll} w_i &{} \quad \text { if }1\leqslant i\leqslant N\\ w_{N+1} &{} \quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Note that if $\{u_{i_1},\ldots , u_{i_k}\}\subseteq U$, then $V_v((Px u_{i_1}\cdots u_{i_k}))=V_v'((P x u_{i_1} \cdots u_{i_k}))$. Since V and $V'$ agree on all other predicates in ${\mathscr {L}}$, and all sentential operators in ${\mathscr {L}}$, by Lemma 4.2.4 we have that

$$\begin{aligned} {{\mathcal {A}}}^*\models \varphi \Leftrightarrow ({{\mathcal {A}}}')^*\models \varphi . \end{aligned}$$

However, by construction

$$\begin{aligned} {{\mathcal {A}}}\not \models \forall x \forall u (Pxu) \text { and } {{\mathcal {A}}}'\models \forall x \forall u (Pxu), \end{aligned}$$

which contradicts (C’). $\square$

Corollary 4.4.3

Assume ${\mathscr {L}}$ is a language in ${{\mathcal {L}}}_2$ such that for some $n\in {{\mathbb {N}}}$ and some $k>0$, there is $P\in Pred _{n,k}^{\mathscr {L}}$. Then ${\mathscr {L}}_{\square }^{\dagger }$ has no less or equal expressive power than ${\mathscr {L}}^*$ with respect to the function $\pi$ sending an ${\mathscr {L}}^*$-structure to its corresponding ${\mathscr {L}}$-structure (which is an ${\mathscr {L}}_{\square }^{\dagger }$-structure), in symbols, ${\mathscr {L}}_\square ^\dagger \not \leqslant _{\pi } {\mathscr {L}}^*$.

Proof

Suppose for a contradiction that ${\mathscr {L}}_\square ^\dagger \leqslant _{\textrm{id}} {\mathscr {L}}^*$. By Proposition 4.1.1 we have then that

$$\begin{aligned} {\mathscr {L}}\leqslant _\textrm{id}{\mathscr {L}}_\square ^\dagger \leqslant _\pi {\mathscr {L}}^*. \end{aligned}$$

By transitivity (Remark 2.2.6), it follows that ${\mathscr {L}}\leqslant _\pi {\mathscr {L}}^*$, contradicting Proposition 4.4.1. $\square$

5.5 Adding a $\square$ to ${\mathscr {L}}^*$

Let ${\mathscr {L}}$ be a language in ${{\mathcal {L}}}_2$. Consider the following extension ${\mathscr {L}}_\square ^*$ of ${\mathscr {L}}^*$. The formulas of ${\mathscr {L}}_ \square ^*$ are built using rules analogous to (R13)-(R16) together with the additional rule

• (R17) if $\varphi$ is an ${\mathscr {L}}_\square ^*$-formula, so is $\square \varphi$.

The class of structures of ${\mathscr {L}}_\square ^*$ is simply the class of ${\mathscr {L}}^*$-structures, that is, each ${\mathscr {L}}_\square ^*$-structure comes from a given ${\mathscr {L}}$-structure as we explained in Sect. 3.4. Given a ${\mathscr {L}}_\square ^*$-structure ${{\mathcal {A}}}^*=(W^*,D,V^*,v^*,w^*)$, we extend the function $V_{v^*}^*$ to ${\mathscr {L}}_\square ^*$-formulas by setting

$$\begin{aligned} \sigma \in V_{v^*}^*(\square \varphi ) \Leftrightarrow \text { for all } w\in W, w\sigma \in V_{v^*}^*(\varphi ). \end{aligned}$$

The modelling relation of ${\mathscr {L}}_\square ^*$ is defined, as for ${\mathscr {L}}^*$, by

$$\begin{aligned} {{\mathcal {A}}}^*\models \varphi \Leftrightarrow w^* \in V_{v^*}^*(\varphi ). \end{aligned}$$

Proposition 4.5.1

It holds that ${\mathscr {L}}_\square ^\dagger \leqslant _\pi {\mathscr {L}}_\square ^*$.

Proof

It suffices to modify the proof of Proposition 4.3.1 to include in the inductive step the case of an ${\mathscr {L}}_\square ^\dagger$-formula of the form $\square \varphi$. So suppose there is an ${\mathscr {L}}_\square ^*$-formula $\varphi ^*$ such that for all ${\mathscr {L}}_\square ^\dagger$-structures ${{\mathcal {A}}}$ it holds that

$$\begin{aligned} {{\mathcal {A}}}\models \varphi \Leftrightarrow {{\mathcal {A}}}^*\models \varphi ^*. \end{aligned}$$

We claim that for all ${\mathscr {L}}_\square ^\dagger$-structures ${{\mathcal {A}}}=(W,D,V,v,w_0)$ it holds that

$$\begin{aligned} {{\mathcal {A}}}\models \square \varphi \Leftrightarrow {{\mathcal {A}}}^*\models \square \varphi ^*. \end{aligned}$$

Indeed,

$$\begin{aligned} {{\mathcal {A}}}\models \square \varphi&\Leftrightarrow w\in V_v(\varphi ) \text { for all } w\in W \\ {}&\Leftrightarrow (W,D,V,v,w) \models \varphi \text { for all } w\in W\\ {}&\Leftrightarrow (W^*,D,V^*,v^*,w\sigma _v) \models \varphi ^* \text { for all } w\in W \\ {}&\Leftrightarrow (W^*,D,V^*,v^*,w_0\sigma _v) \models \square \varphi ^*. \\ {}&\Leftrightarrow {{\mathcal {A}}}^*\models \square \varphi ^*. \end{aligned}$$

$\square$

The following Corollary follows from transitivity (Remark 2.2.6) using Propositions 4.1.1 and 4.5.1.

Corollary 4.5.2

It holds that ${\mathscr {L}}\leqslant _{\pi }{\mathscr {L}}_\square ^*$.

6 Final Remarks

In this section we give a summary of our main argument together with a discussion of Yanovich’s account.

6.1 Assessing Cresswell’s Argument

At the beginning of this paper, we reconstructed Cresswell’s argument for (MEP) as an argument based on two premises.

• Premise 1 Modal discourse in natural language requires the semantics of a modal language equipped (at the very least) with the operators ${{\varvec{actually}}}_n$ and $Ref _n$.

• Premise 2 Any modal language equipped with the operators ${{\varvec{actually}}}_n$ and $Ref _n$ is as expressive as a corresponding two-sorted language with full quantification over worlds.

• Conclusion (MEP): Modal discourse in natural language requires a semantics with the expressive power of full quantification over worlds.

In Sect. 1 we looked at Cresswell’s justification of Premise 1. We sketched the arguments that Cresswell offers to show that modal English discourse requires the semantics of a modal language ${\mathscr {L}}^*$, which, as we saw in Sect. 3.4, is a modal language equipped with the operators ${{\varvec{actually}}}_n$ and $Ref _n$. Every such a language ${\mathscr {L}}^*$ has an associated language ${\mathscr {L}}$ from the family ${{\mathcal {L}}}_2$ with full quantification over worlds (see Sects. 3.2 and 3.4). To justify Premise 2, Cresswell offers a formal proof which seeks to show that ${\mathscr {L}}^*$ is as expressive as its counterpart ${\mathscr {L}}$ (see Cresswell, 1990, pp. 49–59). However, in Cresswell (1990), he does not provide a formal definition of the relation being as expressive as. In Sect. 2, we filled this gap by giving a precise definition of this relation, which reflects Cresswell’s use of it in Cresswell (1990). Section 4 was devoted to compare Cresswell’s formal languages and assess his claim that ${\mathscr {L}}^*$ is as expressive as its counterpart ${\mathscr {L}}$. Our Proposition 4.4.1 shows that ${\mathscr {L}}^*$ is not as expressive as ${\mathscr {L}}$. This result refutes Premise 2 and, hence, undermines the argument for (MEP).

It is worth noting that Corollary 4.5.2 does show that a language ${\mathscr {L}}^*_\square$, which is just like ${\mathscr {L}}^*$ but has the operator of logical necessity $\square$, is as expressive as its associated quantificational language ${\mathscr {L}}$ from the family ${{\mathcal {L}}}_2$. This result opens the way for a reformulation of Cresswell’s argument for (MEP). One could reformulate the two premises of the argument as follows:

• Premise 1’ Modal discourse in natural language requires the semantics of a modal language equipped with the operators $\square$, ${{\varvec{actually}}}_n$ and $Ref _n$.

• Premise 2’ Any modal language equipped with the operators $\square$, ${{\varvec{actually}}}_n$ and $Ref _n$ is as expressive as a corresponding two-sorted language with full quantification over worlds.

Corollary 4.5.2 provides a justification of Premise 2’. Nevertheless, Premise 1’ now requires a justification. We will not examine possible arguments for Premise 1’ here. For the moment, we just want to point out that the burden of proof is on the advocates of (MEP). In order to justify Premise 1’, they would have to show that the operator of logical necessity $\square$ is required for symbolizing modal English discourse. An argument of this sort is not present in Cresswell (1990). None of the examples he provides in Cresswell (1990) in order to substantiate Premise 1 shows that we need an operator like $\square$ to symbolize modal English discourse.

We conclude that Cresswell’s does not provide a convincing argument for (MEP) in Cresswell (1990). Under our original formulation, Cresswell’s argument is unsound because Premise 2 is false. Under the alternative formulation just considered, the argument is unpersuasive because Premise 1’ lacks a proper justification. As we pointed out in the introduction, (MEP) has been a very influential thesis both in linguistics and the philosophy of language. Several theorists have relied on it to draw substantive conclusions about the contents and grammatical structures of modal discourse. Our discussion in this paper suggests that more caution is necessary when invoking facts about the expressive power of formal languages to support linguistic or philosophical theories about natural language.

Before closing this section we will briefly consider a few questions that our discussion raises.

6.1.1 What did Cresswell Prove in the Fourth Chapter of Entities and Indices?

The proof presented in Cresswell (1990, Chapter 4) has two parts. Cresswell concludes that a given language ${\mathscr {L}}^*$ is as expressive as its quantificational counterpart ${\mathscr {L}}$ by seeking to show, first, that his language ${\mathscr {L}}^\dagger$ is as expressive as ${\mathscr {L}}$ and, second, that ${\mathscr {L}}^*$ is as expressive as ${\mathscr {L}}^\dagger$.

When Cresswell introduces the language ${\mathscr {L}}^\dagger$, he describes it as a language which is like ${\mathscr {L}}$ except that it employs the operators $\textrm{Ref}u$ and [u] in place of the universal quantifier for world variables ( Cresswell 1990, p. 48). He does not say that the operator $\square$ is a logical constant (or an improper symbol) of ${\mathscr {L}}^\dagger$. Nonetheless, in the key step of the first part of his proof, he assumes that $\square$ is among the operators of ${\mathscr {L}}^\dagger$:

“It is not difficult to shew that ${\mathscr {L}}^\dagger$ is as powerful as ${\mathscr {L}}$. For we may define $\forall {{\textbf {u}}} \alpha$ as

(6) $(\textrm{Ref}{{\textbf {w}}}) \ \square \ (\textrm{Ref}{{\textbf {u}}})[{{\textbf {w}}}] \alpha$

where ${{\textbf {w}}}$ is the first variable in some standard ordering distinct from ${{\textbf {u}}}$ and not in $\alpha$, and $\square$ is the logical necessity operator whose meaning is given by (15) on p. 9.” Cresswell (1990, p. 49).

Observe that the definition of $\forall {{\textbf {u}}} \alpha$ given in Clause (6) of the previous passage only makes sense if we assume that $\square$ is a logical constant of Cresswell’s language ${\mathscr {L}}^\dagger$. Otherwise, there will be different interpretations of ${\mathscr {L}}^\dagger$ in which $\square$ denotes operations on propositions different from the intended one.¹⁵

In the second part of the proof, on the other hand, there is no mention of the operator $\square$. The key step of that part of the proof is the specification of a translation of ${\mathscr {L}}^\dagger$ into ${\mathscr {L}}^*$. In the definition of this translation there is no clause that tells us how to translate an ${\mathscr {L}}^\dagger$-formula of the form $\square \alpha$ into ${\mathscr {L}}^*$ (Cresswell, 1990, pp. 55–56). There is only a clause that specifies the translation of an ${\mathscr {L}}^\dagger$-formula of the form $(\delta \alpha _1 \ldots \alpha _n)$, where $\delta$ is a regular functor of ${\mathscr {L}}^\dagger$ and $\alpha _1 \ldots \alpha _n$ are ${\mathscr {L}}^\dagger$-formulas.

The above suggests that Cresswell’s proof involves four languages rather than three. What the first part of the proof really shows is that the language ${\mathscr {L}}_\square ^\dagger$ that we defined in Sect. 3.3 is as expressive as its counterpart ${\mathscr {L}}$ from the family ${{\mathcal {L}}}_2$. What the second part of the proof shows is that the language ${\mathscr {L}}^*$ defined in Sect. 3.4 is as expressive as the language ${\mathscr {L}}^\dagger$ defined in Sect. 3.3. From these two claims it does not follow that ${\mathscr {L}}^*$ is as expressive as ${\mathscr {L}}$. Our Proposition 4.4.1, as we have already stressed, establishes that this is not so.

There is, nonetheless, an alternative way of understanding Cresswell’s proof. Since Cresswell clearly assumes that $\square$ is one of the operators of his language ${\mathscr {L}}^\dagger$, one may think that, in fact, he is implicitly supposing that the logical necessity operator $\square$ is a logical constant of the languages ${\mathscr {L}}$, ${\mathscr {L}}^\dagger$, and ${\mathscr {L}}^*$ that he considers in the course of the proof. In the terminology of our Sects. 3 and 4, this would mean that Cresswell’s languages ${\mathscr {L}}^*$ and ${\mathscr {L}}^\dagger$ are in fact our languages ${\mathscr {L}}_\square ^*$ and ${\mathscr {L}}_\square ^\dagger$. It would also mean that what the second part of the proof seeks to show is that ${\mathscr {L}}_\square ^*$ is as expressive as ${\mathscr {L}}_\square ^\dagger$, while the first part seeks to show that ${\mathscr {L}}_\square ^\dagger$ is as expressive as ${\mathscr {L}}$. This is precisely what our Proposition 4.5.1 and our Corollary 4.5.2 state.

Although Cresswell’s proof can reasonably be understood in the way suggested in the previous paragraph, the comment that we made concerning Premise 1’ and Premise 2’ applies here, too. Even though Premise 2’ is correct, the examples that Cresswell offered to justify Premise 1 (see Sect. 1) do not support Premise 1’. The burden of proof is on the advocates of Premise 1’. They have to show that the operator of logical necessity $\square$ is required to symbolize modal English discourse.

6.1.2 Which Modal Operators are Needed in Order to Account for Modal English Discourse?

Although this question is not easy to answer, it is quite relevant for the purposes of our discussion. For, depending on its answer, one could possibly save Cresswell’s argumentative strategy to substantiate (MEP). For example, as mentioned above, if one showed that the operator of logical necessity was needed to account for modal English discourse, then one would have a justification of Premise 1’.

One may thus wonder whether there are examples modal natural-language sentences which can support Premise 1’. One important complication that one must face if one wants to address this question is that the very notion of natural-language modal discourse requires a proper delimitation. Although Cresswell refers frequently to such notion, there is no explicit characterization of it in Cresswell (1990).¹⁶

6.1.3 Are There Operator-Based Extensions ${\mathscr {L}}^{**}$ of ${\mathscr {L}}^*$ for Which ${\mathscr {L}}\leqslant _\pi {\mathscr {L}}^{**}$ Holds?

Corollary 4.5.2 shows that one such extension is ${\mathscr {L}}_\square ^*$, namely the extension which adds the operator of logical necessity $\square$ to ${\mathscr {L}}^*$. It would be interesting to know if there are other modal operators with this property. This is of course related to Question 5.1.1: if one can show that the added operators are required in order to account for modal English discourse, then one can resuscitate Cresswell’s argumentative strategy with variants of Premises 1 and 2 that entail (MEP).

6.2 Yanovich’s Account

Despite the wide acceptance of Cresswell’s claim that modal discourse in natural language requires a semantics with the expressive power of explicit quantification over possible worlds, Yanovich (2015) argued against that claim. Although Yanovich does not follow Cresswell’s formalism he provides examples of first order modal languages which are not as expressive as quantified languages with generalized backwards-looking operators (see Yanovich 2015, Sections 3–6) for details about his formalism). We will not present here his results nor discuss their validity. Nevertheless, it is worth making a couple of comments on Yanovich’s paper.

The first comment concerns Yanovich’s strategy to show that first order modal languages (in his notation ${\textbf {ML}}^{FO}$) and quantified languages with generalized backwards-looking operators (in his notation, ${\textbf {Cr}}^{FO}$) differ in expressive power. His approach is based on an adapted notion of bisimulation for ${\textbf {ML}}^{FO}$. Bisimulation is a binary relation between ${\textbf {ML}}^{FO}$-structures which implies, in particular, that bisimular structures satisfy exactly the same modal ${\textbf {ML}}^{FO}$-sentences (see Yanovich, 2015, Theorem 2). His proof strategy consists in showing that there exist bisimular ${\textbf {ML}}^{FO}$-structures for which their associated ${\textbf {Cr}}^{FO}$-structures do not satisfy the same ${\textbf {Cr}}^{FO}$ sentences. In contrast with Yanovich’s strategy, our proof of Proposition 4.4.1 shows that we do not need to consider bisimular ${\mathscr {L}}^*$-structures in order to show that ${\mathscr {L}}\not \leqslant _\pi {\mathscr {L}}^*$. The main ingredient of our proof strategy is the local charachter of the modelling relation $\models _{{\mathscr {L}}^*}$ (resp. $\models _{{\mathscr {L}}^\dagger }$) in the sense of Lemma 4.4.2 (resp. Lemma 4.2.4). Such a local character allows us to show that ${\mathscr {L}}^*$ cannot capture universal quantification over worlds (without using a notion of bisimulation for ${\mathscr {L}}^*$-structures).

Secondly, Yanovich’s comparison between ${\textbf {ML}}^{FO}$ and ${\textbf {Cr}}^{FO}$ depends on whether such languages include a symbol for identity (see, for example, the discussion that follows Yanovich 2015, Example 3 and Proposition 4).¹⁷ It is worth noticing that our results do not depend on the logical status of the identity predicate. This reveals an interesting difference between our approaches.

In his paper, Yanovich compares the expressive power of ${\textbf {ML}}^{FO}$ with the expressive power of other formal languages, including hybrid-logic languages. In the framework that we adopted in this paper (see Sect. 3), a comparison of expressive power can only be carried out once we have chosen a specific function $\pi$ that relates the structures of the relevant languages. For reasons of space, we did not explore the relationship between the languages defined in Cresswell (1990, Chapters 1–4) and other families of languages that have been discussed in the literature. We hope the framework laid out in this paper can serve as a base to address these issues.