TEMPORAL INTERPRETATION OF MONADIC INTUITIONISTIC QUANTIFIERS

Abstract

We show that monadic intuitionistic quantifiers admit the following temporal interpretation: “always in the future” (for $\forall $ ) and “sometime in the past” (for $\exists $ ). It is well known that Prior’s intuitionistic modal logic ${\sf MIPC}$ axiomatizes the monadic fragment of the intuitionistic predicate logic, and that ${\sf MIPC}$ is translated fully and faithfully into the monadic fragment ${\sf MS4}$ of the predicate ${\sf S4}$ via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension ${\sf TS4}$ of ${\sf S4}$ and provide a full and faithful translation of ${\sf MIPC}$ into ${\sf TS4}$ . We compare this new translation of ${\sf MIPC}$ with the Gödel translation by showing that both ${\sf TS4}$ and ${\sf MS4}$ can be translated fully and faithfully into a tense extension of ${\sf MS4}$ , which we denote by ${\sf MS4.t}$ . This is done by utilizing the relational semantics for these logics. As a result, we arrive at the diagram of full and faithful translations shown in Figure 1 which is commutative up to logical equivalence. We prove the finite model property (fmp) for ${\sf MS4.t}$ using algebraic semantics, and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered.

1 Introduction

It is well known that, unlike classical quantifiers, the interpretation of intuitionistic quantifiers is non-symmetric in that $\forall x A$ is true at a world w iff A is true at every object a in the domain $D_v$ of every world $v$ accessible from $w$ , while $\exists x A$ is true at $w$ iff A is true at some object a in the domain $D_w$ of $w$ . This non-symmetry is also evident in the Gödel translation of the intuitionistic predicate calculus ${\sf IQC}$ into the predicate ${\sf S4}$ , denoted ${\sf QS4}$ , since $\forall xA$ is translated as $\Box \forall x A^t$ while $\exists x A$ as $\exists x A^t$ , where $A^t$ is the translation of A.

Fig. 1 Diagram of translations.

Our aim is to provide a more symmetric temporal interpretation of intuitionistic quantifiers as “always in the future” (for $\forall $ ) and “sometime in the past” (for $\exists $ ). In this paper we restrict our attention to monadic quantifiers (that quantify over one fixed variable), but in Section 7 we discuss the connection to the full predicate case, which is treated in detail in [4]. One of the main reasons to treat the monadic case separately is because it gives rise to a new temporal system ${\sf TS4}$ (see below).

It is well known that the monadic fragment of ${\sf IQC}$ is axiomatized by Prior’s monadic intuitionistic propositional calculus ${\sf MIPC}$ [8, 25]. The monadic fragment of ${\sf QS4}$ was studied by Fischer-Servi [14] who showed that the Gödel translation of ${\sf IQC}$ into ${\sf QS4}$ restricts to the monadic case. We denote the monadic fragment of ${\sf QS4}$ by ${\sf MS4}$ . One of our main contributions is to introduce a tense counterpart of ${\sf MS4}$ , denoted by ${\sf TS4}$ , and prove that a modification of the Gödel translation embeds ${\sf MIPC}$ into ${\sf TS4}$ fully and faithfully. This allows us to give the desired temporal interpretation of monadic intuitionistic quantifiers as “always in the future” (for $\forall $ ) and “sometime in the past” (for $\exists $ ).

While ${\sf MS4}$ and ${\sf TS4}$ are not comparable, we introduce a common extension, which we denote by ${\sf MS4.t}$ . The system ${\sf MS4.t}$ can be thought of as a tense extension of ${\sf MS4}$ . We prove that there exist full and faithful translations of ${\sf MIPC}$ , ${\sf MS4}$ , and ${\sf TS4}$ into ${\sf MS4.t}$ , yielding the diagram in Figure 1. The Gödel translation is denoted by $(\:)^t$ , our new translation by $(\:)^\natural $ , and the three translations into ${\sf MS4.t}$ by $(\:)^\# $ , $(\:)^\flat$ , and $(\:)^\dagger $ , respectively.

We prove these results by employing relational semantics. In addition, we utilize algebraic semantics to prove that ${\sf MS4.t}$ has the fmp. It is then an easy consequence of the fullness and faithfulness of the translations considered that the other systems also have the fmp. That ${\sf MIPC}$ has the fmp was proved in [7], but the proof contained a gap, which was corrected in [15, 23]. The fmp for ${\sf MS4}$ follows from the results in [17, Sec. 12]. An advantage of our approach is in that it provides a uniform means for proving the fmp for all four systems in Figure 1.

In [4] we extended the translation of ${\sf MIPC}$ into ${\sf MS4.t}$ to the predicate setting. ¹ We showed that the same interpretation of intuitionistic quantifiers can be realized via a full and faithful translation of ${\sf IQC}$ into a version of predicate ${\sf S4.t}$ that can be thought of as a predicate analogue of ${\sf MS4.t}$ . We conclude this paper by comparing the translations investigated here with those studied in [4].

2 Logics of interest

In this section we present our four main logics of interest (see Figure 1). We start by recalling the monadic intuitionistic propositional calculus ${\sf MIPC}$ . This system was introduced by Prior [26, p. 38] and it was shown by Bull [8] that ${\sf MIPC}$ axiomatizes the monadic fragment of the intuitionistic predicate calculus ${\sf IQC}$ . ² For this reason we denote the modalities of ${\sf MIPC}$ by $\forall $ and $\exists $ . Let $\mathcal L$ be the propositional language and let $\mathcal L_{\forall \exists }$ be the extension of $\mathcal L$ with two modalities $\forall $ and $\exists $ .

Definition 2.1. The monadic intuitionistic propositional calculus ${\sf MIPC}$ is the intuitionistic modal logic in the propositional modal language $\mathcal L_{\forall \exists }$ containing

1. 1. all theorems of the intuitionistic propositional calculus ${\sf IPC}$ ;

2. 2. the ${\sf S4}$ -axioms for $\forall $ :

   1. (a) $\forall (p\land q)\leftrightarrow (\forall p\land \forall q)$ ,

   2. (b) $\forall p \rightarrow p$ ,

   3. (c) $\forall p \rightarrow \forall \forall p$ ;

3. 3. the ${\sf S5}$ -axioms for $\exists $ :

   1. (a) $\exists (p\vee q)\leftrightarrow (\exists p\vee \exists q)$ ,

   2. (b) $p \rightarrow \exists p$ ,

   3. (c) $\exists \exists p \rightarrow \exists p$ ,

   4. (d) $(\exists p \land \exists q) \rightarrow \exists (\exists p \land q)$ ;

4. 4. the axioms connecting $\forall $ and $\exists $ :

   1. (a) $\exists \forall p\leftrightarrow \forall p$ ,

   2. (b) $\exists p \leftrightarrow \forall \exists p$ ;

and closed under the rules of modus ponens, substitution, and necessitation $(\varphi / \forall \varphi )$ .

Remark 2.2. There are several equivalent axiomatizations of ${\sf MIPC}$ (see, e.g., [2, Lemma 2]).

We next recall the monadic extension of ${\sf S4}$ studied by Fischer-Servi [14] who showed that it axiomatizes the monadic fragment of the predicate ${\sf S4}$ . Let $\mathcal L_{\Box \forall }$ be the propositional bimodal language with two modal operators $\Box $ and $\forall $ . As usual, $\Diamond $ is an abbreviation for $\neg \Box \neg $ and $\exists $ is an abbreviation for $\neg \forall \neg $ .

Definition 2.3. The monadic $\sf S4$ , denoted ${\sf MS4}$ , is the smallest bimodal logic containing all theorems of the classical propositional calculus ${\sf CPC}$ , the $\sf S4$ -axioms for $\Box $ , the $\sf S5$ -axioms for $\forall $ , the left commutativity axiom

$$ \begin{align*} \Box \forall p \to \forall \Box p, \end{align*} $$

and closed under modus ponens, substitution, $\Box $ -necessitation, and $\forall $ -necessitation.

Remark 2.4. Recalling the definition of fusion of two logics (see [16]), ${\sf MS4}$ is obtained from the fusion ${\sf S4}\otimes {\sf S5}$ by adding the left commutativity axiom $\Box \forall p \rightarrow \forall \Box p$ which is the monadic version of the converse Barcan formula. The monadic version of the Barcan formula is the right commutativity axiom $\forall \Box p \rightarrow \Box \forall p$ . Adding it to ${\sf MS4}$ yields the product logic $\sf S4 \times \sf S5$ ; see [16, Chap. 5] for details.

The following lemma will be useful in Section 3.

Lemma 2.5. An equivalent axiomatization of ${\sf MS4}$ is obtained by replacing the left commutativity axiom $\Box \forall p \to \forall \Box p$ by $\exists \Box p \to \Box \exists p$ .

Proof. We show that ${\sf MS4}\vdash \exists \Box p \to \Box \exists p$ . That $\exists \Box p \to \Box \exists p$ together with the other axioms of ${\sf MS4}$ implies $\Box \forall p \to \forall \Box p$ is proved similarly. Since $\forall $ is an ${\sf S5}$ -modality, $\Box \exists p \to \Box \forall \exists p$ is a theorem of ${\sf MS4}$ . By the left commutativity axiom, $\Box \forall \exists p \to \forall \Box \exists p$ is also a theorem of ${\sf MS4}$ . So ${\sf MS4}\vdash \Box \exists p \to \forall \Box \exists p$ , and hence ${\sf MS4}\vdash \exists \Box \exists p \to \exists \forall \Box \exists p$ . But $\exists \Box p \to \exists \Box \exists p$ , $\exists \forall \Box \exists p \to \forall \Box \exists p$ , and $\forall \Box \exists p \to \Box \exists p$ are all theorems of ${\sf MS4}$ because $\forall $ is an ${\sf S5}$ -modality. Thus, ${\sf MS4}\vdash \exists \Box p \to \Box \exists p$ , concluding the proof.

To introduce our main tense logic ${\sf TS4}$ , we first need to recall the tense logic ${\sf S4.t}$ . Let $\mathcal L_T$ be the propositional tense language with two modalities ${[F]}$ and ${[P]}$ . As usual, ${[F]}$ is interpreted as “always in the future” and ${[P]}$ as “always in the past.” We use the following standard abbreviations: ${\langle F \rangle }$ for $\neg {[F]} \neg $ and ${\langle P \rangle }$ for $\neg {[P]} \neg $ . Then ${\langle F \rangle }$ is interpreted as “sometime in the future” and ${\langle P \rangle }$ as “sometime in the past.”

Definition 2.6. Let ${\sf S4.t}$ be the smallest bimodal logic containing all theorems of the classical propositional calculus ${\sf CPC}$ , the ${\sf S4}$ -axioms for ${[F]}$ and ${[P]}$ , the tense axioms

$$ \begin{align*} \begin{array}{c} p \to {[P]} {\langle F \rangle} p, \\ p \to {[F]} {\langle P \rangle} p, \end{array} \end{align*} $$

and closed under modus ponens, substitution, ${[F]}$ -necessitation, and ${[P]}$ -necessitation.

Remark 2.7. The system ${\sf S4.t}$ is the extension of the least tense logic $\sf K.t$ in which both tense modalities satisfy the ${\sf S4}$ -axioms. It was studied by several authors. Esakia [10] showed that the Gödel translation can be extended to embed the Heyting–Brouwer logic $\sf HB$ of Rauszer [28] into ${\sf S4.t}$ fully and faithfully. The language of $\sf HB$ is obtained by enriching the language of $\sf IPC$ by an additional connective of coimplication, and the logic $\sf HB$ is the extension of $\sf IPC$ by the axioms for coimplication, which are dual to the axioms for implication. Wolter [31] extended the celebrated Blok–Esakia theorem to this setting.

We are ready to define ${\sf TS4}$ by combining ${\sf S4}$ and ${\sf S4.t}$ . In Section 4 we will translate ${\sf MIPC}$ into ${\sf TS4}$ fully and faithfully. We will use ${\sf S4}$ to translate intuitionistic connectives and ${\sf S4.t}$ to translate monadic intuitionistic quantifiers. Let $\mathcal {ML}$ be the multimodal propositional language with three modalities $\Box $ , ${[F]}$ , and ${[P]}$ . We use $\Diamond $ , ${\langle F \rangle }$ , and ${\langle P \rangle }$ as usual abbreviations.

Definition 2.8. The logic ${\sf TS4}$ is the least multimodal logic containing all theorems of the classical propositional calculus ${\sf CPC}$ , the ${\sf S4}$ -axioms for $\Box $ , ${[F]}$ , and ${[P]}$ , the tense axioms for ${[F]}$ and ${[P]}$ , the connecting axioms

$$ \begin{align*} \begin{array}{l} \Diamond p \to {\langle F \rangle} p, \\ {\langle F \rangle} p \to \Diamond( {\langle F \rangle} p \land {\langle P \rangle} p), \end{array} \end{align*} $$

and closed under modus ponens, substitution, and three necessitation rules (for $\Box $ , ${[F]}$ , and ${[P]}$ ).

Our final logic of interest is the monadic tense logic ${\sf MS4.t}$ which is obtained by combining ${\sf MS4}$ and ${\sf S4.t}$ . As we will see, both ${\sf MS4}$ and ${\sf TS4}$ translate fully and faithfully into ${\sf MS4.t}$ . In order to avoid confusion between the tense modalities of ${\sf TS4}$ and ${\sf MS4.t}$ , we denote the tense modalities of ${\sf MS4.t}$ by $\Box _F$ and $\Box _P$ . Let $\mathcal L_{T\forall }$ be the propositional language with the tense modalities $\Box _F$ and $\Box _P$ , and the monadic modality $\forall $ .

Definition 2.9. The tense ${\sf MS4}$ , denoted ${\sf MS4.t}$ , is the least multimodal logic containing all theorems of the classical propositional calculus ${\sf CPC}$ , the ${\sf S4.t}$ -axioms for $\Box _F$ and $\Box _P$ , the ${\sf S5}$ -axioms for $\forall $ , the left commutativity axiom

$$ \begin{align*} \Box_F \forall p \to \forall \Box_F p, \end{align*} $$

and closed under modus ponens, substitution, and the necessitation rules (for $\Box _F$ , $\Box _P$ , and $\forall $ ).

Remark 2.10. We can think of ${\sf MS4.t}$ as a tense extension of ${\sf MS4}$ . It is worth stressing that ${\sf MS4.t}$ is not the monadic fragment of the standard predicate extension ${\sf QS4.t}$ of ${\sf S4.t}$ . To see this, it is well known that the Barcan formula $\forall x \Box _F \varphi \to \Box _F \forall x \varphi $ and the converse Barcan formula $\Box _F \forall x \varphi \to \forall x \Box _F \varphi $ are both theorems of any tense predicate logic containing the standard axioms of first-order logic. Therefore, both are theorems of ${\sf QS4.t}$ . Thus, the monadic fragment of ${\sf QS4.t}$ contains both the left commutativity axiom $\Box _F \forall p \to \forall \Box _F p$ and the right commutativity axiom $\forall \Box _F p \to \Box _F \forall p$ . On the other hand, it is easy to see (e.g., using the Kripke semantics for ${\sf MS4.t}$ which we will define in the next section) that, while ${\sf MS4.t}$ contains the left commutativity axiom, the right commutativity axiom is not provable in ${\sf MS4.t}$ .

3 Relational semantics

In this section we investigate the relational semantics for ${\sf MIPC}$ , ${\sf MS4}$ , ${\sf TS4}$ , and ${\sf MS4.t}$ . The relational semantics for ${\sf MIPC}$ and ${\sf MS4}$ have already been studied in the literature, and the relational semantics for ${\sf TS4}$ and ${\sf MS4.t}$ are obtained by a straightforward adaptation.

Definition 3.1. Let R be a quasi-order (reflexive and transitive relation) on a set X. As usual, for $x\in X$ , we write

$$ \begin{align*} R[x]=\{y\in X\mid xRy\} \mbox{ and } R^{-1}[x]=\{y\in X\mid yRx\}, \end{align*} $$

and for $A\subseteq X$ , we write

$$ \begin{align*} R[A]=\bigcup\{R[x]\mid x\in A\} \mbox{ and } R^{-1}[A]=\bigcup\{R^{-1}[x]\mid x\in A\}. \end{align*} $$

We say that $A \subseteq X$ is an R-upset if $R[A]=A$ and that it is an R-downset if $R^{-1}[A]=A$ .

We first describe the relational semantics for ${\sf MIPC}$ . There are several such (see, e.g., [3]), but we concentrate on the one introduced by Ono [23].

Definition 3.2. An ${\sf MIPC}$ -frame is a triple $\mathfrak F=(X,R,Q)$ where X is a set, R is a partial order, Q is a quasi-order, and the following two conditions are satisfied:

1. (O1) $R \subseteq Q$ ,

2. (O2) $x Q y \Rightarrow (\exists z)(x R z \; \& \; z E_Q y)$ (see Figure 2).

Here $E_Q$ is the equivalence relation defined by $x E_Q y$ iff $x Q y$ and $y Q x$ .

Remark 3.3. If U is an R-upset of an ${\sf MIPC}$ -frame $\mathfrak {F}$ , then Condition (O2) implies that $E_Q[U]=Q[U]$ . This motivates our interpretation of $\exists $ as “sometime in the past.” Indeed, taking $Q[U]$ is the standard way to associate an operator on $\wp (X)$ to the tense modality “sometime in the past” (see, e.g., [30, p. 151]).

Fig. 2 Condition (O2).

Definition 3.4. A valuation on an ${\sf MIPC}$ -frame $\mathfrak F=(X,R,Q)$ is a map $v$ associating an R-upset of $\mathfrak F$ to any propositional letter of $\mathcal L_{\forall \exists }$ . The connectives $\wedge ,\vee ,\to ,\neg $ are then interpreted as in intuitionistic Kripke frames, and $\forall ,\exists $ are interpreted as

$$ \begin{align*} \begin{array}{l c l} x \vDash_v \forall \varphi & \text{ iff } & (\forall y \in X)(x Q y \Rightarrow y \vDash_v \varphi ), \\ x \vDash_v \exists \varphi & \text{ iff } & (\exists y \in X)(x E_Q y \;\& \; y \vDash_v \varphi ). \end{array} \end{align*} $$

As usual, we say that $\varphi $ is valid in $\mathfrak F$ , and write $\mathfrak F \vDash \varphi $ , if $x \vDash _v \varphi $ for every valuation $v$ and every $x \in X$ .

It is well known that ${\sf MIPC}$ is a canonical logic (see, e.g., [3, Sec. 6]). Thus, we have:

Theorem 3.5. ${\sf MIPC}$ is a canonical logic, and hence it is sound and complete with respect to its relational semantics. Therefore,

$$ \begin{align*} {\sf MIPC} \vdash \varphi \ \mbox{ iff } \ \mathfrak F \vDash \varphi \mbox{ for every}\ {\sf MIPC}\mbox{-frame } \mathfrak F. \end{align*} $$

Remark 3.6. In addition, ${\sf MIPC}$ has the fmp [7, 15, 23] and hence is decidable. As we will see in Section 6, the fmp of ${\sf MIPC}$ can be derived from the fmp of ${\sf MS4.t}$ .

The relational semantics for ${\sf MS4}$ was introduced by Esakia [12].

Definition 3.7. An ${\sf MS4}$ -frame is a triple $\mathfrak F=(X,R,E)$ where X is a set, R is a quasi-order, E is an equivalence relation, and the following commutativity condition is satisfied (see Figure 3 ):

(E) $$ \begin{align} (\forall x,y,z \in X)(x E y \; \& \; y R z) \Rightarrow (\exists u \in X)(x R u \; \& \; u E z). \end{align} $$

Fig. 3 Condition (E).

Definition 3.8. A valuation on an ${\sf MS4}$ -frame $\mathfrak F=(X,R,E)$ is a map $v$ associating a subset of X to each propositional letter of $\mathcal L_{\Box \forall }$ . The Boolean connectives are interpreted as usual, and

$$ \begin{align*} \begin{array}{l c l} x \vDash_v \Box \varphi & \text{ iff } & (\forall y \in X) (x R y \, \Rightarrow \, y \vDash_v \varphi ), \\ x \vDash_v \forall \varphi & \text{ iff } & (\forall y \in X) (x E y \, \Rightarrow \, y \vDash_v \varphi ). \end{array} \end{align*} $$

By Lemma 2.5, in the axiomatization of ${\sf MS4}$ , the left commutativity axiom $\Box \forall p \to \forall \Box p$ can be replaced by $\exists \Box p \to \Box \exists p$ . Therefore, ${\sf MS4}$ can be axiomatized by Sahlqvist formulas (see, e.g., [5, Sec. 3.6]). Thus, by the Sahlqvist completeness theorem (see, e.g., [5, Theorem 4.42]), it is sound and complete with respect to its relational semantics:

Theorem 3.9. ${\sf MS4}$ is a Sahlqvist logic, and hence it is sound and complete with respect to its relational semantics. Therefore,

$$ \begin{align*} {\sf MS4} \vdash \varphi \ \mbox{ iff } \ \mathfrak F \vDash \varphi \mbox{ for every}\ {\sf MS4}\mbox{-frame}\ \mathfrak F. \end{align*} $$

Remark 3.10. In addition, ${\sf MS4}$ has the fmp and is decidable. This can be derived from the results in [17, Sec. 12] (see also [16, Theorems 6.52 and 9.12]). As we will see in Section 6, the fmp of ${\sf MS4}$ can also be derived from the fmp of ${\sf MS4.t}$ .

We next recall the relational semantics for ${\sf S4.t}$ .

Definition 3.11. An ${\sf S4.t}$ -frame is a pair $\mathfrak {F}=(X,Q)$ where X is a set and Q is a quasi-order on X.

Remark 3.12. While ${\sf S4.t}$ -frames coincide with ${\sf S4}$ -frames, the difference is in the interpretation of the modalities as we use Q to interpret ${[F]}$ and its inverse relation to interpret ${[P]}$ .

Definition 3.13. A valuation on an ${\sf S4.t}$ -frame $\mathfrak F=(X,Q)$ is a map $v$ associating a subset of X to each propositional letter of $\mathcal L_T$ . The Boolean connectives are interpreted as usual, and the tense modalities are interpreted as

$$ \begin{align*} \begin{array}{l c l} x \vDash_v {[F]} \varphi & \text{ iff } & (\forall y \in X) (x Q y \, \Rightarrow \, y \vDash_v \varphi ), \\ x \vDash_v {[P]} \varphi & \text{ iff } & (\forall y \in X) (y Q x \, \Rightarrow \, y \vDash_v \varphi ). \end{array} \end{align*} $$

Remark 3.14. It is straightforward to see that all the axioms of ${\sf S4.t}$ are Sahlqvist formulas. Therefore, ${\sf S4.t}$ is a Sahlqvist logic, and hence it is sound and complete with respect to its relational semantics. That ${\sf S4.t}$ has the fmp follows from [29, pp. 313–314] (see also [18, p. 44] and Remark 6.18(3)).

We now introduce the relational semantics for ${\sf TS4}$ .

Definition 3.15. A ${\sf TS4}$ -frame is a triple $\mathfrak F=(X,R,Q)$ where X is a set and $R,Q$ are quasi-orders on X satisfying Conditions (O1) and (O2).

It follows that ${\sf TS4}$ -frames are a version of ${\sf MIPC}$ -frames, where the relation R is a quasi-order. We interpret $\Box $ using R, and ${[F]}$ , ${[P]}$ using Q and its inverse .

Definition 3.16. A valuation on a ${\sf TS4}$ -frame $\mathfrak F=(X,R,Q)$ is a map $v$ associating a subset of X to each propositional letter of $\mathcal {ML}$ . The Boolean connectives are interpreted as usual, and the modalities $\Box $ , ${[F]}$ , and ${[P]}$ are interpreted as

$$ \begin{align*} \begin{array}{l c l} x \vDash_v \Box \varphi & \text{ iff } & (\forall y \in X) (x R y \Rightarrow y \vDash_v \varphi ), \\ x \vDash_v {[F]} \varphi & \text{ iff } & (\forall y \in X) (x Q y \Rightarrow y \vDash_v \varphi ), \\ x \vDash_v {[P]} \varphi & \text{ iff } & (\forall y \in X) (y Q x \Rightarrow y \vDash_v \varphi ). \end{array} \end{align*} $$

Consequently,

$$ \begin{align*} \begin{array}{l c l} x \vDash_v \Diamond \varphi & \text{ iff } & (\exists y \in X) (x R y \; \& \; y \vDash_v \varphi ),\\ x \vDash_v {\langle F \rangle} \varphi & \text{ iff } & (\exists y \in X) (x Q y \; \& \; y \vDash_v \varphi ),\\ x \vDash_v {\langle P \rangle} \varphi & \text{ iff } & (\exists y \in X) (y Q x \; \& \; y \vDash_v \varphi ). \end{array} \end{align*} $$

Remark 3.17. It is straightforward to check that if $(X,R,Q)$ is a ${\sf TS4}$ -frame, then $(X,R,E_Q)$ is an ${\sf MS4}$ -frame, and that if $(X,R,E)$ is an ${\sf MS4}$ -frame, then $(X,R,Q_E)$ is a ${\sf TS4}$ -frame, where $Q_E$ is defined by

$$ \begin{align*} x Q_E y \mbox{ iff } (\exists z \in X )(xRz \ \& \ zEy). \end{align*} $$

If $(X,R,Q)$ is a ${\sf TS4}$ -frame, by definition we have that

$$ \begin{align*} x Q y \mbox{ iff } (\exists z \in X) (x R z \ \& \ z E_Q y). \end{align*} $$

Thus, $Q=Q_{E_Q}$ . On the other hand, there exist ${\sf MS4}$ -frames $(X,R,E)$ such that $E\ne E_{Q_E}$ (see [3, p. 24]). Therefore, this correspondence is not a bijection.

Since all ${\sf TS4}$ -axioms are Sahlqvist formulas, we have:

Theorem 3.18. ${\sf TS4}$ is a Sahlqvist logic, and hence it is sound and complete with respect to its relational semantics. Therefore,

$$ \begin{align*} {\sf TS4} \vdash \varphi \ \mbox{ iff } \ \mathfrak F \vDash \varphi \mbox{ for every}\ {\sf TS4}\mbox{-frame}\ \mathfrak F. \end{align*} $$

Remark 3.19. In Section 6 we will see that ${\sf TS4}$ has the fmp and hence is decidable.

Finally, we introduce the relational semantics for ${\sf MS4.t}$ . As with $\sf S4$ and $\sf S4.t$ , we have that ${\sf MS4.t}$ -frames are simply ${\sf MS4}$ -frames, the difference is in interpreting tense modalities.

Definition 3.20. A valuation on an ${\sf MS4.t}$ -frame $\mathfrak F=(X,R,E)$ is a map $v$ associating a subset of X to each propositional letter of $\mathcal L_{T\forall }$ . The Boolean connectives are interpreted as usual, and

$$ \begin{align*} \begin{array}{l c l} \mathfrak{F}, x \vDash_v \Box_F \varphi & \text{ iff } & (\forall y \in X) (x R y \Rightarrow y \vDash_v \varphi ), \\ \mathfrak{F}, x \vDash_v \Box_P \varphi & \text{ iff } & (\forall y \in X) (y R x \Rightarrow y \vDash_v \varphi ), \\ \mathfrak{F}, x \vDash_v \forall \varphi & \text{ iff } & (\forall y \in X) (x E y \Rightarrow y \vDash_v \varphi). \end{array} \end{align*} $$

Since both ${\sf MS4}$ and ${\sf S4.t}$ are Sahlqvist logics, so is ${\sf MS4.t}$ . Thus, we have:

Theorem 3.21. ${\sf MS4.t}$ is a Sahlqvist logic, and hence it is sound and complete with respect to its relational semantics. Therefore,

$$ \begin{align*} {\sf MS4.t} \vdash \varphi \ \mbox{ iff } \ \mathfrak F \vDash \varphi \mbox{ for every}\ {\sf MS4.t}\mbox{-frame}\ \mathfrak F. \end{align*} $$

In Section 6 we will prove that ${\sf MS4.t}$ has the fmp and hence is decidable.

4 The four translations

In this section we define the translations of Figure 1 and show that they are full and faithful by using relational semantics. We start by recalling that the Gödel translation of ${\sf MIPC}$ into ${\sf MS4}$ is defined by

$$ \begin{align*} \bot^t &= \bot,\\ p^t &= \Box p \quad \mbox{for each propositional letter } p, \\ (\varphi \land \psi)^t &= \varphi^t \land \psi^t, \\ (\varphi \lor \psi)^t &= \varphi^t \lor \psi^t, \\ (\varphi \to \psi)^t &= \Box (\neg \varphi^t \lor \psi^t), \\ (\forall \varphi)^t &= \Box \forall \varphi^t, \\ (\exists \varphi)^t &= \exists \varphi^t. \end{align*} $$

Definition 4.1. The translation $(-)^\natural : {\sf MIPC} \to {\sf TS4}$ is defined as $(-)^t$ on propositional letters, $\bot $ , $\wedge $ , $\vee $ , and $\to $ ; and for $\forall $ and $\exists $ we set:

$$ \begin{align*} (\forall \varphi)^\natural &= {[F]} \varphi^\natural,\\ (\exists \varphi)^\natural &= {\langle P \rangle} \varphi^\natural. \end{align*} $$

Remark 4.2. Thus, $(-)^\natural $ realizes the desired temporal interpretation of the intuitionistic monadic quantifiers: $\forall $ as “always in the future” and $\exists $ as “sometime in the past.”

Definition 4.3. The translation $(-)^\dagger : {\sf TS4} \to {\sf MS4.t}$ is defined by

$$ \begin{align*} p^\dagger &=p \quad \mbox{ for each propositional letter}\ p,\\ (\varphi \circ \psi)^\dagger &= \varphi^\dagger \circ \psi^\dagger \quad \mbox{for } \circ= \wedge, \vee,\\ (\neg \varphi)^\dagger & =\neg \varphi^\dagger,\\ (\Box \varphi)^\dagger &=\Box_F \varphi^\dagger,\\ ({[F]} \varphi)^\dagger &=\Box_F \forall \varphi^\dagger,\\ ({[P]} \varphi)^\dagger &=\forall \Box_P \varphi^\dagger. \end{align*} $$

Remark 4.4.

1. 1. The translation $(-)^\dagger : {\sf TS4} \to {\sf MS4.t}$ is suggested by the correspondence between ${\sf TS4}$ -frames and ${\sf MS4}$ -frames described in Remark 3.17. Each ${\sf MS4.t}$ -frame $\mathfrak {F}=(X,R,E)$ is an ${\sf MS4}$ -frame, the relation $Q_E$ on the corresponding ${\sf TS4}$ -frame is the composition of R and E, and the inverse relation is the composition of E and . Therefore, the modalities ${[F]}$ and ${[P]}$ are translated as $\Box _F \forall $ and $\forall \Box _P$ , respectively. Notice that, since ${\sf MS4}$ lacks a modality corresponding to the relation , we are not able to translate ${\sf TS4}$ into ${\sf MS4}$ .

2. 2. It is natural to consider a modification of $(-)^\dagger :{\sf TS4} \to {\sf MS4.t}$ where ${[P]}$ is translated as $\Box _P \forall $ . However, such a modification does not result in a faithful translation. Nevertheless, as we will see in Section 5, its composition with $(-)^\natural :{\sf MIPC} \to {\sf TS4}$ is full and faithful.

The translation of ${\sf MS4}$ into ${\sf MS4.t}$ reflects that ${\sf MS4.t}$ is a tense extension of ${\sf MS4}$ .

Definition 4.5. The translation $(-)^\# : {\sf MS4} \to {\sf MS4.t}$ replaces in each formula $\varphi $ of $\mathcal {L}_{\Box \forall }$ every occurrence of $\Box $ with $\Box _F$ .

We show that these four translations are full and faithful by using relational semantics. For this we first need to generalize the well-known notion of the skeleton of an ${\sf S4}$ -frame (see, e.g., [9, Sec. 3.9]).

Definition 4.6.

1. 1. If R is a quasi-order on a set X, we define $\sim $ to be the equivalence relation on X given by

   $$ \begin{align*} x \sim y \mbox{ iff } xRy \mbox{ and } yRx. \end{align*} $$
   We let $X'$ be the set of equivalence classes of $\sim $ , and define $R'$ on $X'$ by
   $$ \begin{align*} [x] R' [y] \mbox{ iff } xRy. \end{align*} $$

2. 2. Let $\mathfrak F=(X,R,E)$ be an ${\sf MS4}$ -frame. Recall that $Q_E$ is defined on X by $x Q_E y$ iff $(\exists z \in X )(xRz \ \& \ zEy)$ (see Remark 3.17). Define $Q'$ on $X'$ by

   $$ \begin{align*} [x] Q' [y] \mbox{ iff } xQ_Ey. \end{align*} $$
   We call $\mathfrak {F}^t=(X',R',Q')$ the skeleton of the ${\sf MS4}$ -frame $\mathfrak {F}$ .

3. 3. Let $\mathfrak F=(X,R,Q)$ be a ${\sf TS4}$ -frame. Define $Q'$ on $X'$ by

   $$ \begin{align*} [x] Q' [y] \mbox{ iff } xQy. \end{align*} $$
   We call $\mathfrak {F}^\natural =(X',R',Q')$ the skeleton of the ${\sf TS4}$ -frame $\mathfrak {F}$ .

4. 4. For an ${\sf MS4.t}$ -frame $\mathfrak F=(X,R,E)$ let $\mathfrak {F}^\dagger = (X,R,Q_E)$ where $Q_E$ is defined as in (2).

Remark 4.7. If a ${\sf TS4}$ -frame $\mathfrak {F}=(X,R,Q)$ is such that R is a partial order, then $\mathfrak {F}$ is isomorphic to its skeleton $\mathfrak {F}^\natural $ . However, we cannot always recover an ${\sf MS4}$ -frame $\mathfrak {F}=(X,R,E)$ from its skeleton $\mathfrak {F}^t$ even if R is a partial order. Indeed, it is not always the case that $E=E_{Q_E}$ . Nonetheless, if $\mathfrak {F}$ is canonical (and in particular finite) and R is a partial order, then $E=E_{Q_E}$ ; see [3, Sec. 2] for details.

Proposition 4.8.

1. 1. If $\mathfrak F$ is an ${\sf MS4}$ -frame, then $\mathfrak {F}^t$ is an ${\sf MIPC}$ -frame.

2. 2. If $\mathfrak F$ is a ${\sf TS4}$ -frame, then $\mathfrak {F}^\natural $ is an ${\sf MIPC}$ -frame.

3. 3. If $\mathfrak F$ is an ${\sf MS4.t}$ -frame, then $\mathfrak {F}^\dagger $ is a ${\sf TS4}$ -frame.

Proof. (1). It is well known that $(X',R')$ is an intuitionistic Kripke frame. That $Q'$ is well defined follows from Condition (E). Showing that $Q'$ is a quasi-order and that (O1), (O2) hold in $\mathfrak {F}^t$ is straightforward.

(2). We have that $Q'$ is well defined on $X'$ because $R \subseteq Q$ in $\mathfrak {F}$ . Showing that $Q'$ is a quasi-order and that (O1), (O2) hold in $\mathfrak {F}^\natural $ is straightforward.

(3). Since ${\sf MS4.t}$ -frames coincide with ${\sf MS4}$ -frames, it follows from Remark 3.17 that $\mathfrak {F}^\dagger $ is a ${\sf TS4}$ -frame.

It is well known that an ${\sf S4}$ -frame validates the Gödel translation of an intuitionistic formula $\varphi $ iff its skeleton validates $\varphi $ (see, e.g., [9, Corollary 3.82]). We will prove in Proposition 4.11 that analogous results hold for the four translations defined in this section. For this we need the following technical lemma.

Lemma 4.9. For any formula $\chi $ of $\mathcal L_{\forall \exists }$ , we have

$$ \begin{align*} {\sf MS4} \vdash \chi^t \to \Box \chi^t. \end{align*} $$

Consequently, if $\mathfrak {F}=(X,R,E)$ is an ${\sf MS4}$ -frame and $v$ a valuation on $\mathfrak {F}$ , then the set of points $x\in X$ such that $\mathfrak {F}, x \vDash _v \chi ^t$ forms an R-upset of $\mathfrak {F}$ .

Proof. We prove that ${\sf MS4} \vdash \chi ^t \to \Box \chi ^t$ by induction on the complexity of $\chi $ . This is obvious when $\chi =\bot $ . The cases when $\chi $ is p, $\varphi \to \psi $ , or $\forall \varphi $ follow from the axiom $\Box \varphi \to \Box \Box \varphi $ . We next consider the cases when $\chi $ is $\varphi \wedge \psi $ or $\varphi \vee \psi $ . Suppose that the claim is true for $\varphi $ and $\psi $ , so $\varphi ^t \to \Box \varphi ^t$ and $\psi ^t \to \Box \psi ^t$ are theorems of ${\sf MS4}$ . Then $\varphi ^t \wedge \psi ^t \to \Box (\varphi ^t \wedge \psi ^t)$ and $\varphi ^t \vee \psi ^t \to \Box (\varphi ^t \vee \psi ^t)$ are also theorems of ${\sf MS4}$ . Finally, if $\chi $ is $\exists \varphi $ and ${\sf MS4} \vdash \varphi ^t\to \Box \varphi ^t$ , then ${\sf MS4} \vdash \exists \varphi ^t \to \exists \Box \varphi ^t$ . Therefore, since ${\sf MS4} \vdash \exists \Box \varphi ^t \to \Box \exists \varphi ^t$ by Lemma 2.5, we conclude that ${\sf MS4} \vdash \exists \varphi ^t \to \Box \exists \varphi ^t$ .

Let $\mathfrak {F}=(X,R,E)$ be an ${\sf MS4}$ -frame, $v$ a valuation of $\mathfrak {F}$ , and $x\in X$ . Since ${\sf MS4} \vdash \chi ^t \to \Box \chi ^t$ , if $\mathfrak {F}, x \vDash _v \chi ^t$ , then $\mathfrak {F}, x \vDash _v \Box \chi ^t$ . Therefore, for each y such that $xRy$ we have $\mathfrak {F}, y \vDash _v \chi ^t$ . Thus, $\{ x \in X \mid \mathfrak {F}, x \vDash _v \chi ^t \}$ is an R-upset.

The next result generalizes to our setting a well-known correspondence result [9, Lemma 3.81] between ${\sf IPC}$ -models and ${\sf S4}$ -models.

Proposition 4.10.

1. 1. For each valuation $v$ on an ${\sf MS4}$ -frame $\mathfrak {F}$ there is a valuation $v'$ on $\mathfrak {F}^t$ such that for each $x\in \mathfrak F$ and $\mathcal {L}_{\forall \exists }$ -formula $\varphi $ , we have

   $$ \begin{align*} \mathfrak{F}^t, [x] \vDash_{v'} \varphi \ \mbox{ iff } \ \mathfrak{F}, x \vDash_v \varphi^t. \end{align*} $$

2. 2. For each valuation $v$ on a ${\sf TS4}$ -frame $\mathfrak {F}$ there is a valuation $v'$ on $\mathfrak {F}^\natural $ such that for each $x \in \mathfrak {F}$ and $\mathcal {L}_{\forall \exists }$ -formula $\varphi $ , we have

   $$ \begin{align*} \mathfrak{F^\natural}, [x] \vDash_{v'} \varphi \ \mbox{ iff } \ \mathfrak{F},x \vDash_{v} \varphi^\natural. \end{align*} $$

3. 3. Each valuation $v$ on an ${\sf MS4.t}$ -frame $\mathfrak {F}$ is also a valuation on $\mathfrak {F}^\dagger $ and for each $x \in \mathfrak {F}$ and $\mathcal {ML}$ -formula $\varphi $ , we have

   $$ \begin{align*} \mathfrak{F^\dagger}, x \vDash_v \varphi \ \mbox{ iff } \ \mathfrak{F},x \vDash_v \varphi^\dagger. \end{align*} $$

4. 4. Each valuation $v$ on an ${\sf MS4.t}$ -frame $\mathfrak {F}$ is also a valuation on $\mathfrak {F}$ as an ${\sf MS4}$ -frame and for each $x \in \mathfrak {F}$ and $\mathcal L_{\Box \forall }$ -formula $\varphi $ , we have

   $$ \begin{align*} \mathfrak{F}, x \vDash_v \varphi \ \mbox{ iff } \ \mathfrak{F},x \vDash_v \varphi^\#. \end{align*} $$

Proof. (1). Define $v'$ on $\mathfrak {F}^t$ by $v'(p)=\{ [x] \in X' \mid R[x] \subseteq v(p)\}$ . It is easy to see that $v'$ is well defined. We show that $\mathfrak {F}^t, [x] \vDash _{v'} \varphi $ iff $\mathfrak {F},x \vDash _v \varphi ^t$ by induction on the complexity of $\varphi $ . Since $v'(p)= \{ [x] \mid \mathfrak {F},x \vDash _v \Box p \}$ , the claim is obvious when $\varphi $ is a propositional letter. We prove the claim for $\varphi $ of the form $\forall \psi $ and $\exists \psi $ since the other cases are well known (see, e.g., [9, Lemma 3.81]). Suppose $\varphi =\forall \psi $ . By the definition of $Q'$ and induction hypothesis, we have

$$ \begin{align*} \mathfrak{F}^t, [x] \vDash_{v'} \forall \psi &\mbox{ iff } (\forall [y] \in X')([x]Q'[y] \, \Rightarrow \, \mathfrak{F}^t, [y] \vDash_{v'} \psi)\\ &\mbox{ iff } (\forall y \in X)(xQ_Ey \, \Rightarrow \, \mathfrak{F}^t, [y] \vDash_{v'} \psi)\\ &\mbox{ iff } (\forall y \in X)(xQ_Ey \, \Rightarrow \, \mathfrak{F}, y \vDash_v \psi^t). \end{align*} $$

On the other hand,

$$ \begin{align*} \mathfrak{F}, x \vDash_v (\forall \psi)^t &\mbox{ iff } \mathfrak{F}, x \vDash_v \Box \forall \psi^t\\ &\mbox{ iff } (\forall z \in X)(xRz \, \Rightarrow \, (\forall y \in X)(zEy \, \Rightarrow \, \mathfrak{F}, y \vDash_v \psi^t))\\ &\mbox{ iff } (\forall y \in X)(xQ_Ey \, \Rightarrow \, \mathfrak{F}, y \vDash_v \psi^t). \end{align*} $$

Thus, $\mathfrak {F}^t, [x] \vDash _{v'} \forall \psi $ iff $\mathfrak {F}, x \vDash _v (\forall \psi )^t$ .

Suppose $\varphi =\exists \psi $ . As noted in Remark 3.3, $Q'$ and $E_{Q'}$ coincide on $R'$ -upsets, and it is straightforward to see by induction that the set $\{ [y] \mid \mathfrak {F}^t, [y] \vDash _{v'} \psi \}$ is an $R'$ -upset. Therefore, by the induction hypothesis,

$$ \begin{align*} \mathfrak{F}^t, [x] \vDash_{v'} \exists \psi &\mbox{ iff } (\exists [y] \in X')([x]E_{Q'}[y] \; \& \; \mathfrak{F}^t, [y] \vDash_{v'} \psi)\\ &\mbox{ iff } [x] \in E_{Q'}[\{ [y] \mid \mathfrak{F}^t, [y] \vDash_{v'} \psi \}]\\ &\mbox{ iff } [x] \in Q'[\{ [y] \mid \mathfrak{F}^t, [y] \vDash_{v'} \psi \}]\\ &\mbox{ iff } x \in Q_E[\{ y \mid \mathfrak{F}^t, [y] \vDash_{v'} \psi \}]\\ &\mbox{ iff } x \in Q_E[\{ y \mid \mathfrak{F}, y \vDash_v \psi^t \}]. \end{align*} $$

On the other hand, since $\{ y \mid \mathfrak {F}, y \vDash _v \psi ^t \}$ is an R-upset (see Lemma 4.9), and E and $Q_E$ coincide on R-upsets, we have

$$ \begin{align*} \mathfrak{F}, x \vDash_v (\exists \psi)^t &\mbox{ iff } \mathfrak{F}, x \vDash_v \exists \psi^t \\ &\mbox{ iff } (\exists y \in X)(xEy \; \& \; \mathfrak{F}, y \vDash_v \psi^t)\\ &\mbox{ iff } x \in E[\{ y \mid \mathfrak{F}, y \vDash_v \psi^t \}]\\ &\mbox{ iff } x \in Q_E[\{ y \mid \mathfrak{F}, y \vDash_v \psi^t \}]. \end{align*} $$

Thus, $\mathfrak {F}^t, [x] \vDash _{v'} \exists \psi $ iff $\mathfrak {F}, x \vDash _v (\exists \psi )^t$ .

(2). As in (1) we define $v'$ by $v'(p)=\{ [x] \in X' \mid R[x] \subseteq v(p)\}$ . We show that $\mathfrak {F^\natural }, [x] \vDash _{v'} \varphi $ iff $\mathfrak {F},x \vDash _{v} \varphi ^\natural $ by induction on the complexity of $\varphi $ . It is sufficient to only consider the cases when $\varphi $ is of the form $\forall \psi $ or $\exists \psi $ . Suppose $\varphi =\forall \psi $ . Then by the definition of $Q'$ and induction hypothesis,

$$ \begin{align*} \mathfrak{F}^\natural, [x] \vDash_{v'} \forall \psi &\mbox{ iff } (\forall [y] \in X')([x]Q'[y] \, \Rightarrow \, \mathfrak{F}^\natural, [y] \vDash_{v'} \psi)\\ &\mbox{ iff } (\forall y \in X)(xQy \, \Rightarrow \, \mathfrak{F}^\natural, [y] \vDash_{v'} \psi) \\ &\mbox{ iff } (\forall y \in X)(xQy \, \Rightarrow \, \mathfrak{F}, y \vDash_v \psi^\natural) \\ &\mbox{ iff } \mathfrak{F}, x \vDash_v {[F]} \psi^\natural \\ &\mbox{ iff } \mathfrak{F}, x \vDash_v (\forall \psi)^\natural. \end{align*} $$

Suppose $\varphi =\exists \psi $ . As noted in Remark 3.3, $Q'$ and $E_{Q'}$ coincide on $R'$ -upsets. Since the set $\{ [y] \mid \mathfrak {F}^\natural , [y] \vDash _{v'} \psi \}$ is an $R'$ -upset, by the induction hypothesis, we have

$$ \begin{align*} \mathfrak{F}^\natural, [x] \vDash_{v'} \exists \psi &\mbox{ iff } (\exists [y] \in X')([x]E_{Q'}[y] \; \& \; \mathfrak{F}^\natural, [y] \vDash_{v'} \psi)\\ &\mbox{ iff } [x] \in E_{Q'}[\{ [y] \mid \mathfrak{F}^\natural, [y] \vDash_{v'} \psi \}]\\ &\mbox{ iff } [x] \in Q'[\{ [y] \mid \mathfrak{F}^\natural, [y] \vDash_{v'} \psi \}]\\ &\mbox{ iff } x \in Q[\{ y \mid \mathfrak{F}^\natural, [y] \vDash_{v'} \psi \}]\\ &\mbox{ iff } x \in Q[\{ y \mid \mathfrak{F}, y \vDash_v \psi^\natural \}] \\ &\mbox{ iff } (\exists y \in X)(yQx \; \& \; \mathfrak{F}, y \vDash_v \psi^\natural) \\ &\mbox{ iff } \mathfrak{F}, x \vDash_v {\langle P \rangle} \psi^\natural \\ &\mbox{ iff } \mathfrak{F}, x \vDash_v (\exists \psi)^\natural. \end{align*} $$

(3). It is clear that if v is a valuation on $\mathfrak {F}$ , then v is also a valuation on $\mathfrak {F}^\dagger $ . We show that $\mathfrak {F}^\dagger , x \vDash _v \varphi $ iff $\mathfrak {F},x \vDash _v \varphi ^\dagger $ by induction on the complexity of $\varphi $ . The only nontrivial cases are when $\varphi $ is of the form $\Box \psi $ , ${[F]} \psi $ , and ${[P]} \psi $ . Suppose $\varphi =\Box \psi $ . Then, by the induction hypothesis,

$$ \begin{align*} \mathfrak{F}^\dagger, x \vDash_v \Box \psi &\mbox{ iff } (\forall y \in X)(xRy \, \Rightarrow \, \mathfrak{F}^\dagger, y \vDash_v \psi)\\ &\mbox{ iff } (\forall y \in X)(xRy \, \Rightarrow \, \mathfrak{F}, y \vDash_v \psi^\dagger) \\ &\mbox{ iff } \mathfrak{F}, x \vDash_v \Box_F \psi^\dagger\\ &\mbox{ iff } \mathfrak{F}, x \vDash_v (\Box \psi)^\dagger. \end{align*} $$

Suppose $\varphi ={[F]} \psi $ . Then, by the induction hypothesis,

$$ \begin{align*} \mathfrak{F}^\dagger, x \vDash_v {[F]} \psi &\mbox{ iff } (\forall y \in X)(x Q_E y \, \Rightarrow \, \mathfrak{F}^\dagger, y \vDash_v \psi)\\&\mbox{ iff } (\forall z \in X)(xRz \, \Rightarrow \, (\forall y \in X)(zEy \, \Rightarrow \, \mathfrak{F}^\dagger, y \vDash_v \psi))\\&\mbox{ iff } (\forall z \in X)(xRz \, \Rightarrow \, (\forall y \in X)(zEy \, \Rightarrow \, \mathfrak{F}, y \vDash_v \psi^\dagger)) \\&\mbox{ iff } (\forall z \in X)(xRz \, \Rightarrow \, \mathfrak{F}, z \vDash \forall \psi^\dagger)\\&\mbox{ iff } \mathfrak{F}, x \vDash_v \Box_F \forall \psi^\dagger\\&\mbox{ iff } \mathfrak{F}, x \vDash_v ({[F]} \psi)^\dagger. \end{align*} $$

Suppose $\varphi ={[P]} \psi $ . Then, by the induction hypothesis,

$$ \begin{align*} \mathfrak{F}^\dagger, x \vDash_v {[P]} \psi &\mbox{ iff } (\forall y \in X)(y Q_E x \, \Rightarrow \, \mathfrak{F}^\dagger, y \vDash_v \psi)\\&\mbox{ iff } (\forall y,z \in X)(yRz \ \& \ zEx \, \Rightarrow \, \mathfrak{F}^\dagger, y \vDash_v \psi)\\&\mbox{ iff } (\forall z \in X)(zEx \, \Rightarrow \, (\forall y \in X)(yRz \, \Rightarrow \, \mathfrak{F}^\dagger, y \vDash_v \psi))\\&\mbox{ iff } (\forall z \in X)(zEx \, \Rightarrow \, (\forall y \in X)(yRz \, \Rightarrow \, \mathfrak{F}, y \vDash_v \psi^\dagger)) \\&\mbox{ iff } (\forall z \in X)(xEz \, \Rightarrow \, \mathfrak{F}, z \vDash \Box_P \psi^\dagger)\\&\mbox{ iff } \mathfrak{F}, x \vDash_v \forall \Box_P \psi^\dagger\\&\mbox{ iff } \mathfrak{F}, x \vDash_v ({[P]} \psi)^\dagger. \end{align*} $$

(4). This is an immediate consequence of the definition of $(-)^\#$ and the relational semantics for ${\sf MS4}$ and ${\sf MS4.t}$ .

Proposition 4.11.

1. 1. For each ${\sf MS4}$ -frame $\mathfrak {F}$ and $\mathcal {L}_{\forall \exists }$ -formula $\varphi $ , we have

   $$ \begin{align*} \mathfrak{F}^t \vDash \varphi \ \mbox{ iff } \ \mathfrak{F} \vDash \varphi^t. \end{align*} $$

2. 2. For each ${\sf TS4}$ -frame $\mathfrak {F}$ and $\mathcal {L}_{\forall \exists }$ -formula $\varphi $ , we have

   $$ \begin{align*} \mathfrak{F}^\natural \vDash \varphi \ \mbox{ iff } \ \mathfrak{F} \vDash \varphi^\natural. \end{align*} $$

3. 3. For each ${\sf MS4.t}$ -frame $\mathfrak {F}$ and $\mathcal {ML}$ -formula $\varphi $ , we have

   $$ \begin{align*} \mathfrak{F}^\dagger \vDash \varphi \ \mbox{ iff } \ \mathfrak{F} \vDash \varphi^\dagger. \end{align*} $$

4. 4. For each ${\sf MS4.t}$ -frame $\mathfrak {F}$ and $\mathcal L_{\Box \forall }$ -formula $\varphi $ , we have

   $$ \begin{align*} \mathfrak{F} \vDash \varphi \ \mbox{ iff } \ \mathfrak{F} \vDash \varphi^\#. \end{align*} $$

Proof. We only prove (2) since the proofs of (1), (3), and (4) are similar. If $\mathfrak {F} \nvDash \varphi ^\natural $ , then there is a valuation $v$ on $\mathfrak {F}$ such that $\mathfrak {F}, x \nvDash _v \varphi ^\natural $ for some $x \in X$ . By Proposition 4.10(2), $v'$ is a valuation on $\mathfrak {F}^\natural $ such that $\mathfrak {F}^\natural , [x] \nvDash _{v'} \varphi $ . Therefore, $\mathfrak {F}^\natural \nvDash \varphi $ . If $\mathfrak {F}^\natural \nvDash \varphi $ , then there is a valuation $w$ on $\mathfrak {F}^\natural $ and $[x] \in X'$ such that $\mathfrak {F}^\natural , [x] \nvDash _w \varphi $ . Let $v$ be the valuation on $\mathfrak {F}$ given by $v(p)=\{x \mid [x] \in w(p)\}$ . Since $\mathfrak {F}^\natural $ is an ${\sf MIPC}$ -frame, $w(p)$ is an $R'$ -upset of $\mathfrak {F}^\natural $ for each p. So $v(p)$ is an R-upset of $\mathfrak {F}$ for each p. Therefore, $w=v'$ because

$$ \begin{align*} v'(p)=\{ [x] \in X' \mid R[x] \subseteq v(p) \}=\{ [x] \in X' \mid x \in v(p) \}=w(p). \end{align*} $$

Thus, $\mathfrak {F}^\natural , [x] \nvDash _{v'} \varphi $ . By Proposition 4.10(2), $\mathfrak {F}, x \nvDash _v \varphi ^\natural $ . Consequently, $\mathfrak {F} \nvDash \varphi ^\natural $ .

In order to show that the translations are full, we also need the following result, which generalizes to our setting a well-known fact that each ${\sf IPC}$ -frame is the skeleton of an ${\sf S4}$ -frame.

Proposition 4.12.

1. 1. For each ${\sf MIPC}$ -frame $\mathfrak {G}$ there is an ${\sf MS4}$ -frame $\mathfrak {F}$ such that $\mathfrak {G}$ is isomorphic to $\mathfrak {F}^t$ .

2. 2. Each ${\sf MIPC}$ -frame $\mathfrak {G}$ is also a ${\sf TS4}$ -frame and $\mathfrak {G}^\natural $ is isomorphic to $\mathfrak {G}$ .

3. 3. For each ${\sf TS4}$ -frame $\mathfrak {G}$ there is an ${\sf MS4.t}$ -frame $\mathfrak {F}$ such that $\mathfrak {G}=\mathfrak {F}^\dagger $ .

Proof. (1). Let $\mathfrak {G}=(X,R,Q)$ be an ${\sf MIPC}$ -frame. We show that $\mathfrak {F}=(X,R,E_Q)$ is an ${\sf MS4}$ -frame. If $xE_Qy$ and $yRz$ , then by definition of $E_Q$ and Condition (O1), $xQy$ and $yQz$ . Since Q is transitive, $xQz$ . Condition (O2) then implies that there is $u \in X$ with $xRu$ and $u E_Q z$ . Thus, $\mathfrak F$ is an ${\sf MS4}$ -frame. Since R is a partial order, it is an immediate consequence of Definition 4.6(1) that $\sim $ is the identity relation. It follows from Condition (O2) that $Q=Q_{E_Q}$ . Hence, $\mathfrak {G}$ is isomorphic to $\mathfrak {F}^t$ .

(2). Let $\mathfrak {G}=(X,R,Q)$ be an ${\sf MIPC}$ -frame. It is clear from the definition of ${\sf TS4}$ -frames that $\mathfrak {G}$ is also a ${\sf TS4}$ -frame. Since R is a partial order, $\sim $ is the identity relation. Therefore, $\mathfrak {G}$ is isomorphic to $\mathfrak {G}^\natural $ .

(3). Let $\mathfrak {G}=(X,R,Q)$ be a ${\sf TS4}$ -frame. As we observed in Remark 3.17, $\mathfrak {F}=(X,R,E_Q)$ is an ${\sf MS4}$ -frame, and so an ${\sf MS4.t}$ -frame. By definition of ${\sf TS4}$ -frames we have that $Q=Q_{E_Q}$ , and hence $\mathfrak {G}=\mathfrak {F}^\dagger $ .

We are ready to prove the main result of this section that the four translations are full and faithful. That the Gödel translation of ${\sf MIPC}$ into ${\sf MS4}$ is full and faithful was first observed by Fischer-Servi [14] using the translations of ${\sf MIPC}$ and ${\sf MS4}$ into ${\sf IQC}$ and ${\sf QS4}$ , respectively, and the predicate version of the Gödel translation. In [15] she gave a different proof of this result, using that ${\sf MIPC}$ has the fmp. Our proof utilizes the relational semantics and generalizes the semantic proof that the Gödel translation of ${\sf IPC}$ into ${\sf S4}$ is full and faithful (see, e.g., [9, Sec. 3.9]).

Theorem 4.13.

1. 1. The Gödel translation $(-)^t$ of ${\sf MIPC}$ into ${\sf MS4}$ is full and faithful; that is,

   $$ \begin{align*} {\sf MIPC} \vdash\varphi \quad \mbox{ iff } \quad {\sf MS4} \vdash \varphi^t. \end{align*} $$

2. 2. The translation $(-)^\natural $ of ${\sf MIPC}$ into ${\sf TS4}$ is full and faithful; that is,

   $$ \begin{align*} {\sf MIPC} \vdash\varphi \quad \mbox{ iff } \quad {\sf TS4} \vdash\varphi^\natural. \end{align*} $$

3. 3. The translation $(-)^\dagger $ of ${\sf TS4}$ into ${\sf MS4.t}$ is full and faithful; that is,

   $$ \begin{align*} \begin{array}{l c l} {\sf TS4} \vdash \varphi & \mbox{ iff } & {\sf MS4.t} \vdash \varphi^\dagger. \end{array} \end{align*} $$

4. 4. The translation $(-)^\#$ of ${\sf MS4}$ into ${\sf MS4.t}$ is full and faithful; that is,

   $$ \begin{align*} \begin{array}{l c l} {\sf MS4} \vdash \varphi & \mbox{ iff } & {\sf MS4.t} \vdash \varphi^\#. \end{array} \end{align*} $$

Proof. We prove (2). For faithfulness, suppose that ${\sf TS4} \nvdash \varphi ^\natural $ . By Theorem 3.18, there is a ${\sf TS4}$ -frame $\mathfrak {F}$ such that $\mathfrak {F} \nvDash \varphi ^\natural $ . By Propositions 4.8(2) and 4.11(2), $\mathfrak {F}^\natural $ is an ${\sf MIPC}$ -frame and $\mathfrak {F}^\natural \nvDash \varphi $ . Thus, by Theorem 3.5, ${\sf MIPC} \nvdash \varphi $ . For fullness, let ${\sf MIPC} \nvdash \varphi $ . Then there is an ${\sf MIPC}$ -frame $\mathfrak {G}$ such that $\mathfrak {G} \nvDash \varphi $ . By Proposition 4.12(2), there is a ${\sf TS4}$ -frame $\mathfrak{F}$ such that $\mathfrak {G}$ is isomorphic to $\mathfrak {F}^\natural $ . Therefore, $\mathfrak {F}^\natural \nvDash \varphi $ . Proposition 4.11(2) implies that $\mathfrak {F} \nvDash \varphi ^\natural $ . Thus, ${\sf TS4} \nvdash \varphi ^\natural $ .

The proofs of (1), (3), and (4) are obtained analogously using Theorems 3.5, 3.9, 3.18, and 3.21, and Propositions 4.8, 4.11, and 4.12.

5 Compositions of the translations

In this section we show that the translations introduced in the previous section form a commutative diagram up to logical equivalence.

We denote the composition of $(-)^\#$ and $(-)^t$ by $(-)^{t \#}$ , and the composition of $(-)^\dagger $ and $(-)^\natural $ by $(-)^{\natural \dagger }$ . Since we proved that all these four translations are full and faithful, we also have that $(-)^{t \#}$ and $(-)^{\natural \dagger }$ are full and faithful translations of ${\sf MIPC}$ into ${\sf MS4.t}$ . We have thus obtained the following diagram of full and faithful translations. We next show that this diagram is commutative up to logical equivalence in ${\sf MS4.t}$ .

Lemma 5.1. For each formula $\varphi $ of $\mathcal L_{\forall \exists }$ , we have

$$ \begin{align*} {\sf MS4.t} \vdash \varphi^{t \#} \leftrightarrow \Diamond_P \varphi^{t \#}. \end{align*} $$

Proof. By Lemma 4.9 and Theorem 4.13(4), ${\sf MS4.t} \vdash \varphi ^{t \#} \to \Box _F \varphi ^{t \#}$ . Therefore, ${\sf MS4.t} \vdash \Diamond _P \varphi ^{t \#} \to \Diamond _P \Box _F \varphi ^{t \#}$ . The tense axiom then gives ${\sf MS4.t} \vdash \Diamond _P \varphi ^{t \#} \to \varphi ^{t \#}$ . Thus, ${\sf MS4.t} \vdash \varphi ^{t \#} \leftrightarrow \Diamond _P \varphi ^{t \#}$ .

Theorem 5.2. For each $\mathcal L_{\forall \exists }$ -formula $\chi $ we have

$$ \begin{align*} {\sf MS4.t} \vdash \chi^{t \#} \leftrightarrow \chi^{\natural \dagger}. \end{align*} $$

Proof. The two compositions compare as follows:

$$ \begin{align*} &\bot^{t \#}=\bot && \bot^{\natural \dagger}=\bot \\ &p^{t \#}=\Box_F p && p^{\natural \dagger}=\Box_F p \\ &(\varphi \wedge \psi)^{t \#}= \varphi^{t \#} \wedge \psi^{t \#} \qquad && (\varphi \wedge \psi)^{\natural \dagger}= \varphi^{\natural \dagger} \wedge \psi^{\natural \dagger} \\ &(\varphi \vee \psi)^{t \#}= \varphi^{t \#} \vee \psi^{t \#} \qquad && (\varphi \vee \psi)^{\natural \dagger}= \varphi^{\natural \dagger} \vee \psi^{\natural \dagger} \\ &(\varphi \to \psi)^{t \#}= \Box_F(\neg \varphi^{t \#} \vee \psi^{t \#}) \qquad && (\varphi \to \psi)^{\natural \dagger}= \Box_F(\neg \varphi^{\natural \dagger} \vee \psi^{\natural \dagger}) \\ &(\forall \varphi)^{t \#}= \Box_F \forall \varphi^{t \#} && (\forall \varphi)^{\natural \dagger}= \Box_F \forall \varphi^{\natural \dagger} \\ &(\exists \varphi)^{t \#}= \exists \varphi^{t \#} && (\exists \varphi)^{\natural \dagger}= ({\langle P \rangle} \varphi^\natural)^\dagger= (\neg {[P]} \neg \varphi^\natural)^\dagger \\ & && \hphantom{(\exists \varphi)^{\natural \dagger}} = \neg \forall \Box_P \neg \varphi^{\natural \dagger}. \end{align*} $$

Thus, they are identical except the $\exists $ -clause. Therefore, to prove that ${\sf MS4.t} \vdash \chi ^{t \#} \leftrightarrow \chi ^{\natural \dagger }$ it is sufficient to prove that ${\sf MS4.t} \vdash \varphi ^{t \#} \leftrightarrow \varphi ^{\natural \dagger }$ implies ${\sf MS4.t} \vdash \exists \varphi ^{t \#} \leftrightarrow \neg \forall \Box _P \neg \varphi ^{\natural \dagger }$ . Since ${\sf MS4.t} \vdash \neg \forall \Box _P \neg \varphi ^{\natural \dagger } \leftrightarrow \exists \Diamond _P \varphi ^{\natural \dagger }$ , it is enough to prove that ${\sf MS4.t} \vdash \exists \varphi ^{t \#} \leftrightarrow \exists \Diamond _P \varphi ^{\natural \dagger }$ . From the assumption ${\sf MS4.t} \vdash \varphi ^{t \#} \leftrightarrow \varphi ^{\natural \dagger }$ it follows that ${\sf MS4.t} \vdash \exists \Diamond _P \varphi ^{t \#} \leftrightarrow \exists \Diamond _P \varphi ^{\natural \dagger }$ . By Lemma 5.1, ${\sf MS4.t} \vdash \varphi ^{t \#} \leftrightarrow \Diamond _P \varphi ^{t \#}$ and hence ${\sf MS4.t} \vdash \exists \varphi ^{t \#} \leftrightarrow \exists \Diamond _P \varphi ^{t \#}$ . Thus, ${\sf MS4.t} \vdash \exists \varphi ^{t \#} \leftrightarrow \exists \Diamond _P \varphi ^{\natural \dagger }$ .

As we pointed out in Remark 4.4(2), there is another natural translation of ${\sf MIPC}$ into ${\sf MS4.t}$ .

Definition 5.3. Let $(-)^\flat :{\sf MIPC} \to {\sf MS4.t}$ be the translation that differs from $(-)^{t \#}$ and $(-)^{\natural \dagger }$ only in the $\exists $ -clause:

$$ \begin{align*} &(\exists \varphi)^\flat= \Diamond_P \exists \varphi^\flat. \end{align*} $$

The translation $(-)^\flat $ provides a temporal interpretation of intuitionistic monadic quantifiers that is similar to the translation $(-)^\natural $ (see also Section 7).

Theorem 5.4. For each $\mathcal L_{\forall \exists }$ -formula $\chi $ we have

$$ \begin{align*} {\sf MS4.t} \vdash \chi^\flat \leftrightarrow \chi^{t \#}. \end{align*} $$

Consequently, the translation $(-)^\flat $ of ${\sf MIPC}$ into ${\sf MS4.t}$ is full and faithful.

Proof. The translations $(-)^\flat $ and $(-)^{t \#}$ are identical except the $\exists $ -clause. Therefore, to prove that ${\sf MS4.t} \vdash \chi ^\flat \leftrightarrow \chi ^{t \#}$ it is sufficient to prove that ${\sf MS4.t} \vdash \varphi ^\flat \leftrightarrow \varphi ^{t \#}$ implies ${\sf MS4.t} \vdash \Diamond _P \exists \varphi ^\flat \leftrightarrow \exists \varphi ^{t \#}$ . By Lemma 5.1, ${\sf MS4.t} \vdash (\exists \varphi )^{t \#} \leftrightarrow \Diamond _P (\exists \varphi )^{t \#}$ which means ${\sf MS4.t} \vdash \exists \varphi ^{t \#} \leftrightarrow \Diamond _P \exists \varphi ^{t \#}$ . From the assumption ${\sf MS4.t} \vdash \varphi ^\flat \leftrightarrow \varphi ^{t \#}$ it follows that ${\sf MS4.t} \vdash \Diamond _P \exists \varphi ^\flat \leftrightarrow \exists \varphi ^{t \#}$ . Since $(-)^{t \#}$ is full and faithful, we conclude that $(-)^\flat $ is full and faithful as well.

As a result, we obtain the following diagram of full and faithful translations that is commutative up to logical equivalence in ${\sf MS4.t}$ :

6 Finite model property

In this section we give a uniform proof of the fmp for the four logics studied in this paper. Our strategy is to first establish the fmp for ${\sf MS4.t}$ via algebraic methods, and then use the full and faithful translations to conclude that the other three logics also have the fmp.

The algebraic semantics for ${\sf MS4.t}$ is given by ${\sf MS4.t}$ -algebras. To define these algebras, we first recall the definition of ${\sf S4}$ -algebras, ${\sf S5}$ -algebras, and ${\sf S4.t}$ -algebras, which provide algebraic semantics for ${\sf S4}$ , ${\sf S5}$ , and ${\sf S4.t}$ , respectively. ${\sf S4}$ -algebras are known under various names. They were first introduced by McKinsey and Tarski [22] under the name of closure algebras. Rasiowa and Sikorski [27] call them topological Boolean algebras and Blok [6] calls them interior algebras. ${\sf S5}$ -algebras were first introduced by Halmos [20] under the name of monadic algebras, and ${\sf S4.t}$ -algebras by Esakia [11] under the name of ${\sf S4}^2$ -algebras.

Definition 6.1. Let B be a Boolean algebra.

1. 1. A unary function $\Box :B\to B$ is an interior operator on B if

   $$ \begin{align*} \Box(a\wedge b) = \Box a \wedge \Box b, \qquad \Box 1 =1, \qquad \Box a \leq a, \qquad \Box a \leq \Box \Box a, \end{align*} $$
   for all $a,b \in B$ .

2. 2. An ${\sf S4}$ -algebra is a pair $\mathfrak {B}=(B, \Box )$ where B is a Boolean algebra and $\Box $ is an interior operator on B.

3. 3. An ${\sf S5}$ -algebra is an ${\sf S4}$ -algebra $\mathfrak {B}=(B, \forall )$ that in addition satisfies $a \leq \forall \exists a$ for all $a \in B$ , where $\exists $ denotes the dual operator $\neg \forall \neg $ .

4. 4. An ${\sf S4.t}$ -algebra is a triple $\mathfrak {B}=(B, \Box _F, \Box _P)$ where B is a Boolean algebra and $\Box _F, \Box _P$ are interior operators on B such that

   (PF) $$ \begin{align} a \le \Box_P \Diamond_F a, \end{align} $$
   (FP) $$ \begin{align} a \le \Box_F \Diamond_P a, \end{align} $$
   for all $a \in B$ , where $\Diamond _F=\neg \Box _F \neg $ and $\Diamond _P=\neg \Box _P \neg $ .

${\sf MS4.t}$ -algebras are obtained by combining ${\sf S4.t}$ -algebras and ${\sf S5}$ -algebras.

Definition 6.2. An ${\sf MS4.t}$ -algebra is a tuple $\mathfrak B=(B, \Box _F, \Box _P, \forall )$ where

1. 1. $(B, \Box _F, \Box _P)$ is an ${\sf S4.t}$ -algebra,

2. 2. $(B, \forall )$ is an ${\sf S5}$ -algebra,

3. 3. $\Box _F \forall a \leq \forall \Box _F a$ for each $a\in B$ .

Validity of $\mathcal L_{T\forall }$ -formulas in ${\sf MS4.t}$ -algebras is defined in the usual way (see, e.g., [9, 27]). If a formula $\varphi $ is valid in an ${\sf MS4.t}$ -algebra $\mathfrak {B}$ , we write $\mathfrak B \vDash \varphi $ . The standard Lindenbaum–Tarski construction (see, e.g., [27, Chap. VI]) yields the following:

Theorem 6.3. ${\sf MS4.t}$ is sound and complete with respect to its algebraic semantics. Therefore,

$$ \begin{align*} {\sf MS4.t} \vdash \varphi \ \mbox{ iff } \ \mathfrak B \vDash \varphi \mbox{ for every}\ {\sf MS4.t}\text{-}{algebra}\ \mathfrak B. \end{align*} $$

Definition 6.4. Let $\mathfrak B=(B, \Box _F, \Box _P, \forall )$ be an ${\sf MS4.t}$ -algebra. We define

1. 1. $H_F:=\{a \in B \mid \Box _F a=a\}$ to be the set of $\Box _F$ -fixpoints,

2. 2. $H_P:=\{a \in B \mid \Box _P a=a\}$ to be the set of $\Box _P$ -fixpoints,

3. 3. $B_0:=\{a \in B \mid \forall a=a\}$ to be the set of $\forall $ -fixpoints.

Remark 6.5.

1. 1. It is well known (see, e.g., [13, Proposition 2.2.4]) that $H_F$ and $H_P$ with the restricted order from B are both Heyting algebras that are bounded sublattices of B. Moreover, it follows from Definition 6.1(4) that $H_F$ coincides with the set of $\Diamond _P$ -fixpoints and $H_P$ with the set of $\Diamond _F$ -fixpoints. Furthermore, $\neg $ maps $H_F$ to $H_P$ and vice versa. Indeed, if $a \in H_F$ , then $a = \Box _F a$ . By (PF), $\Diamond _P a = \Diamond _P \Box _F a \le a$ , so $\Diamond _P a = a$ , and hence $\Box _P \neg a =\neg \Diamond _P a= \neg a$ . Therefore, $\neg a \in H_P$ . Similarly, if $a \in H_P$ , then $\neg a \in H_F$ . Thus, $\neg $ is a dual isomorphism between $H_F$ and $H_P$ .

2. 2. It is easy to see that $B_0$ is an ${\sf S4}$ -subalgebra of $(B, \Box _F)$ because the inequality $\Box _F \forall a \leq \forall \Box _F a$ , which corresponds to the left commutativity axiom, is equivalent to the equality $\forall \Box _F \forall a = \Box _F \forall a$ .

We now prove that ${\sf MS4.t}$ has the fmp. For this we must show that if ${\sf MS4.t}\not \vdash \varphi $ , then $\varphi $ is refuted on a finite ${\sf MS4.t}$ -algebra.

Definition 6.6. Let $\mathfrak {B}=(B, \Box _F, \Box _P, \forall )$ be an ${\sf MS4.t}$ -algebra and $S \subseteq B$ a finite subset. Then $(B, \forall )$ is an ${\sf S5}$ -algebra. Let $(B', \forall ')$ be the ${\sf S5}$ -subalgebra of $(B, \forall )$ generated by S. It is well known (see [1]) that $(B', \forall ')$ is finite. Define $\Box _F'$ and $\Box _P'$ on $B'$ by

$$ \begin{align*} \Box_F' a &= \bigvee \{ b \in B'\cap H_F \mid b \leq a \}, \\ \Box_P' a &= \bigvee \{ b \in B'\cap H_P \mid b \leq a \}. \end{align*} $$

We denote $(B', \Box _F', \Box _P', \forall ')$ by $\mathfrak {B}_S$ .

Lemma 6.7. $\mathfrak {B}_S$ is an ${\sf MS4.t}$ -algebra.

Proof. $(B',\forall ')$ is an ${\sf S5}$ -algebra by definition. Since $(B, \Box _F)$ and $(B, \Box _P)$ are both ${\sf S4}$ -algebras, a standard argument (see [22, Lemma 4.14]) shows that $(B', \Box _F')$ and $(B', \Box _P')$ are also ${\sf S4}$ -algebras. We show that $(B', \Box _F', \Box _P')$ is an ${\sf S4.t}$ -algebra. As noted in Remark 6.5(1), $\neg $ is a dual isomorphism between the algebras $H_F$ and $H_P$ of $\Box _F$ -fixpoints and $\Box _P$ -fixpoints of $\mathfrak {B}$ . Therefore,

$$ \begin{align*} \Diamond_F' a := \neg \Box_F' \neg a & = \neg \bigvee \{ b \in B'\cap H_F \mid b \leq \neg a \} \\ & = \neg \bigvee \{ b \in B'\cap H_F \mid a \leq \neg b \} \\ & = \bigwedge \{ \neg b \mid b \in B'\cap H_F, \; a \leq \neg b \} \\ & = \bigwedge \{ c \in B' \cap H_P \mid a \leq c \}. \end{align*} $$

Since this meet is finite and $\Box _P$ commutes with finite meets, we obtain

$$ \begin{align*} \Box_P \Diamond_F' a & = \Box_P \left( \bigwedge \{ c \in B' \cap H_P \mid a \leq c \} \right) \\ & = \bigwedge \{ \Box_P c \mid c \in B' \cap H_P , \; a \leq c \} \\ & = \bigwedge \{ c \in B' \cap H_P \mid a \leq c \} \\ & = \Diamond_F' a. \end{align*} $$

Thus, $\Diamond _F' a \in B' \cap H_P $ which yields

$$ \begin{align*} \Box_P' \Diamond_F' a = \bigvee \{ b \in B' \cap H_P \mid b \leq \Diamond_F' a \} = \Diamond_F' a. \end{align*} $$

Similarly, we have that $\Diamond _P' a = \bigwedge \{ c \in B' \cap H_F \mid a \leq c \}$ from which we deduce that $\Box _F' \Diamond _P' a = \Diamond _P' a$ . This implies that $a \le \Box _P' \Diamond _F' a$ and $a \le \Box _F' \Diamond _P' a$ . Consequently, $(B, \Box _F', \Box _P')$ is an ${\sf S4.t}$ -algebra.

It remains to show that $\Box _F' \forall ' a \le \forall ' \Box _F' a$ holds in $\mathfrak {B}_S$ . For this it is sufficient to show that the set $B_0':=B' \cap B_0$ of the $\forall '$ -fixpoints of $B'$ is an ${\sf S4}$ -subalgebra of $(B', \Box _F')$ because then $\Box _F' \forall ' a=\forall ' \Box _F' \forall ' a \le \forall ' \Box _F' a$ . Suppose that $d \in B_0'$ . By definition, $\Box _F' d = \bigvee \{ b \in B'\cap H_F \mid b \leq d \}$ . Let $b \in B'\cap H_F$ . It follows from Lemma 2.5 that $\exists b = \exists \Box _F b =\Box _F \exists \Box _F b = \Box _F \exists b$ . Therefore, $\exists b \in B' \cap H_F$ . Moreover, $b \le \exists b$ and $b \le d$ imply $\exists b \le \exists d = d$ . Thus, $\Box _F' d = \bigvee \{ \exists b \mid b \in B'\cap H_F, \; b \leq d \}$ . Since $(B', \forall ')$ is an ${\sf S5}$ -algebra, $B_0'$ is the set of $\exists '$ -fixpoints of $B'$ and is closed under finite joins. Consequently, $\Box _F' d \in B_0'$ .

Theorem 6.8. ${\sf MS4.t}$ has the fmp.

Proof. By Theorem 6.3, it is sufficient to prove that each $\mathcal L_{T\forall }$ -formula $\varphi $ refuted on some ${\sf MS4.t}$ -algebra is also refuted on a finite ${\sf MS4.t}$ -algebra. Let $t(x_1, \ldots , x_n)$ be the term in the language of ${\sf MS4.t}$ -algebras that corresponds to $\varphi $ , and suppose there is an ${\sf MS4.t}$ -algebra $\mathfrak {B}=(B, \Box _F, \Box _P, \forall )$ and $a_1, \ldots , a_n \in B$ such that $t(a_1, \ldots , a_n) \neq 1$ in $\mathfrak {B}$ . Let

$$ \begin{align*} S=\{ t'(a_1, \ldots, a_n) \mid t' \mbox{ is a subterm of } t \}. \end{align*} $$

Then S is a finite subset of B. Therefore, by Lemma 6.7, $\mathfrak {B}_S=(B', \Box _F', \Box _P', \forall )$ is a finite ${\sf MS4.t}$ -algebra. It follows from the definition of $\Box _F'$ that, for each $b \in B'$ , if $\Box _F b \in B'$ , then $\Box _F' b= \Box _F b$ . Similarly, if $\Box _P b \in B'$ , then $\Box _P' b= \Box _P b$ . Thus, for each subterm $t'$ of t, the computation of $t'$ in $\mathfrak {B}_S$ is the same as that in $\mathfrak {B}$ . Consequently, $t(a_1, \ldots , a_n) \neq 1$ in $\mathfrak {B}_S$ , and we have found a finite ${\sf MS4.t}$ -algebra refuting $\varphi $ .

We conclude this section by showing that the fmp for ${\sf TS4}$ , ${\sf MS4}$ , and ${\sf MIPC}$ can be obtained as a consequence of Theorem 6.8 via the full and faithful translations into ${\sf MS4.t}$ described in Section 4. In order to do so, we state the fmp of ${\sf MS4.t}$ in terms of ${\sf MS4.t}$ -frames thanks to the correspondence between finite ${\sf MS4.t}$ -algebras and finite ${\sf MS4.t}$ -frames. In fact, we will obtain such a correspondence as a consequence of a representation result for ${\sf MS4.t}$ -algebras.

Definition 6.9. Let R be a quasi-order on a set X and $A \subseteq X$ . We define

$$ \begin{align*} \Box_R(A)=X\setminus R^{-1}[X\setminus A]. \end{align*} $$

If

is the inverse relation of R, we have

If E is an equivalence relation on X, we use the notation

$$ \begin{align*} \forall_E(A)=X\setminus E^{-1}[X\setminus A]=X\setminus E[X\setminus A]. \end{align*} $$

Proposition 6.10. For each ${\sf MS4.t}$ -frame $\mathfrak F=(X,R,E)$ we have that is an ${\sf MS4.t}$ -algebra.

Proof. Since R is a quasi-order, so is , and hence $(\wp (X),\Box _R)$ and are ${\sf S4}$ -algebras; and since E is an equivalence relation, $(\wp (X),\forall _E)$ is an ${\sf S5}$ -algebra (see [21, Theorem 3.5]). In addition, the commutativity condition yields that $\Box _R \forall _E (A) \leq \forall _E \Box _R (A)$ for each $A \in \wp (X)$ . A standard argument (see [21, Theorem 3.6]) gives that $\Box _R$ and satisfy (PF) and (FP). Therefore, $\mathfrak F^+$ is an ${\sf MS4.t}$ -algebra.

Remark 6.11. If $\mathfrak B=\mathfrak F^+$ , then the elements of $H_F$ and $H_P$ are, respectively, the R-upsets and R-downsets of $\mathfrak {F}$ , and the elements of $B_0$ are the E-saturated subsets of $\mathfrak {F}$ (that is, unions of E-equivalence classes).

We next prove that each ${\sf MS4.t}$ -algebra is represented as a subalgebra of $\mathfrak F^+$ for some ${\sf MS4.t}$ -frame $\mathfrak F$ .

Definition 6.12. Let $\mathfrak B=(B, \Box _F, \Box _P, \forall )$ be an ${\sf MS4.t}$ -algebra. The canonical frame of $\mathfrak B$ is the frame $\mathfrak B_+=(X_{\mathfrak B},R_{\mathfrak B},E_{\mathfrak B})$ where $X_{\mathfrak B}$ is the set of ultrafilters of B, $x R_{\mathfrak B} y$ iff $x\cap H_F \subseteq y$ iff $y\cap H_P \subseteq x$ , and $x E_{\mathfrak B} y$ iff $x\cap B_0 = y\cap B_0$ .

Lemma 6.13. If $\mathfrak B$ is an ${\sf MS4.t}$ -algebra, then $\mathfrak B_+$ is an ${\sf MS4.t}$ -frame.

Proof. Since $(B,\Box _F)$ is an ${\sf S4}$ -algebra, we have that $R_{\mathfrak B}$ is a quasi-order (see [21, Theorem 3.14]); and since $(B,\forall )$ is an ${\sf S5}$ -algebra, $E_{\mathfrak B}$ is an equivalence relation (see [21, Theorem 3.18]). It remains to show that Definition 3.7(E) is satisfied. Let $x,y,z \in X_{\mathfrak B}$ be such that $x E_{\mathfrak B} y$ and $y R_{\mathfrak B} z$ . This means that $x \cap B_0=y \cap B_0$ and $y \cap H_F \subseteq z$ . Let F be the filter of $\mathfrak B$ generated by $(x \cap H_F) \cup (z \cap B_0)$ . We show that F is proper. Otherwise, since $x \cap H_F$ and $z \cap B_0$ are closed under finite meets, there are $a \in x \cap H_F$ and $b \in z \cap B_0$ such that $a \land b =0$ . Therefore, $a \le \neg b$ . Thus, $a = \Box _F a \le \Box _F \neg b$ , so $\Box _F \neg b \in x$ . Since $B_0$ is an ${\sf S4}$ -subalgebra of $(B, \Box _F)$ (see Remark 6.5(2)) and $b \in B_0$ , we have $\Box _F \neg b \in B_0$ . This yields $\Box _F \neg b \in x \cap B_0=y \cap B_0$ , which implies $\Box _F \neg b \in y \cap H_F \subseteq z$ . Therefore, $\neg b \in z$ which contradicts $b \in z$ . Thus, F is proper, and so there is an ultrafilter u of B such that $F\subseteq u$ . Consequently, $x\cap H_F\subseteq u$ and $z\cap B_0\subseteq u\cap B_0$ . Since $z \cap B_0$ and $u \cap B_0$ are both ultrafilters of $B_0$ , we conclude that $z\cap B_0= u\cap B_0$ . Thus, there is $u\in X_{\mathfrak B}$ with $xR_{\mathfrak B}u$ and $uE_{\mathfrak B}z$ .

Definition 6.14. Let $\mathfrak {B}$ be an ${\sf MS4.t}$ -algebra. The Stone map $\beta :\mathfrak B\to (\mathfrak B_+)^+$ is defined by

$$ \begin{align*} \beta(a)=\{ x \in X_{\mathfrak B} \mid a \in x \}. \end{align*} $$

It follows from Jónsson-Tarski duality that $\beta$ is a 1-1 homomorphism of $\mathsf{MS4.t}$ -algebras, and that it is an isomorphism if $\mathfrak{B}$ is finite. Thus, we obtain the following representation theorem for ${\sf MS4.t}$ -algebras.

Theorem 6.15 (Representation theorem)

Let $\mathfrak {B}$ be an ${\sf MS4.t}$ -algebra.

1. 1. $\mathfrak {B}$ is isomorphic to a subalgebra of $(\mathfrak B_+)^+$ .

2. 2. When $\mathfrak {B}$ is finite, its embedding into $(\mathfrak B_+)^+$ is an isomorphism.

Remark 6.16. As usual, to recover the image of the embedding of $\mathfrak {B}$ into $(\mathfrak B_+)^+$ we need to endow $\mathfrak B_+$ with a Stone topology (see, e.g., [13, Definition 3.3.3]). This leads to the notion of perfect ${\sf MS4.t}$ -frames and a duality between the categories of ${\sf MS4.t}$ -algebras and perfect ${\sf MS4.t}$ -frames.

Thanks to the representation theorem, the fmp of ${\sf MS4.t}$ can be equivalently stated as follows: if $\varphi $ is not a theorem of ${\sf MS4.t}$ , then it is refuted in a finite ${\sf MS4.t}$ -frame. We now obtain the fmp of ${\sf TS4}$ , ${\sf MS4}$ , and ${\sf MIPC}$ as a consequence of the fmp of ${\sf MS4.t}$ .

Theorem 6.17.

1. 1. ${\sf TS4}$ has the fmp.

2. 2. ${\sf MS4}$ has the fmp.

3. 3. ${\sf MIPC}$ has the fmp.

Proof. (1). Suppose that ${\sf TS4} \nvdash \varphi $ . By Theorem 4.13(3), ${\sf MS4.t} \nvdash \varphi ^\dagger $ . Since ${\sf MS4.t}$ has the fmp, there is a finite ${\sf MS4.t}$ -frame $\mathfrak {F}$ such that $\mathfrak {F} \nvDash \varphi ^\dagger $ . By Proposition 4.11(3), $\mathfrak {F}^\dagger \nvDash \varphi $ . We have thus obtained a finite ${\sf TS4}$ -frame $\mathfrak {F}^\dagger $ refuting $\varphi $ .

(2). Similar to the proof of (1) but uses the translation $(-)^\# : {\sf MS4} \to {\sf MS4.t}$ instead of $(-)^\dagger $ .

(3). Similar to the proof of (1) but uses the composition $(-)^{t \#}: {\sf MIPC} \to {\sf MS4.t}$ instead of $(-)^\dagger $ . Alternatively, we can use the translation $(-)^{\natural \dagger }$ of ${\sf MIPC}$ into ${\sf MS4.t}$ .

Remark 6.18.

1. 1. That ${\sf MIPC}$ has the fmp was first established by Bull [7] using algebraic semantics. His proof contained a gap, which was corrected independently by Fischer-Servi [15] and Ono [23]. A semantic proof is given in [16], which is based on a technique developed by Grefe [19].

2. 2. The fmp for ${\sf MS4}$ can be derived from the results in [17, Sec. 12] (see also [16, Theorems 6.52 and 9.12]). The proof given above is more direct.

3. 3. The proof of the fmp for ${\sf MS4.t}$ contains the proof of the fmp for ${\sf S4.t}$ , but the latter is known (see [29, pp. 313–314] or [18, p. 44]). In fact, ${\sf MS4.t}$ is a conservative extension of ${\sf S4.t}$ .

7 Connection with the full predicate case

In [4] we studied a translation that is the predicate analogue of the translation $(-)^\flat: \mathsf{MIPC} \to \mathsf{MS4.t}$ . We proved that this translation embeds ${\sf IQC}$ fully and faithfully into a weakening of the tense predicate logic ${\sf QS4.t}$ . This weakening is necessary since ${\sf QS4.t}$ proves the Barcan formula for both $\Box _F$ and $\Box _P$ , so Kripke frames of ${\sf QS4.t}$ have constant domains, and hence they validate the translation of the constant domain axiom $\forall x(A\vee B)\to (A\vee \forall x B)$ , where x is not free in A. Since this is not provable in ${\sf IQC}$ , the translation cannot be full. Instead we considered the tense predicate logic ${\sf Q^\circ S4.t}$ in which the universal instantiation axiom $\forall x A \to A(y/x)$ is replaced by its weakened version $\forall y(\forall x A \to A(y/x))$ . The main result of [4] proves that ${\sf IQC}$ translates fully and faithfully into ${\sf Q^\circ S4.t}$ (provided the translation is restricted to sentences).

It is natural to investigate the relationship between ${\sf MS4.t}$ and predicate extensions of ${\sf S4.t}$ . As we already pointed out in Remark 2.10, ${\sf MS4.t}$ is not the monadic fragment of ${\sf QS4.t}$ . In addition, ${\sf MS4.t}$ cannot be the monadic fragment of ${\sf Q^\circ S4.t}$ either since the formula $\forall x A \to A$ is not in general provable in ${\sf Q^\circ S4.t}$ , whereas $\forall \varphi \to \varphi $ is provable in ${\sf MS4.t}$ . On the other hand, call a formula $\varphi $ (in the language of ${\sf MS4.t}$ ) bounded if each occurrence of a propositional letter in $\varphi $ is under the scope of $\forall $ . Bounded formulas play the same role as sentences of ${\sf Q^\circ S4.t}$ containing only one fixed variable. It is quite plausible that for a bounded formula $\varphi $ we have ${\sf MS4.t}\vdash \varphi $ iff ${\sf Q^\circ S4.t}$ proves the translation of $\varphi $ where each occurrence of a propositional letter p is replaced with the unary predicate $P(x)$ and $\forall $ is replaced with $\forall x$ (for a similar translation of $\sf MIPC$ and its extensions into ${\sf IQC}$ and its extensions, see [24]). If true, this would yield that the monadic sentences provable in ${\sf Q^\circ S4.t}$ are exactly the bounded formulas $\varphi $ provable in ${\sf MS4.t}$ . It would also yield that restricting the translation ${\sf IQC} \to {\sf Q^\circ S4.t}$ of [4] to the monadic setting gives the translation $(-)^\flat :{\sf MIPC} \to {\sf MS4.t}$ for bounded formulas.

It is natural to seek an axiomatization of the full monadic fragment of ${\sf Q^\circ S4.t}$ . Note that in this fragment $\forall $ does not behave like an ${\sf S5}$ -modality. For example, $\forall \varphi \to \varphi $ is not in general a theorem of this fragment.

Finally, the translation $(-)^\#: {\sf MS4} \to {\sf MS4.t}$ suggests a translation of ${\sf QS4}$ into ${\sf Q^\circ S4.t}$ which replaces each occurrence of $\Box $ with $\Box _F$ . It is easy to see that for sentences this translation is full and faithful. Composing it with the standard Gödel translation of ${\sf IQC}$ into ${\sf QS4}$ yields a translation ${\sf IQC} \to {\sf Q^\circ S4.t}$ which is different from the translation of [4]. This translation restricts to the translation $(-)^{t \#}: {\sf MIPC} \to {\sf MS4.t}$ for bounded formulas. Thus, the upper part of the diagram of Section 4 extends to the predicate case.

On the other hand, we do not see a natural way to interpret the tense modalities of ${\sf TS4}$ as monadic quantifiers, and hence we cannot think of a natural predicate logic which could take the role of ${\sf TS4}$ in the diagram of Section 4. Thus, the lower part of the diagram does not seem to have a natural extension to the predicate case. Nevertheless, we can consider the predicate analogue of the translation $(-)^{\natural \dagger }:{\sf MIPC} \to {\sf MS4.t}$ . Arguing as in Theorems 5.2 and 5.4 yields a translation of ${\sf IQC}$ into ${\sf Q^\circ S4.t}$ that is full and faithful on sentences and coincides, up to logical equivalence in ${\sf Q^\circ S4.t}$ , with the other two predicate translations described in this section.

We thus obtain the following diagram in the predicate case which is commutative up to logical equivalence in ${\sf Q^\circ S4.t}$ .