Visser frames for sublogics of $\IL$

Throughout the introduction, let $T$ stand for consistent computably enumerable theories extending Peano Arithmetic $\mathbf{PA}$. The notion of interpretability has been investigated through the framework of modal logic. The language of interpretability logics is obtained from that of modal propositional logic by adding the binary modal operator $\rhd$, where the modal formula $A\rhd B$ is intended to mean “$T+B$ is interpretable in $T+A$”. The logic $\IL$ of interpretability is a base logic to investigate several interpretability logics. Berarducci [1] and Shavrukov [9] independently proved that the extension $\mathbf{ILM}$ of $\IL$ is arithmetically complete with respect to arithmetical interpretations based on any $\Sigma_{1}$-sound theory $T$. It is known that the notions of interpretability and $\Pi_{1}$-conservativity coincide for computably enumerable extensions of $\mathbf{PA}$, so the logic $\mathbf{ILM}$ is also the logic of $\Pi_{1}$-conservativity (cf. Hájek and Montagna [3]), where the formula $A\rhd B$ is read as “$T+B$ is $\Pi_{1}$-conservative over $T+A$”. In this context, Ignatiev [4] further investigated the logic of $\Gamma$-conservativity for $\Gamma\in\{\Sigma_{n},\Pi_{n}:n\geq 1\}$ and introduced the logic $\mathbf{CL}$ of conservativity, that is a sublogic of $\IL$, as a base logic to investigate several conservativity logics.

The logics $\IL$ and $\mathbf{CL}$ are complete with respect to a relational semantics, which is called Veltman semantics. We say that a triple $(W,R,\{S_{x}\}_{x\in W})$ is an $\IL$-frame if $W$ is a non-empty set, $R$ is a transitive and conversely well-founded binary relation on $W$, and for each $x\in W$, $S_{x}$ is a transitive and reflexive binary relation on $R[x]:=\{y\in W\mid xRy\}$ satisfying the condition ($\dagger$): $(\forall x,y,z\in W)(xRy\,\&\,yRz\Rightarrow yS_{x}z)$. Also, a $\mathbf{CL}$-frame is obtained by removing the condition $(\dagger)$ from the definition of an $\IL$-frame. Then, de Jongh and Veltman [2] proved that $\IL$ is sound and complete with respect to the class of all finite $\IL$-frames. Moreover, Ignatiev [4] proved that $\mathbf{CL}$ is sound and complete with respect to the class of all finite $\mathbf{CL}$-frames.

Visser [11] introduced and investigated the notion of Visser frames, which are also called simplified Veltman frames. An $\IL$-Visser frame is a triple $(W,R,S)$ where $(W,R)$ satisfies the same condition as in the definition of Veltman frames and $S$ is a transitive and reflexive binary relation on $W$ satisfying the condition ($\dagger$’): $R\subseteq S$. A $\mathbf{CL}$-Visser frame is obtained by removing the condition ($\dagger$’) from the definition of an $\IL$-Visser frame. Then, Visser [11] proved that $\IL$ is sound and complete with respect to the class of all $\IL$-Visser frames. On the other hand, unlike the case of Veltman frames, Visser [12] proved that $\IL$ does not have finite frame property with respect to $\IL$-Visser frames. Ignatiev [4] proved that the modal completeness theorem of $\mathbf{CL}$, that is, $\mathbf{CL}$ is sound and complete with respect to the class of all $\mathbf{CL}$-Visser frames⁵⁵5Ignatiev actually proved a stronger result that $\mathbf{CL}$ is sound and complete with respect to the class of all $\mathbf{CL}$-Visser frames where $S$ is symmetric.. Then, he proposed the following problem.

By using the modal completeness of $\mathbf{CL}$ with respect to Visser frames, Ignatiev also proved the arithmetical completeness of $\mathbf{CL}$.

However, the authors at least think that there is one fault in his proof of Theorem 1.2. His proof is proceeded by embedding a counter $\mathbf{CL}$-Visser model $(W,R,S,\Vdash)$ obtained by his modal completeness theorem into arithmetic as in the proof of Solovay [10], where $W$ is an infinite set. Then, Ignatiev stated that for each $x\in W$, finite set $X\subseteq W$, and modal formula $C$, the following relation is represented by a $\Delta_{0}$ formula:

However, since $W$ is an infinite set, the formula “$\exists{z}\in W$” is not a $\Delta_{0}$ formula in general. In addition, the author at least think that this formula is not represented by any $\Delta_{0}$ formula as long as we use his model. On the other hand, if $\mathbf{CL}$ has the finite frame property with respect to Visser frames, then the above formula is obviously represented by a $\Delta_{0}$ formula because $W$ is a finite set. Therefore, if Problem 1.1 is affirmatively solved, then Ignatiev’s proof of Theorem 1.2 is completely correct.

Veltman semantics was also introduced for other sublogics of $\IL$ than $\mathbf{CL}$. Visser [11] introduced $\IL^{-}\!$-frames (or Veltman prestructures) which are general notions of $\IL$-frames and $\mathbf{CL}$-frames. Kurahashi and Okawa [8] introduced the sublogic $\IL^{-}$ of $\IL$, and proved that $\IL^{-}$ is sound and complete with respect to the class of all finite $\IL^{-}\!$-frames. In [8], several sublogics of $\IL$ were introduced by adding several axioms to $\IL^{-}$, and their completeness with respect to $\IL^{-}\!$-frames was investigated.

In this context, we investigate finite frame property with respect to Visser frames for several sublogics of $\IL$. Table 1 summarizes the results obtained in this paper on the soundness and completeness, and the finite frame property of these sublogics with respect to Visser frames. In particular, Theorem 3.2 answers Problem 1.1 affirmatively.

In this section, we introduce the logic $\IL$, sublogics of $\IL$, and two relational semantics: Veltman frames and Visser frames. Then, we introduce and prove several basic properties for these logics and frames.

The language of interpretability logics consists of propositional variables $p,q,r,\cdots$, the logical constants $\top$, $\bot$, the connectives $\to$, $\lor$, $\land$, and the modal operators $\Box$ and $\rhd$. The modal formulas are given by

where $\circ\in\{\to,\lor,\land\}$. We define $\Diamond A:\equiv\lnot\Box\lnot A$. The binary modal operator $\rhd$ binds stronger than $\to$, and weaker than $\lnot$, $\land$, $\lor$, $\Box$, and $\Diamond$.

The authors introduced in [8] the logic $\IL^{-}$ and proved that $\IL^{-}$ is sound and complete with respect to the class of all (finite) $\IL^{-}\!$-frames.

Here, we introduce the following axioms $\mathbf{J2}_{+}$ and $\mathbf{J4}_{+}$ which were introduced by Kurahashi and Okawa [8] and Visser [11], respectively.

$\mathbf{J2}_{+}$:

$\bigl{(}A\rhd(B\lor C)\bigr{)}\land B\rhd C\to A\rhd C$.

$\mathbf{J4}_{+}$:

$\Box(A\to B)\to(C\rhd A\to C\rhd B)$.

Let $L(\Sigma_{1},\ldots,\Sigma_{n})$ denote the logic obtained by adding the axioms $\Sigma_{1},\ldots,\Sigma_{n}$ to the logic $L$. Then, the following facts hold (for a proof, see [8]).

In the next section, we investigate the modal completeness of several sublogics of $\IL$ in Figure 1 with respect to Visser frames which were originally introduced by Visser [11]. In this paper, we consider the following $\IL^{-}(\mathbf{J4}_{+})$-Visser frames as the basis, which are general notion of Visser’s Visser frames.

Proposition 2.13 shows that each logic $L$ as in Definition 2.14 is sound with respect to all $L$-Visser frames, that is, every $L$-Visser frame validates all axioms of $L$. Conversely, the modal completeness of the logics $\IL$ and $\mathbf{CL}$ with respect to corresponding Visser frames is already known. On the other hand, Visser [12] proved that $\IL$ does not have finite frame property.

Our main purpose of this paper is to prove the finite frame property for $\mathbf{CL}$ with respect to Visser frames (Theorem 3.2).

In this section, we study modal completeness of each extension of $\IL^{-}(\mathbf{J4}_{+})$ with respect to Visser frames. Among other things, we prove that the logic $\mathbf{CL}$ has finite frame property.