Countermodel Construction via Optimal Hypersequent Calculi for Non-normal Modal Logics

Abstract

We develop semantically-oriented calculi for the cube of non-normal modal logics and some deontic extensions. The calculi manipulate hypersequents and have a simple semantic interpretation. Their main feature is that they allow for direct countermodel extraction. Moreover they provide an optimal decision procedure for the respective logics. They also enjoy standard proof-theoretical properties, such as a syntactical proof of cut-admissibility.

Supported by WWTF project MA16-28 and by Project TICAMORE ANR-16-CE91-0002-01.

Appendix

Theorem 2

If $\mathcal H_{\mathsf E^{\star }}$ contains $\Box _\mathsf R$, then the rules $\mathsf {cut}$ and $\mathsf {sub}$ are admissible in $\mathcal H_{\mathsf E^{\star }}$, otherwise $\mathsf {cut}$ and $\mathsf {sub_M}$ are admissible in $\mathcal H_{\mathsf E^{\star }}$.

Proof

We prove that $\mathsf {cut}$ and $\mathsf {sub}$ are admissible in non-monotonic $\mathcal H_{\mathsf E^{\star }}$; the proof in the monotonic cases is analogous. Recall that, for an application of $\mathsf {cut}$, the cut formula is the formula which is deleted by that application, while the cut height is the sum of the heights of the derivations of the premisses of $\mathsf {cut}$.

The theorem is a consequence of the following claims, where Cut(c, h) means that all applications of $\mathsf {cut}$ of height h on a cut formula of weight c are admissible, and Sub(c) means that all applications of $\mathsf {sub}$ where A has weight c are admissible (for all $\varSigma , \varPi _1, ..., \varPi _k$): (A) $\forall c. Cut(c, 0)$. (B) $\forall h. Cut(0, h)$. (C) . (D) .

(A) deals with applications of $\mathsf {cut}$ to initial sequents and is trivial.

(B) If the cut formula has weight 0, then it is $\bot$, $\top$, or a propositional variable p. In both situations the proof is by complete induction on h. The basic case $h=0$ is a particular case of (A). For the inductive step, we distinguish three cases.

(i) The cut formula $\bot$, $\top$, or p is not principal in the last rule applied in the derivation of the left premiss. By examining all possible rule applications, we show that the application of $\mathsf {cut}$ can be replaced by one o more applications of $\mathsf {cut}$ at a smaller height. For instance, assume that the last rule applied is $\Box _\mathsf L$.

The derivation is transformed as follows, with a hp-application of $\mathsf {wk}$ and an application of $\mathsf {cut}$ of smaller height.

The situation is similar if the last rule in the derivation of the left premiss is applied to some sequent in $G$.

(ii) The cut formula $\bot$, $\top$, or p is not principal in the last rule applied in the derivation of the right premiss. The case is analogous to (i). As an example, suppose that the last rule applied is ${}_{\mathsf M}\Box _\mathsf R$.

The derivation is converted into

where $\mathsf {cut}$ is applied at a smaller height.

(iii) The cut formula $\bot$, $\top$, or p is principal in the last rule applied in the derivation of both premisses. Then the cut formula is p, as $\bot$ (resp. $\top$) is never principal on the right-hand side (resp. left-hand side) of the conclusion of any rule application. This means that both premisses are derived by $\mathsf {init}$, which implies $h=0$. Then we are back to case (A).

(C) Assume $\forall h Cut(c,h)$. The proof is by induction on the height m of the derivation of . Here we only consider the case where $m>0$ and the last rule applied in the derivation is $\Box _\mathsf R$, with one block among $\langle A, \varPi _1 \rangle , ..., \langle A, \varPi _k \rangle$ principal in the rule application:

The derivation is converted as follows. First we derive:

Moreover, by applying $\mathsf {ew}$ to $G\mid \varSigma \Rightarrow A$ we obtain . By auxiliary applications of $\mathsf {wk}$ we can cut A and get . Then with further applications of cut (each time with auxiliary applications of $\mathsf {wk}$) we obtain By doing the same with the other premisses of $\Box _\mathsf R$ in the initial derivation we obtain also and Then by $\Box _\mathsf R$ we derive the conclusion of $\mathsf {sub}$

(D) Assume $\forall c'<c.\, (Sub(c') \wedge \forall h'.\, Cut(c', h'))$ and $\forall h''<h.\, Cut (c, h'')$. We show that all applications of $\mathsf {cut}$ of height h on a cut formula of weight c can be replaced by different applications of $\mathsf {cut}$, either of smaller height or on a cut formula of smaller weight. We can assume $c, h > 0$ as the cases $c=0$ and $h=0$ have been considered already in (B) and (A). We distinguish two cases.

(i) The cut formula is not principal in the last rule application in the derivation of at least one of the two premisses of $\mathsf {cut}$. This case is analogous to (i) or (ii) in (B).

(ii) The cut formula is principal in the last rule application in the derivation of both premisses. Then the cut formula is either , or $B \wedge C$, or $\Box B$.

— If the cut formula is we have

The derivation is converted into the following one:

— If the cut formula is $B\wedge C$ the situation is similar.

— If the cut formula is $\Box B$ we have

The derivation is converted as follows, with several applications of $\mathsf {cut}$ of smaller height.

   $\square$

Theorem 8

If $\mathcal H_{\mathsf {E(T/P/D)}^{\star }}$ contains $\Box _\mathsf R$, then the rules $\mathsf {cut}$ and $\mathsf {sub}$ are admissible in $\mathcal H_{\mathsf {E(T/P/D)}^{\star }}$, otherwise $\mathsf {cut}$ and $\mathsf {sub_M}$ are admissible in $\mathcal H_{\mathsf {E(T/P/D)}^{\star }}$.

Proof

We extend point (C) (ii) in the proof of Theorem 2 to the cases where the last rule applied in the derivation of is $\mathsf T$, $\mathsf P$, $\mathsf D$, or $\mathsf D_{\mathsf {aux}}$ (resp. $\mathsf T$, $\mathsf P$, or $\mathsf D_{\mathsf M}$ in monotonic case). We consider as examples the following two cases.

— The last rule is $\mathsf T$:

By applying the inductive hypothesis to the premiss we obtain . Then, from this and $G\mid \varSigma \Rightarrow A$, by several applications of $\mathsf {cut}$ (each time with auxiliary applications of $\mathsf {wk}$) we obtain . Finally, by $\mathsf T$ we derive .

— The last rule is $\mathsf P$:

By applying the inductive hypothesis to the premiss (aftar auxiliary applications of $\mathsf {ew}$ to the other premisses of $\mathsf {sub}$) we obtain . Then, from this and $G\mid \varSigma \Rightarrow A$, by several applications of $\mathsf {cut}$ (each time with auxiliary applications of $\mathsf {wk}$) we obtain . Finally, by $\mathsf P$ we derive .    $\square$