# Hypersequent Calculi for Lewis’ Conditional Logics with Uniformity and Reflexivity

## Abstract

We present the first internal calculi for Lewis’ conditional logics characterized by uniformity and reflexivity, including non-standard internal hypersequent calculi for a number of extensions of the logic \(\mathbb {V}\mathbb {T}\mathbb {U}\). These calculi allow for syntactic proofs of cut elimination and known connections to \(\mathsf {S5}\). We then introduce standard internal hypersequent calculi for all these logics, in which sequents are enriched by additional structures to encode plausibility formulas as well as diamond formulas. These calculi provide both a decision procedure for the respective logics and constructive countermodel extraction from a failed proof search attempt.

## 1 Introduction

*Ramsey’s Rule*:

where the operator \(\circ \) is any\(A \circ B \rightarrow C\) holds if and only if Open image in new window holds

*update*operator satisfying Katsuno and Mendelzon’s postulates. The relation means that

*C*is entailed by “

*A*updated by

*B*” if and only if the conditional Open image in new window is entailed by

*A*. In this sense it can be said that the conditional Open image in new window expresses an hypothetical update of a piece of information

*A*.

The family of logics studied by Lewis in [13] is semantically characterized by *sphere models*, where each world *x* is equipped with a set of nested sets of worlds Open image in new window. Each set in Open image in new window is called a *sphere*: the intuition is that concerning *x*, worlds in inner spheres are more plausible than worlds belonging only to outer spheres. Lewis takes as primitive the *comparative plausibility* connective \(\preccurlyeq \), with a formula \(A \preccurlyeq B\) meaning “*A* is at least as plausible as *B*”. The conditional Open image in new window is then defined as “*A* is impossible or \(A \wedge \lnot B\) is less plausible than *A*”. Vice versa, \(\preccurlyeq \) can be defined in terms of Open image in new window.

*Uniformity*: all worlds have the same set of accessible worlds, where the worlds accessible from a world

*x*are those belonging to any sphere Open image in new window; (ii)

*Total reflexivity*: every world

*x*belongs to some sphere Open image in new window. The basic logic is \(\mathbb {V}\mathbb {T}\mathbb {U}\); we will then consider some of its extensions, including the above mentioned \(\mathbb {V}\mathbb {C}\mathbb {U}\). It is worth mentioning that equivalent logics are those of Comparative Concept Similarity studied in the context of ontologies [17]. These logics contain a connective \(\leftleftarrows \), which allows to express, e.g.,

Here we investigate *internal calculi* for logics extending \(\mathbb {V}\mathbb {T}\mathbb {U}\), i.e., calculi where each configuration of a derivation corresponds to a formula of the corresponding logic, in contrast to external calculi which make use of extra-logical elements (such as labels, terms and relations on them). Ideally, we seek calculi with the following features: (i) they should be *standard*, i.e., each connective is handled by a finite set of rules with a finte and fixed set of premises; (ii) they should be *modular*, i.e., it should be possible to obtain calculi for stronger logics by adding independent rules to a base calculus; (iii) they should have good proof-theoretical properties, such as a syntactic proof of cut admissibility; finally (iv) they should provide a decision procedure for the respective logics. In our opinion requirement (i) is particularly important: a standard calculus provides a self-explanatory presentation of the logic, thus a kind of proof-theoretic semantics.

In previous work [7], we defined calculi with many of these properties for weaker logics of the Lewis’ family. For the logics with *uniformity* to the best of our knowledge no internal calculi are known; the only known external calculi for these adopt a hybrid language and a relational semantics [6]. We also consider logics with *absoluteness*, a property stronger than uniformity stating that all worlds have the same system of spheres. It is unlikely that sequents, even extended as in [7], are sufficient to capture logics with uniformity: Since modal logic \(\mathsf {S5}\) can be embedded into \(\mathbb {V}\mathbb {T}\mathbb {U}\), a sequent calculus for the latter would most probably also yield a sequent calculus for \(\mathsf {S5}\). The existence of such a calculus, however, would be very surprising. We therefore adopt the framework of *hypersequents* [2], where the basic objects are multisets of sequents.

We first provide a non-standard hypersequent calculus for \(\mathbb {V}\mathbb {T}\mathbb {U}\) and its extensions and syntactically prove cut-elimination and hence completeness. We then show that by translating \(\Box A\) as \(\bot \preccurlyeq \lnot A\) the calculi - restricted to such formulas - correspond to known hypersequent calculi for \(\mathsf {S5}\). Further, we construct standard calculi for all the logics by enriching the hypersequents with additional structural connectives encoding plausibility and “possible” formulas respectively. The obtained standard calculi provide decision procedures for the respective logics. Finally, we give a direct semantic completeness proof for the logics without absoluteness, by considering the invertible version of the rules and constructing a countermodel from a failed attempt at proof search. Thus, the calculi can also be used for countermodel generation, a task of independent interest.

## 2 Preliminaries

We consider the *conditional logics* of [13]. The set of *conditional formulae* is given by \( A {:}{:} = p \mid \bot \mid A \rightarrow A \mid A \preccurlyeq A \), where \(p \in \mathcal {V}\) is a propositional variable. We define the boolean connectives \(\wedge , \vee , \top \) in terms of \( \bot \) and \( \rightarrow \) as usual. Intuitively, a formula \(A \preccurlyeq B\) is interpreted as “*A* is at least as plausible as *B*”. Lewis’ *counterfactual implication*Open image in new window is defined by Open image in new window, whereas the *outer modality*\(\Box \) is defined by \(\Box A \equiv (\bot \preccurlyeq \lnot A)\). The logics we consider are defined as follows:

### Definition 1

*universal sphere model*(or

*model*) is a triple Open image in new window, consisting of a non-empty set

*W*of elements, called

*worlds*, a mapping Open image in new window, and a propositional valuation Open image in new window. Elements of Open image in new window are called

*spheres*. We assume the following conditions:

The valuation Open image in new window is extended to all formulae by: \([\![{\bot }]\!]= \emptyset \); \([\![{A \rightarrow B}]\!]= (W - [\![{A}]\!]) \cup [\![{B}]\!]\); Open image in new window. We also write \(w \Vdash A\) instead of \(w \in [\![{A}]\!]\) as well as \( \alpha \Vdash ^{\forall }A \) for \( \forall x \in \alpha .\; x \Vdash A\) and \( \alpha \Vdash ^{\exists } A \) for \( \exists x \in \alpha .\; x \Vdash A \)^{1}. Validity and satisfiability of formulae in a class of models are defined as usual. *Conditional logic*\(\mathbb {V}\mathbb {T}\mathbb {U}\) is the set of formulae valid in all universal sphere models.

*weak centering*: for all Open image in new window we have \(w \in \alpha \);*centering*: for all \( w \in W\;\) we have Open image in new window;*absoluteness*: for all \( w,v \in W\;\) we have Open image in new window.

^{2}:

Lewis’ logics and axioms.

## 3 Hypersequent Calculi

In this section we introduce calculi for \( \mathbb {V}\mathbb {T}\mathbb {U}\) and extensions. We call a calculus *standard* if (a) it has a finite number of rules and (b) each rule has a finite and fixed number of premisses. With respect to this definition, the calculi introduced in this section are *non-standard*, whereas the calculi we introduce in Sect. 6 are standard.

*sequent*is a pair consisting of two multisets of formulae, written as \(\varGamma \Rightarrow \varDelta \).

### Definition 2

*hypersequent*is a non-empty multiset of sequents, written \(\varGamma _1 \Rightarrow \varDelta _1 \mid \ldots \mid \varGamma _n \Rightarrow \varDelta _n\), where \(n \ge 1\) is the cardinality of the multiset. The

*conditional formula interpretation*of a hypersequent is

*outer modality*defined by \(\Box A \equiv (\bot \preccurlyeq \lnot A)\).

The rules of the calculi \(\mathsf {H}_{\mathcal {L}}\) extend the calculi from [12] to the hypersequent setting and are given in Fig. 1. These calculi are *non-standard*, meaning that the rules have an unbounded number of premisses. We abbreviate multisets of formulae \(A_k,\ldots , A_n\) to \(\varvec{[ A ]}_{k}^{n}\), and \(C_k\preccurlyeq D_k, \ldots , C_n \preccurlyeq D_n\) to \(\varvec{[ C \preccurlyeq D ]}_{k}^{n}\) with the convention that \(\varvec{[ A ]}_{k}^{n}\) is empty if \(k > n\). The crucial rule for uniformity is the rule \(\mathsf {trf}_m\). Intuitively it unpacks a number of comparative plausibility formulae behaving like boxed formulae on the left hand side of a component in the conclusion into a different component in the rightmost premiss, most clearly seen in the case of \(n = 1\). The leftmost set of premisses ensures that the comparative plausibility formulae indeed behave like boxed formulae. The rule \(\mathsf {T}_m\) is the local version of \(\mathsf {trf}_m\), and essentially captures total reflexivity.

### Lemma 3

For \(\mathcal {L}\) any of the considered logics, the calculus \(\mathsf {H}_{\mathcal {L}}\) is sound for \(\mathcal {L}\).

### Proof

This follows from validity of \(\Box A \rightarrow A\) in all the logics and the fact that the rules preserve soundness with respect to \(\iota \). The latter is shown for each rule by constructing a countermodel for one of the premisses from a countermodel for the conclusion, using that the sphere system is universal. For all the rules apart from \(\mathsf {trf},\mathsf {abs}_L,\mathsf {abs}_R,\mathsf {spl}\) this follows as in [12], using that \(\Box A \rightarrow \Box \Box A\) is valid. For \(\mathsf {abs}_L,\mathsf {abs}_R\) this follows straightforwardly from absoluteness, and for \(\mathsf {spl}\) this follows from the fact that frames for \(\mathbb {V}\mathbb {C}\mathbb {A}\) are degenerate in the sense that Open image in new window for every world *w* (see footnote 2).

By induction on the formula complexity we straightforwardly obtain:

### Lemma 4

For every formula *A* we have \(\mathsf {H}_{\mathcal {L}}\vdash \mathcal {G} \mid \varGamma , A \Rightarrow A, \varDelta \).

*admissible*in \(\mathsf {H}_{\mathcal {L}}\) if whenever the premisses are derivable in \(\mathsf {H}_{\mathcal {L}}\), then so is its conclusion. It is

*depth-preserving admissible*, if the depth of the derivation of its conclusion is at most the maximal depth of the derivations of its premisses.

### Lemma 5

The rules \(\mathsf {IW}, \mathsf {EW}, \mathsf {mrg}\) from Fig. 2 are depth-preserving admissible in \(\mathsf {H}_{\mathcal {L}}\).

### Proof

By induction on the depth of the derivation in all cases. For \(\mathsf {mrg}\), if the last applied rule was \(\mathsf {trf}_m\), we might need to replace it with \(\mathsf {T}_m\). \(\square \)

*cut rule*:Cut-free completeness then will follow from cut elimination. In the following we write \(\mathsf {H}_{\mathcal {L}}\mathsf {cut}\) for the system \(\mathsf {H}_{\mathcal {L}}\) with the cut rule.

### Lemma 6

**(Completeness with cut).** For \(\mathcal {L}\) one of the considered logics the calculus \(\mathsf {H}_{\mathcal {L}}\mathsf {cut}\) is complete for \(\mathcal {L}\), i.e., whenever \(A \in \mathcal {L}\), then \(\mathsf {H}_{\mathcal {L}}\mathsf {cut}\vdash \;\Rightarrow A\).

### Proof

## 4 Cut Elimination

### Definition 7

The *cut rank* of a \(\mathsf {H}_{\mathcal {L}}^*\)-derivation \(\mathcal {D}\) is the maximal complexity of a cut formula occurring in \(\mathcal {D}\), written \(\rho (\mathcal {D})\). A rule is *cut-rank preserving admissible* in \(\mathsf {H}_{\mathcal {L}}^*\) if whenever its premiss(es) are derivable in \(\mathsf {H}_{\mathcal {L}}^*\) with cut-rank *n*, then so is its conclusion.

### Lemma 8

The rules \(\mathsf {EW},\mathsf {IW}\) are depth- and cut-rank preserving admissible in \(\mathsf {H}_{\mathcal {L}}^*\).

### Proof

Standard induction on the depth of the derivation. \(\square \)

### Lemma 9

**(Shift Right).** Suppose that for \(k>0\) and \(n_1,\ldots , n_k > 0\) there are \(\mathsf {H}_{\mathcal {L}}^*\)-derivations \(\mathcal {D}_1\) and \(\mathcal {D}_2\) of \(\mathcal {G} \mid \varOmega \Rightarrow \varTheta , A\) and \(\mathcal {H} \mid A^{n_1},\varXi _1 \Rightarrow \varUpsilon _1 \mid \ldots \mid A^{n_k}, \varXi _k \Rightarrow \varUpsilon _k\) with \(\rho (\mathcal {D}_1) < |A| > \rho (\mathcal {D}_2)\) and such that the displayed occurrence of *A* is principal in the last rule application in \(\mathcal {D}_1\). Then there is a \(\mathsf {H}_{\mathcal {L}}^*\)-derivation \(\mathcal {D}\) with endhypersequent \( \mathcal {G} \mid \mathcal {H} \mid \varOmega , \varXi _1 \Rightarrow \varTheta , \varUpsilon _1 \mid \ldots \mid \varOmega , \varXi _k \Rightarrow \varTheta , \varUpsilon _k \) and \(\rho (\mathcal {D}) < |A|\).

### Proof

*A*is principal in the last rule in \(\mathcal {D}_2\), we apply the induction hypothesis on the premiss(es) of that rule, followed by the same rule (and possibly structural rules). If at least one of the displayed occurrences is principal in the last rule in \(\mathcal {D}_2\), we distinguish cases according to the last applied rule in \(\mathcal {D}_1\), with subcases according to the last rule in \(\mathcal {D}_2\). For space reasons we only consider an exemplary case, the remaining cases are similar. Suppose the last rules in \(\mathcal {D}_1\) and \(\mathcal {D}_2\) are \(R_{m,n+1}\) and \(\mathsf {trf}_s\) respectively, that

*A*is the formula \(E \preccurlyeq F\) and that \(\mathcal {D}_1\) ends in:First we apply the induction hypothesis to the conclusion of this and the premisses of \(\mathsf {trf}_s\) to eliminate all the occurrences of \(E \preccurlyeq F\) from the context. Hence we assume that the only occurrences of \(E \preccurlyeq F\) in the conclusion of \(\mathsf {trf}_s\) are principal and that \(\mathcal {D}_2\) ends in:with \(E \preccurlyeq F\) not occurring in \(\varvec{[ G\preccurlyeq H ]}_{1}^{r}\). Cuts on the formulae

*E*and

*F*then yield:Admissibility of internal weakening (Lemma 8) and an application of \(R_{m+s, n+t}\) then gives:

### Lemma 10

**(Shift Left).** Suppose that for \(k>0\) and \(n_1,\ldots , n_k > 0\) there are \(\mathsf {H}_{\mathcal {L}}^*\)-derivations \(\mathcal {D}_1\) and \(\mathcal {D}_2\) of the hypersequents \(\mathcal {G} \mid \varOmega _1 \Rightarrow \varTheta _1, A^{n_1} \mid \ldots \mid \varOmega _k \Rightarrow \varTheta _k, A^{n_k}\) and \(\mathcal {H} \mid A,\varXi \Rightarrow \varUpsilon \) with \(\rho (\mathcal {D}_1) < |A| > \rho (\mathcal {D}_2)\). Then there is a \(\mathsf {H}_{\mathcal {L}}^*\)-derivation \(\mathcal {D}\) with endsequent \( \mathcal {G} \mid \mathcal {H} \mid \varOmega _1, \varXi \Rightarrow \varTheta _1, \varUpsilon \mid \ldots \mid \varOmega _k, \varXi \Rightarrow \varTheta _k, \varUpsilon \) and \(\rho (\mathcal {D}) < |A|\).

### Proof

By induction on the depth of \(\mathcal {D}_1\). If none of the displayed occurrences of *A* is principal in the last rule in \(\mathcal {D}_1\) or the active formula of \(\mathsf {abs}_R\), we apply the induction hypothesis on the premiss(es) of the last rule in \(\mathcal {D}_1\) followed by the same rule and possibly admissibility of weakening and contraction. If one of the occurrences of *A* is active in \(\mathsf {abs}_R\), we use admissibility of \(\mathsf {EW}\) (Lemma 8) and \(\mathsf {abs}_L\) on \(\mathcal {D}_2\) to obtain \(\mathcal {H} \mid \varXi \Rightarrow \varUpsilon \mid A \Rightarrow \;\). Then the induction hypothesis on this and the premiss of \(\mathsf {abs}_R\) followed by \(\mathsf {mrg}\) and \(\mathsf {IW}\) yields the result. If an occurrence of *A* is principal in the last rule in \(\mathcal {D}_1\), we use the induction hypothesis to remove all the occurrences of *A* in the context of that rule. Then, in case this rule is \(R_{m,n},W_{m,n}, W_{m,n}^\mathsf {abs}\), we apply contraction in the premisses and apply the same rule, so that only one occurrence of *A* is principal. Now Lemma 9 yields the result. \(\square \)

### Theorem 11

**(Cut Elimination).** Let \(\mathcal {L} \in \{ \mathbb {V}\mathbb {T}\mathbb {U}, \mathbb {V}\mathbb {W}\mathbb {U}, \mathbb {V}\mathbb {C}\mathbb {U}, \mathbb {V}\mathbb {T}\mathbb {A}, \mathbb {V}\mathbb {W}\mathbb {A}, \mathbb {V}\mathbb {C}\mathbb {A}\}\). If a hypersequent is derivable in \(\mathsf {H}_{\mathcal {L}}^*\), then it is derivable in \(\mathsf {H}_{\mathcal {L}}\).

### Proof

First we eliminate all applications of \(\mathsf {cut}\) by induction on the tuples \(\langle \rho (\mathcal {D}), \sharp ( \mathcal {D} ) \rangle \) under the lexicographic ordering, where \(\sharp ( \mathcal {D} )\) is the number of applications of \(\mathsf {cut}\) in \(\mathcal {D}\) with cut formula of complexity \(\rho (\mathcal {D})\). Then applications of \(W_{m,n}^\mathsf {abs},R_C^\mathsf {abs},R_W^\mathsf {abs}\) are replaced with the \(W_{m,n},R_C, R_W\) and \(\mathsf {abs}_L,\mathsf {abs}_R\), and \(\mathsf {mrg}\) is eliminated using Lemma 5. It is straightforward to check that applications of \(W_{m,n}^\mathsf {abs}, R_C^\mathsf {abs}, R_W^\mathsf {abs}\) are only introduced in systems including the absoluteness rules. \(\square \)

### Corollary 12

**(Cut-free completeness).** If \(A \in \mathcal {L}\), then \(\mathsf {H}_{\mathcal {L}}\vdash \;\Rightarrow A\).

## 5 Connections to Modal Logic

*modal fragment*of a conditional logic \(\mathcal {L}\), i.e., the fragment where comparative plausibility formulae are restricted to the shape \((\bot \preccurlyeq \lnot A)\), and we write \(A^\Box \) for the result of replacing every subformula \(\bot \preccurlyeq \lnot B\) of

*A*with \(\Box B\). The proofs use the fact that the hypersequent calculus \(\mathsf {H}_{\mathsf {S5}}\) with the propositional rules of Fig. 1, the structural rules and the rulesis cut-free complete for \(\mathsf {S5}\) [16], see also [11].

### Lemma 13

If \(A^\Box \in \mathsf {S5}\), then \(A \in \mathcal {L}^\Box \) for each of the logics \(\mathcal {L}\) considered here.

### Proof

*modal splitting rule*from [2]: It is straightforward to check that the resulting calculus is sound for \(\mathsf {S5}\).

### Lemma 14

If \(\mathcal {L} \ne \mathbb {V}\mathbb {C}\mathbb {A}\) and \(A \in \mathcal {L}^\Box \), then \(A^\Box \in \mathsf {S5}\).

### Proof

*m*applications of \(\Box _L\) and \(\mathsf {T}\) respectively. Applications of \(W_{m,n}\) and \(R_C\) are translated by \(\mathsf {T}\), and \(R_W\) is replaced with weakening, using that whenever \(\mathcal {G} \mid \varGamma \Rightarrow \varDelta , \bot \) is derivable in the system for \(\mathsf {S5}\), then so is \(\mathcal {G} \mid \varGamma \Rightarrow \varDelta \). Finally, \(\mathsf {abs}_L,\mathsf {abs}_R\) are replaced with the modalised splitting rule

*MS*. \(\square \)

### Theorem 15

[13, Sect. 6.3]**.** Let \(\mathcal {L} \ne \mathbb {V}\mathbb {C}\mathbb {A}\). Then \(A \in \mathcal {L}^\Box \) iff \(A^\Box \in \mathsf {S5}\).

The proof of the previous theorem is immediate from the preceeding lemmas. It is then also straightforward to derive the known collapses of the counterfactual implication Open image in new window in \(\mathbb {V}\mathbb {W}\mathbb {A}\) and \(\mathbb {V}\mathbb {C}\mathbb {A}\). Recall that Open image in new window.

### Proposition 16

- 1.
- 2.
\(\mathsf {H}_{\mathbb {V}\mathbb {C}\mathbb {A}} \vdash \;\Rightarrow A \leftrightarrow \Box A\)

- 3.
Open image in new window and \(\mathsf {H}_{\mathbb {V}\mathbb {C}\mathbb {A}} \vdash \;\Rightarrow (A \preccurlyeq B) \leftrightarrow (B \rightarrow A)\).

## 6 Standard Calculi

To convert the non-standard calculi \(\mathsf {H}_{\mathcal {L}}\) into standard calculi, we consider an extended notion of sequents, where the succedent contains additional structural connectives. These sequents extend those of [7, 15] with a connective \(\left\langle . \right\rangle \) interpreting possible formulae.

### Definition 17

A *conditional block* is a tuple \(\left[ \varSigma \vartriangleleft C \right] \) containing a multiset \(\varSigma \) of formulae and a single formula *C*. A *transfer block* is a multiset of formulae, written \(\left\langle \varTheta \right\rangle \). An *extended sequent* is a tuple \(\varGamma \Rightarrow \varDelta \) consisting of a multiset \(\varGamma \) of formulae and a multiset \(\varDelta \) containing formulae, conditional blocks, and transfer blocks. An *extended hypersequent* is a multiset containing extended sequents, written \(\varGamma _1 \Rightarrow \varDelta _1 \mid \ldots \mid \varGamma _n \Rightarrow \varDelta _n\).

*formula interpretation*of an extended sequent is (all blocks shown explicitly):

*formula interpretation*of an extended hypersequent is given by

### Theorem 18

**(Soundness).** If \(\mathsf {SH}_{\mathcal {L}}\vdash \mathcal {G}\), then \(\vdash _{\mathcal {L}} \iota _e(\mathcal {G})\), and if \(\mathsf {SH}_{\mathcal {L}} \vdash \;\Rightarrow A\), then \(A \in \mathcal {L}\).

### Proof

As for Lemma 3, by showing that the rules preserve validity under \(\iota _e\) and using validity of \(\Box A \rightarrow A\). For the rules \(\preccurlyeq _L,\preccurlyeq _R, \mathsf {com}, \mathsf {jump}, \mathsf {W}, \mathsf {C}\) this is similar as in [7]. For rule \(\mathsf {T}\), if the interpretation of the conclusion is falsified in \(\mathfrak {M},w\), then there is a world Open image in new window with \(\mathfrak {M}, v \Vdash \bigwedge \varGamma \wedge (A \preccurlyeq B) \wedge \lnot \bigvee \varDelta \wedge \Box \lnot \bigvee \varTheta \). If \([\![{B}]\!]= \emptyset \), then in particular \(\mathfrak {M}, v \Vdash \Box \lnot (\bigvee \varTheta \vee B)\), and the formula interpretation of the second premiss is falsified in \(\mathfrak {M},w\). Otherwise, from \(\mathfrak {M},v \Vdash A \preccurlyeq B\) we obtain a world Open image in new window with \(\mathfrak {M},x \Vdash A\), and from \(\mathfrak {M},v \Vdash \Box \lnot \bigvee \varTheta \) we also get that \(\mathfrak {M},x \Vdash \lnot \bigvee \varTheta \). Hence the formula interpretation of the first premiss is falsified at \(\mathfrak {M},w\). The remaining cases are similar. \(\square \)

### Theorem 19

**(Completeness).** If \(A \in \mathcal {L}\) then \(\mathsf {SH}_{\mathcal {L}} \vdash \;\Rightarrow A\).

### Proof

By simulating derivations in \(\mathsf {H}_{\mathcal {L}}\). Most of the rules are simulated as in [7], except for the rules \(\mathsf {trf}_m, \mathsf {T}_m\). For \(\mathsf {T}_m\) the derivation is given in Fig. 4. The derivation of \(\mathsf {trf}_m\) only replaces \(\mathsf {jump}_T\) with \(\mathsf {jump}_U\). \(\square \)

## 7 Semantic Completeness via Invertible Calculi

### Lemma 20

The rules \(\mathsf {IW},\mathsf {EW},\mathsf {CW},\mathsf {CIW},\mathsf {TW}\) are admissible in \(\mathsf {SH}_{\mathcal {L}}\).

### Lemma 21

The rules \(\mathsf {ICL},\mathsf {ICR},\mathsf {Con}_B,\mathsf {Con}_S,\mathsf {mrg}\) are admissible in \(\mathsf {SH}_{\mathcal {L}}^i\).

From Lemmas 20 and 21 it immediately follows that:

### Proposition 22

The invertible and non-invertible calculi are equivalent.

### Definition 23

*-saturated*if it satisfies all of the following conditions:

- 1.
(\(\preccurlyeq _R\)) if \(\varGamma \Rightarrow \varDelta , A \preccurlyeq B \in \mathcal {G}\), then \(\left[ \varSigma , A \vartriangleleft B \right] \in \varDelta \) for some \(\varSigma \);

- 2.
(\(\preccurlyeq _L\)) if \(\varGamma , C \preccurlyeq D \Rightarrow \varDelta , \left[ \varSigma \vartriangleleft A \right] \in \mathcal {G}\), then \(D \in \varSigma \) or \(\left[ \varSigma \vartriangleleft C \right] \in \varDelta \);

- 3.
(\(\mathsf {com}\)) if \(\varGamma \Rightarrow \varDelta , \left[ \varSigma \vartriangleleft A \right] , \left[ \varPi \vartriangleleft B \right] \in \mathcal {G}\), then \(\varSigma \subseteq \varPi \) or \(\varPi \subseteq \varSigma \);

- 4.
(\(\mathsf {jump}\)) if \(\varGamma \Rightarrow \varDelta , \left[ \varSigma \vartriangleleft A \right] \in \mathcal {G}\), then \(A, \varTheta \Rightarrow \varSigma , \varPi \in \mathcal {G}\) for some \(\varTheta ,\varPi \);

- 5.
(\(\mathsf {T}\)) if \(\varGamma , C \preccurlyeq D \Rightarrow \varDelta , \left\langle \varTheta \right\rangle \in \mathcal {G}\), then \(D \in \varTheta \) or \(C, \varSigma \Rightarrow \varTheta , \varPi \in \mathcal {G}\) for some \(\varSigma , \varPi \);

- 6.
(\(\mathsf {in}_{\mathsf {trf}}\)) if \(\varGamma \Rightarrow \varDelta \in \mathcal {G}\), then \(\left\langle \varTheta \right\rangle \in \varDelta \) for some \(\varTheta \);

- 7.
(\(\mathsf {jump}_U, \mathsf {jump}_T\)) if \(\varGamma \Rightarrow \varDelta , \left\langle \varTheta \right\rangle \in \mathcal {G}\) and \(\varSigma \Rightarrow \varPi \in \mathcal {G}\), then \(\varTheta \subseteq \varPi \);

- 8.
(\(\rightarrow _L\)) if \(\varGamma , A \rightarrow B \Rightarrow \varDelta \in \mathcal {G}\), then \(B \in \varGamma \) or \(A \in \varDelta \);

- 9.
(\(\rightarrow _R\)) if \(\varGamma \Rightarrow A \rightarrow B, \varDelta \in \mathcal {G}\), then \(A \in \varGamma \) and \(B \in \varDelta \);

*-saturated*(resp. \(\mathbb {V}\mathbb {C}\mathbb {U}\)

*-saturated*) if it also satisfies (

*W*) (resp. (

*C*)) below:

- 1.
(\(\mathsf {W}\)) if \(\varGamma \Rightarrow \varDelta , \left[ \varSigma \vartriangleleft A \right] \in \mathcal {G}\), then \(\varSigma \subseteq \varDelta \);

- 2.
(\(\mathsf {C}\)) if \(\varGamma , C \preccurlyeq D \Rightarrow \varDelta \in \mathcal {G}\), then \(C\in \varGamma \) or \(D \in \varDelta \);

*unprovable*if it is not an instance of (init) or (\(\bot _L\)). We construct a countermodel from an unprovable \(\mathbb {V}\mathbb {T}\mathbb {U}\)-saturated extended hypersequent \(\mathcal {G} = \varGamma _1 \Rightarrow \varDelta _1 \mid \ldots \mid \varGamma _n \Rightarrow \varDelta _n\) as follows:

\(W := \{ 1, \ldots , n \}\)

\([\![{p}]\!]:= \{ i \le n : p \in \varGamma _i \}\)

### Lemma 24

For a \(\mathbb {V}\mathbb {T}\mathbb {U}\)-saturated hypersequent \(\mathcal {G}\) the structure \(\mathfrak {M}_{\mathcal {G}}\) is a \(\mathbb {V}\mathbb {T}\mathbb {U}\)-model.

### Proof

Nesting of spheres is obvious from the fact that \( \{ m_k\} \subseteq \{ m_k, m_{k-1}\} \subseteq \ldots \subseteq \{ m_k, \ldots ,m_1\} \subseteq W\); reflexivity and uniformity follow from the fact that Open image in new window. \(\square \)

### Lemma 25

*i*associated to component \(\varGamma _i \Rightarrow \varDelta _i\). Then:

- 1.
given a formula

*A*, if \(A\in \varGamma _i\) then \(\mathfrak {M}_{\mathcal {G}}, i \Vdash A\) - 2.
given a formula

*A*, if \(A\in \varDelta _i\) then \(\mathfrak {M}_{\mathcal {G}}, i \not \Vdash A\) - 3.
given a block \(\left[ \varSigma \vartriangleleft C \right] \), if \(\left[ \varSigma \vartriangleleft C \right] \in \varDelta _i\), then \(\mathfrak {M}_{\mathcal {G}}, i \not \Vdash \bigvee _{B \in \varSigma } (B \preccurlyeq C)\)

- 4.
given a formula

*B*, if \(\left\langle \varTheta ,B \right\rangle \in \varDelta _i\) for some \(\varTheta \), then \(\mathfrak {M}_{\mathcal {G}}, i \not \Vdash \Diamond B\)

### Proof

We prove statements 1 and 2 by mutual induction on the complexity of *A*. The base case and the propositional case are straightforward, hence we consider \(A = E \preccurlyeq F\). Let \(i\in W\) be associated to \(\varGamma _i \Rightarrow \varDelta _i\) with \(\varDelta _i = \varDelta '_i, \left[ \varSigma _1 \vartriangleleft D_1 \right] , \ldots , \left[ \varSigma _k \vartriangleleft D_k \right] ,\left\langle \varTheta \right\rangle \), where \(\varDelta '_i\) contains no conditional block and \(\varSigma _1\subseteq \varSigma _2 \subseteq \ldots \subseteq \varSigma _k\).

Suppose \(E \preccurlyeq F\in \varGamma _i\). For Open image in new window, we have to show that \(\alpha \Vdash ^{\forall } \lnot F\) or \(\alpha \Vdash ^{\exists } E\).

In case \(\alpha \ne W\) we have \(\alpha = \{ m_k,\ldots , m_{j}\}\) for some \(j\le k\) and each \(m_\ell \in \alpha \) comes from a block \(\left[ \varSigma _\ell \vartriangleleft D_\ell \right] \) and is associated to a component \(D_\ell ,\Lambda _\ell \Rightarrow \varPi _\ell , \varSigma _\ell \) of \(\mathcal {G}\). By saturation condition (\(\preccurlyeq _L\)), either \(F\in \varSigma _j\) or \(E = D_j\). In the former case with \(\varSigma _j \subseteq \varSigma _{j+1} \subseteq \ldots \varSigma _k\) and the induction hypothesis we have \(\mathfrak {M}_{\mathcal {G}}, m_\ell \not \Vdash F\), for \(\ell = j, \ldots , k\), showing that \(\alpha \Vdash ^{\forall } \lnot F\). If \(E = D_j\), by induction hypothesis on the component \(E,\Lambda _j \Rightarrow \varPi _j, \varSigma _j\), we get \(\mathfrak {M}_{\mathcal {G}}, m_j\Vdash E\), showing \(\alpha \Vdash ^{\exists } E\).

In case \(\alpha = W\), by saturation condition (T) either \(F\in \left\langle \varTheta \right\rangle \), or \(E, \Lambda \Rightarrow \varPi , \varTheta \in \mathcal {G}\) for some \(\Lambda ,\varPi \). In the latter case for the world

*j*associated to the component \(E, \Lambda \Rightarrow \varPi , \varTheta \) by induction hypothesis on*E*we get \(\mathfrak {M}_{\mathcal {G}}, j \Vdash E\), whence \(W\Vdash ^{\exists } E\). In the former case we have \(F\in \left\langle \varTheta \right\rangle \). Any \(k\in W\) (including k=i) is associated to a component \(\varGamma _k \Rightarrow \varDelta _k\), but by saturation condition (jump\(_T\), jump\(_U\)) we have \(\varTheta \subseteq \varDelta _k\), whence \(F\in \varDelta _k\); by induction hypothesis on*F*we have \(\mathfrak {M}_{\mathcal {G}}, k \not \Vdash F\), showing \(W\Vdash ^{\forall } \lnot F\).Suppose \(E \preccurlyeq F\in \varDelta _i\). Recall that Open image in new window with each \(m_\ell \) associated to a sequent \(D_\ell ,\Lambda _\ell \Rightarrow \varPi _\ell , \varSigma _\ell \in \mathcal {G}\) coming from a block \(\left[ \varSigma _\ell \vartriangleleft D_\ell \right] \in \varDelta _i\), for \(\ell = j, \ldots , k\). By saturation, there is \(j\le k\) with \(D_j = F\) and \(E\in \varSigma _j\). Consider \(m_j\) associated to the component \(F, \Lambda _j \Rightarrow \varSigma _j, \varPi \). By induction hypothesis we get \(\mathfrak {M}_{\mathcal {G}}, m_j \Vdash F\). Since \(\varSigma _j\subseteq \varSigma _{j+1} \subseteq \ldots \subseteq \varSigma _k\), we also get \(\mathfrak {M}_{\mathcal {G}}, m_\ell \not \Vdash E\), for \(\ell = j,\ldots ,k\). Thus for Open image in new window we get \(\alpha \not \Vdash ^{\forall } \lnot F\) and \(\alpha \not \Vdash ^{\exists } E\), showing \(\mathfrak {M}_{\mathcal {G}}, i \not \Vdash E\preccurlyeq F\).

The proof of 3 uses 2, recalling that a block is a disjunction of \(\preccurlyeq \)-formulas. The proof of 4 uses 2 with an argument as in the proof of 1 for the case of \(\alpha = W\) with \(B\in \left\langle \varTheta \right\rangle \). \(\square \)

The countermodel construction described above can be extended to \( \mathbb {V}\mathbb {W}\mathbb {U}\) and \( \mathbb {V}\mathbb {C}\mathbb {U}\) by modifying the definition of the model as follows. For \( \mathbb {V}\mathbb {W}\mathbb {U}\), let Open image in new window. For \( \mathbb {V}\mathbb {C}\mathbb {U}\), we add \(\{i\}\) to Open image in new window for any *i*. The proof of Lemma 25 can be easily extended to both cases (statements 1 and 2), using the specific saturation conditions for these systems. We leave the details to the reader; the case of Absoluteness will be handled in future work. From Lemma 25 we obtain:

### Lemma 26

for any \(i\in W\) associated to sequent \(\varGamma _i \Rightarrow \varDelta _i\) we have \(\mathfrak {M}_{\mathcal {G}}, i \not \Vdash \iota _e(\varGamma _i \Rightarrow \varDelta _i)\)

for any \(i\in W\) we have \(\mathfrak {M}_{\mathcal {G}}, i \not \Vdash \iota _e(\mathcal {G})\)

To use these results in a decision procedure, we consider *local loop checking*: rules are not applied if there is a premiss from which the conclusion is derivable using structural rules. Since these are all admissible in \(\mathsf {SH}_{\mathcal {L}}^i\), this does not jeopardise completeness.

### Proposition 27

Backwards proof search with local loop checking terminates and every leaf of the resulting derivation is an axiom or a saturated sequent.

### Proof

By Lemmas 20 and 21, we may assume that the proof search only considers *duplication-free* sequents, i.e., sequents containing duplicates neither of formulae nor of blocks. By the subformula property, the number of duplication-free sequents possibly relevant to a derivation of a sequent is bounded in the number of subformulae of that sequent, and hence backwards proof search for \(\mathcal {G}\) terminates. Furthermore, every leaf is either an axiom or a saturated sequent, since otherwise another rule could be applied. \(\square \)

### Theorem 28

**(Completeness).** If \(\iota _{e}(\mathcal {G}) \in \mathcal {L}\), then \(\mathsf {SH}_{\mathcal {L}} \vdash \mathcal {G}\) for \(\mathcal {L} \in \{ \mathbb {V}\mathbb {T}\mathbb {U}, \mathbb {V}\mathbb {W}\mathbb {U}, \mathbb {V}\mathbb {C}\mathbb {U}\}\).

### Proof

By Proposition 27 backwards proof search with root \(\mathcal {G}\) terminates and every leaf of it is an axiom or a saturated sequent. By invertibility of the rules each sequent \(\mathcal {G'}\) occurring as a leaf is valid. But then \(\mathcal {G'}\) must an axiom, since otherwise, by Lemma 26 we can bulid a countermodel \(\mathfrak {M}_{\mathcal {G'}}\) falsifying \(\iota _e(\mathcal {G'})\) and hence by monotonicity also \(\iota _e(\mathcal {G})\). \(\square \)

We note that Proposition 27 gives rise to a (non-optimal) co-NEXPTIME-decision procedure for validity: Since applying backwards proof search with local loop checking to an input sequent \(\;\Rightarrow G\) terminates and every leaf of the resulting derivation is an instance of \(\mathsf {init}\) or \(\bot _L\) or a saturated sequent, in order to check whether \(\;\Rightarrow G\) is derivable is suffices to non-determinstically choose a duplication-free \(\mathcal {L}\)-saturated extended hypersequent containing only subformulas of *G* and containing a component \(\varGamma \Rightarrow \varDelta , G\). If this is not possible, then backwards proof search will produce a proof of \(\;\Rightarrow G\). But if it is possible, then by Lemma 26 this hypersequent gives rise to a countermodel for *G*. Since the size of duplication-free extended hypersequents consisting of subformulae of *G* is bounded exponentially in the number of subformulae of *G*, this gives the co-NEXPTIME complexity bound. Of course it is known that the logics of this section are EXPTIME-complete [5].

## 8 Conclusion

In this work we have introduced to our knowledge the first internal hypersequent calculi for Lewis’ conditional logics with uniformity and reflexivity, both in non-standard and in standard form. While the former lend themselves to syntactic cut elimination, the latter are amenable to a semantic completeness proof via countermodel construction from a failed proof search and give rise to decision procedures for the considered logics.

While the treatment of these logics is an important step towards a comprehensive proof-theoretic treatment of the whole family of Lewis’ logics, many interesting questions are still open. In particular, we plan to extend the semantic completeness proof also to the logics with absoluteness. Further, by moving to the framework of *grafted hypersequents* [10] we expect to be able to extend our results to the logics \(\mathbb {V}\mathbb {U}\) and \(\mathbb {V}\mathbb {N}\mathbb {U}\). Concerning Lewis’ conditional logics, this would leave only the logics satisfying *Stalnaker’s assumption* [13] lacking a satisfactory internal proof system. Their proof-theoretic investigation will be subject of future research. Finally, we aim at providing complexity-optimal proof methods for the logics under consideration. In particular, for logics with absoluteness, one could make the blocks “global” to the whole hypersequent. We conjecture that such calculi could yield complexity-optimal decision procedures.

## Footnotes

- 1.
Using this notation we thus have: \(x \Vdash A\preccurlyeq B \) iff for all Open image in new window. \( \alpha \Vdash ^{\forall } \lnot B \) or \( \alpha \Vdash ^{\exists } A \).

- 2.
Observe that \( \mathbb {V}\mathbb {T}\mathbb {A}\)+

*weak centering*collapses to S5, since in any model over a set of worlds*W*it must be for all \(w\in W\), Open image in new window. Furthermore, \( \mathbb {V}\mathbb {T}\mathbb {A}\) +*centering*collapses to Classical Logic, as in any model the set of worlds must be a singleton \(\{w\}\) and Open image in new window, so that \(A\preccurlyeq B\) is equivalent to the material implication \(B \rightarrow A\). See also Proposition 16 below..

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