Studia Logica (2011) 97: 351–383DOI: 10.1007/s11225-011-9315-5 © Springer 2011

Takahiro Seki The γ-admissibility of RelevantModal Logics II — The Methodusing Metavaluations

Abstract. The γ-admissibility is one of the most important problems in the realm

of relevant logics. To prove the γ-admissibility, either the method of normal models or

the method using metavaluations may be employed. The γ-admissibility of a wide class

of relevant modal logics has been discussed in Part I based on a former method, but the

γ-admissibility based on metavaluations has not hitherto been fully considered. Sahlqvist

axioms are well known as a means of expressing generalized forms of formulas with modal

operators. This paper shows that γ is admissible for relevant modal logics with restricted

Sahlqvist axioms in terms of the method using metavaluations.

Keywords: γ-admissibility, relevant modal logic, metavaluation, Sahlqvist axiom.

1. Introduction

In Part I ([9]), the γ-admissibility of relevant modal logics has been discussedusing the method of normal models. Routley et al. [6] were able to establishwhich relevant logics admit γ using the method of normal models, but adetermination of such logics by a method using metavaluations has not yetbeen achieved. This paper will discuss the same problem, but in terms of amethod using metavaluations.

The concept of metavaluations was introduced in [4] for positive relevantlogics, and was extended in [11] and [12] for contractionless relevant logicsand relevant logics without WI, A ∧ (A → B) → B, respectively. However,using these approaches, the γ-admissibility of stronger relevant logics suchas R cannot be shown. By modifying the definition of metavaluations, itbecomes possible to prove that the relevant logic R admits γ in [1]. In orderto consider which relevant logics this method can be applied to, upon closelyexamining the proof in [1], we notice that both

A∨ ∼ A andC ∨ A

C∨ ∼ (A →∼ A)

(and further WI, in some cases) play an essential role. The logic obtained

Presented by Robert Goldblatt; Received August 27, 2008; Revised January 27, 2010

352 T. Seki

from B by adding the above axiom and disjunctive rule is called Gg in thispaper.

Here we consider the corresponding question for relevant modal logics.As a means of extending the approach in [1], the following methods can beconsidered. One is a simple extension, that is, metavaluations for modalformulas �A and ♦A are defined by considering whether they belong to reg-ular prime theories, as adopted for implication and negation. This approachcan be applied for regular relevant modal logics, in the sense of [7], with thefollowing disjunctive rules:

C ∨ A

C ∨ ♦· A andC ∨ A

C ∨ ♦A,

where ♦· A is the abbreviation of ∼ � ∼ A. The other is a relational ex-tension, that is, the construction of a model similar to a Kripke model forclassical modal logics, which was proposed in [2] and [3]. In this model,worlds are restricted to regular prime theories, so we adopt regular modallogics with the rule of necessitation as relevant modal logics admitting γ.Furthermore, we notice that the following axioms play an important role:

�(A ∨ B) → ♦· A ∨ �B and � A ∧ ♦B → ♦(A ∧ B),

where �A is the abbreviation of ∼ ♦ ∼ A.In addition, we consider when γ is admissible in logics with various kinds

of postulates containing modal operators. A Sahlqvist axiom is a generalizedform of a formula with modal operators. Kripke-completeness of every clas-sical modal logic with Sahlqvist axioms is a fundamental general result oncompleteness of classical modal logics. As described in [7], relevant modallogics with Sahlqvist axioms are also complete with respect to Routley-Meyersemantics. By utilizing the proof in [7], the γ-admissibility of relevant modallogics with some Sahlqvist axioms can be proved. Fortunately, our resultscover all works on the γ-admissibility of relevant modal logics by employingthe method using metavaluations following the earlier studies [2] and [3].

2. Preliminaries

In this section, we present the basic notions of relevant modal logics. We use&,⇒,∀ and ∃ to denote respectively conjunction, implication, universal andexistential quantifiers in the metalanguage. We omit parentheses by assum-ing that ∀,∃ bind more strongly than &, and that & binds more stronglythan ⇒.

The γ-admissibility of Relevant Modal Logics II 353

The language of relevant modal logics consists of (i) propositional vari-ables; (ii) logical connectives →,∧,∨ and ∼; and (iii) modal operators �and ♦. Formulas are defined in the usual way, and are denoted by capitalletters A,B,C. When necessary, we use ′ or subscripts. A formula A whosevariables are listed among p1, · · · , pn is denoted by A[p1, · · · , pn]. Prop andWff will denote the set of all propositional variables and formulas, respec-tively. Further, we introduce the following abbreviations:

�Adef=∼ ♦ ∼ A, ♦· A def=∼ � ∼ A,

For a non-negative integer n, �n and �n are defined inductively as follows.

(i) �0A is A. (ii) For n ≥ 1, �nA is either ��n−1A or ��n−1A.(iii) �0A is A. (iv) For n ≥ 1, �nA is either ♦�n−1A or ♦· �n−1A.

Especially, �nA, �nA, ♦nA and ♦nA denote

� · · ·�︸ ︷︷ ︸n times

A, � · · ·�︸ ︷︷ ︸n times

A, ♦ · · ·♦︸ ︷︷ ︸n times

A, and ♦· · · ·♦·︸ ︷︷ ︸n times

A,

respectively. Further, �nA denotes either �nA or �nA, and ♦× nA denoteseither ♦nA or ♦· nA. When n = 1, we omit the superscript n.

Though the definitions of the relevant logics Gg, L1, L2 and L3 aregiven in Part I, here we present them again. For more information, seePart I ([9]).

The relevant logic Gg is defined as follows:

(a) Axioms

(A1) A → A

(A2) A ∧ B → A

(A3) A ∧ B → B

(A4) (A → B) ∧ (A → C) → (A → B ∧ C)(A5) A → A ∨ B

(A6) B → A ∨ B

(A7) (A → C) ∧ (B → C) → (A ∨ B → C)(A8) A ∧ (B ∨ C) → (A ∧ B) ∨ C

(A9) ∼∼ A → A

(A10) A∨ ∼ A

354 T. Seki

(b) Rules of inference

(R1)A → B A

B(R2)

A B

A ∧ B

(R3)A → B

(B → C) → (A → C)(R4)

A → B

(C → A) → (C → B)

(R5)A →∼ B

B →∼ A(R6)

C ∨ A

C∨ ∼ (A →∼ A)

L1 denotes any logic obtained from Gg by adding any set of the axiomsand rules of inference listed below.

(B1) A ∧ (A → B) → B

(B2) (A → B) ∧ (B → C) → (A → C)

(B3) (A → (A → B)) → (A → B)

(B4) A → ((A → B) → B)

(B5) A → (B → B)

(B6) A → (B → A)

(B7) (A → B) → ((A → C) → (A → B ∧ C))

(B8) A → (A → A)

(B9) A ∨ B → ((A → B) → B)

(B10) (A ∧ B → C) → (A∧ ∼ C →∼ B)

(B11) A →∼ (A →∼ A)

(B12) (A →∼ A) →∼ A

(B13) (A →∼ B) → (B →∼ A)

(B14) A → B∨ ∼ B

(B15) A → (∼ A → B)

(Q1)C ∨ (A → B) C ∨ A

C ∨ B

(Q2)C ∨ (A →∼ B)C ∨ (B →∼ A)

(Q3)C ∨ (∼ A → A)

C ∨ A

(Q4)A

(A → B) → B

The γ-admissibility of Relevant Modal Logics II 355

L2 denotes any logic obtained from L1 with (Q1) by adding any set ofthe axioms and rules of inference listed below. (The set may be empty.)Moreover, any logic L2 with (Q2) is called L3.

(B16) (A → B) → ((B → C) → (A → C))

(B17) (A → B) → ((C → A) → (C → B))

(B18) (A → (B → C)) → (B → (A → C))

(B19) (A → (B → C)) → ((A → B) → (A → C))

(B20) (A → B) → ((A → (B → C)) → (A → C))

(B21) (A ∧ B → C) → (A → (B → C))

(Q5)E ∨ (A → B) E ∨ (C → D)

E ∨ ((B → C) → (A → D))

Next, we present the relevant modal logics discussed in this paper. Therelevant modal logic L.Cg

�♦is defined as follows, where L is L1, L2 or L3.

(a) Axioms consist of all axioms of L together with the following.

(A11) �A ∧ �B → �(A ∧ B)(A12) ♦(A ∨ B) → ♦A ∨ ♦B

(b) Rules of inference consist of all the rules of inference of L togetherwith the following.

(R7)A → B

�A → �B(R8)

A → B

♦A → ♦B(R9)

C ∨ A

C ∨ ♦· A (R10)C ∨ A

C ∨ ♦A

In particular, if L is Gg, then L.Cg�♦ is called G.Cg

�♦.Let LR denote logics obtained from L.Cg

�♦ by adding any set of thefollowing axioms and rules of inference, provided that if (B22) is containedthen (Q1) must be contained:

(B22) �(A → B) → (�A → �B)

(Q6)C ∨ (A → B)

C ∨ (�A → �B)

(Q7)C ∨ (A → B)

C ∨ (♦A → ♦B)

(Q8)B ∨ �A

B ∨ A

(Q9)B ∨ �A

B ∨ A

356 T. Seki

(Q10)B ∨ A

B ∨ �A

(Q11)A

�A

(Q12)A

�A

The relevant modal logic L.Ng�♦ is defined as follows, where L is L1, L2

or L3.

(a) Axioms consist of all axioms of L together with (A11), (A12) and thefollowing axioms:

(A14) �(A ∨ B) → ♦· A ∨ �B

(A15) �A ∧ ♦B → ♦(A ∧ B)(b) Rules of inference consist of all rules of inference of L together with

(R7), (R8), (Q11) and (Q12).

Let LM denote logics obtained from L.Ng�♦ by adding any set of the

following axioms and rules of inference (R9), (R10), (Q8) and (Q9) (the setmay be empty):

(B23) �(A → B) → (♦A → ♦B)(B24) �(A → B) → (♦· A → ♦· B)

In particular, if L is Gg, then L.Ng�♦ is called G.Ng

�♦.We often use the notation L A to denote that A is a theorem of L.As additional modal axioms, we introduce (restricted) Sahlqvist formu-

las. Here we give a general definition of Sahlqvist formulas, following [7].A formula A is positive if it can be constructed using no connectives otherthan ∧,∨,�,♦, � and ♦· . A formula of the form �m1p1 ∧ · · · ∧ �mkpk withnot necessarily distinct propositional variables p1, · · · , pk is called a stronglypositive formula. A given formula A is negative (in a logic L) if it is equiv-alent in L to ∼ B for a positive formula B. A modal formula A is untied(in L) if it can be constructed from strongly positive formulas and negativeformulas (in L) using only ∧,♦ and ♦· . A formula, A, is called Sahlqvist if itis a conjunction of the form �k(B → C), where k ≥ 0, B is untied in L andC is positive. Moreover, Sahlqvist formulas which are adopted as axioms ofthe logic under consideration are called Sahlqvist axioms.

In the following, we present key notions used in our proof of the γ-admissibility. Let L be LR or LM defined above, and a and b are non-emptysets of formulas.

The γ-admissibility of Relevant Modal Logics II 357

• L a → b iff there exist A1, · · · , Am ∈ a (m ≥ 1) and B1, · · · , Bn ∈b (n ≥ 1) such that A1 ∧ · · · ∧ Am → B1 ∨ · · · ∨ Bn is a theorem of L.L � a → b denotes that L a → b does not hold.

• a is an L-theory iff (a) if A ∈ a and B ∈ a then A ∧ B ∈ a, and (b) ifA → B is a theorem of L and A ∈ a then B ∈ a.

• a is regular iff (a) a contains all theorems of L, and (b) a conforms tothe following closure conditions:

– C ∨ A ∈ a ⇒ C∨ ∼ (A →∼ A) ∈ a

– C ∨ A ∈ a ⇒ C ∨ ♦A ∈ a

– C ∨ A ∈ a ⇒ C ∨ ♦· A ∈ a

Moreover, if L contains the left-hand side disjunctive rules, then a con-forms to the right-hand side closure conditions:

(Q1) C ∨ A ∈ a & C ∨ (A → B) ∈ a ⇒ C ∨ B ∈ a(Q2) C ∨ (A →∼ B) ∈ a ⇒ C ∨ (B →∼ A) ∈ a(Q3) C ∨ (∼ A → A) ∈ a ⇒ C ∨ A ∈ a(Q5) E ∨ (A → B) ∈ a & E ∨ (C → D) ∈ a

⇒ E ∨ ((B → C) → (A → D)) ∈ a(Q6) C ∨ (A → B) ∈ a ⇒ C ∨ (�A → �B) ∈ a(Q7) C ∨ (A → B) ∈ a ⇒ C ∨ (♦A → ♦B) ∈ a(Q8) C ∨ �A ∈ a ⇒ C ∨ A ∈ a(Q9) C ∨ �A ∈ a ⇒ C ∨ A ∈ a

(Q10) C ∨ A ∈ a ⇒ C ∨ �A ∈ a

• a is prime iff A ∨ B ∈ a implies either A ∈ a or B ∈ a (or both).

• a is consistent iff ∼ A /∈ a for any theorem A of L.

• a is normal iff a is regular, prime and consistent.

Note that for every normal a, there exists no A ∈ Wff such that both A ∈ aand ∼ A ∈ a. Thus, for a normal a, if A ∈ a then ∼ A /∈ a.

The following proposition, which has been proved in Section 4.8 of [6](for non-modal cases) and [8] (for modal cases), will be used in later sections.

Proposition 2.1. (1) If L � a → b, then there exists a prime L-theorya′ such that a ⊆ a′ and L � a′ → b.

(2) If A is not a theorem of L, then there exists a regular prime L-theorys such that A /∈ s.

358 T. Seki

Henceforth, we denote the set of all prime L-theories and the set of allregular prime L-theories by W and O, respectively. (Note that O ⊆ W .)Furthermore, a and b denote elements of W , while s, t and u denote elementsof O. When necessary, we use ′ or subscripts.

For a ∈ W , define a∗ = {A | ∼ A /∈ a}. Then it is easy to show thefollowing: for every a, b ∈ W , (s1) a∗ ∈ W , (s2) a∗∗ = a and (s3) a ⊆ b iffb∗ ⊆ a∗.

3. Proof of the γ-admissibility of relevant logics

Though we will define some metavaluations for modal operators in later sec-tions, the definitions of metavaluations for (non-modal) logical connectivesare identical to each other. This section shows the γ-admissibility of relevantlogics introduced in Section 2. Throughout this section, L denotes L1, L2and L3.

First, we introduce the notion of metavaluations. An L-metavaluation v(a function from Wff × O to {t, f}) is defined as follows.

1. for every p ∈ Prop, v(p, s) = t iff p ∈ s

2. v(A ∧ B, s) = t iff v(A, s) = t & v(B, s) = t

3. v(A ∨ B, s) = t iff v(A, s) = t or v(B, s) = t

4. v(A → B, s) = t iff A → B ∈ s & (v(A, s) = t ⇒ v(B, s) = t)

5. v(∼ A, s) = t iff ∼ A ∈ s & v(A, s) = f

Further, for any s ∈ O,

Tr(s) = {A | v(A, s) = t}.

Here we note that Tr(s) is prime, that is, A ∨ B ∈ Tr(s) implies eitherA ∈ Tr(s) or B ∈ Tr(s) (or both).

The following lemma is called the mediating lemma in [2].

Lemma 3.1. For any s ∈ O, s∗ ⊆ Tr(s) ⊆ s.

This lemma is proved in the same way as in [3]. The proof comprisesthree parts showing that (a) s∗ ⊆ s, (b) Tr(s) ⊆ s and (c) s∗ ⊆ Tr(s). Claim(a) is based on the fact that (A10), i.e., A∨ ∼ A, is an axiom, while both (b)and (c) are proved by induction on the length of formula A. Furthermore,we should take note of the case in which A is of the form B → C in theproof of (c).

The γ-admissibility of Relevant Modal Logics II 359

Suppose that B → C ∈ s∗. By (a), B → C ∈ s. To show thatB ∈ Tr(s) implies C ∈ Tr(s), assume otherwise. Then B ∈ s by (b)and C /∈ s∗ by the hypothesis of induction. So we have B∧ ∼ C ∈ sand hence ∼ (B → C) ∨ (B∧ ∼ C) ∈ s. Since s ∈ O satisfies the clo-sure condition (later, we simply say “by the closure condition”), we have∼ (B → C)∨ ∼ (B∧ ∼ C →∼ (B∧ ∼ C)) ∈ s. Since L ∼ (B →C)∨ ∼ (B∧ ∼ C →∼ (B∧ ∼ C)) →∼ (B → C)∨ ∼ (B → C), we have∼ (B → C)∨ ∼ (B → C) ∈ s, so ∼ (B → C) ∈ s. This contradicts ourassumption. Thus, B ∈ Tr(s) implies C ∈ Tr(s).

Then, we have the following corollary.

Corollary 3.2. If A ∈ Tr(s) for all s ∈ O, then A is a theorem of L.

Proof. Suppose that A is not a theorem of L. By (2) of Proposition 2.1,there exists s ∈ O such that A /∈ s. Then we have A /∈ Tr(s) by Lemma 3.1.

As a corollary, we have the following.

Corollary 3.3. For any s ∈ O:

(1) for all A ∈ Wff, A ∈ Tr(s) or ∼ A ∈ Tr(s).

(2) there exists no A ∈ Wff such that both A ∈ Tr(s) and ∼ A ∈ Tr(s).

(3) for all A ∈ Wff, ∼ A ∈ Tr(s) iff A /∈ Tr(s).

Proof. (3) is derived from (1) and (2), so we give proofs for (1) and (2).(1) Suppose that A /∈ Tr(s). By Lemma 3.1, we have A /∈ s∗, and hence

∼ A ∈ s. Thus, we have ∼ A ∈ Tr(s) as desired.(2) Suppose otherwise, that is, there exists A such that both A ∈ Tr(s)

and ∼ A ∈ Tr(s). From the latter, we have A /∈ Tr(s), which contradictsthe former. Therefore, there exists no A such that both A ∈ Tr(s) and∼ A ∈ Tr(s).

The following lemma implies that Tr(s) is regular.

Lemma 3.4. If A is a theorem of L, then A ∈ Tr(s) for all s ∈ O.

Proof. This lemma is proved by the induction on the length of proof ofthe theorem. Here we give a proof of typical cases. First, let L be Gg.

(R3) Suppose that A → B ∈ Tr(s) for all s ∈ O. By Corollary 3.2,L A → B. Then L (B → C) → (A → C), so (B → C) → (A → C) ∈ s.Next, to see that B → C ∈ Tr(s) implies A → C ∈ Tr(s), suppose that

360 T. Seki

B → C ∈ Tr(s). Then we have B → C ∈ s, so A → C ∈ s. Further,to see that A ∈ Tr(s) implies C ∈ Tr(s), suppose that A ∈ Tr(s). By theassumption A → B ∈ Tr(s), we have B ∈ Tr(s). Further, by the assumptionB → C ∈ Tr(s), we have C ∈ Tr(s), which is the desired result.

(R6) Suppose that C ∨ A ∈ Tr(s) for all s ∈ O. When C ∈ Tr(s), thedesired result is shown easily, so we consider the case in which A ∈ Tr(s).By Corollary 3.3, we have ∼ A /∈ Tr(s), and hence A →∼ A /∈ Tr(s). Againby Corollary 3.3, we have ∼ (A →∼ A) ∈ Tr(s), so C∨ ∼ (A →∼ A) ∈ Tr(s)as desired.

Next, let L be L2 and in particular contain (B16) as an axiom. It sufficesto show that both (a) (A → B) → ((B → C) → (A → C)) ∈ s and (b)A → B ∈ Tr(s) implies (B → C) → (A → C) ∈ Tr(s). Since s ∈ O, (a)is trivial. To show (b), suppose that A → B ∈ Tr(s) and show that (c)(B → C) → (A → C) ∈ s and (d) B → C ∈ Tr(s) implies A → C ∈ Tr(s).Since (B16) is a theorem of L and A → B ∈ s by Lemma 3.1, we have (c).To show (d), suppose that B → C ∈ Tr(s) and show that (e) A → C ∈ sand (f) A ∈ Tr(s) implies C ∈ Tr(s). By Lemma 3.1, we have B → C ∈ s.Then (A → C) ∨ ((B → C) → (A → C)) ∈ s and (A → C) ∨ (B → C) ∈ s,so we have (A → C)∨ (A → C) ∈ s by the closure condition. Thus, we have(e). To show (f), suppose that A ∈ Tr(s). Then we have B ∈ Tr(s) by theassumption A → B ∈ Tr(s), so C ∈ Tr(s) by the assumption B → C ∈ Tr(s).This completes a proof.

Here, note that if L is L2, then the closure condition corresponding to(Q1) is required in the proof.

By Corollary 3.3 and Lemma 3.4, Tr(s) is a normal L-theory.

Theorem 3.5. Ackermann’s rule γ is admissible for L.

Proof. Suppose that both ∼ A∨B and A are theorems of L and that B isnot a theorem of L. By Corollary 3.2, there exists s ∈ O such that B /∈ Tr(s).Then we have ∼ A ∨ B ∈ Tr(s) and A ∈ Tr(s) by Lemma 3.4. In the lightof Corollary 3.3, we obtain a contradiction.

4. Proof of the γ-admissibility I: simple extension

In this section, we discuss a proof of the γ-admissibility of LR and some ofits extensions by a simple definition of metavaluations for modal operators,which is defined with the help of regular prime theories such as those for →and ∼. An LR-metavaluation v (a function from Wff×O to {t, f}) is definedinductively as follows.

The γ-admissibility of Relevant Modal Logics II 361

1. For p(∈ Prop), A ∧ B,A ∨ B, A → B and ∼ A, definitions are similar toSection 3.

2. v(�A, s) = t iff �A ∈ s & v(A, s) = t.

3. v(♦A, s) = t iff ♦A ∈ s∗ or v(A, s) = t.

Furthermore, the definition of Tr(s) is similar in Section 3.An outline of a proof of the γ-admissibility of LR is the same as in Section

3. Here we focus on the modal extensions and metavaluations defined above.

Lemma 4.1. (1) For any s ∈ O, s∗ ⊆ Tr(s) ⊆ s.

(2) If A ∈ Tr(s) for all s ∈ O, then A is a theorem of LR.

(3) For any s ∈ O and all A ∈ Wff, ∼ A ∈ Tr(s) iff A /∈ Tr(s).

Proof. Since (2) and (3) are proved in same way as in Section 3, we givea proof of (1). The proof comprises three parts showing that (a) s∗ ⊆ s, (b)Tr(s) ⊆ s and (c) s∗ ⊆ Tr(s). Claim (a) is based on the fact that (A10) isan axiom, while both (b) and (c) are proved by induction on the length offormula A. Here it suffices to consider the cases in which A is of the forms�B and ♦B.

(b1) A is of the form �B. By definition, we have �B ∈ s as desired.(b2) A is of the form ♦B. Suppose that ♦B ∈ Tr(s). Then ♦B ∈ s∗ orB ∈ Tr(s). (i) When ♦B ∈ s∗, we have ♦B ∈ s by (a). (ii) When B ∈ Tr(s),we have B ∈ s by the hypothesis of induction. Then ♦B ∨B ∈ s, and hence♦B ∨ ♦B ∈ s by the closure condition. So, we have ♦B ∈ s. From (i) and(ii), we have the desired result.

(c1) A is of the form �B. Suppose that �B ∈ s∗. By (a), we have �B ∈ s.Further, since ♦· ∼ B /∈ s, we have ♦· ∼ B ∨ ♦· ∼ B /∈ s. By the closurecondition, we have ♦· ∼ B∨ ∼ B /∈ s. Then ∼ B /∈ s, so B ∈ s∗. By thehypothesis of induction, we have B ∈ Tr(s) . Thus, we have �B ∈ Tr(s) asdesired.(c2) A is of the form ♦B. By definition, we have ♦B ∈ Tr(s) as desired.

By the definition of �A and ♦· A, and (3) of Lemma 4.1, we can easilyshow the following.

Proposition 4.2. For any s ∈ O,

(1) v(�A, s) = t iff �A ∈ s & v(A, s) = t

(2) v(♦· A, s) = t iff ♦· A ∈ s∗ or v(A, s) = t

362 T. Seki

Lemma 4.3. If A is a theorem of LR, then A ∈ Tr(s) for all s ∈ O.

Proof. This lemma is proved by induction on the length of the proof of thetheorem. Here we give proofs of the following axiom and rules of inference.

(A12) It is obvious that ♦(A∨B) → ♦A∨♦B ∈ s. To show ♦(A∨B) ∈Tr(s) implies ♦A ∨ ♦B ∈ Tr(s), suppose ♦(A ∨ B) ∈ Tr(s). (i) When♦(A ∨ B) ∈ s∗, ♦A ∨ ♦B ∈ s∗. Since s∗ is prime by (s1), ♦A ∈ s∗ or♦B ∈ s∗. By the definition of LR-metavaluations, we have ♦A ∈ Tr(s) or♦B ∈ Tr(s), and hence ♦A∨♦B ∈ Tr(s). (ii) When A∨B ∈ Tr(s), A ∈ Tr(s)or B ∈ Tr(s). By the definition of LR-metavaluations, we have ♦A ∈ Tr(s)or ♦B ∈ Tr(s), and hence ♦A ∨ ♦B ∈ Tr(s). From (i) and (ii), we have thedesired result.

(R7) Let A → B ∈ Tr(s) for all s ∈ O. By (2) of Lemma 4.1, LR A →B, and hence LR �A → �B. Then it is obvious that �A → �B ∈ s.Next, to show �A ∈ Tr(s) implies �B ∈ Tr(s), suppose �A ∈ Tr(s). Since�A ∈ s and LR �A → �B, we have �B ∈ s. Further, since A ∈ Tr(s)and A → B ∈ Tr(s), we have B ∈ Tr(s) Thus, we have the desired result.

(R10) Let C ∨ A ∈ Tr(s) for all s ∈ O. (i) When C ∈ Tr(s), it isobvious that C ∨ ♦· A ∈ Tr(s). (ii) When A ∈ Tr(s), we have A ∈ s by (1)of Lemma 4.1. Then we have ♦· A ∨ A ∈ s, and hence ♦· A ∨ ♦· A ∈ s by theclosure condition. So, we have ♦· A ∈ s. Therefore, we have ♦· A ∈ Tr(s), soC ∨ ♦· A ∈ Tr(s). From (i) and (ii), we have the desired result.

Note that the following claims, for example, are impossible to prove ingeneral: �(A → B) → (♦· A → ♦· B) ∈ Tr(s) (unless (B13) is derivable inLR) and �(A ∨ B) → ♦· A ∨ �B ∈ Tr(s) for all s ∈ O. This fact indicatesthat the approach described in this section does not work well in LM .

By similar argument in Section 3, we have the following.

Theorem 4.4. Ackermann’s rule γ is admissible for LR.

In a similar manner, we can prove the γ-admissibility of the logic obtainedfrom LR by adding �A → �2A and �2A → �A as axioms. More generally,we will consider the γ-admissibility of LR with some Sahlqvist axioms. Thefollowing proposition is easily proved by induction on n.

Proposition 4.5. For any s ∈ O and any non-negative integer n,

(1) v(�nA, s) = t implies �nA ∈ s & v(A, s) = t

(2) �nA ∈ s∗ or v(A, s) = t implies v(�nA, s) = t

Note that the converse also holds for n = 0, 1 and does not hold for n ≥ 2in general. Later, we will discuss the case in which the converse always holds.

The γ-admissibility of Relevant Modal Logics II 363

Lemma 4.6. Let B[p1, · · · , pn] be a strongly positive formula and C be anyformula constructed from p1, · · · , pn by applying ∧, ∨, ♦ and ♦· . Then ifB ∈ Tr(s), then C ∈ Tr(s) for any s ∈ O.

Proof. By induction on the construction of C. Suppose that B ∈ Tr(s).Here we give proofs for the cases in which C is of the forms pi (∈ Prop)and ♦C1.

(i) C is of the form pi. Here we omit the subscript. Since p appears in thestrongly positive formula B, �mp ∈ Tr(s). By (1) of Proposition 4.5, wehave p ∈ Tr(s).

(ii) C is of the form ♦C1. By the hypothesis of induction, C1 ∈ Tr(s), fromwhich we have ♦C1 ∈ Tr(s) as desired.

We refer to a Sahlqvist formula that is equivalent to a conjunction offormulas of the form �lB → �mC as Type I:

• l is either 0 or 1, and m is either 0 or 1;

• B[p1, · · · , pn] is a strongly positive formula;

• C is constructed from p1, · · · , pn by applying ∧, ∨, ♦ and ♦· .Examples of Type I Sahlqvist formulas are �mp → �np, ��mp → �np,�mp → ��np and ��mp → ��np, where m,n are non-negative integers.

Thus, we have the following lemma, which implies that Tr(s) is regular.

Lemma 4.7. If A is a theorem of LR with Type I Sahlqvist axioms, thenA ∈ Tr(s) for all s ∈ O.

Proof. It suffices to show that A is a Type I Sahlqvist axiom of the form�lB → �mC. (Note that l and m are either 0 or 1.) Since �lB → �mC isa theorem, we have �lB → �mC ∈ s. To show that �lB ∈ Tr(s) implies�mC ∈ Tr(s), suppose that �lB ∈ Tr(s). Then �lB ∈ s∗ or B ∈ Tr(s).(i) When �lB ∈ s∗, we have �mC ∈ s∗ and hence �mC ∈ Tr(s) by (1) ofLemma 4.1. (ii) When B ∈ Tr(s), we see C ∈ Tr(s) by Lemma 4.6. Further,we have �lB ∈ Tr(s) and hence �lB ∈ s by (1) of Lemma 4.1. Then wehave �mC ∈ s. Thus, �mC ∈ Tr(s).

By similar argument in Section 3, we have the following.

Theorem 4.8. Ackermann’s rule γ is admissible for LR with Type I Sahlqvistaxioms.

364 T. Seki

Note that the γ-admissibility of the relevant modal logic NR has beenproved by the method of normal models in [5]. The logic NR is obtainedfrom R.Cg

�♦ by removing the modal operator ♦ and by adding (Q11) asthe rule of inference and (B22) and Type I Sahlqvist axioms �A → A and�A → �2A as axioms. Theorem 4.8 establishes that the γ-admissibility ofNR can also be proved by the method using metavaluations.

To consider more generalized formulas with modal operators, we mustassume the disjunctive rules

(Q8)B ∨ �A

B ∨ Aand (Q9)

B ∨ �A

B ∨ A.

In the rest of this section, let the logic under consideration contain (Q8)and (Q9). These disjunctive rules play an important role in the proof of thefollowing proposition.

Proposition 4.9. For any s ∈ O and any non-negative integer n,

(1) v(�nA, s) = t iff �nA ∈ s & v(A, s) = t

(2) v(�nA, s) = t iff �nA ∈ s∗ or v(A, s) = t

Proof. (1) The ‘only if’ part is just (1) of Proposition 4.5, so we give aproof for the ‘if’ part by induction on n. For n = 0, 1, it is trivial, so we givea proof of the case in which n ≥ 2 and �nA = ��n−1A. The case in whichn ≥ 2 and �nA = ��n−1A is proved similarly. Suppose that ��n−1A ∈ sand v(A, s) = t. Then we have �n−1A∨��n−1A ∈ s, so �n−1A∨�n−1A ∈ sby the closure condition. Since �n−1A ∈ s and v(A, s) = t, v(�n−1A, s) = tby the hypothesis of induction. Thus, we have v(��n−1A, s) = t as desired.

(2) The ‘if’ part is just (2) of Proposition 4.5, so we give a proof for the‘only if’ part by induction on n. For n = 0, 1, it is trivial, so we give a proofof the case in which n ≥ 2 and �nA = ♦�n−1A. The case in which n ≥ 2and �nA = ♦· �n−1A is proved similarly. Suppose that v(♦�n−1A, s) = t.Then ♦�n−1A ∈ s∗ or v(�n−1A, s) = t. (i) When ♦�n−1A ∈ s∗, this leadsto the desired result. (ii) When v(�n−1A, s) = t, �n−1A ∈ s∗ or v(A, s) = tby the hypothesis of induction. (a) When �n−1A ∈ s∗, ∼ �n−1A /∈ s. Then∼ �n−1A∨ ∼ �n−1A /∈ s, and hence ∼ �n−1A ∨ � ∼ �n−1A /∈ s by theclosure condition. Thus we have � ∼ �n−1A /∈ s, so ♦�n−1A ∈ s∗. Thisleads to the desired result. (b) When v(A, s) = t, this leads to the desiredresult. From (i) and (ii), ♦�n−1A ∈ s∗ or v(A, s) = t.

We refer to a Sahlqvist formula that is equivalent to a conjunction offormulas of the form �k(�lB → �mC) as Type II:

The γ-admissibility of Relevant Modal Logics II 365

• B[p1, · · · , pn] is a strongly positive formula;

• C is constructed from p1, · · · , pn by applying ∧, ∨, ♦ and ♦· .Examples of Type II Sahlqvist formulas are �k�lp → �m�np, p∧�p∧ · · · ∧�n−1p → �np, where k, l,m, n are non-negative integers.

By an argument similar to that described above, we have the following.

Lemma 4.10. Let L be a logic obtained from LR by adding Type II Sahlqvistformulas as axioms and (Q8) and (Q9) as rules of inference. If A is atheorem of L, then A ∈ Tr(s) for all s ∈ O.

Proof. It suffices to show that A is a Sahlqvist axiom of the form�k(�lB → �mC). In this proof, �lB → �mC is abbreviated by A′. SinceL �kA′, it is trivial that �kA′ ∈ s. The remainder is to show A′ ∈ Tr(s).Since L �kA′, L A′ ∨ �kA′. By applying (Q8) and (Q9), we haveL A′∨A′. Therefore, L A′, and hence A′ ∈ s. To show that �lB ∈ Tr(s)implies �mC ∈ Tr(s), suppose that �lB ∈ Tr(s). By (2) of Proposition 4.9,�lB ∈ s∗ or B ∈ Tr(s). (i) When �lB ∈ s∗, we have �mC ∈ s∗ becauseL A′ as described above. By (1) of Lemma 4.1, �mC ∈ Tr(s). (ii) WhenB ∈ Tr(s), we see C ∈ Tr(s) by Lemma 4.6. Further, we have �lB ∈ Tr(s)and hence �lB ∈ s by (1) of Lemma 4.1. Since L A′, we have �mC ∈ s.Thus, �mC ∈ Tr(s) by (1) of Proposition 4.9.

By an argument similar to that in Section 3, we have the following.

Theorem 4.11. Let L be the logic obtained from LR by adding (Q8), (Q9)and Type II Sahlqvist axioms. Ackermann’s rule γ is admissible for L.

5. Proof of the γ-admissibility II: relational extension

In this section, we prove the γ-admissibility of LM and some its extensions bythe method following [2] and [3]. The key to this method is to use the notionof relations between possible worlds, prime theories in our discussion, as inKripke models. Throughout this section, let the logics under considerationbe LM .

First, we present some notions which are needed for defining metavalu-ations. For convenience, we use the following notations. For all a ∈ W andany non-negative integer n:

�−na = {A | �nA ∈ a} �−na = {A | �nA ∈ a}

366 T. Seki

Binary relations S� and S♦ on W are defined as follows. For all a, b ∈ W :

S�abdef⇐⇒ �−1a ⊆ b S♦ab

def⇐⇒ b ⊆ ♦−1a

To simplify the notation, define binary relations Sn� and Sn

♦ on W , where nis a non-negative integer, inductively as follows. For all a, b ∈ W :

(S�-i) S0�ab iff a ⊆ b

(S�-ii) for n > 0, Sn�ab iff ∃c ∈ W (S�ac & Sn−1

� cb)(S♦-i) S0

♦ab iff b ⊆ a

(S♦-ii) for n > 0, Sn♦ab iff ∃c ∈ W (S♦ac & Sn−1

♦ cb)

For the binary relations Sn� and Sn

♦, we have the following by inductionon n.

Proposition 5.1. For all a, b ∈ W and any non-negative integer n,

(1) Sn�ab iff �−na ⊆ b

(2) Sn�a∗b∗ iff b ⊆ ♦· −na

(3) Sn♦ab iff b ⊆ ♦−na

(4) Sn♦a∗b∗ iff �−na ⊆ b

By using ∗, S� and S♦, binary relations S′� and S′

♦ on W are defined asfollows. For all a, b ∈ W :

S′�ab

def⇐⇒ S�ab & S�a∗b∗ S′♦ab

def⇐⇒ S♦ab & S♦a∗b∗

Note that a binary relation S′� corresponds to TL in [2]. Further, to simplify

the notation, define binary relations S′n� and S′n

♦ on W , where n is a non-negative integer, inductively as follows. For all a, b ∈ W :

(S′�-i) S′0

�ab iff a = b

(S′�-ii) for n > 0, S′n

�ab iff ∃c ∈ W (S′�ac & S′n−1

� cb)(S′

♦-i) S′0♦ab iff a = b

(S′♦-ii) for n > 0, S′n

♦ ab iff ∃c ∈ W (S′♦ac & S′n−1

♦ cb)

The following proposition is easy to prove. Note that (Q11) and (Q12),necessitation with respect to � and �, respectively, are used in the proof.

Proposition 5.2. For any s ∈ O, a ∈ W and any non-negative integer n,

(1) if S′n�sa then a ∈ O

(2) if S′n♦ sa then a ∈ O

The γ-admissibility of Relevant Modal Logics II 367

For the binary relations S′n� and S′n

♦ , we have the following.

Proposition 5.3. For all a, b ∈ W and any non-negative integer n,

(1) if S′n�ab, then �−na ⊆ b ⊆ ♦· −na; and the converse holds for n = 0, 1.

(2) if S′n♦ ab, then �−na ⊆ b ⊆ ♦−na; and the converse holds for n = 0, 1.

(3) S′n�ab iff S′n

�a∗b∗

(4) S′n♦ ab iff S′n

♦ a∗b∗

Proof. Here we give proofs of (1) and (3) by induction on n, and those of(2) and (4) are similar.

(1) (i) For n = 0, it is trivial by definition of S′0�. (ii) For n = 1, it

is trivial by definition of S′� and (1) and (2) of Proposition 5.1. (iii) For

n ≥ 2, suppose that S′n�ab. Then there exists c ∈ W such that S′

�ac andS′n−1

� cb. By (ii) and the hypothesis of induction, we have �−1a ⊆ c ⊆ ♦· −1a

and �−(n−1)c ⊆ b ⊆ ♦· −(n−1)c. First, taking any A ∈ �−na, �nA ∈ a. Thenwe have �n−1A ∈ c and hence A ∈ b. Thus, �−na ⊆ b. Next, take anyA ∈ b. Then we have ♦· n−1A ∈ c and hence ♦· nA ∈ a, i.e., A ∈ ♦· −na. Thus,b ⊆ ♦· −na. Therefore, �−na ⊆ b ⊆ ♦· −na.

(3) (i) For n = 0, it is trivial by definition of S′0� and (s2). (ii) For n ≥ 1,

S′n�ab

⇐⇒ ∃c ∈ W (S′�ac & S′n−1

� cb) (definition of S′n� )

⇐⇒ ∃c ∈ W (S′�a∗c∗ & S′n−1

� c∗b∗) (definition of S′�, (s2) and IH)

⇐⇒ S′n�a∗b∗ (definition of S′n

� )

Note that the converses of (1) and (2) do not necessarily hold for anypositive integer n; we will discuss later the case in which they always hold.

Proposition 5.4. For any a ∈ W ,

(1) �A ∈ a iff ∀b ∈ W (S′�ab ⇒ A ∈ b)

(2) ♦A ∈ a iff ∃b ∈ W (S′♦ab & A ∈ b)

(3) �A ∈ a iff ∀b ∈ W (S′♦ab ⇒ A ∈ b)

(4) ♦· A ∈ a iff ∃b ∈ W (S′�ab & A ∈ b)

Proof. (1) and (4) have been proved in [2] or [3]. Here we give a proof of(2) and (3).

368 T. Seki

(2) For the ‘only if’ part, suppose that ♦A ∈ a. Assume that LM �−1a ∪ {A} → (Wff − ♦−1a). Then there exist B1, · · · , Bm ∈ �−1a andC1, · · · , Cn /∈ ♦−1a such that

LM B1 ∧ · · · ∧ Bm ∧ A → C1 ∨ · · · ∨ Cn.

Then �B1, · · · , �Bm ∈ a; ♦C1, · · · , ♦Cn /∈ a and

LM �B1 ∧ · · · ∧ �Bm ∧ ♦A → ♦C1 ∨ · · · ∨ ♦Cn

(Note that (A15) is an axiom of LM .) This is a contradiction. Thus, LM ��−1a ∪ {A} → (Wff − ♦−1a). By (1) of Proposition 2.1, there exists b ∈ Wsuch that �−1a∪{A} ⊆ b and LM � b → (Wff−♦−1a). Then it is trivial that�−1a ⊆ b and A ∈ b. Further, taking any B ∈ b, we have B /∈ Wff−♦−1a. SoB ∈ ♦−1a, and hence ♦B ∈ a. Therefore, b ⊆ ♦−1a. By (2) of Proposition5.3, there exists b ∈ W such that S′

♦ab and A ∈ b.For the ‘if’ part, suppose that there exists b ∈ W such that S′

♦ab andA ∈ b. Since b ⊆ ♦−1a by (2) of Proposition 5.3, we have A ∈ ♦−1a, andhence ♦A ∈ a.

(3) By contraposition.

�A /∈ a⇐⇒ ♦ ∼ A ∈ a∗

⇐⇒ ∃b ∈ W (S′♦a∗b∗ & ∼ A ∈ b∗) (by (2))

⇐⇒ ∃b ∈ W (S′♦ab & A /∈ b) (by (4) of Proposition 5.3)

Note that the axioms (A14) and (A15) play an important role in theproofs for (1) and (2), respectively.

An LM -metavaluation v (a function from Wff ×O to {t, f}) is defined asfollows.

1. For p(∈ Prop), A ∧ B,A ∨ B, A → B and ∼ A, definitions are similar toSection 3.

2. v(�A, s) = t iff ∀t ∈ O(S′�st ⇒ v(A, t) = t)

3. v(♦A, s) = t iff ∃t ∈ O(S′♦st & v(A, t) = t)

Further, for any s ∈ O, Tr(s) = {A | v(A, s) = t} as in Section 3.Lemma 3.1 and Corollaries 3.2 and 3.3 also hold in the above definition

of metavaluations.

The γ-admissibility of Relevant Modal Logics II 369

Lemma 5.5. (1) For any s ∈ O, s∗ ⊆ Tr(s) ⊆ s.

(2) If A ∈ Tr(s) for all s ∈ O, then A is a theorem of LM .

(3) For any s ∈ O and all A ∈ Wff, ∼ A ∈ Tr(s) iff A /∈ Tr(s).

Proof. Since (2) and (3) are proved in same way as in Section 3, we givea proof of (1). The proof comprises three parts showing that (a) s∗ ⊆ s, (b)Tr(s) ⊆ s and (c) s∗ ⊆ Tr(s). Claim (a) is based on the fact that (A10) isan axiom, while both (b) and (c) are proved by induction on the length offormula A. Here it suffices to consider the cases in which A is of the forms�B and ♦B.

(b1) A is of the form �B. Suppose that �B /∈ s. By (1) of Proposition5.4, there exists a ∈ W such that S′

�sa and B /∈ a. By (1) of Proposition5.2, a ∈ O. Further, by the hypothesis of induction, we have B /∈ Tr(a).Therefore, we have �B /∈ Tr(s), which is the desired result.(b2) A is of the form ♦B. Suppose that ♦B ∈ Tr(s). Then there existst ∈ O such that S′

♦st and B ∈ Tr(t). By the hypothesis of induction, wehave B ∈ t. Thus, by (2) of Proposition 5.4, we have ♦B ∈ s as desired.

(c1) A is of the form �B. Suppose that �B /∈ Tr(s). Then there existst ∈ O such that S′

�st and B /∈ Tr(t). We have S′�s∗t∗ by (3) of Proposition

5.3, and B /∈ t∗ by the hypothesis of induction. Thus, by (1) of Proposition5.4, we have �B /∈ s∗ as desired.(c2) A is of the form ♦B. Suppose that ♦B ∈ s∗. By (2) of Proposition5.4, there exists a ∈ W such that S′

♦s∗a∗ and B ∈ a∗. So S′♦sa holds by (4)

of Proposition 5.3, and we have a ∈ O by (2) of Proposition 5.2. Further,we have B ∈ Tr(a) by the hypothesis of induction. Therefore, we have♦B ∈ Tr(s), which is the desired result.

The following lemma is proved by induction on n for (1) and (2), andby using Lemma 5.5 for (3) and (4). Note that Proposition 5.2 makes itpossible to apply induction.

Lemma 5.6. For all s ∈ O and any non-negative integer n,

(1) �nA ∈ Tr(s) iff ∀t ∈ O(S′n�st ⇒ A ∈ Tr(t))

(2) ♦nA ∈ Tr(s) iff ∃t ∈ O(S′n♦ st & A ∈ Tr(t))

(3) �nA ∈ Tr(s) iff ∀t ∈ O(S′n♦ st ⇒ A ∈ Tr(t))

(4) ♦· nA ∈ Tr(s) iff ∃t ∈ O(S′n�st & A ∈ Tr(t)).

370 T. Seki

By using the results in [8], we have the following correspondences betweenthe disjunctive rules (Q8), (Q9), (R9), (R10) and the frame postulates: Forall s ∈ O,

(Q8) S�ss (Q9) S♦s∗s∗ (R9) S�s∗s∗ (R10) S♦ss

Then we have the following.

Lemma 5.7. If LM is the logic in which all of the disjunctive rules (Q8),(Q9), (R9) and (R10) are derived, then S′

�ss and S′♦ss hold for all s ∈ O.

The following lemma implies that Tr(s) is regular.

Lemma 5.8. If A is a theorem of LM , then A ∈ Tr(s) for all s ∈ O.

Proof. This lemma is proved by induction on the length of the proof of thetheorem. Here we give proofs of the following axiom and rule of inference.

(A15) It is obvious that �A ∧ ♦B → ♦(A ∧ B) ∈ s. To see that�A ∧ ♦B ∈ Tr(s) implies ♦(A ∧ B) ∈ Tr(s), suppose �A ∧ ♦B ∈ Tr(s).Then there exists t ∈ O such that S′

♦st and B ∈ Tr(t). Further, we haveA ∈ Tr(t). So, A ∧ B ∈ Tr(t), and hence ♦(A ∧ B) ∈ Tr(s) as desired.

(Q11) Suppose that A ∈ Tr(s) for all s ∈ O. Take any s ∈ O. Further,suppose S′

�st for any t ∈ O. We have A ∈ Tr(t) by the assumption, so�A ∈ Tr(s).

Further, by Lemmas 5.5 and 5.8, Tr(s) is a normal LM -theory. Thus, wehave the following. The proof is the same as in Theorem 3.5.

Theorem 5.9. Ackermann’s rule γ is admissible for LM .

We consider the γ-admissibility of logics with Sahlqvist axioms in thepresent method. Here we present some terms and notations, which wereintroduced in Part I ([9]), for use in later discussions. For a ∈ W andnon-negative integer n, we write a ↑n

�= {b ∈ W | Sn�ab} and a ↑n

♦= {b ∈W | Sn

♦a∗b∗}. A frame-theoretic term a1↑n1 ∪ · · · ∪ ak↑nk , where ↑ denoteseither ↑� or ↑♦, with (not necessarily distinct) a1, · · · , ak ∈ W will be calledan S-term for brevity. Likewise, for a ∈ W and non-negative integer n, wedefine a↑′n�= {b ∈ W | S′n

�ab} and a↑′n♦ = {b ∈ W | S′n♦ ab}.

A Sahlqvist theorem in usual Routley-Meyer semantics was shown in [7].A procedure for obtaining the first-order sentence corresponding to a givenSahlqvist formula appears in a proof of Theorem 26 in [7] (p. 406). Herewe show the procedure for the first-order sentence from a given Sahlqvistformula �k(B → C) without providing definitions of frames, general frames,

The γ-admissibility of Relevant Modal Logics II 371

descriptive frames or other notations. For their precise definitions, see [7].Let F = 〈O,W,R, S�, S♦,∗ , e, P 〉 be a descriptive B.C�♦-frame or a B.C�♦-frame. Then the following 1 – 5 are mutually equivalent:

1. (F, a) |= �k(B[p1, · · · , pn] → C[p1, · · · , pn, q1, · · · , ql]),where (F, a) |= A means that A is true at a in F under any valuation.

2. ∀X1, · · · , Xn, Y1, · · · , Yl ∈ P(a ∈ �k(B → C)[X1, · · · , Xn, Y1, · · · , Yl]

)

3. ∀X1, · · · , Xn, Y1, · · · , Yl ∈ P∀b1, b2, b3 ∈ W(Sk

�ab1 & Rb1b2b3 &b2 ∈ B[X1, · · · , Xn] ⇒ b3 ∈ C[X1, · · · , Xn, Y1, · · · , Yl]

),

where Sk�ab1 is a conjunction of k formulas of the form either S�bc or

S♦b∗c∗, which is uniquely determined by the construction of a given �k.

4. ∀X1, · · · , Xn, Y1, · · · , Yl ∈ P∀b1, b2, b3, b4, · · · , bt ∈ W(Sk

�ab1 &Rb1b2b3 & D &

∧i≤n

Ti ⊆ Xi &∧

j≤m

cj ∈ Nj [X1, · · · , Xn] &∧

h≤l

u ∈ Yh

⇒ b3 ∈ C[X1, · · · , Xn, Y1, · · · , Yl]),

where D is a conjunction of formulas of the form either S♦bc or S�b∗c∗,Ti are suitable S-terms and Nj [p1, · · · , pn] are negative formulas.

5. ∀b1, · · · , bt ∈ W(Sk

�ab1 & Rb1b2b3 & D

⇒∨

j≤m+1

dj ∈ Kj [T1, · · · , Tn, u↑0, · · · , u↑0]),

where dj = c∗j for j ≤ m, Kj is a positive formula such that Nj isequivalent to ∼ Kj for j ≤ m, dm+1 = b3, Km+1 is C and ↑ denoteseither ↑� or ↑♦.

Thus, the first-order sentence corresponding to �k(B → C) can be writtenby

∀a ∈ O ∀b1, · · · , bt ∈ W(Sk

�ab1 & Rb1b2b3 & D ⇒∨j≤m+1

dj ∈ Kj [T1, · · · , Tn, u↑0, · · · , u↑0])

(*)

Since our definitions of S�, S♦ and ∗ are just same as those of the usualcanonical model, if (*) can be described by only S�, S♦ and ∗, then it alsoholds in our setting. However, to interpret modal operators in our definitionof metavaluations, we use S′

� and S′♦, not S� and S♦. We hope to obtain

the first order sentence corresponding to a given Sahlqvist formula in termsof S′

� and S′♦. In the light of the definitions of S′

� and S′♦, the form of the

Sahlqvist formulas which can be dealt seems to be rather restricted. In thefollowing, we consider Sahlqvist formulas and their corresponding first-order

372 T. Seki

sentences in terms of S′� and S′

♦ to which a Sahlqvist theorem in the usualRoutley-Meyer semantics can be applied.

When k = 0, (*) can be described only by Sn�, Sn

♦ and ∗. For, Sk�ab1 &

Rb1b2b3 can be written by b2 ⊆ b3, that is, S0�b2b3 since a ∈ O and Rab2b3.

Further, we do not assume the existence of u, so all propositional variablesappearing in C also appear in B. Taking b3 for b2, (*) can be written simplyby

∀b3, · · · , bt ∈ W(D ⇒

∨j≤m+1

dj ∈ Kj [T1, · · · , Tn]).

Let D∗ and T ∗i be obtained from D and Ti by replacing each bj(∈ W ) to b∗j .

Then∀b3, · · · , bt ∈ W

(D∗ ⇒

∨j≤m+1

d∗j ∈ Kj [T ∗1 , · · · , T ∗

n ])

also holds. Further let D′ be obtained from D by replacing each S� and S♦to S′

� and S′♦, respectively. Let T ′

i be obtained from Ti by replacing each↑� and ↑♦ to ↑′� and ↑′♦, respectively. However, we do not find

∀b3, · · · , bt ∈ W(D′ ⇒

∨j≤m+1

dj ∈ Kj [T ′1, · · · , T ′

n])

in general.One of the reasons is that the succedent is a disjunctive form. For ex-

ample, we consider the following case:

∀b, c, d1, d2 ∈ W(D′ ⇒ d1 ∈ p1[b↑′1� ∪c↑′2�, c↑′3♦] or d2 ∈ p2[b↑′1� ∪c↑′2�, c↑′3♦]

),

i.e., ∀b, c, d1, d2 ∈ W (D′ ⇒ d1 ∈ b↑′1� ∪c↑′2� or d2 ∈ c↑′3♦). Suppose that D′.By Propositions 5.3 and 5.1, we have D and D∗. Then “d1 ∈ b↑1

� ∪c↑2� or

d2 ∈ c↑3♦” and “d∗1 ∈ b∗↑1

� ∪c∗↑2� or d∗2 ∈ c∗↑3

♦”. But we cannot obtain thedesired result in general. To avoid this situation, we may assume that m = 0,which means that there is no subformula of the form negative formula in B.

Further, as another example, we consider the following case:

∀b, c, d ∈ W(D′ ⇒ d ∈ p[b↑′1� ∪c↑′2�]

),

i.e., ∀b, c, d ∈ W (D′ ⇒ d ∈ b↑′1� ∪c↑′2�). Suppose that D′. By Propositions5.3 and 5.1, we have D and D∗. Then d ∈ b↑1

� ∪c↑2� and d∗ ∈ b∗↑1

� ∪c∗↑2�,

i.e., “S�bd or S2�cd” and “S�b∗d∗ or S2

�c∗d∗”. But we cannot obtain thedesired result. To avoid this situation, we may assume that S-terms arerestricted to the form b ↑n

� or b ↑n♦, which means that each propositional

variable in B appears only once.

The γ-admissibility of Relevant Modal Logics II 373

The other reason is that dj ∈ Kj [T ′1, · · · , T ′

n] may be written in a formsuch as ∃x(S′m

� cx & S′n♦ dx) which cannot be reduced to a single S′

�- orS′

♦-sentence. For example, we consider the following case:

∀b, c ∈ W(D′ ⇒ c ∈ ♦kp[b↑′l�]

),

i.e., ∀b, c ∈ W (D′ ⇒ ∃x(S′k♦ cx & x ∈ b ↑′l�)). Suppose that D′. By

Propositions 5.3 and 5.1, we have D and D∗. Then ∃x(Sk♦cx & Sl

�bx) and∃y(Sk

♦c∗y & Sl�b∗y). We cannot obtain the desired result in general, except

for k = 0 or l = 0. This suggests that �lp appearing in B and �kp appearingin C satisfies kl = 0.

Similar things can be said for k ≥ 1 in logics under consideration inwhich disjunctive rules (Q1) and (Q2) are derivable. In this case, the firstpart is rather different from the argument described above. The outlineis as follows. For a logic under consideration in which (Q1) and (Q2) arederivable, the frame postulates Rbbb and Rbb∗b∗ hold for any b ∈ O. (Formore information, see [6] or [8].) Since b1 ∈ O by Proposition 5.2, taking b1

or b∗1 for both b2 and b3, the R-sentence in (*) can be removed.

We will examine some Sahlqvist formulas based on the above considera-tion.

Lemma 5.10. Let B[p1, · · · , pn] be a formula constructed from strongly posi-tive formulas by using only ∧, ♦ and ♦· and each pi appears once in B. Thenfor every s ∈ O,

B[p1, · · · , pn] ∈ Tr(s) iff ∃t1, · · · , th ∈ O(D′ &∧

1≤i≤n

�lipi ∈ Tr(tj)),

where the right-hand side sentence has one free individual variable s, D′ isa conjunction of sentences of the form either S′

�titj or S′♦titj.

Proof. By induction on the construction of B.

(i) B is a strongly positive formula. For mutually distinct p1, · · · , pn, wehave �l1p1 ∧ · · · ∧ �lnpn ∈ Tr(s) iff

∧1≤i≤n �lipi ∈ Tr(s)

(ii) B is of the form B1 ∧ B2. By the hypotheses of induction,

B1 ∧ B2 ∈ Tr(s)iff B1 ∈ Tr(s) & B2 ∈ Tr(s)iff ∃t1, · · · , th ∈ O(D′

1 &∧

1≤i≤m �lipi ∈ Tr(tj))& ∃t1, · · · , th ∈ O(D′

2 &∧

m+1≤i≤n �lipi ∈ Tr(tj))iff ∃t1, · · · , th ∈ O(D′

1 & D′2 &

∧1≤i≤n �lipi ∈ Tr(tj))

374 T. Seki

(iii) B is of the form ♦B1. By the hypothesis of induction,

♦B1 ∈ Tr(s)iff ∃t ∈ O(S′

♦st & B1 ∈ Tr(t))iff ∃t, t1, · · · , th ∈ O(S′

♦st & D′1 &

∧1≤i≤n �lipi ∈ Tr(tj))

(iv) B is of the form ♦· B1. By (4) of Lemma 5.6 and the hypothesis ofinduction,

♦· B1 ∈ Tr(s)iff ∃t ∈ O(S′

�st & B1 ∈ Tr(t))iff ∃t, t1, · · · , th ∈ O(S′

�st & D′1 &

∧1≤i≤n �lipi ∈ Tr(tj))

Though we have used the notations such as A[X1, · · · , Xn] without defi-nition, we give its formal definition in the below argument. Let A[p1, · · · , pn]be a formula constructed from p1, · · · , pn by using only ∧, �, ♦, � and ♦· .For c ∈ W , let Ti denote either c↑l

� or c↑l♦, and T ′

i denote either c↑′l� or c↑′l♦.Then A[T1, · · · , Tn] is defined as follows.

• pi[T1, · · · , Tn] = Ti

• (B ∧ C)[T1, · · · , Tn] = B[T1, · · · , Tn] ∩ C[T1, · · · , Tn]• (�B)[T1, · · · , Tn] = {a ∈ W | ∀b(S�ab ⇒ b ∈ B[T1, · · · , Tn])}• (♦B)[T1, · · · , Tn] = {a ∈ W | ∃b(S♦ab & b ∈ B[T1, · · · , Tn])}• (�B)[T1, · · · , Tn] = {a ∈ W | ∀b(S♦a∗b∗ ⇒ b ∈ B[T1, · · · , Tn])}• (♦· B)[T1, · · · , Tn] = {a ∈ W | ∃b(S�a∗b∗ & b ∈ B[T1, · · · , Tn])}

Further, A[T ′1, · · · , T ′

n] is defined as follows.

• pi[T ′1, · · · , T ′

n] = T ′i

• (B ∧ C)[T ′1, · · · , T ′

n] = B[T ′1, · · · , T ′

n] ∩ C[T ′1, · · · , T ′

n]• (�B)[T ′

1, · · · , T ′n] = {a ∈ W | ∀b(S′

�ab ⇒ b ∈ B[T ′1, · · · , T ′

n])}• (♦B)[T ′

1, · · · , T ′n] = {a ∈ W | ∃b(S′

♦ab & b ∈ B[T ′1, · · · , T ′

n])}• (�B)[T ′

1, · · · , T ′n] = {a ∈ W | ∀b(S′

♦ab ⇒ b ∈ B[T ′1, · · · , T ′

n])}• (♦· B)[T ′

1, · · · , T ′n] = {a ∈ W | ∃b(S′

�ab & b ∈ B[T ′1, · · · , T ′

n])}Lemma 5.11. For i = 0, · · · , n, let non-negative integers li and mi satisfylimi = 0. Let C be a formula constructed from ♦× m1p1, · · · , ♦× mnpn by usingonly ∧, � and �, and s, t1, · · · , tn ∈ O. If s ∈ C[t1 ↑′l1 , · · · , tn ↑′ln ] and�lipi ∈ Tr(ti) for i = 1, · · · , n, then C ∈ Tr(s), where ↑′ denotes ↑′� if� = � and denotes ↑′♦ if � = �.

The γ-admissibility of Relevant Modal Logics II 375

Proof. By induction on the construction of C. For simplicity, we prove thecase k = 1 and omit the subscripts.

(i) C is of the form ♦mp. Since lm = 0, l = 0 or m = 0. (1) When l = 0,suppose that s ∈ ♦mt↑′0 and p ∈ Tr(t). Then there exists u ∈ O such thatS′m

♦ su and u ∈ t↑′0. Since u ∈ t↑′0 iff t = u even if ↑′ is either ↑′� or ↑′♦,we have S′m

♦ st. By (2) of Lemma 5.6, ♦mp ∈ Tr(s), which is the desiredresult. (2) When m = 0, the following cases must be considered. (a) When� is �, suppose that s ∈ p[t↑′l�] and �lp ∈ Tr(t). Since s ∈ t↑′l�, i.e., S′l

�ts,we have p ∈ Tr(s) by (1) of Lemma 5.6. (b) When � is �, suppose thats ∈ p[t↑′l♦] and �lp ∈ Tr(t). Since s ∈ t↑′l♦, i.e., S′l

♦ts, we have p ∈ Tr(s) by(3) of Lemma 5.6.

(ii) C is of the form ♦· mp. Since lm = 0, l = 0 or m = 0. (1) When l = 0,suppose that s ∈ ♦· mt↑′0 and p ∈ Tr(t). Then there exists u ∈ O such thatS′m

� su and u ∈ t↑′0. Since t = u as in (1) of (i), we have S′m� st. By (4) of

Lemma 5.6, ♦· mp ∈ Tr(s), which is the desired result. (2) When m = 0, thediscussion is the same as in (2) of (i).

(iii) C is of the form C1 ∧C2. It is trivial from the hypotheses of induction.

(iv) C is of the form �C1. Suppose that s ∈ �C1[t↑′l] and �lp ∈ Tr(t)and take any u ∈ O satisfying S′

�su. Then we have u ∈ C1[t↑′l]. By thehypothesis of induction, C1 ∈ Tr(u), which is the desired result.

(v) C is of the form �C1. Suppose that s ∈ �C1[t ↑′l] and �lp ∈ Tr(t)and take any u ∈ O satisfying S′

♦su. Then we have u ∈ C1[t↑′l]. By thehypothesis of induction, C1 ∈ Tr(u), which is the desired result.

Lemma 5.12. For i = 0, · · · , n, let non-negative integers li and mi satisfyli+mi ≤ 1. Let C be a formula constructed from ♦× m1p1, · · · , ♦× mkpk by usingonly ∧, � and �. For d, c1, · · · , cn ∈ W , if both d ∈ C[c1↑l1 , · · · , cn↑ln ] andd∗ ∈ C[c∗1↑l1 , · · · , c∗n↑ln ], then d ∈ C[c1↑′l1 , · · · , cn↑′ln ], where each ↑′ denoteseither ↑′� or ↑′♦.

Proof. By induction on the construction of C. For simplicity, we prove thecase k = 1, and omit the subscripts.

(i) C is of the form ♦mp. (1) When m = 1, note that l = 0. (a) When ↑ is↑�, suppose that d ∈ ♦p[c↑0

�] and d∗ ∈ ♦p[c∗↑0�]. Then there exists b ∈ W

such that S♦db and b ∈ p[c↑0�] = c↑0

�, and hence c ⊆ b ⊆ ♦−1d, which meansthat S♦dc. In a similar way, we have S♦d∗c∗. Thus, we have S′

♦dc. Sincec = c, i.e., c ∈ c↑′0�= p[c↑′0�], we have d ∈ ♦p[c↑′0�] as desired. (b) When ↑ is↑♦, suppose that d ∈ ♦p[c↑0

♦] and d∗ ∈ ♦p[c∗↑0♦]. Then there exists b ∈ W

376 T. Seki

such that S♦db and b ∈ p[c↑0♦] = c↑0

♦, and hence b ⊆ ♦−1d and b∗ ⊆ c∗. So,we have c ⊆ b ⊆ ♦−1d, and hence S♦dc. In a similar way, we have S♦d∗c∗.Thus we have S′

♦dc. Since c = c, i.e., c ∈ c↑′0♦= p[c↑′0♦], we have d ∈ ♦p[c↑′0♦]as desired.

(2) When m = 0, we consider the following cases and note that l iseither 0 or 1. (a) When ↑ is ↑�, suppose that d ∈ p[c↑l

�] and d∗ ∈ p[c∗↑l�].

Then Sl�cd and Sl

�c∗d∗. Since l is either 0 or 1, we have S′l�cd, and hence

d ∈ c↑′l�= p[c↑l�]. (b) When ↑ is ↑♦, suppose that d ∈ p[c↑l

♦] and d∗ ∈ p[c∗↑l♦].

Then Sl♦c∗d∗ and Sl

♦c∗∗d∗∗. Since l is either 0 or 1, we can easily see thatS′l

♦cd, and hence d ∈ c↑′l♦= p[c↑l♦].

(ii) C is of the form ♦· mp. (1) When m = 1, note that l = 0. (a) When ↑ is↑�, suppose that d ∈ ♦· p[c↑0

�] and d∗ ∈ ♦· p[c∗↑0�]. Then there exists b ∈ W

such that S�d∗b∗ and b ∈ p[c↑0�] = c↑0

�. By (2) of Proposition 5.1, we havec ⊆ b ⊆ ♦· −1d, which means that S�d∗c∗. In a similar way, we have S�dc.Thus, we have S′

�dc. Since c = c, i.e., c ∈ c↑′0�= p[c↑′0�], we have d ∈ ♦· p[c↑′0�]as desired. (b) When ↑ is ↑♦, suppose that d ∈ ♦· p[c↑0

♦] and d∗ ∈ ♦· p[c∗↑0♦].

Then there exists b ∈ W such that S�d∗b∗ and b ∈ p[c↑0♦] = c↑0

♦, and henceb ⊆ ♦· −1d by (4) of Proposition 5.1 and b∗ ⊆ c∗. So, we have c ⊆ b ⊆ ♦· −1d,and hence S�d∗c∗. In a similar way, we have S�dc. Thus we have S′

�dc.Since c = c, i.e., c ∈ c↑′0♦= p[c↑′0♦], we have d ∈ ♦· p[c↑′0♦] as desired. (2) Whenm = 0, the argument is same as in (2) of (i).

(iii) C is of the form C1∧C2. This is trivial by the hypotheses of induction.

(iv) C is of the form �C1. Suppose that d ∈ �C1[c↑l] and d∗ ∈ �C1[c∗↑l],and take any b ∈ W satisfying S′

�db. Then S�db and S�d∗b∗. So, we haveb ∈ C1[c↑l] and b∗ ∈ C1[c∗↑l]. By the hypothesis of induction, b ∈ C1[c↑′l],which is the desired result.

(v) C is of the form �C1. Suppose that d ∈ �C1[c↑l] and d∗ ∈ �C1[c∗↑l],and take any b ∈ W satisfying S′

♦db. Then S♦db and S♦d∗b∗. So, we haveb ∈ C1[c↑l] and b∗ ∈ C1[c∗↑l]. By the hypothesis of induction, b ∈ C1[c↑′l],which is the desired result.

We refer to Sahlqvist formulas that are equivalent to a conjunction offormulas of the forms B → C and �k(B → C), where k is any non-negativeinteger, as Type III and Type IV, respectively:

• B is a formula constructed from �l1p1, · · · , �lnpn by using only ∧, ♦ and♦· , where p1, · · · , pn are mutually distinct.

• C is a formula constructed from ♦× m1p1, · · · , ♦× mnpn by using only ∧, �and �.

The γ-admissibility of Relevant Modal Logics II 377

• For every i (1 ≤ i ≤ n), li + mi ≤ 1.

Examples of Type III Sahlqvist formulas are �p → p and ♦· p → �♦· p, andType IV Sahlqvist formulas are the above two formulas, �(♦2p → ♦p) and��(♦ � p → p).

Lemma 5.13. Let L be a logic obtained from LM by adding Type III Sahlqvistformulas as axioms. If A is a theorem of L, then A ∈ Tr(s) for all s ∈ O.

Proof. It suffices to show the case in which A is a Type III Sahlqvist axiomof the form (B → C)[p1, · · · , pn]. Since B → C is an axiom of L and s ∈ O,B → C ∈ s. Further, we have ∀b1, · · · , bt ∈ W (D ⇒ d ∈ C[c1↑l1 , · · · , cn↑ln ])by a Sahlqvist theorem. Then ∀b1, · · · , bt ∈ W (D∗ ⇒ d∗ ∈ C[c∗1 ↑l1

, · · · , c∗n↑ln ]) also holds. To see that B ∈ Tr(s) implies C ∈ Tr(s), supposethat B ∈ Tr(s). By Lemma 5.10,

∃t1, · · · , th ∈ O(D′ &∧

1≤i≤n

�lipi ∈ Tr(ti)),

where D′ is a conjunction of sentences of the form either S′�titj or S′

♦titj .Then we have D and D∗, so s ∈ C[c1 ↑l1 , · · · , cn ↑ln ] and s∗ ∈C[c∗1↑l1 , · · · , c∗n↑ln ]. By Lemma 5.12, s ∈ C[c1↑′l1 , · · · , cn↑′ln ]. Thus we havethe desired result, that is, C ∈ Tr(s) by Lemma 5.11.

By similar argument in Section 3, we have the following.

Theorem 5.14. Ackermann’s rule γ is admissible for logics LM with TypeIII Sahlqvist axioms.

To prove the γ-admissibility of relevant modal logics with TypeIV Sahlqvist axioms using our method, the derivability of (Q1), (Q2), (Q8)and (Q9) in the logic under consideration must be assumed. Below, LSM

denotes a logic obtained from LM by adding (Q1), (Q2), (Q8) and (Q9) asrules of inference.

Lemma 5.15. Let L be a logic obtained from LSM by adding TypeIV Sahlqvist formulas as axioms. If A is a theorem of L, then A ∈ Tr(s) forall s ∈ O.

Proof. It suffices to show the case in which A is a Type IV Sahlqvistaxiom of the form �k(B → C)[p1, · · · , pn]. Suppose that S′k

�st for anyt ∈ O and show that B → C ∈ Tr(t), where S′k

�st is a conjunction of kformulas of the form either S′

�ab or S′♦ab, which is uniquely determined by

378 T. Seki

the construction of a given �k. First, L (B → C) ∨ �k(B → C), so L (B → C) ∨ (B → C) by applying (Q8) and (Q9). Thus, L B → C, fromwhich B → C ∈ t. Next, to show that B ∈ Tr(t) implies C ∈ Tr(t), supposethat B ∈ Tr(t). By a Sahlqvist theorem, we have ∀a ∈ O∀b1, · · · , bt ∈W (Sk

�ab1 & Rb1b2b3 & D ⇒ d ∈ C[c1↑l1 , · · · , cn↑ln ]). Since Sk�st, Rttt and

Rtt∗t∗ holds, we have ∀b4, · · · , bt ∈ W (D ⇒ d ∈ C[c1 ↑l1 , · · · , cn ↑ln ]) and∀b4, · · · , bt ∈ W (D∗ ⇒ d∗ ∈ C[c∗1↑l1 , · · · , c∗n↑ln ]). By Lemma 5.10,

∃t1, · · · , th ∈ O(D′ &∧

1≤i≤n

�lipi ∈ Tr(ti)),

where D′ is a conjunction of sentences of the form either S′�titj or S′

♦titj .Then we have D and D∗, so t ∈ C[c1 ↑l1 , · · · , cn ↑ln ] and t∗ ∈C[c∗1↑l1 , · · · , c∗n↑ln ]. By Lemma 5.12, t ∈ C[c1↑′l1 , · · · , cn↑′ln ]. Thus we havethe desired result, that is, C ∈ Tr(t) by Lemma 5.11. This completes theproof.

By similar argument in Section 3, we have the following.

Theorem 5.16. Ackermann’s rule γ is admissible for logics LSM with TypeIV Sahlqvist axioms.

However, the method used thus far cannot successfully be applied to rele-vant modal logics with a wider class of Sahlqvist axioms. For all appearancesof �n and �n in the antecedent and those of ♦n and ♦· n in the succedent ofa Type III or Type IV Sahlqvist formula, every n is restricted to a value ofeither 0 or 1. It is thus natural to consider, by extending our method, whichconditions make relevant modal logics with such Sahlqvist axioms admit γ.Examining in detail the proof described above, we notice that if (1) and (2)of Proposition 5.3 hold for all non-negative integers n, then our problem issolved; it suffices to construct a condition to ensure that

(**) S′n�ab iff �−na ⊆ b ⊆ ♦· −na

(***) S′n♦ ab iff �−na ⊆ b ⊆ ♦−na

hold for any n. Here we consider simpler conditions to ensure that (**) and(***) hold.

Below, the logic obtained from LM (LSM ) by adding

�A → A, A → ♦A, �A → �2A and ♦2A → ♦A

as axioms is called LT4M (LT4SM ), and the logic obtained from LM (LSM )by adding

♦A → �♦A and ♦· A → �♦· A

The γ-admissibility of Relevant Modal Logics II 379

as axioms is called L5M (L5SM ). Note that all axioms listed above are TypeIII Sahlqvist formulas. The following Lemma holds in LT4M , LT4SM , L5M

and L5SM .

Lemma 5.17. Let a logic under consideration be LT4M , LT4SM , L5M orL5SM . For all a, b ∈ W and any non-negative integer n,

(1) S′n�ab iff �−na ⊆ b ⊆ ♦· −na

(2) S′n♦ ab iff �−na ⊆ b ⊆ ♦−na.

Proof. (1) Here we give a proof only for the case when the logic underconsideration is LT4M .

The ‘only if’ part is just (1) of Proposition 5.3. For the ‘if’ part, theproof is by induction on n. We have already proved the cases n = 0, 1 in(1) of Proposition 5.3, so we give a proof for n ≥ 2. Suppose that �−na ⊆b ⊆ ♦· −na. Since �p → p is an axiom of LT4M , S′

�aa holds for all a ∈ W .To show that �−(n−1)a ⊆ b, suppose that A ∈ �−(n−1)a. Then �n−1A ∈ a.Since LT4M �n−1A → �nA, we have �nA ∈ a. By assumption, A ∈ bas desired. Next, to show that b ⊆ ♦· −(n−1)a, suppose that A ∈ b. Byassumption, ♦· nA ∈ a. Since LT4M ♦· nA → ♦· n−1A, we have ♦· n−1A ∈ a,and hence A ∈ ♦· −(n−1)a as desired. Thus, by the hypothesis of induction,S′n−1

� ab. Therefore, there exists an exact a ∈ W such that S′�aa and S′n−1

� ab,which means that S′n

�ab.

(2) Here we give a proof only for the case when the logic under considerationis L5M .

The ‘only if’ part is just (2) of Proposition 5.3. For the ‘if’ part, theproof is by induction on n. We have already proved the cases n = 0, 1 in(2) of Proposition 5.3, so we give a proof for n ≥ 2. Suppose that �−na ⊆b ⊆ ♦−na. Let c = {A | � A ∈ a} ∪ {♦n−1B | B ∈ b} and d = {C | ♦C /∈a} ∪ {�n−1D | D /∈ b}. To show that L5M � c → d, suppose otherwise.Then there exist �A1, · · · , �Al ∈ a; B1, · · · , Bp ∈ b; ♦C1, · · · ,♦Cq /∈ a andD1, · · · , Dr /∈ b such that

A1∧· · ·∧Al∧♦n−1B1∧· · ·∧♦n−1Bp → C1∨· · ·∨Cq∨�n−1D1∨· · ·∨�n−1Dr

is a theorem of L5M . It follows that

�(A1∧· · ·∧Al)∧�♦n−1(B1∧· · ·∧Bp) → ♦(C1∨· · ·∨Cq)∨�n(D1∨· · ·∨Dr)

is a theorem of L5M . Since L5M ♦n(B1∧· · ·∧Bp) → �♦n−1(B1∧· · ·∧Bp),

�(A1 ∧ · · · ∧Al)∧♦n(B1 ∧ · · · ∧Bp) → ♦(C1 ∨ · · · ∨Cq)∨�n(D1 ∨ · · · ∨Dr)

380 T. Seki

is a theorem of L5M . (Note that L5M ♦2A → �♦A.) Further, we have�(A1∧· · ·∧Al) ∈ a, B1∧· · ·∧Bp ∈ b, ♦(C1∨· · ·∨Cq) /∈ a and D1∨· · ·∨Dr /∈ b.By assumption, we have ♦n(B1 ∧ · · · ∧ Bp) ∈ a and �n(D1 ∨ · · · ∨ Dr) /∈ a.This is a contradiction, so L5M � c → d.

By (1) of Proposition 2.1, there exists c′ ∈ W such that c ⊆ c′ andL5M � c′ → d. Since �A ∈ a implies A ∈ c ⊆ c′, we have �−1a ⊆ c′. IfA ∈ c′, then A /∈ d, and hence ♦A ∈ a. Thus we have c′ ⊆ ♦−1a. Next, if�n−1A ∈ c′, then �n−1A /∈ d, and hence A ∈ b. Thus �−(n−1)c′ ⊆ b. SinceA ∈ b implies ♦n−1A ∈ c ⊆ c′, we have b ⊆ ♦−(n−1)c′. By the hypothesis ofinduction, there exists c′ ∈ W such that S′

♦ac′ and S′n−1♦ c′b.

The following lemma can be proved in a similar way to Lemma 5.12except that Lemma 5.17 is used.

Lemma 5.18. For i = 0, · · · , k, let non-negative integers li and mi satisfylimi = 0. Let C be a formula constructed from ♦× m1p1, · · · ,♦× mkpk by usingonly ∧, � and �. For d, c1, · · · , ck ∈ W , if both d ∈ C[c1↑l1 , · · · , ck↑lk ] andd∗ ∈ C[c∗1↑l1 , · · · , c∗k↑lk ], then d ∈ C[c1↑′l1 , · · · , ck↑′lk ], where each ↑′ denoteseither ↑′� or ↑′♦.

We refer to Sahlqvist formulas that are equivalent to a conjunction offormulas of the forms B → C and �k(B → C), where k is any non-negativeinteger, as Type V and Type VI, respectively:

• B is a formula constructed from �l1p1, · · · , �lnpn by using only ∧, ♦ and♦· , where p1, · · · , pn are mutually distinct.

• C is a formula constructed from ♦× m1p1, · · · , ♦× mnpn by using only ∧, �and �.

• For i = 1, · · · , n, limi = 0.

It is clear that all Type III (and Type IV) Sahlqvist formulas are Type V(and Type VI, respectively) Sahlqvist formulas. Examples of Type V andType IV Sahlqvist formulas are �k �l p → �mp and �k(�lp → �m♦× np),respectively, for any non-negative integers k, l, m, n.

The following lemma can be also proved in a similar way to Lemmas 5.13and 5.15.

Lemma 5.19. (1) Let L be a logic obtained from LT4M or L5M by addingType V Sahlqvist formulas as axioms. If A is a theorem of L, thenA ∈ Tr(s) for all s ∈ O.

The γ-admissibility of Relevant Modal Logics II 381

(2) Let L be a logic obtained from LT4SM or L5SM by adding Type VISahlqvist formulas as axioms. If A is a theorem of L, then A ∈ Tr(s)for all s ∈ O.

Thus, we have the following.

Theorem 5.20. (1) Let L be a logic obtained from LT4M or L5M byadding Type V Sahlqvist formulas as axioms. Ackermann’s rule γ isadmissible for L.

(2) Let L be a logic obtained from LT4SM or L5SM by adding TypeVI Sahlqvist formulas as axioms. Ackermann’s rule γ is admissiblefor L.

6. Conclusions and Remarks

This paper shows the γ-admissibility of relevant modal logics with Sahlqvistaxioms of the following forms in terms of the method using metavaluations.

(1) �k(�lB → �mC), where B is a strongly positive formula and C is a�-free positive formula, and

(2) �k(B → C), where B is a formula constructed from �l1p1, · · · ,�lnpn

by using only ∧, ♦ and ♦· , where each pi must appear only once, andC is a formula constructed from ♦× m1p1, · · · , ♦× mnpn by using only ∧,� and �, and for i = 1, · · · , n, limi = 0.

Concerning the definitions of metavaluations for modal formulas, it can besaid to be essential whether the formula belongs to a given regular primetheory for case (1), and that the relations between regular prime theories areessential for case (2). For technical reasons related to our proposed methods,we have considered relevant modal logics over G.Cg

�♦ and G.Ng�♦ for the

case of (1) and (2), respectively. Especially, in the case of (2), the existenceof (A14) and (A15) is assumed; the γ-admissibility of relevant modal logicswith (A14) and (A15) cannot be proved by the method described in Part I [9].

We summarize the correspondence between Types of Sahlqvist formulasand underlying relevant modal logics in Figure 1, in which the inclusionrange of each Type Sahlqvist formulas are expressed.

Note that there may be several individual logics besides those discussedin this paper whose γ-admissibility can be proved by the method usingmetavaluations. In fact, the γ-admissibility of LM with �mA → ♦· nA and�mA → ♦nA can be proved by the method described in Section 5. Note

382 T. Seki

Types of Sahlqvist formulas

Type V�

�

�

�Type III

�

�

�

�

Type VI�

�

�

�

Type IV�

�

�

�Type I

�

�

�

�

Type II�

�

�

�

LR

LM

LT4M , L5M

LR with (Q8) and (Q9)

LSM

LT4SM , L5SM

Underlying relevant modal logics

Figure 1. Correspondence between Types of Sahlqvist formulas and underlying logics

that the γ-admissibility of LM with �mA → �nA cannot be proved bythe method described in Section 5, but rather by the method described inSection 4.

On the other hand, it remains to be seen whether γ is admissible inrelevant modal logics to which present method cannot be applied. In fact,we do not know whether γ is admissible for relevant modal logics with anySahlqvist axioms. Our method also cannot be applied to logics with a fullrange of Sahlqvist formulas. For example, a Sahlqvist formula �p∧ ∼ �p →�2p cannot be dealt with using our method, but it can be handled usingthe method of normal models proposed in Part I [9]. Also, it is still unclearwhether γ holds for logics without disjunctive rules such as (R6), (R9) and(R10). This case will be discussed in [10]. These facts may indicate somelimitations to the method using metavaluations discussed in this paper.

Acknowledgements. The author would like to thank Professors RobertK. Meyer and Hiroakira Ono for their suggestions regarding an earlier versionof this paper. Further, the author acknowledges the anonymous referee forher/his many useful and detailed suggestions and corrections.

The γ-admissibility of Relevant Modal Logics II 383

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Takahiro SekiUniversity Evaluation CenterHeadquarters for Strategy and PlanningNiigata University8050 Ikarashi 2-no-choNishi-ku, Niigata City, 950-2181, Japantseki@adm.niigata-u.ac.jp