Studia Logica (2011) 97: 199–231DOI: 10.1007/s11225-011-9306-6 © Springer 2011

Takahiro Seki The γ-admissibility of Relevant

Modal Logics I — The Method

of Normal Models

Abstract. The admissibility of Ackermann’s rule γ is one of the most important prob-

lems in relevant logic. While the γ-admissibility of normal modal logics based on the

relevant logic R has been previously discussed, the case for weaker relevant modal logics

has not yet been considered. The method of normal models has often been used to prove

the γ-admissibility. This paper discusses which relevant modal logics admit γ from the

viewpoint of the method of normal models.

Keywords: γ-admissibility, relevant modal logic, Routley-Meyer semantics, normal model,

Sahlqvist axiom.

1. Introduction

The purpose of this paper is to investigate which relevant modal logics admitAckermann’s rule γ in terms of the method of normal models, introducedin [8]. The rule γ, sometimes called material detachment or disjunctive

syllogism, is given by∼ A ∨ B A

B

The name γ originates in Ackermann’s [1] pioneering study of relevant logic,and was so named because this rule was adopted as the third rule of inference.Generally the rule γ is not adopted as a rule of inference in present relevantlogics. Since γ is a non-trivial rule in relevant logics whenever it does notappear explicitly, the γ-admissibility is one of the most important problemsin relevant logics. For example, γ is admissible for the relevant logics E

and R ([6]), but not for the relevant logic LR, which is obtained from R

by omitting the distribution axiom ([7]). Further, it has previously beenestablished that the following relevant modal logics admit γ: NR ([8]), R4

([4]), RD, RT, RBr, R4 and R5 ([3]). It should be noted that these logicsare normal modal logics based on the relevant logic R.

There are several methods that have been used to show theγ-admissibility, with most of these semantical: the algebraic method as in [6];

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

200 T. Seki

the method of normal models as in [8] and [9]; and the method using metaval-uations as in [3, 4, 5, 15] and [16]. Thus far, the γ-admissibility has onlybeen established for a handful of particular relevant modal logics, and theγ-admissibility of a wider class of relevant modal logics has not yet beendiscussed. The problem with respect to the second method will be discussedin this paper, and that with respect to the third method will be studiedin [13] and [14]. Thus, this paper concentrates on the method of normalmodels based on Routley-Meyer (or relational) semantics, in which relevantdisjunction is interpreted as classical disjunction, while relevant negation isnot interpreted as classical negation. Roughly speaking, the method is toconstruct a new model, called a normal model, by adding a new base world toa given model, and then to interpret relevant negation as classical negationat some base world.

To ascertain which relevant modal logics admit γ, we must first con-sider logics that can satisfy the above proposed method. For the method ofnormal models, Section 5.6 of [10] has previously considered this question.The answer is that γ is admissible for the following relevant logics, calledconventionally normal in [10]:

(i) G, the logic obtained from the basic relevant logic B by adding theexcluded middle axiom A∨ ∼ A;

(ii) any logic extending G by any combination of additional axioms andrules of inference whose frame postulates are conjunctions of R andO statements or else are implications whose antecedents involve onlya single R statement;

(iii) many extensions of G which also include additional axioms and rulesof inference whose implicational frame postulates involve two R state-ments in the antecedent, provided the extensions enable the derivationof the disjunctive rules

(JR1)C ∨ A C ∨ (A → B)

C ∨ Band (JR4)

C ∨ A

C∨ ∼ (A →∼ A);

(iv) G + {(JR1), (JR4)} in the sense of [10].

However, it is impossible to prove the γ-admissibility of logics (i) and (ii)above in terms of the proposed method unless the disjunctive rule (JR4) isprovided. We refer to this problem in a later section. Consequently, γ isadmissible for the relevant logics mentioned in (i) – (iv), replacing G withG + {(JR4)}.

The γ-admissibility of Relevant Modal Logics I 201

A similar problem naturally arises in relevant modal logics. The basicrelevant modal logic B.C�♦ was introduced in [11]. In comparison withthe relevant modal logics for which the γ-admissibility has been proved, thecharacteristics of this logic are as follows: (1) It is non-normal (more pre-cisely, regular in the sense of [11]), and (2) the necessary operator � and thepossibility operator ♦ are independent (that is, the definition ♦A =∼ � ∼ Ais not adopted). Our interest is when γ is admissible in modal logics basedon relevant logics weaker than R, but with similar characteristics to thoselisted above. We show that γ is admissible for regular modal logics basedon relevant logics admitting γ with the following disjunctive rules:

C ∨ A

C∨ ∼ � ∼ Aand

C ∨ A

C ∨ ♦A.

Further, we discuss the γ-admissibility of relevant modal logics with gener-alized modal axioms such as Sahlqvist axioms. This paper gives a proof forthe γ-admissibility of relevant modal logics with restricted Sahlqvist axioms,including ♦k�lA → �m♦nA, where k, l,m, n are non-negative integers, interms of the method of normal models.

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 ⇒.

2.1. Relevant modal logics

The language of relevant modal logics consists of (i) propositional variables;(ii) logical connectives →,∧,∨ and ∼; and (iii) modal operators � and ♦.Formulas are defined in the usual way, and are denoted by capital lettersA,B,C,D,E. Prop and Wff will denote the set of all propositional variablesand formulas, respectively. We also introduce the following abbreviations:

�Adef=∼ ♦ ∼ A, ♦· A

def=∼ � ∼ A.

Further, for a non-negative integer n, �n and �n are defined inductively asfollows.

(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.

202 T. Seki

First, we present relevant logics admitting γ, where this admissibility isensured by the method of normal models. The relevant logic Gg is definedas 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

(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)

Note that the important logics weaker than Gg are B and G, where B

consists of (A1) – (A9) as axioms and (R1) – (R5) as rules of inference, thatis, B is obtained from Gg by omitting (A10) and (R6); G is obtained fromB by adding (A10) as an axiom.

L1 denotes any logic obtained from Gg by adding any set of the axiomsand the 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)

The γ-admissibility of Relevant Modal Logics I 203

(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

Below, let L with (Q1), for example, denote a logic L in which (Q1) isderivable. L2 denotes any logic obtained from L1 with (Q1) by adding anyset of the axioms and the rules of inference listed below. (The set may beempty.) 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))

204 T. Seki

It may be noted that the important logics that are stronger than Gg areas follows:

TW : obtained from Gg by adding (B13), (B16) and (B17)T : obtained from TW by adding (B3) and (B12)EW : obtained from TW by adding (Q4)E : obtained from T by adding (Q4)RW : obtained from EW by adding (B4)R : obtained from RW by adding (B3)

These logics are examples of L3, and hence of L2.

We now present the relevant modal logics discussed in this paper. Therelevant modal logic L.C

g

�♦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.C

g

�♦.

Let LR denote a logic obtained from L.Cg

�♦defined above by adding

any set of the following rules of inference, provided that if (B22) or (Q7) iscontained then (Q1) or (Q2), respectively, 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

The γ-admissibility of Relevant Modal Logics I 205

(Q10)B ∨ A

B ∨ �A

(Q11)A

�A

(Q12)A

�A

Note that LR is a regular relevant modal logic in the sense of [11].

The relevant modal logic L.Kg

�♦is defined as follows, where L is L3.

(a) Axioms consist of all axioms of L.Cg

�♦together with (B22) and the

following axiom:

(A13) �(A → B) → (♦A → ♦B)

(b) Rules of inference consist of all rules of inference of L.Cg

�♦together

with (Q11).

In particular, if L.Cg

�♦is G.C

g

�♦with (Q1) and (Q2), then L.K

g

�♦is called

G.Kg

�♦. Note that we cannot take both L1 without (Q1) and (Q2), and L2

without (Q2) for L.

Let LN denote logics obtained from L.Kg

�♦defined above by adding any

set of the following axioms and the rules of inference (Q6) – (Q12):

(B23) �(A → B) → (♦A → ♦B)

(B24) �(A → B) → (♦· A → ♦· B)

Note that LN is a normal relevant modal logic in the sense of [11]. The casefor additional modal axioms will be discussed later.

2.2. Routley-Meyer semantics

A G.Cg

�♦-frame is a 7-tuple 〈O,W,R, S�, S♦,∗ , e〉 where (a) W is the set of

all worlds, (b) O is a non-empty subset of W , (c) R is a ternary relation onW , (d) S� and S♦ are binary relations on W , (e) ∗ is an unary operationon W , and (f) e is an element of W , called the null world. To simplify thenotation, we define binary relations ≤, S�· and S♦· on W , and an element uof O, called the universal world, as follows. For all a, b ∈ W :

a ≤ bdef⇐⇒ ∃c(c ∈ O & Rcab), S�· ab

def⇐⇒ S♦a∗b∗,

S♦· abdef⇐⇒ S�a∗b∗, u

def= e∗.

206 T. Seki

For a non-negative integer n, binary relations Sn� and Sn

� on W , respec-tively associated with �n and �n, are defined inductively as follows. For alla, b ∈ W :

(i) S0�ab iff a ≤ b

(ii) for n ≥ 1,

Sn�ab iff

{∃c ∈ W (S�ac & Sn−1

�cb), if �n denotes ��n−1

∃c ∈ W (S�· ac & Sn−1�

cb), if �n denotes ��n−1

(iii) S0�ab iff b ≤ a

(iv) for n ≥ 1,

Sn�ab iff

{∃c ∈ W (S♦ac & Sn−1

� cb), if �n denotes ♦�n−1

∃c ∈ W (S♦· ac & Sn−1� cb), if �n denotes ♦· �n−1

Later on we use the following informal definitions (ii)’ and (iv)’ instead of(ii) and (iv) respectively, which is possible since the discussion is unaffectedby this replacement.

(ii)’ for n ≥ 1, Sn�ab iff ∃c ∈ W (S�ac & Sn−1

�cb)

(iv)’ for n ≥ 1, Sn�ab iff ∃c ∈ W (S�ac & Sn−1

� cb)

A G.Cg

�♦-frame 〈O,W,R, S�, S♦,∗ , e〉 satisfies the following postulates.

For all a, b, c, d ∈ W :

(p1) a ≤ a

(p2) a ≤ b & Rbcd ⇒ Racd

(p3) a ≤ b ⇒ b∗ ≤ a∗

(p4) a∗∗ = a

(p5) a ≤ b & S�bc ⇒ S�ac

(p6) a ≤ b & S♦ac ⇒ S♦bc

(p7) a ≤ b & a ∈ O ⇒ b ∈ O

(p8) a ∈ O ⇒ a∗ ≤ a

(p9) a ∈ O ⇒ Ra∗aa∗

(p10) a ∈ O ⇒ S♦· aa

(p11) a ∈ O ⇒ S♦aa

(p12) Ruab ⇒ a = e or b = u

The γ-admissibility of Relevant Modal Logics I 207

(p13) Reue

(p14) e �= u

(p15) S�ee

(p16) S�ua ⇒ a = u

(p17) S♦ea ⇒ a = e

(p18) S♦uu

Every G.Cg

�♦-frame 〈O,W,R, S�, S♦,∗ , e〉 satisfies e ≤ a ≤ u for all a ∈ W .

It may be remarked that the relation ≤ is not always quasi-order. Note thatthe postulates (p8), (p9), (p10) and (p11) correspond to the axioms or rulesof inference (A10), (R6), (R9) and (R10), respectively.

It should be noted that we introduce enlarged frames with the null world eand the universal world u because the logic obtained from B.C�♦ by addinga Sahlqvist axiom p∧ ∼ p → q, for example, is incomplete with respect tothe frames without these worlds e and u, i.e., frames in the usual sense,while the logic is complete with respect to the frames with them. For moreinformation, see [11] (p.409).

A G.Kg

�♦-frame 〈O,W,R, S�, S♦,∗ , e〉 is a G.C

g

�♦-frame satisfying the

postulates (q22), (r1), (r2) and (r11) mentioned later in addition to thefollowing postulates. For all a, b, c, d ∈ W :

(p19) Rabc & S♦bd ⇒ ∃x ∈ W∃y ∈ W (S�ax & S♦cy & Rxdy)

Postulate (p19) corresponds to the axiom (A13).

All the axioms and rules of inference we have mentioned have correspond-ing frame conditions. That is, if a logic LR (LN ) contains an axiom or arule of inference, then any LR-frame (LN -frame) F = 〈O,W,R, S�, S♦,∗ , e〉satisfies certain postulates. Here is a list of such correspondences. Thepostulate (qi) corresponds to the axiom (Bi) for 1 ≤ i ≤ 24, and the pos-tulate (rj) corresponds to the rule of inference (Qj) for 1 ≤ j ≤ 12. For alla, b, c, d, f ∈ W :

(q1) Raaa

(q2) Rabc ⇒ ∃x ∈ W (Rabx & Raxc)

(q3) Rabc ⇒ ∃x ∈ W (Rabx & Rxbc)

(q4) Rabc ⇒ Rbac

(q5) a �= e & Rabc ⇒ b ≤ c

208 T. Seki

(q6) b �= e & Rabc ⇒ a ≤ c

(q7) Rabc & Rcdf ⇒ Radf & Rbdf

(q8) Rabc ⇒ a ≤ c or b ≤ c

(q9) Rabc ⇒ Rbac & a ≤ c

(q10) Rabc ⇒ ∃x ∈ W (b ≤ x & c∗ ≤ x & Raxb∗)

(q11) Ra∗aa∗

(q12) Raa∗a

(q13) Rabc ⇒ Rac∗b∗

(q14) a �= e ⇒ a∗ ≤ a

(q15) c �= e & Rabc ⇒ a ≤ b∗

(q16) Rabc & Rcdf ⇒ ∃x ∈ W (Radx & Rbxf)

(q17) Rabc & Rcdf ⇒ ∃x ∈ W (Rbdx & Raxf)

(q18) Rabc & Rcdf ⇒ ∃x ∈ W (Radx & Rxbf)

(q19) Rabc & Rcdf ⇒ ∃x ∈ W∃y ∈ W (Radx & Rbdy & Rxyf)

(q20) Rabc & Rcdf ⇒ ∃x ∈ W∃y ∈ W (Radx & Rbdy & Ryxf)

(q21) Rabc & Rcdf ⇒ ∃x(b ≤ x & d ≤ x & Raxf)

(q22) Rabc & S�cd ⇒ ∃x ∈ W∃y ∈ W (S�ax & S�by & Rxyd)

(q23) Rabc & S♦bd ⇒ ∃x ∈ W∃y ∈ W (S�· ax & S♦cy & Rxdy)

(q24) Rabc & S♦· bd ⇒ ∃x ∈ W∃y ∈ W (S�ax & S♦· cy & Rxdy)

(r1) a ∈ O ⇒ Raaa

(r2) a ∈ O & Rabc ⇒ Rac∗b∗

(r3) a ∈ O ⇒ Raa∗a

(r4) ∃x ∈ O(Raxa)

(r5) a ∈ O & Rabc & Rcdf ⇒ ∃x ∈ W∃y ∈ W (Radx & Rbxy & Rayf)

(r6) a ∈ O & Rabc & S�cd ⇒ ∃x ∈ W (Raxd & S�bx)

(r7) a ∈ O & Rabc & S♦bd ⇒ ∃x ∈ W (Radx & S♦cx)

(r8) a ∈ O ⇒ S�aa

(r9) a ∈ O ⇒ S�· aa

(r10) a ∈ O & S�ab ⇒ a ≤ b

(r11) a ∈ O & S�ab ⇒ b ∈ O

(r12) a ∈ O & S�· ab ⇒ b ∈ O

The γ-admissibility of Relevant Modal Logics I 209

We can easily show the following.

Proposition 2.1. Let F = 〈O,W,R, S�, S♦,∗ , e〉 be a G.Cg

�♦-frame.

(1) For all a ∈ O and any non-negative integer n, (a) Sn�a∗a∗ and

(b) Sn�aa.

(2) For all a, b ∈ W and any integer n ≥ 1,

(a) a ≤ b & Sn�bc ⇒ Sn

�ac (b) a ≤ b & Sn�ac ⇒ Sn

�bc.

(3) If F satisfies (r8) and (r9), then for all a ∈ O and any non-negative

integer n, (a) Sn�aa and (b) Sn

�a∗a∗.

We call an 8-tuple 〈O,W,R, S�, S♦,∗ , e, v〉 a G.Cg

�♦-model on a G.C

g

�♦-

frame (or more simply, a G.Cg

�♦-model) F = 〈O,W,R, S�, S♦,∗ , e〉, where

F is a G.Cg

�♦-frame and v is a mapping from Prop × W to {t, f}, called

a valuation on F, which satisfies the following (1) hereditary condition, (2)E-condition and (3) U-condition, respectively. For all a, b ∈ W and allp ∈ Prop:

(1) a ≤ b & v(p, a) = t ⇒ v(p, b) = t, (2) v(p, e) = f, and (3) v(p, u) = t.

Given a G.Cg

�♦-model 〈O,W,R, S�, S♦,∗ , e, v〉, we define the interpretation

I associated with v. A mapping I from Wff×W to {t, f} is defined inductivelyas follows. For a ∈ W :

i. for any p ∈ Prop, I(p, a) = t iff v(p, a) = t

ii. I(A ∧ B, a) = t iff I(A, a) = t & I(B, a) = t

iii. I(A ∨ B, a) = t iff I(A, a) = t or I(B, a) = t

iv. I(A → B, a) = t iff∀b ∈ W∀c ∈ W (Rabc & I(A, b) = t ⇒ I(B, c) = t)

v. I(∼ A, a) = t iff I(A, a∗) = f

vi. I(�A, a) = t iff ∀b ∈ W (S�ab ⇒ I(A, b) = t)

vii. I(♦A, a) = t iff ∃b ∈ W (S♦ab & I(A, b) = t)

It is then easy to show the following. For all a ∈ W and any non-negativeinteger n:

(a) I(�A, a) = t iff ∀b ∈ W (S�· ab ⇒ I(A, b) = t)

(b) I(♦· A, a) = t iff ∃b ∈ W (S♦· ab & I(A, b) = t)

210 T. Seki

(c) I(�nA, a) = t iff ∀b ∈ W (Sn�ab ⇒ I(A, b) = t)

(d) I(�nA, a) = t iff ∃b ∈ W (Sn�ab & I(A, b) = t)

Let A ∈ Wff. Then we can say (a) A holds in a G.Cg

�♦-model M =

〈O,W,R, S�, S♦,∗ , e, v〉 iff I(A, a) = t for every a ∈ O, and (b) A is valid ina G.C

g

�♦-frame F = 〈O,W,R, S�, S♦,∗ , e〉 iff A holds in every G.C

g

�♦-model

M on F.

G.Kg

�♦-models, LR-models and LN -models are defined similarly to

G.Cg

�♦-models.

The rules of inference (R6), (R9), (R10), (Q1) – (Q3) and (Q5) – (Q10)are called disjunctive rules. If LR (or LN ) is a logic in which all disjunctiverules are derivable from axioms and non-disjunctive rules, then we obtaincompleteness by using the usual method of the canonical model (see [11]).Otherwise completeness can be proved by using the modified method of thecanonical model, which is discussed in [12].

Proposition 2.2. Let L be a logic LR or LN defined above and A ∈ Wff.

Then A is a theorem of L iff A is valid in every L-frame.

2.3. A Sahlqvist theorem

A formula A is positive if it can be constructed using no connectives otherthan ∧,∨,�,♦,� and ♦· . A positive formula of the form �m1p1∧· · ·∧�mkpk

with not necessarily distinct propositional variables p1, . . . , pk is called astrongly positive formula. A given formula A is negative (in a logic L) if itis equivalent in L to ∼ B for a positive formula B. A modal formula A isuntied (in L) if it can be constructed from strongly positive formulas andnegative formulas (in L) using only ∧,♦ and ♦· . A formula, A, is calledSahlqvist if it is a conjunction of the form �k(B → C), where k ≥ 0, B isuntied in L and C is positive.

The following completeness result, called a Sahlqvist theorem, can beshown as in [11].

Theorem 2.3. Let L be a logic obtained from LR or LN by adding a set

of Sahlqvist formulas as axioms. Then L is complete with respect to all

L-frames.

The point of this theorem is that the logic L is D�-elementary, that is,there exists a set Φ of first-order sentences in the predicates O,R, S�, S♦,∗

and the constant e such that for every descriptive L-frame or L-frame F, F isan L-frame iff F satisfies each sentence in Φ. Since the logics LR and LN are

The γ-admissibility of Relevant Modal Logics I 211

D�-elementary, our interest is how to show the logic L is D�-elementary, inother words, what is a first-order sentence corresponding to a given Sahlqvistformula. Following [11], the construction will be sketched. Though the no-tion of general frames is essentially used, we present some additional notionsthat are required for our discussion. For the precise definition of generalframes and descriptive frames, see [11].

Below, A[p1, . . . , pn] denotes a formula A whose variables are listedamong p1, . . . , pn. Let 〈O,W,R, S�, S♦,∗ , e〉 be an L-frame. ForA[p1, . . . , pn] ∈ Wff and X1, . . . ,Xn ⊆ W , A[X1, . . . ,Xn] is defined asfollows.

• For pi ∈ Prop, pi[X1, . . . ,Xn] = Xi

• (B ∧ C)[X1, . . . ,Xn] = B[X1, . . . ,Xn] ∩ C[X1, . . . ,Xn]

• (B ∨ C)[X1, . . . ,Xn] = B[X1, . . . ,Xn] ∪ C[X1, . . . ,Xn]

• (B → C)[X1, . . . ,Xn] ={a ∈ W | ∀b∀c(Rabc & b ∈ B[X1, . . . ,Xn] ⇒ c ∈ C[X1, . . . ,Xn])}

• (∼ B)[X1, . . . ,Xn] = {a ∈ W | a∗ /∈ B[X1, . . . ,Xn]}

• (�B)[X1, . . . ,Xn] = {a ∈ W | ∀b(S�ab ⇒ b ∈ B[X1, . . . ,Xn])}

• (♦B)[X1, . . . ,Xn] = {a ∈ W | ∃b(S♦ab & b ∈ B[X1, . . . ,Xn])}

For a ∈ W and a non-negative integer n, we write a↑n�= {b ∈ W | Sn

�ab}.The frame-theoretic term a1↑

n1

�∪ · · · ∪ ak↑

nk

�with (not necessarily distinct)

a1, . . . , ak ∈ W will be called an S�-term for brevity.A procedure for obtaining the first-order sentence corresponding to a

given Sahlqvist formula appears in a proof of the following theorem, whichis Theorem 26 in [11] (p.406).

Theorem 2.4. Let L be a logic LR or LN defined above. For any Sahlqvist

formula A, there exists a first order formula φ(a) in the predicates O, R,

S�, S♦, ∗ and the constant e having a as its only free variable and such that

the following holds for every descriptive L-frame or L-frame F and every

a ∈ W ,

(F, a) |= A iff F satisfies φ(a),

where (F, a) |= A means that I(A, a) = t under any valuation on F.

It suffices to consider a conjunct �k(B → C) of a formula equivalentto the given Sahlqvist formula A. Let F = 〈O,W,R, S�, S♦,∗ , e, P 〉 be adescriptive B.C�♦-frame or a B.C�♦-frame. Then the following statements1 – 5 are mutually equivalent:

212 T. Seki

1. (F, a) |= �k(B[p1, . . . , pn] → C[p1, . . . , pn, q1, . . . , ql])

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]

)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♦· bc, Ti

are suitable S�-terms and Nj [p1, . . . , pn] are negative formulas in L.

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 and Km+1 is C.

Note that for equivalence between 3 and 4, we use the convention that u isin Yh ∈ P and the following, which is Lemma 25 in [11] (p.405): For everya ∈ W and all X1, . . . ,Xn ∈ P ,

a ∈ B[X1, . . . ,Xn] iff∃b1, . . . , bt ∈ W

(D &

∧i≤n Ti ⊆ Xi &

∧j≤m cj ∈ Nj[X1, . . . ,Xn]

).

Taking∀b1, . . . , bt ∈ W

(Sk

�ab1 & Rb1b2b3 & D

⇒∨

j≤m+1

dj ∈ Kj[T1, . . . , Tn, u↑0�, . . . , u↑0

�])

for φ(a), the first-order sentence corresponding to �k(B → C) can be writtenby ∀a ∈ O

(φ(a)

).

3. The γ-admissibility of logics without Sahlqvist axiom

The γ-admissibility cannot be proved in general by standard completeness.If we suppose that both ∼ A∨B and A are theorems while B is not, then itis impossible to reach a contradiction unless we assume that a frame meetsthe condition that a∗ = a for some a ∈ O. Such a frame is called normal.In other words, by standard completeness results one knows that any non-theorem is falsifiable in some model, but one does not know it is falsifiablein a normal model, i.e., a model on a normal frame. The proof consists

The γ-admissibility of Relevant Modal Logics I 213

of demonstrating that it is, from which the admissibility of γ will follow.In this section, following the approach of Section 5.6 of [10], we prove theγ-admissibility by the method of normal models. Throughout this section,L denotes either LR or LN .

An L-frame F = 〈O,W,R, S�, S♦,∗ , e〉 is normal if it satisfies the follow-ing postulate:

a = a∗, for some a ∈ O.

Further, a normal L-model (on a normal L-frame) is a pair of a normalL-frame with a valuation on it.

Borrowing the terminologies of the method of the canonical model, thedifference between a (usual) L-model and a normal L-model is that we maytake the set of regular prime L-theories and the set of regular prime L-theories containing a normal L-theory, respectively, as O. For the definitionsof L-theory, regular, prime and normal, see [10]. For every normal L-theoryΣ, the following proposition holds: A ∈ Σ iff ∼ A /∈ Σ. Thus, there existsa ∈ O such that I(∼ A, a) = t iff I(A, a) = f in every normal L-model.

By Proposition 2.2, if A is not a theorem of L, then there exists the(canonical) L-frame F = 〈O,W,R, S�, S♦,∗ , e〉, a valuation v on F and theinterpretation I associated with v satisfying I(A, o) = f for o ∈ O. However,it is not certain that the model is normal, so we must construct a normalmodel from this model.

Let F = 〈O,W,R, S�, S♦,∗ , e〉, o ∈ O and v be as above and be fixed. Inwhat follows, we impose the condition that 0 /∈ W . (Note that o and 0 inthis paper correspond to T and T ′, respectively, in [10].) The normalization

of an L-frame F at 0 for o ∈ O is the structure F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉satisfying the following conditions:

(a) W ′ = W ∪ {0};

(b) O′ = O ∪ {0};

(c) R′ is a ternary relation on W ′, which is defined by R on elements ofW (= W ′ − {0}) and satisfies

R′000, R′00a iff R′ooa, R′0a0 iff R′oao∗, R′a00 iff R′aoo∗,

R′0ab iff R′oab, R′a0b iff R′aob, R′ab0 iff R′abo∗,

for a, b ∈ W ;

(d) S′� is a binary relation on W ′ that is defined by S� on elements ofW and satisfies

214 T. Seki

S′�00, S′�0a iff S′�oa, S′�a0 iff S′�ao∗,

for a ∈ W ;

(e) S′♦ is a binary relation on W ′ that is defined by S♦ on elements of Wand satisfies

S′♦00, S′♦0a iff S′♦o∗a, S′♦a0 iff S′♦ao,

for a ∈ W ;

(f) ∗′ is an unary operator on W ′ that satisfies 0∗′ = 0 and a∗′ = a∗, fora ∈ W .

Binary relations ≤′, S′�· and S′♦· are defined as in Section 2.

From the above definitions, it is easy to see that relations R′, S′� and S′♦reduce to relations R, S� and S♦, respectively, except for R′000, S′�00 andS′♦00. Thus, we have the following lemma.

Lemma 3.1. Let F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 be the normalization of a

given L-frame F = 〈O,W,R, S�, S♦,∗ , e〉 at 0 for o ∈ O.

(1) The relations R′, S′� and S′♦ are well-defined.

(2) For all a, b ∈ W :

(a) a ≤′ b iff a ≤ b (b) 0 ≤′ b iff o ≤ b (c) a ≤′ 0 iff a ≤ o∗

(d) S′�· ab iff S�· ab (e) S′�· 0b iff S�· ob (f) S′�· a0 iff S�· ao∗

(g) S′♦· ab iff S♦· ab (h) S′♦· 0b iff S♦· o∗b (i) S′♦· a0 iff S♦· ao

This lemma is proved as in [10] (pp.387–388), and the following corollaryis also derived.

Corollary 3.2. Let F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 be the normalization of a

given L-frame F = 〈O,W,R, S�, S♦,∗ , e〉 at 0 for o ∈ O. Then o∗ ≤′ 0 ≤′ o.

The following two lemmas correspond to Lemma 5.4 of [10] (p.388).

Lemma 3.3. Let F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 be the normalization of a

given LR-frame F = 〈O,W,R, S�, S♦,∗ , e〉 at 0 for o ∈ O. Then F′ is also

an LR-frame.

Proof. We must check each frame postulate. Here we give a proof for somespecific cases. We first consider the case in which LR is G.C

g

�♦.

(p2) Most of the cases are clear, so we give a proof for the followingproblematic case. For all a ∈ W (= W ′ − {0}):

a ≤′ 0 & R′000 ⇒ R′a00.

The γ-admissibility of Relevant Modal Logics I 215

Suppose that a ≤′ 0 and R′000. Then we have a ≤ o∗ and Ro∗oo∗ by (p9).By (p2), we have Raoo∗, and hence R′a00.

(p5) Most of the cases are clear, so we give proofs for the followingproblematic cases. For all a, b ∈ W :

(i) a ≤′ 0 & S′�0b ⇒ S′�ab; (ii) a ≤′ 0 & S′�00 ⇒ S′�a0.

Case (i): Suppose that a ≤′ 0 and S′�0b. Then we have a ≤ o∗ and S�ob.Since o∗ ≤ o by (p8), we have S�o∗b by (p5). Again by (p5), we have S�ab,and hence S′�ab. Case (ii): Suppose that a ≤′ 0 and S′�00. Then we havea ≤ o∗ and S♦· oo, i.e., S�o∗o∗, by (p10). By (p5), we have S�ao∗, and henceS′�a0.

(p6) Most of the cases are clear, so we give proofs for the followingproblematic cases. For all a, b ∈ W :

(i) 0 ≤′ a & S′♦0b ⇒ S′♦ab; (ii) 0 ≤′ a & S′♦00 ⇒ S′♦a0.

Case (i): Suppose that 0 ≤′ a and S′♦0b. Then we have o ≤ a and S♦o∗b.Since o∗ ≤ o by (p8), we have S♦ob by (p6). Again by (p6), we have S♦ab,and hence S′♦ab. Case (ii): Suppose that 0 ≤′ a and S′♦00. Then we haveo ≤ a and S♦oo by (p11). By (p6), we have S♦ao, and hence S′♦a0.

(p12) We give a proof for the following case. Other cases are clear orcan be proved in a similar manner.

R′u00 ⇒ 0 = e or 0 = u.

Suppose that R′u00. Then we have Ruoo∗. By (p12), we have o = e oro∗ = u. For the former case, we have 0 ≤′ e by Corollary 3.2, and hence0 = e since e is the minimum element of W ′ with respect to ≤′. For thelatter case, we have u ≤′ 0 by Corollary 3.2, and hence 0 = u since u is themaximum element of W ′ with respect to ≤′. Proofs for (p16) and (p17) aresimilar.

Next, we consider the case in which LR contains (B22), provided with(Q1). Most of the cases can be proved easily, so we give proofs for thefollowing problematic cases. For all a, b, c ∈ W :

(i) R′ab0 & S′�0c ⇒ ∃x∃y(S′�ax & S′�by & R′xyc)

(ii) R′ab0 & S′�00 ⇒ ∃x∃y(S′�ax & S′�by & R′xy0)

(iii) R′000 & S′�0c ⇒ ∃x∃y(S′�0x & S′�0y & R′xyc)

(iv) R′0b0 & S′�00 ⇒ ∃x∃y(S′�0x & S′�by & R′xy0)

216 T. Seki

Case (i): Suppose that R′ab0 and S′�0c. Then we have Rabo∗ and S�oc.Since o∗ ≤ o by (p8), S�o∗c by (p5). By (q22), there exist x, y ∈ W suchthat S�ax, S�by and Rxyc. Therefore, there exist x, y ∈ W ′ such thatS′�ax, S′�by and R′xyc. Case (ii): Suppose that R′ab0 and S′�00. Then wehave Rabo∗ and S�o∗o∗ by (p10). By (q22), there exist x, y ∈ W such thatS�ax, S�by and Rxyo∗. Thus, there exist x, y ∈ W ′ such that S′�ax, S′�byand R′xy0. Case (iii): Suppose that R′000 and S′�0c. Then we have S�ocand Rooo by (r1). (Note that we assume LR contains (Q1).) By (q22), thereexist x, y ∈ W such that S�ox, S�oy and Rxyc. Thus, there exist x, y ∈ W ′

such that S′�0x, S′�0y and R′xyc. Case (iv): Suppose that R′0b0 and S′�00.Then we have Robo∗ and S�o∗o∗ by (p10). By (q22), there exist x, y ∈ Wsuch that S�ox, S�by and Rxyo∗. Thus, there exist x, y ∈ W ′ such thatS′�0x, S′�by and R′xy0.

The above proof shows that the disjunctive rules (R6), (R9) and (R10)must be imposed. Note that for a non-modal case, the above method workswell for Gg, but not for G. That is, it is impossible to establish a proof forG using the present approach.

Lemma 3.4. Let F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 be the normalization of a

given LN -frame F = 〈O,W,R, S�, S♦,∗ , e〉 at 0 for o ∈ O. Then F′ is also

an LN -frame.

Proof. We must check each frame postulate. Here we give a proof for aspecific case concerning S′� and S′♦. We consider the case in which LN isG.K

g

�♦.

(p19) Here, we give a proof only for the following case:

R′000 & S′♦0d ⇒ ∃x∃y(S′�0x & S′♦0y & R′xdy).

Suppose that R′000 and S′♦0d. Then we have S♦o∗d. Further, we have Roooby (r1), so Roo∗o∗ by (r2). Applying (p19), there exist x, y ∈ W such thatS�ox, S♦o∗y and Rxdy. Thus, there exist x, y ∈ W ′ such that S′�0x, S′♦0yand R′xdy.

Finally, we remark that a situation as in (p19) arises in a proof for (r7).Unless L contains (Q2), it is not possible to establish a proof for (r7) usingthe present approach. So we impose the condition that L contains (Q2)whenever it contains (Q7).

The above proof shows the further need for the disjunctive rules (Q1)and (Q2).

The γ-admissibility of Relevant Modal Logics I 217

Thus, we see that F′ is a normal L-frame since 0 ∈ O′ and 0∗′ = 0. Thenthe following lemma is easily proved.

Lemma 3.5. Let A ∈ Wff. If A is a theorem of L, then A is valid in every

normal L-frame.

For an L-frame F and some o ∈ O, we can construct the normalizationF′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 of F at 0 for that o. By Lemmas 3.3 and 3.4,F′ is an L-frame. We define a function v′ from Prop×W ′ to {t, f} as follows.For all p ∈ Prop:

v′(p, a) = v(p, a), for a ∈ W ; v′(p, 0) = v(p, o).

Then the following lemma is proved as in [10] (p.390).

Lemma 3.6. The function v′ is a valuation on F′. That is, for all a, b ∈ W ′

and all p ∈ Prop, if a ≤′ b and v′(p, a) = t, then v′(p, b) = t.

The interpretation I ′ associated with v′ is defined as in Section 2. Thenby induction on the construction of formulas, we obtain the following hered-itary lemma.

Lemma 3.7. For all a, b ∈ W ′ and all A ∈ Wff, if a ≤′ b and I ′(A, a) = t,

then I ′(A, b) = t.

Lemma 3.8. For all a ∈ W and all A ∈ Wff, I ′(A, a) = I(A, a).

Proof. This is proved by induction on the construction of A. Here we givea proof for two cases; for the other cases including the base case, see [10](p.391).

(i) A is of the form �B. First, suppose that I ′(�B, a) = t. To showI(�B, a) = t, take any b ∈ W satisfying S�ab. Since a, b ∈ W , we haveS′

�ab, and hence I ′(B, b) = t. By the hypothesis of induction, I(B, b) = t,

which is the desired result.For the converse, suppose that I(�B, a) = t. To show I ′(�B, a) = t,

take any b ∈ W ′ satisfying S′�ab. Since a ∈ W , the following two cases arise.(a) When b ∈ W , we have S�ab, and so I(B, b) = t. By the hypothesisof induction, we have I ′(B, b) = t, which is the desired result. (b) Whenb = 0, S�ao∗, and so I(B, o∗) = t. By the hypothesis of induction, we haveI ′(B, o∗) = t. Since o∗ ≤′ 0 by Corollary 3.2, I ′(B, 0) = t by Lemma 3.7;this is the desired result.

(ii) A is of the form ♦B. First, suppose that I ′(♦B, a) = t. Then thereexists b ∈ W ′ such that S′♦ab and I ′(B, b) = t. Since a ∈ W , the following

218 T. Seki

two cases arise. (a) When b ∈ W , S♦ab and I(B, b) = t by the hypothesis ofinduction, and hence we have I(♦B, a) = t. (b) When b = 0, we have S♦ao.Furthermore, since 0 ≤′ o by Corollary 3.2, I ′(B, o) = t by Lemma 3.7, andby the hypothesis of induction, we have I(B, o) = t; therefore I(♦B, a) = t.

For the converse, suppose that I(♦B, a) = t. Then there exists b ∈ Wsuch that S♦ab and I(B, b) = t. So, we have S′♦ab and I ′(B, b) = t by thehypothesis of induction. Therefore I ′(♦B, a) = t.

Thus, we obtain completeness with respect to the class of normal frames.

Theorem 3.9. Let A ∈ Wff. A is a theorem of L iff A is valid in every

normal L-frame.

Proof. The ‘only if’ part is simply Lemma 3.5, so we give a proof of the ‘if’part. Suppose that A is not a theorem of L. By Proposition 2.2, there existsthe (canonical) L-frame F = 〈O,W,R, S�, S♦,∗ , e〉, a valuation v on F andthe interpretation I associated with v such that I(A, o) = f for some o ∈ O.For 0 /∈ W , we can construct the normalization F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉of F at 0 for o ∈ O as described above. By Lemma 3.3 or 3.4, F′ is a normalL-frame. Further, we define a valuation v′ on F′ and the interpretation I ′

associated with v′ as described above. By Lemma 3.8, we have I ′(A, o) = f.Since 0 ≤′ o by Corollary 3.2, I ′(A, 0) = f by Lemma 3.7. This means thatthere exists a normal L-frame F′ in which A is not valid.

As a corollary, we obtain the γ-admissibility of L.

Corollary 3.10. Let L be either LR or LN defined above. Ackermann’s

rule γ is admissible for L.

Proof. Suppose that both ∼ A ∨ B and A are theorems of L. Further, as-sume that B is not a theorem of L. By Theorem 3.9, there exists a normal L-frame F′ = 〈O′,W ′, R′, S′

�, S′♦, ∗′, e〉, a valuation v′ on F′, the interpretation

I ′ associated with v′, and 0 ∈ O′ satisfying 0∗′ = 0 such that I ′(B, 0) = f.Since 0 ∈ O′, I ′(∼ A ∨ B, 0) = I ′(A, 0) = t. Then ‘I ′(∼ A, 0) = t orI ′(B, 0) = t’ and I ′(A, 0) = t, and hence ‘I ′(A, 0∗′) = f or I ′(B, 0) = t’ andI ′(A, 0) = t. Since 0∗′ = 0, ‘I ′(A, 0) = f or I ′(B, 0) = t’ and I ′(A, 0) = t.Thus we have I ′(B, 0) = t, which is a contradiction. Therefore, B is atheorem of L.

4. The γ-admissibility of logics with some Sahlqvist axioms

In this section, we discuss the γ-admissibility of logics with some Sahlqvistformulas as axioms. The γ-admissibility of individual logics with Sahlqvist

The γ-admissibility of Relevant Modal Logics I 219

axioms such as �A → A can be also proved by the method mentionedin Section 3. We consider an extension of this method in order to applyto collective logics with a wider class of Sahlqvist axioms. Proof of theγ-admissibility in the present approach succeeds if we restrict the form ofSahlqvist formulas. In order to achieve the proof, it suffices to show thatthe frame postulates corresponding to given Sahlqvist formulas also hold inthe normalization of the frame under consideration, which corresponds toLemmas 3.3 and 3.4 in Section 3.

For a non-negative integer n, binary relations S′n� and S′n� on W ′ aredefined as in Section 2. Then we have the following.

Lemma 4.1. Let F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 be the normalization of a

given L-frame F = 〈O,W,R, S�, S♦,∗ , e〉 at 0 for o ∈ O.

(1) For all a, b ∈ W and any integer n ≥ 1:

(a) S′n�ab iff Sn�ab (b) S′n�a0 iff Sn

�ao∗

(c) S′n�

0b if Sn�ob (d) S′n

�0b only if Sn

�o∗b

(e) S′n� ab iff Sn�ab (f) S′n� a0 iff Sn

�ao

(g) S′n� 0b if Sn�o∗b (h) S′n� 0b only if Sn

�ob

Also, (a) – (c) and (e) – (g) hold for n = 0.

(2) For all b ∈ W and n = 0, 1: (a) S′n�0b iff Sn�ob, (b) S′n� 0b iff Sn

�o∗b.

Proof. (2) has been proved in Lemma 3.1. (1) is proved simultaneously byinduction on n. The cases n = 0, 1 of (1a) – (1c) and of (1e) – (1g) havebeen proved in Lemma 3.1, and the case n = 1 of (1d) and (1h) is provedeasily. So, we consider the inductive step, that is, the case n ≥ 2.

(1a) The ‘if’ part is proved as follows: Suppose that Sn�ab. Then there

exists c ∈ W such that S�ac and Sn−1�

cb. We have S′�ac and S′n−1�

cb bythe hypothesis of induction, and hence S′n�ab as desired. The ‘only if’ partis proved as follows: Suppose that S′n�ab. Then there exists c ∈ W ′ suchthat S′�ac and S′n−1

�cb. The following two cases arise. (i) When c ∈ W (=

W ′ − {0}), S�ac and Sn−1�

cb by the hypothesis of induction, and henceSn

�ab as desired. (ii) When c = 0, S�ao∗ and Sn−1�

o∗b by the hypothesis ofinduction of (1d), and hence Sn

�ab as desired. The proof for (1e) is similar.(1b) The ‘if’ part is proved as follows: Suppose that Sn

�ao∗. Then thereexists c ∈ W such that S�ac and Sn−1

�co∗. We have S′�ac and S′n−1

�c0 by

the hypothesis of induction, and hence S′n�a0 as desired. The ‘only if’ partis proved as follows: Suppose that S′n�a0. Then there exists c ∈ W ′ such

220 T. Seki

that S′�ac and S′n−1�

c0. The following two cases arise. (i) When c ∈ W , wehave S�ac and Sn−1

�co∗ by the hypothesis of induction, and hence Sn

�ao∗.(ii) When c = 0, we have S�ao∗ and Sn−1

�o∗o∗ by (1a) of Proposition 2.1,

and so we have Sn�ao∗ as desired. The proof for (1f) is similar.

(1c) Suppose that Sn�ob. Then there exists c ∈ W such that S�oc and

Sn−1�

cb. We have S′�0c and S′n−1�

cb by the hypothesis of induction of (1a),and hence S′n�0b as desired. The proof for (1g) is similar.

(1d) Suppose that S′n�0b. Then there exists c ∈ W ′ such that S′�0c andS′n−1

�cb. The following two cases arise. (i) When c ∈ W , we have S�oc and

Sn−1�

cb by hypothesis of induction of (1a). Since o∗ ≤ o by (p8), we haveS�o∗c by (p5). Thus, Sn

�o∗b. (ii) When c = 0, we have S�o∗o∗ and Sn−1�

o∗bby (1a) of Proposition 2.1 and the hypothesis of induction, and hence Sn

�o∗bas desired. The proof for (1h) is similar.

Note that the converse of (1c) and (1g) does not hold in general. Thecase in which the converse holds will be discussed later.

For an L-frame F = 〈O,W,R, S�, S♦,∗ , e〉 and some o ∈ O, let F′ =〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 be the normalization of F at 0 for that o. Below,a S�-term of the form a ↑n1

�∪ · · · ∪ a ↑nk

�is denoted by T (a). Further,

T ′(b) denotes the S′�-term b↑′n1

�∪ · · · ∪ b↑′nk

�corresponding to T (a), where

b↑′ni

�= {c ∈ W ′ | S′ni

�bc}.

A positive formula A is �-free if it contains neither � nor �, and it isnon-modal if it contains only ∧ and ∨.

Lemma 4.2. Let A[p1, . . . , pq] be a positive formula, Ti(c) (1 ≤ i ≤ q) be a

S�-term and T ′i (d) be its corresponding S′�-term. For a, b ∈ W :

(1) If a ∈ A[T1(b), . . . , Tq(b)], then a ∈ A[T ′1(b), . . . , T′q(b)].

(2) If o∗ ∈ A[T1(b), . . . , Tq(b)], then 0 ∈ A[T ′1(b), . . . , T′q(b)].

(3) If a ∈ A[T1(o), . . . , Tq(o)], then a ∈ A[T ′1(0), . . . , T′q(0)].

(4) If o∗ ∈ A[T1(o), . . . , Tq(o)], then 0 ∈ A[T ′1(0), . . . , T′q(0)].

Proof. This lemma is proved simultaneously by induction on the construc-tion of the positive formula A. Here we give a proof for the case with A of theform pi(∈ Prop), �B and �B of only (2) and (3). Moreover, for simplicity,we prove the case q = 1, and omit the subscripts.

(2) (i) A is of the form p. Suppose that o∗ ∈ p[T (b)](= T (b)). It can bewritten by

∨i S

ni

�bo∗. By (1b) of Lemma 4.1,

∨i S′ni

�b0, i.e., 0 ∈ p[T ′(b)] as

desired.

The γ-admissibility of Relevant Modal Logics I 221

(ii) A is of the form �B. Suppose that o∗ ∈ (�B)[T (b)]. To show0 ∈ (�B)[T ′(b)], suppose that S′�0x for any x ∈ W ′. (a) When x ∈ W (=W ′−{0}), we have S�o∗x by (1d) of Lemma 4.1. Then we have x ∈ B[T (b)].By the hypothesis of induction of (1), x ∈ B[T ′(b)], which is the desiredresult. (b) When x = 0, o∗ ∈ B[T (b)] since S�o∗o∗ by (1) of Proposition 2.1.By the hypothesis of induction, 0 ∈ B[T ′(b)], which is the desired result.

(iii) A is of the form �B. Suppose that o∗ ∈ (�B)[T (b)]. Then thereexists x ∈ W such that S�o∗x and x ∈ B[T (b)]. By (1g) of Lemma 4.1 andthe hypothesis of induction of (1), S′�0x and x ∈ B[T ′(b)]. Thus we have0 ∈ (�B)[T ′(b)], which is the desired result.

(3) (i) A is of the form p. Suppose that a ∈ p[T (o)](= T (o)). This can bewritten as

∨i S

ni

�oa. By (1c) of Lemma 4.1,

∨i S′ni

�0a, i.e., a ∈ p[T ′(0)] as

desired.

(ii) A is of the form �B. Suppose that a ∈ (�B)[T (o)]. To showa ∈ (�B)[T ′(0)], suppose that S′�ax for any x ∈ W ′. (a) When x ∈ W ,we have S�ax, and hence x ∈ B[T (o)]. By the hypothesis of induction,x ∈ B[T ′(0)], which is the desired result. (b) When x = 0, we have S�ao∗,and hence o∗ ∈ B[T (o)]. By the hypothesis of induction of (4), 0 ∈ B[T ′(0)],which is the desired result.

(iii) A is of the form �B. Suppose that a ∈ (�B)[T (o)]. Then thereexists x ∈ W such that S�ax and x ∈ B[T (o)]. By (1e) of Lemma 4.1and the hypothesis of induction, S′�ax and x ∈ B[T ′(0)]. Thus we havea ∈ (�B)[T ′(0)], which is the desired result.

Lemma 4.3. Let A[p1, . . . , pq] be a �-free positive formula and T ′i (0) (1 ≤i ≤ q) be S′�-terms. Then 0 ∈ A[T ′1(0), . . . , T

′q(0)].

Proof. By induction on the construction of �-free positive formula A. Herewe give a proof for the case that A is of the form pi(∈ Prop) and �B.

(i) A is of the form pi. Since S′�00 holds, S′n�00 holds for any non-negativeinteger n. So, 0 ∈ 0↑′n1

�∪ · · ·∪0↑′nk

�for any non-negative integers n1, . . . , nk.

Thus, we have 0 ∈ pi[T′1(0), . . . , T

′q(0)].

(ii) A is of the form �B. By the hypothesis of induction, 0 ∈B[T ′1(0), . . . , T

′q(0)]. Since S′�00 holds, we have 0 ∈ (�B)[T ′1(0), . . . , T

′q(0)] as

desired.

We consider Sahlqvist axioms to which the discussion in Section 3 applies.The following lemmas correspond to Lemmas 3.3 and 3.4. In Types 1 – 3,there is no condition for underlying relevant modal logics LR and LN .

222 T. Seki

First, we consider a Sahlqvist formula that is equivalent to to a conjunc-tion of formulas of the following form, which will be called Type 1:

�k(�l1p1 ∧ . . . �lrpr∧ ∼ �m1B1 ∧ · · · ∧ ∼ �msBs) → �n1C1 ∨ · · · ∨ �ntCt,

where Bi (1 ≤ i ≤ s) and Cj (1 ≤ j ≤ t) are �-free positive formulasconstructed from p′1, . . . , p

′q which denotes mutually different variables

in p1, . . . , pr; and k, mi (1 ≤ i ≤ s) and nj (1 ≤ j ≤ t) are either 0or 1.

Examples of Type 1 Sahlqvist formulas are as follows:

�p → �p, �p → p, p → �p, p → ��p, ��p → p, �p → ��p, ��p → �p.

Lemma 4.4. Let L be a logic obtained from LR or LN by adding Type 1

Sahlqvist formulas as axioms. Let F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 be the nor-

malization of a given L-frame F = 〈O,W,R, S�, S♦,∗ , e〉 at 0 for o ∈ O.

Then F′ is also an L-frame.

Proof. We must check the following frame postulate. For all a, b ∈ W ′:

S′k� ab ⇒∨

1≤j≤t

a ∈ �njCj[T′1(b), . . . , T

′q(b)]

or∨

1≤i≤s

b∗′ ∈ �miBi[T′1(b), . . . , T

′q(b)]

Then we consider the following four cases. Excepting (iv), for simplicity, weassume q = s = t = 1 and omit the subscripts.

(i) a, b ∈ W (= W ′ − {0}). Suppose that S′k� ab. By (1e) of Lemma 4.1,

Sk�ab. From the frame postulate, we have a ∈ �nC[T (b)] or b∗ ∈ �mB[T (b)].

By (1) of Lemma 4.2, a ∈ �nC[T ′(b)] or b∗′ ∈ �mB[T ′(b)], which is thedesired result.

(ii) a ∈ W and b = 0. Suppose that S′k� a0. By (1f) of Lemma 4.1, Sk�ao.

From the frame postulate, we have a ∈ �nC[T (o)] or o∗ ∈ �mB[T (o)]. By(3) and (4) of Lemma 4.2, a ∈ �nC[T ′(0)] or 0∗′ ∈ �mB[T ′(0)]. Since0∗′ = 0, this is the desired result.

(iii) a = 0 and b ∈ W . Suppose that S′k� 0b. By (2b) of Lemma 4.1,

Sk�o∗b. (Remark that k is either 0 or 1.) From the frame postulate, we

have o∗ ∈ �nC[T (b)] or b∗ ∈ �mB[T (b)]. By (2) and (1) of Lemma 4.2,0 ∈ �nC[T ′(b)] or b∗′ ∈ �mB[T ′(b)], which is the desired result.

The γ-admissibility of Relevant Modal Logics I 223

(iv) a = b = 0. Suppose that S′k� 00. (1) At least one of m and nis 0. Without loss of generality, suppose n = 0. By Lemma 4.3, we have0 ∈ C[T ′(0)], and hence 0 ∈ �nC[T ′(0)] or 0 ∈ �mB[T ′(0)]. (2) Otherwise,i.e., all m and n are 1. To see 0 ∈ �C[T ′(0)] or 0∗′ ∈ �B[T ′(0)], suppose thatS′

�0x for x ∈ W ′. (a) If x ∈ W , then S�ox. Since S�oo holds, o ∈ �C[T (o)]

or o∗ ∈ �B[T (o)] from the frame postulate. Since S�ox and S�o∗x by(p8) and (p5), we have x ∈ C[T (o)] or x ∈ B[T (o)]. By (3) of Lemma 4.2,x ∈ C[T ′(0)] or x ∈ B[T ′(0)]. (b) If x = 0, then x ∈ C[T ′(0)] or x ∈ B[T ′(0)]by Lemma 4.3, because C and B are �-free positive. From (a) and (b), wehave x ∈ C[T ′(0)] or x ∈ B[T ′(0)]. Therefore, we have the desired result.

We call a Sahlqvist formula that is equivalent to a conjunction of formulasof the following form Type 2:

�l1p1 ∧ . . . �lrpr∧ ∼ �m1B1 ∧ · · · ∧ ∼ �msBs → �n1C1 ∨ · · · ∨ �ntCt,

where Bi (1 ≤ i ≤ s) and Cj (1 ≤ j ≤ t) are constructed fromp′1, . . . , p

′q which denotes mutually different variables in p1, . . . , pr,

using only ∧ and ∨, andmax{m1, . . . ,ms, n1, . . . , nt}−1 ≤ min

{max{li | �lip′j} | 1 ≤ j ≤ q

}.

Examples of Type 2 Sahlqvist formulas are �p → �2p and p ∧ �p ∧ · · · ∧�np → �n+1p.

Lemma 4.5. Let L be a logic obtained from LR or LN by adding Type 2

Sahlqvist formulas as axioms. Let F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 be the nor-

malization of a given L-frame F = 〈O,W,R, S�, S♦,∗ , e〉 at 0 for o ∈ O.

Then F′ is also an L-frame.

Proof. We must check the following corresponding frame postulate. Forall a ∈ W ′:∨

1≤j≤t

a ∈ �njCj [T′1(a), . . . , T ′q(a)] or

∨1≤i≤s

a∗′ ∈ �miBi[T′1(a), . . . , T ′q(a)]

Then we consider the following two cases. For simplicity, we assume q = s =t = 1 and omit the subscripts.

(i) a ∈ W (= W ′ − {0}). From the frame postulate, we have a ∈�nC[T (a)] or a∗ ∈ �mB[T (a)]. By (1) of Lemma 4.2, a ∈ �nC[T ′(a)] or

a∗′ ∈ �mB[T ′(a)], which is the desired result.

(ii) a = 0. We will show that 0 ∈ �nC[T ′(0)] or 0∗′ ∈ �mB[T ′(0)].From the frame postulate, we have o ∈ �nC[T (o)] or o∗ ∈ �mB[T (o)]. When

224 T. Seki

o ∈ �nC[T (o)], suppose that S′n�0x for x ∈ W ′. (a) When x ∈ W , the follow-ing two cases can be considered. If n = 0, 1 or there exist y1, . . . , yn−1 ∈ Wsuch that S′�0y1 & . . . & S′�yn−1x, then Sn

�ox. So we have x ∈ C[T (o)]. By(3) of Lemma 4.2, x ∈ C[T ′(0)], which leads the desired result. Otherwise,there exists q such that 1 ≤ q ≤ n − 1 and S′q

�0x. Since S′

�00 holds, S′w

�0x,

i.e., x ∈ 0 ↑′w� holds for any w ≥ n−1. In the light of the condition n−1 ≤ Mj

for 1 ≤ j ≤ q, where Mj = max{li | �lip′j}, we have x ∈ 0↑′Mj

�⊆ T ′j(0) for

every j. So, we can easily see that x ∈ C[T ′(0)], which leads the desiredresult. (b) When x = 0, 0 ∈ C[T ′(0)] by Lemma 4.3, because C is �-freepositive. This leads the desired result. When o∗ ∈ �mB[T (o)], the proof issimilar. (Note that 0∗′ = 0 and that (p8) is used.)

We call a Sahlqvist formula that is equivalent to a conjunction of formulasof the following form Type 3:

�k(p1 ∧ · · · ∧ pr) → �n1C1 ∨ · · · ∨ �ntCt,

where Cj (1 ≤ j ≤ t) is constructed from p1, . . . , pr using only ∧ and∨, and k − 1 ≤ max{n1, . . . , nt}.

An example of Type 3 Sahlqvist formulas is �2p → �p.

Lemma 4.6. Let L be a logic obtained from LR or LN by adding Type 3

Sahlqvist formulas as axioms. Let F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 be the nor-

malization of a given L-frame F = 〈O,W,R, S�, S♦,∗ , e〉 at 0 for o ∈ O.

Then F′ is also an L-frame.

Proof. We must check the following corresponding frame postulate. Forall a, b ∈ W ′:

S′k� ab ⇒∨

1≤j≤t

a ∈ �njCj[T′1(b), . . . , T

′r(b)]

When k is 0 or 1, Type 3 Sahlqvist formulas are special case of Type 1 andhence a proof has appeared in Lemma 4.4. So, we concentrate on the casek ≥ 2. Then we consider the following four cases. For simplicity, we assumer = t = 1 and omit the subscripts with the exception of some parts of (iii).

(i) a, b ∈ W (= W ′ − {0}). Suppose that S′k� ab. Then Sk�ab by (1e) of

Lemma 4.1. From the frame postulate, we have a ∈ �nC[T (b)]. By (1) ofLemma 4.2, a ∈ �nC[T ′(b)], which is the desired result.

(ii) a ∈ W and b = 0. Suppose that S′k� a0. By (1f) of Lemma 4.1, Sk�ao.

From the frame postulate, we have a ∈ �nC[T (o)]. By (3) of Lemma 4.2,a ∈ �nC[T ′(0)], which is the desired result.

The γ-admissibility of Relevant Modal Logics I 225

(iii) a = 0 and b ∈ W . Suppose that S′k� 0b. Then there existc1, . . . , ck−1 ∈ W ′ such that S′�0c1 & S′�c1c2 & . . . & S′�ck−1b. (a) When

c1, . . . , ck−1 ∈ W , we have Sk�o∗b. From the frame postulate, we have o∗ ∈

�nC[T (b)]. By (2) of Lemma 4.2, 0 ∈ �nC[T ′(b)], which is the desired result.(b) Otherwise, there is l such that 1 ≤ l ≤ k − 1 and S′l�0b. Since S′�00,we have S′m� 0b for any m ≥ k − 1. Let N = max{nj | 1 ≤ j ≤ t} and

J be j taking this N . Since N satisfies k − 1 ≤ N , S′N� 0b. On the otherhand, b ∈ b ↑′0�= T ′i (b) for 1 ≤ i ≤ r since b ≤′ b. Since Cj (1 ≤ j ≤ t)is non-modal, it is clear that b ∈ Cj[T

′1(b), . . . , T

′r(b)]. Thus we have 0 ∈

�NCJ [T ′1(b), . . . , T′r(b)]. Therefore,

∨j 0 ∈ �njCj[T

′1(b), . . . , T

′r(b)], which is

the desired result.

(iv) a = b = 0. By Lemma 4.3, it is clear that 0 ∈ �nC[T ′(0)].

Thus, we have proved the γ-admissibility of logics with Type 1 – 3Sahlqvist formulas.

Theorem 4.7. Let L be a logic obtained from LR or LN by adding Type iSahlqvist formulas as axioms, for i = 1, 2, 3. Then L admits γ.

We have considered the γ-admissibility of logics with various Sahlqvistformulas, but, in general, it is impossible to prove it for logics with Sahlqvistformulas such as �k�lp → �m�np. For example, a Sahlqvist formula��p → �2�p does not belong to Types 1 – 3. The corresponding framepostulate is ∀a∀b(S�ab ⇒ a ∈ �2�p[b ↑1

�]), i.e., ∀a∀b(S�ab & S2

�ac ⇒∃x(S�bx & S�cx)

). Consider a proof of the γ-admissibility of a logic with

the above Sahlqvist formula in terms of the present method. By checkingif the normalization of a frame satisfies the frame postulates, a proof forS′�0b & S′2�0c ⇒ ∃x(S′�bx & S′�cx) (for b, c ∈ W ) does not succeed. But wenotice that the proof does succeed if the frame satisfies the frame conditionS�oo for o ∈ O, which corresponds the disjunctive rules (Q8) and (Q9). So,we consider the cases in which the logic L under consideration contains therules of inference (Q8) and (Q9).

In addition to Lemma 4.1, the following lemma holds by the presence of(r8) and (r9).

Lemma 4.8. Let L be a logic obtained from LR or LN by adding (Q8) and

(Q9) as rules of inference. Let F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 be the normal-

ization of a given L-frame F = 〈O,W,R, S�, S♦,∗ , e〉 at 0 for o ∈ O. For all

b ∈ W and any non-negative integer n:

(a) S′n�0b iff Sn�ob (b) S′n� 0b iff Sn

�o∗b

226 T. Seki

When a logic L contains (Q8) and (Q9), we can show the γ-admissibilityof a logic obtained from L by adding rather richer Sahlqvist formulas. Wecall a Sahlqvist formula that is equivalent to a conjunction of formulas ofthe following form Type 4:

�k(�l1p1 ∧ . . . �lrpr∧ ∼ �m1B1 ∧ · · · ∧ ∼ �msBs) → �n1C1 ∨ · · · ∨ �ntCt,

where Bi (1 ≤ i ≤ s) and Cj (1 ≤ j ≤ t) are �-free positive formulasconstructed from p′1, . . . , p

′q which denotes mutually different variables

in p1, . . . , pr.

We notice that there are no restrictions on k, lh (1 ≤ h ≤ r), mi (1 ≤i ≤ s) and nj (1 ≤ j ≤ t) in Type 4. Thus, Types 1–3 Sahlqvist formulasare all included in Type 4 Sahlqvist formulas. Examples of Type 4 Sahlqvistformulas which do not include Types 1–3 are as follows:

�k�lp → �m�np, �2(p ∧ �p∧ ∼ �2q) → �(p ∧ q), �4(p ∧ q) → p ∨ �q,

where k, l,m, n are any non-negative integers.

The following lemma is proved in a similar manner to Lemma 4.4.

Lemma 4.9. Let L be a logic obtained from LR or LN by adding Type 4

Sahlqvist formulas as axioms and (Q8) and (Q9) as the rules of inference.

Let F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 be the normalization of a given L-frame

F = 〈O,W,R, S�, S♦,∗ , e〉 at 0 for o ∈ O. Then F′ is also an L-frame.

Thus, we have proved the γ-admissibility of logics with Type 4 Sahlqvistformulas.

Theorem 4.10. Let L be a logic obtained from LR or LN by adding and Type

4 Sahlqvist formulas as axioms and (Q8) and (Q9) as the rules of inference.

Then L admits γ.

When Type 4 Sahlqvist formulas are further generalized as the follow-ing Type 5, the γ-admissibility can be proved in LN with (Q8) and (Q9).Though the same method does not work well for the case of LR with (Q8)and (Q9), it does work well when (Q2) is added.

We call a Sahlqvist formula that is equivalent to a conjunction of formulasof the following form Type 5:

�h(�k(�l1p1∧. . . �lrpr∧ ∼ �m1B1∧· · · ∧ ∼ �msBs) → �n1C1∨· · ·∨�ntCt),

The γ-admissibility of Relevant Modal Logics I 227

where Bi (1 ≤ i ≤ s) and Cj (1 ≤ j ≤ t) are �-free positive formulasconstructed from p′1, . . . , p

′q which denotes mutually different variables

in p1, . . . , pr.

Examples of Type 5 Sahlqvist formulas are �2(p → p) and �3(p∧ ∼ p → q).

Lemma 4.11. Let L be a logic obtained from LR with (Q2) or LN by adding

Type 5 Sahlqvist formulas as axioms and (Q8) and (Q9) as the rules of

inference. Let F′ = 〈O′,W ′, R′, S′�, S′♦, ∗′, e〉 be the normalization of a given

L-frame F = 〈O,W,R, S�, S♦,∗ , e〉 at 0 for o ∈ O. Then F′ is also an

L-frame.

Proof. We must check the following corresponding frame postulate. Forall a ∈ O′ and b, c, d, f ∈ W ′:

S′h�ab & R′bcd & S′k� df ⇒∨1≤j≤t

d ∈ �njCj [T′1(f), . . . , T ′q(f)] or

∨1≤i≤s

c∗′ ∈ �miBi[T′1(f), . . . , T ′q(f)]

For simplicity, we assume q = s = t = 1 and omit the subscripts. Fora, b, c ∈ W ′, we define a†, b†, c† ∈ W as follows:

a† =

{a, if a ∈ Wo, if a = 0

b† =

{a, if b ∈ Wo∗, if b = 0

c† =

{c, if c ∈ Wo, if c = 0

(i) d, f ∈ W (= W ′−{0}). (a) When a = b = 0, we have Roc†d and Sk�c†f

by assumption with the help of (p8). Since Sh�oo, d ∈ �nC[T (f)] or f∗ ∈

�mB[T (f)] from the frame postulate. By (1) of Lemma 4.2, d ∈ �nC[T ′(f)]or f∗ ∈ �mB[T ′(f)], which is the desired result. (b) Otherwise, we haveSh

�a†b†, Rb†c†d and Sk�c†f by assumption with the help of (p8). Then d ∈

�nC[T (f)] or f∗ ∈ �mB[T (f)] from the frame postulate. The remainder issimilar to (a).

(ii) d ∈ W,f = 0. (a) When a = b = 0, we have Roc†d and Sk�c†o

by assumption with the help of (1b) of Proposition 2.1. Since Sh�oo, d ∈

�nC[T (o)] or o∗ ∈ �mB[T (o)] from the frame postulate. By (3) and (4) ofLemma 4.2, d ∈ �nC[T ′(0)] or 0∗′ = 0 ∈ �mB[T ′(0)], which is the desiredresult. (b) Otherwise, we have Sh

�a†b†, Rb†c†d and Sk�c†o by assumption

with the help of (p8) and (1b) of Proposition 2.1. Then d ∈ �nC[T (o)] or

o∗ ∈ �mB[T (o)] from the frame postulate. The remainder is similar to (a).

(iii) d = 0, f ∈ W . (a) When a = b = c = 0, we have Sk�o∗f by

assumption. By (3a) of Proposition 2.1, Sh�oo. Moreover, we have Roo∗o∗

228 T. Seki

by applying (r2) to (r1). From the frame postulate, o∗ ∈ �nC[T (f)] or

f∗ ∈ �mB[T (f)]. By (2) and (1) of Lemma 4.2, 0 ∈ �nC[T ′(f)] or f∗′ ∈�mB[T ′(f)], which is the desired result. (b) When a = b = 0 and c ∈ W ,we have Roco∗ and Sk

�cf by assumption. Since Sh�oo by (3a) of Proposition

2.1, o∗ ∈ �nC[T (f)] or f∗ ∈ �mB[T (f)] from the frame postulate. Theremainder is similar to (a). (c) Otherwise, we have Sh

�a†b†, Rb†c†o∗ andSk

�c†f by the assumption with the help of (p8) and (p9). From the framepostulate, o∗ ∈ �nC[T (f)] or f∗ ∈ �mB[T (f)]. The remainder is similarto (a).

(iv) d = f = 0. (a) When a = b = c = 0, we have Sh�oo, Rooo and Sk

�ooby (r1) and (3a) and (1b) of Proposition 2.1. From the frame postulate,o ∈ �nC[T (o)] or o∗ ∈ �mB[T (o)]. If o ∈ �nC[T (o)], to see 0 ∈ �nC[T (0)],take any x ∈ W ′ satisfying that S′n�0x. Then the following two cases arise.If x ∈ W , then we have Sn

�ox by (b) of Lemma 4.8. Then x ∈ C[T (o)],and hence x ∈ C[T ′(0)] by (3) of Lemma 4.2. On the other hand, if x =0, 0 ∈ C[T ′(0)] by Lemma 4.3. In any case, we have x ∈ C[T ′(0)], so0 ∈ �nC[T ′(0)] as desired. If o∗ ∈ �mB[T (o)], we have 0 ∈ �mB[T (0)]by (4) of Lemma 4.2 as desired. (b) When a = b = 0 and c ∈ W , wehave Roco∗ and Sk

�co by assumption. Since Sh�oo by (3a) of Proposition 2.1,

o∗ ∈ �nC[T (o)] or o∗ ∈ �mB[T (o)] from the frame postulate. By (4) ofLemma 4.2, 0 ∈ �nC[T (0)] or 0∗′ = 0 ∈ �mB[T (0)], which is the desiredresult. (c) Otherwise, we have Sh

�a†b†, Rb†c†o∗ and Sk�c†o by the assumption,

(p9) and (1b) of Proposition 2.1. From the frame postulate, o∗ ∈ �nC[T (o)]or o∗ ∈ �mB[T (o)]. The remainder is similar to (b).

Thus, we have proved the γ-admissibility of logics with Type 5 Sahlqvistformulas.

Theorem 4.12. Let L be a logic obtained from LR with (Q2) or LN by

adding Type 5 Sahlqvist formulas as axioms and (Q8) and (Q9) as the rules

of inference. Then L admits γ.

5. Conclusions and Remarks

This paper shows the γ-admissibility of relevant modal logics with Sahlqvistaxioms of the following form in terms of the method of normal models:

�h(�k(�l1p1∧. . . �lrpr∧ ∼ �m1B1∧· · · ∧ ∼ �msBs) → �n1C1∨· · ·∨�ntCt),

where Bi (1 ≤ i ≤ s) and Cj (1 ≤ j ≤ t) are �-free positive formulasconstructed from p1, . . . , pr.

The γ-admissibility of Relevant Modal Logics I 229

Types of Sahlqvist formulas

Type 2

��

��

Type 1

��

��

Type 3

��

��

Type 4�

�

�

�

Type 5�

�

�

�LR, LN

LR with (Q8) and (Q9),LN with (Q8) and (Q9)

LR with (Q2), (Q8) and (Q9),LN with (Q8) and (Q9)

Underlying relevant modal logics

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

We summarize the correspondence between Types of Sahlqvist formulasand underlying relevant modal logics in Figure 1, in which inclusions of rangeof each Type Sahlqvist formulas are expressed. As for the method of normalmodels, the γ-admissibility is related to contraction (or (B1), theorem modusponens) in (non-modal) relevant logics, and to the axiom T, �A → A andA → ♦A, in the modal extensions. This is corroborated by the assumptionof the disjunctive rules related to the above axioms in our logics.

Thus, roughly speaking, the following relevant modal logics may be calledconventionally normal:

(i) G.Cg

�♦and G.K

g

�♦

(ii) any logic extending G.Cg

�♦or G.K

g

�♦by any combination of addi-

tional axioms and rules of inference whose frame postulates are (1)single R, S� or S♦ statements, (2) conjunctions of R and O state-ments, (3) implications whose antecedents involve only a single R,Sn

�, Sn� or O statement, (4) implications whose antecedents involve

‘R and O’, ‘Sn� and O’ or ‘Sn

� and O’ statements, or else (5) implica-tions involving two R, Sn

� and Sn� statements in the antecedent and

a single R, Sn� or Sn

� statement in the succeedent;

(iii) many extensions of G.Cg

�♦or G.K

g

�♦which also include additional

axioms and rules of inference whose implicational frame postulatesinvolve kinds with two R, Sn

� and Sn� statements in the antecedent,

provided the extensions enable the derivation of the disjunctive rules(Q1), (Q2), (Q8) and (Q9).

From the frame-theoretic viewpoint, one may wonder why the followingframe postulates are not adopted.

230 T. Seki

Rabc & c ≤ d ⇒ Rabd a ≤ c & Rbcd ⇒ Rbad

S�ab & b ≤ c ⇒ S�ac S♦ab & c ≤ b ⇒ S♦ac

To achieve a proof of the γ-admissibility in such a case, the existence of thedisjunctive rules (Q1), (Q2), (Q8) and (Q9) must be assumed. So we do notadopt them as the rules of inference.

Note that there may be several individual logics other than those dis-cussed in this paper whose γ-admissibility can be proved by the method ofnormal models. In fact, the γ-admissibility of LR with �(A → A) can beproved by our method.

On the other hand, it remains to be shown whether γ is admissiblefor relevant modal logics with any Sahlqvist axioms. Our method can-not be applied to logics with a full range of Sahlqvist formulas. For ex-ample, a Sahlqvist formula �p ∧ �q → �(p ∧ q) cannot be dealt withusing our method. The corresponding frame postulate is ∀a∀b

(S�ab ⇒

∃c(S�ac & S�ac & b ≤ c)). However, it does not work well for checking

S′�0b ⇒ ∃c(S′�0c & S′�0c & b ≤′ c) for b, c ∈ W . In Part II ([13]), this casewill be discussed. Also, there is the question of whether γ holds for a logicwithout disjunctive rules such as (R6), (R9) and (R10). This case will bediscussed in [14]. These facts may indicate a limitation of the method ofnormal models.

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.

References

[1] Ackermann, W., ‘Begrundung einer strengen Implikation’, Journal of Symbolic Logic

21:113–128, 1956.

[2] Dunn, J.M., and G. Restall, ‘Relevance logic’, in D. Gabbay and F. Guenthner

(eds.), Handbook of Philosophical Logic, 2nd edition, Vol. 6, pp.1–128, Kluwer, Dor-

drecht, 2002.

[3] Mares, E.D., ‘Classically complete modal relevant logics’, Mathematical Logic Quar-

terly 39:165–177, 1993.

[4] Mares, E.D., and R.K. Meyer, ‘The admissibility of γ in R4’, Notre Dame Journal

of Formal Logic 33:197–206, 1992.

[5] Meyer, R.K., ‘Ackermann, Takeuti, and Schnitt; γ for higher-order relevant logics’,

Bulletin of the Section of Logic 5:138–144, 1976.

The γ-admissibility of Relevant Modal Logics I 231

[6] Meyer, R.K., and J. M. Dunn, ‘E, R and γ’, Journal of Symbolic Logic 34:460–474,

1969.

[7] Meyer, R.K., S. Giambrone and R.T. Brady, ‘Where gamma fails’, Studia Logica

43:247–256, 1984.

[8] Routley, R., and R.K. Meyer, ‘The semantics of entailment II’, Journal of Philo-

sophical Logic 1:192–208, 1972.

[9] Routley, R., and R.K. Meyer, ‘The semantics of entailment I’, in H. Leblanc (ed.),

Truth, Syntax and Semantics, pp.194–243, North-Holland, Amsterdam, 1973.

[10] Routley, R., V. Plumwood, R.K. Meyer and R.T. Brady, Relevant Logics and

Their Rivals I, Ridgeview Publishing Company, Atascadero, 1982.

[11] Seki, T., ‘A Sahlqvist theorem for relevant modal logics’, Studia Logica 73:383–411,

2003.

[12] Seki, T., ‘Completeness of relevant modal logics with disjunctive rules’, Reports on

Mathematical Logic 44:3–18, 2009.

[13] Seki, T., ‘The γ-admissibility of relevant modal logics II — the method using metaval-

uations’, Studia Logica, to appear.

[14] Seki, T., ‘Some metacomplete relevant modal logics’, in preparation.

[15] Slaney, J. K., ‘A Metacompleteness theorem for contraction-free relevant logics’,

Studia Logica 43:159–168, 1984.

[16] Slaney, J.K., ‘Reduced models for relevant logics without WI’, Notre Dame Journal

of Formal Logic 28:395–407, 1987.

Takahiro Seki

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