The fixed point and the Craig interpolation properties for sublogics of $\mathbf{IL}$

We study the fixed point property and the Craig interpolation property for sublogics of the interpretability logic $\mathbf{IL}$. We provide a complete description of these sublogics concerning the uniqueness of fixed points, the fixed point property and the Craig interpolation property.


Introduction
De Jongh and Sambin's fixed point theorem [9] for the modal propositional logic GL is one of notable results of modal logical investigation of formalized provability.For any modal formula A, let v(A) be the set of all propositional variables contained in A. A logic L is said to have the fixed point property (FPP) if for any modal formula A(p) in which the propositional variable p appears only in the scope of , there exists a modal formula B such that v(B) ⊆ v(A) \ {p} and L B ↔ A(B).De Jongh and Sambin's theorem states that GL has FPP, and this is understood as a counterpart of the fixed point theorem in formal arithmetic (see [4]).Bernardi [2] also proved the uniqueness of fixed points (UFP) for GL.
A logic L is said to have the Craig interpolation property (CIP) if for any formulas A and B, if L A → B, then there exists a formula C such that v(C) ⊆ v(A) ∩ v(B), L A → C and L C → B. Smoryński [10] and Boolos [3] independently proved that GL has CIP.Smoryński also made an important observation that FPP for GL follows from CIP and UFP.
The interpretability logic IL is an extension of GL in the language of GL equipped with the binary modal operator , where the modal formula A B is read as "T + B is relatively interpretable in T + A".It is natural to ask whether IL also has the properties that hold for GL.Indeed, de Jongh and Visser [6] proved UFP for IL and that IL has FPP.Also Areces, Hoogland and de Jongh [1] proved that IL has CIP.
Ignatiev [7] introduced the sublogic CL of IL as a base logic of the modal logical investigation of the notion of partial conservativity, and proved that CL is complete with respect to relational semantics (that is, regular Veltman semantics).Kurahashi and Okawa [8] also introduced several sublogics of IL, and showed the completeness and the incompleteness of these sublogics with respect to relational semantics.
In this paper, we investigate UFP, FPP and CIP for sublogics of IL shown in Figure 1.Moreover, for technical reasons, we introduce and investigate the notions of UFP and FPP that are restricted versions of UFP and FPP with respect to some particular forms of formulas, respectively.Table 1 summarizes a complete description of these sublogics concerning UFP, UFP, FPP, FPP and CIP.

UFP UFP FPP FPP CIP
Table 1: UFP, UFP, FPP, FPP and CIP for sublogics of IL The paper is organized as follows.In Section 3, we show that UFP holds for extensions of IL − (J4 + ), and that UFP is not the case for sublogics of IL − (J1, J5).We also show that UFP holds for extensions of IL − .In Section 4, we prove that the logic IL − (J2 + , J5) has CIP by modifying a semantical proof of CIP for IL by Areces, Hoogland and de Jongh.We also notice that CIP for IL easily follows from CIP for IL − (J2 + , J5).In Section 5, we observe that FPP for IL − (J2 + , J5) immediately follows from our results in the previous sections.Also we give a syntactical proof of FPP for IL − (J2 + , J5).Moreover, we prove that IL − (J4, J5) has FPP.In Section 6, we provide counter models of FPP for CL and IL − (J1, J5) and a counter model of FPP for IL − (J1, J4 + , J5).As a consequence, we also show that CIP is not the case for these sublogics except for IL − (J2 + , J5) and IL.

IL and its sublogics
The interpretability logic IL is a base logic of modal logical investigations of the notion of relative interpretability (see [12,13]).The language of IL consists of propositional variables p, q, . .., the propositional constant ⊥, the logical connective →, the unary modal operator and the binary modal operator .Other logical connectives, the propositional constant and the modal operator ♦ are introduced as usual abbreviations.The formulas of IL are generated by the following grammar: Definition 2.1.The axioms of the modal propositional logic IL are as follows: L1 All tautologies in the language of IL;

The inference rules of IL are Modus Ponens
A A → B B and Necessitation A A .
The conservativity logic CL is obtained from IL by removing the axiom scheme J5, that was introduced by Ignatiev [7] as a base logic of modal logical investigations of the notion of partial conservativity.Several other sublogics of IL were introduced in [8].The basis for these newly introduced logics is the logic IL − .Definition 2.2.The language of IL − is that of IL, and the axioms of IL − are L1, L2, L3, J3 and J6: A ↔ (¬A) ⊥.The inference rules of For schemata Σ 1 , . . ., Σ n , let IL − (Σ 1 , . . ., Σ n ) be the logic obtained by adding Σ 1 , . . ., Σ n as axiom schemata to IL − .The following schemata J2 + and J4 + were introduced in [8] and [12], respectively: In this paper, we mainly deal with logics having some of the axiom schemata J1, J2 + , J4 + and J5 (see Figure 1 in Section 1).Then we have the following proposition.
Proposition 2.3.Let A, B and C be any formulas.

IL −
¬A → A B.

IL
Proof.In this proof, let B ≡ (A ∧ ¬A).

IL − -frames and models
Definition 2.6.We say that a system W, R, {S w } w∈W is an IL − -frame if it satisfies the following three conditions: 1. W is a non-empty set; 2. R is a transitive and conversely well-founded binary relation on W ; 3. For each w ∈ W , S w is a binary relation on W with ∀x, y ∈ W (xS w y ⇒ wRx).
is a usual satisfaction relation on the Kripke frame W, R with the following additional condition: A formula A is said to be valid in an IL − -frame W, R, {S w } w∈W if for any satisfaction relation on the frame and any w ∈ W , w A.

2.
A is valid in all (finite) IL − -frames in which all axioms of L are valid.

The fixed point and the Craig interpolation properties
For each formula A, let v(A) be the set of all propositional variables contained in A.
Definition 2.9.We say that a formula A is modalized in a propositional variable p if every occurrence of p in A is in the scope of some modal operators or .
Definition 2.10.A logic L is said to have the fixed point property (FPP) if for any propositional variable p and any formula A(p) which is modalized in p, there exists a formula F such that v(F ) ⊆ v(A) \ {p} and L F ↔ A(F ).
Definition 2.11.We say that the uniqueness of fixed points (UFP) holds for a logic L if for any propositional variables p, q and any formula A(p) which is modalized in p and does not contain q, L (p ↔ A(p)) ∧ (q ↔ A(q)) → (p ↔ q).Theorem 2.12 (De Jongh and Visser [6]).

UFP holds for IL.
In particular, de Jongh and Visser showed that a fixed point of a formula A(p) B(p) is A( ) B( ¬A( )).Then a fixed point of every formula A(p) which is modalized in p is explicitly calculable by a usual argument.Definition 2.13.A logic L is said to have the Craig interpolation property (CIP) if for any formulas A and B, there exists a formula Theorem 2.14 (Areces, Hoogland and de Jongh [1]).IL has CIP.

Uniqueness of fixed points
In this section, we investigate the uniqueness of fixed points for sublogics.First, we show that UFP holds for extensions of IL − (J4 + ).Secondly, we prove that UFP is not the case for sublogics of IL − (J1, J5).Then we investigate the newly introduced notion that a formula A(p) is left-modalized in a propositional variable p.We prove that UFP with respect to formulas which are left-modalized in p ( UFP) holds for all extensions of IL − .At last, we discuss Smoryński's implication "CIP + UFP ⇒ FPP" in our framework.

UFP
By adapting Smoryński's argument [11], de Jongh and Visser [6] showed that UFP holds for every logic closed under Modus Ponens and Necessitation, and containing L1, L2, L3, E1 and E2, where Since E1 and E2 are easy consequences of Proposition 2.3.2 and J4 + respectively, we obtain the following theorem.As shown in [6], in the proof of Theorem 3.1, the use of the following substitution principle is essential.Proposition 3.2 (The Substitution Principle).Let A, B and C(p) be any formulas.

IL
Proposition 3.2.2shows that every extension L of IL − (J4 + ) proves (A ↔ B) → (C(A) ↔ C(B)) for any formula C(p) which is modalized in p.We notice that the converse of this statement also holds.Proposition 3.3.Let L be any extension of IL − .Suppose that for any formula Proof.Let A, B and C be any formulas and assume p / ∈ v(C).Then the formula C p is modalized in p.By the supposition, we have On the other hand, we show that UFP does not hold for sublogics of IL − (J1, J5) in general.Proposition 3.4.Let p, q be distinct propositional variables.Then, Proof.We define an IL − -frame F = W, R, {S w } w∈W as follows: • W := {w, x, y}; Obviously, by Proposition 2.7, IL − (J1, J5) is valid in F. Let be a satisfaction relation on F satisfying the following conditions: • w p and w q; • x p and x q; • y p and y q.We prove w (p ↔ ( ¬p)) ∧ (q ↔ ( ¬q)) ∧ ¬(p ↔ q).Since w p and w q, w ¬(p ↔ q) is obvious.We show w (p ↔ ( ¬p)) ∧ (q ↔ ( ¬q)).Since w p and w q, it suffices to prove w ¬p and w ¬( ¬q).
w ¬p: Let z ∈ W be any element with wRz.Then z = x.Since xS w y and y ¬p, we obtain w ¬p.
w ¬( ¬q): Let z ∈ W be any element with xS w z.Then z = x or z = y.In either case, we obtain z q.Since xRy, we conclude w ¬( ¬q).At last, we show w (p ↔ ( ¬p)) ∧ (q ↔ ( ¬q)).Let z ∈ W be such that wRz.Then z = x.Since there is no z ∈ W such that xRz , x ( ¬p) ∧ ( ¬q).Since x p and x q, we have x (p ↔ ( ¬p)) ∧ (q ↔ ( ¬q)).Hence, we obtain w

UFP
Even for extensions of IL − , Proposition 2.3.2 suggests that the uniqueness of fixed points may hold with respect to formulas in some particular forms.From this perspective, we introduce the notion that formulas are left-modalized in p.
Definition 3.5.We say that a formula A is left-modalized in a propositional variable p if A is modalized in p and for any subformula Then we obtain the following version of the substitution principle.
Proposition 3.6.Let A, B and C(p) be any formulas such that for any subformula Proof. 1.This is proved by induction on the construction of C(p).We only prove the case C(p) ≡ D(p) E (By our supposition, p / ∈ v(E)).For any subformula D E of D, it is also a subformula of C, and hence p / ∈ v(E ).Then, by induction hypothesis, we obtain 2. This follows from our proof of 1.
We introduce our restricted versions of UFP and FPP.
Definition 3.7.We say that UFP holds for a logic L if for any formula A(p) Definition 3.8.We say that a logic L has FPP if for any formula A(p) which is left-modalized in p, there exists a formula Then UFP holds for every our sublogic of IL.

Applications of Smoryński's argument
We have shown that UFP and the substitution principle hold for extensions of IL − (J4 + ) (Theorem 3.1 and Proposition 3.2).Then by applying Smoryński's argument [10], we prove that for any appropriate extension of IL − (J4 + ), CIP implies FPP.
Lemma 3.10.Let L be any extension of IL − (J4 + ) that is closed under substituting a formula for a propositional variable.If L has CIP, then L also has FPP.
Proof.Suppose L ⊇ IL − (J4 + ) and L has CIP.Let A(p) be any formula modalized in p. Then by Theorem 3.1, We have Since L has CIP, there exists a formula By substituting A(F ) for p, we get Then Then by applying the axiom scheme L3, we obtain L A(F ) ↔ A(A(F )).From this with (1), we conclude L F ↔ A(F ).Therefore F is a fixed point of A(p) in L.
Also we have shown that UFP and the substitution principle with respect to left-modalized formulas hold for extensions of IL − (Theorem 3.9 and Proposition 3.6).Thus our proof of Lemma 3.10 also works for the following lemma.Lemma 3.11.Let L be any extension of IL − that is closed under substituting a formula for a propositional variable.If L has CIP, then L also has FPP.

The Craig interpolation property
In this section, we prove the following theorem.

Preparations for our proof of Theorem 4.1
In this subsection, we prepare several definitions and prove some lemmas that are used in our proof of Theorem 4.1.Only in this section, we write A instead of IL − (J2 + , J5) A if there is no confusion.Notice that by Proposition 2.3, For a formula A, we define the formula ∼A as follows: For a set X of formulas, by L X we denote the set of all formulas built up from ⊥ and propositional variables occurring in formulas in X.We simply write L A instead of L {A} .For a finite set X of formulas, let X be a conjunction of all elements of X.For the sake of simplicity, only in this section, X → A will be written as X → A.
For a set Φ of formulas, we define Φ := {A : there exists a formula B such that A B ∈ Φ or B A ∈ Φ}.
Definition 4.2.A set Φ of formulas is said to be adequate if it satisfies the following conditions: 1. Φ is closed under taking subformulas and the ∼-operation; Note that for any finite set X of formulas, there exists the smallest finite adequate set Φ containing X.We denote this set by Φ X .Definition 4.3.
1.A pair (Γ 1 , Γ 2 ) of finite sets of formulas is said to be separable if for some formula I ∈ L Γ1 ∩ L Γ2 , Γ 1 → I and Γ 2 → ¬I.A pair is said to be inseparable if it is not separable.
2. A pair (Γ 1 , Γ 2 ) of finite sets of formulas is said to be complete if it is inseparable and We say a finite set is inseparable, then it can be shown that both of Γ 1 and Γ 2 are consistent.
In the rest of this subsection, we fix some sets X and Y of formulas.Put Φ ) is inseparable, then for any formula A ∈ Φ 1 , at least one of (X ∪ {A}, Y ) and (X ∪ {∼A}, Y ) is inseparable.Also a similar statement holds for Φ 2 and Y .Then we obtain the following proposition.
Proposition 4.4.If (X, Y ) is inseparable, then there exists some complete pair let Γ 1 and Γ 2 be the first and the second components of Γ, respectively.Definition 4.5.We define a binary relation Then ≺ is a transitive and conversely well-founded binary relation on We say that ∆ is an A-critical successor of Γ (write Γ ≺ A ∆) if the following conditions are met: From the following claim, Definition 4.6 makes sense.
, then the sets Γ A 1 in clauses 2 and 3 of Definition 4.6 coincide.This is also the case for Γ A 2 .Proof.We prove only for Γ A 1 .It suffices to show that for any formula B, the following are equivalent: , the clause 2 holds by letting I ≡ A.
In order to prove the Truth Lemma (Lemma 4.11), we show the following two lemmas.Lemma 4.9.
We show that (X , Y ) is inseparable.Suppose, for a contradiction, that where κ is an appropriate index set for Y .Then for each j ∈ κ, B j ∈ Φ 2 and there exists a formula Then By (3), J2, J3 and J5, we have On the other hand, from X → J, (By Proposition 2.3.2) By J2, J3 and J5, we have From this and (4), we conclude that ¬(J j∈κ I j ) separates (Γ 1 , Γ 2 ), a contradiction.Therefore (X , Y ) is inseparable.Now let ∆ ∈ K(Φ 1 , Φ 2 ) be a complete pair extending (X , Y ).We have Γ ≺ F ∆ and G, ∼F ∈ ∆ 1 .The other case ¬(G F ) ∈ Γ 2 is proved in a similar way.
we obtain G ∈ ∆ 1 by the inseparability of ∆.We distinguish the following two cases: (Case 1): Let: where κ is an appropriate index set such that for each j ∈ κ, C j ∈ Φ 2 and there exists a formula On the other hand, from X → J, Then by Proposition 2.3.2,we obtain Then by Proposition 2.3.6,we obtain Γ 1 → (( ¬F ∧F ∧¬A) A → G A).
(Case 2): Assume A ∈ Φ 2 .Let: As in Case 1, it can be shown ∼F ∈ Γ 1 ∪ Γ 2 .We prove that (X , Y ) is inseparable.Suppose, for a contradiction, that for some where κ is an appropriate index set such that for each j ∈ κ, B j ∈ Φ 1 and there exists a formula Since G F ∈ Γ 1 , by Lemma 2.5.2, we obtain Γ 1 → G ( ¬F ∧F ).Therefore by Proposition 2.3.6,we obtain On the other hand, from Y → ¬J, From this and ( 6), we conclude ∼G ∈ ∆ 1 because Γ ≺ A ∆.This contradicts the consistency of ∆ 1 .
In both cases, (X , Y ) is inseparable, and hence we can obtain a complete pair Θ ∈ K(Φ 1 , Φ 2 ) which extends (X , Y ) and satisfies the desired conditions.

Proof of Theorem 4.1
We are ready to prove Theorem 4.1.
For finite sequences τ and σ of formulas, let τ ⊆ σ denote that σ is an endextension of τ .Let τ * A be the sequence obtained from τ by adding A as the last element.
We define an IL − -model M = W, R, {S w } w∈W , as follows: Proof.It is clear that R is transitive and conversely well-founded.

Consequences of Theorem 4.1
In this subsection, we prove some consequences of Theorem 4.1 on interpolation properties.First, we prove that IL − (J2 + , J5) has a version of theinterpolation property (see [1]).Secondly, we notice that CIP for IL easily follows from Theorem 4.1.
Before them, we show the generated submodel lemma.Let M = W, R, {S w } w∈W , be any IL − -model such that J4 + is valid in the frame of M .For each r ∈ W , we define an IL − -model M * = W * , R * , {S * w } w∈W * , * as follows: • W * :=↑ (r) ∪ {r}; • xR * y : ⇐⇒ xRy; • yS * x z : ⇐⇒ yS x z; We call M * the submodel of M generated by r.It is easy to show that if J1 is valid in the frame of M , then it is also valid in the frame of M * .This is also the case for J2 + and J5.Also the following lemma is easily obtained.
Lemma 4.12 (The Generated Submodel Lemma).Let M = W, R, {S w } w∈W , be any IL − -model such that J4 + is valid in the frame of M .For any r ∈ W , let M * = W * , R * , {S * w } w∈W * , * be the submodel of M generated by r.Then for any x ∈ W * and formula A, x A if and only if x * A.
Moreover, we give a syntactical proof of FPP for IL − (J2 + , J5) by modifying de Jongh and Visser's proof of FPP for IL.Since the Substitution Principle (Proposition 3.2) holds for extensions of IL − (J4 + ), as usual, it suffices to prove that every formula of the form A(p) B(p) has a fixed point in IL − (J2 + , J5).As a consequence, we show that every formula A(p) which is modalized in p has the same fixed point in IL − (J2 + , J5) as given by de Jongh and Visser.That is, Theorem 5.2.For any formulas A(p) and B(p Lemma 5.3.Let L be any extension of IL − .For any formulas A and B, if Proof.Suppose L ¬A → (A ↔ B).Then, L ¬A → ( ¬A ↔ ¬B) and hence L ¬A → ¬B.By combining this with our supposition, we obtain On the other hand, L ¬B → ( ¬A → ¬A).Hence, by the axiom scheme L3, L ¬B → ¬A.Therefore, by our supposition, Lemma 5.4.For any formulas A and C, . Therefore, by Proposition 3.2.1,we obtain The lemma directly follows from this and Lemma 5.3.
Lemma 5.5.For any formulas A, C and D, Proof.By Lemma 5.4 and R2, we obtain Therefore, by Lemma 2.5.1, we obtain Lemma 5.6.For any formulas B and C, . Therefore, by Proposition 3.2.1,we obtain The lemma is a consequence of this with Lemma 5.3.
Before proving Theorem 5.9, we prepare two lemmas.The following lemma is proved in a usual way by using Proposition 3.6.Therefore, we conclude Proof of Theorem 5.9.Let F :≡ ¬A( ).Since ¬A( ) is left-modalized in p, IL − F ↔ ¬A(F ) by Lemma 5.10.Since IL − (J4) (F ↔ ¬A(F )), by Proposition 3.6.2,we have By Lemma 5.11, A ≡ B: We distinguish the following two cases.
• There exists an n ∈ W such that either x n B or y n B: Then ∀m ≥ n + 1(x m B and y m B).
• For all n ∈ W , x n B and y n B: Then ∀m ≥ 0(x m B and y m B).
A ≡ B C: We distinguish the following five cases.By induction hypothesis, there exists n 0 ∈ ω which is a witness of the statement of the claim for B. We define a natural number n so that for any z ∈ W with the index i, if i ≥ n − 1, then z ¬B ∨ C. We distinguish the following four cases.

• (Case
-∀m ≥ n 0 (x m B and y m B): Then, by (III) and (IV), there are infinitely many odd numbers k such that x k C and y k C. Thus, by induction hypothesis, there exists n 1 ∈ ω such that ∀m ≥ n 1 (x m C and y m C).Then, we define n := max{n 0 , n 1 } + 1.
-∀m ≥ n 0 (x m B and y m B): Then, by (III), there are infinitely many odd numbers k such that x k C. Thus, by induction hypothesis, there exists n 1 ∈ ω such that ∀m ≥ n 1 (x m C).Then, we define n := max{n 0 , n 1 } + 1.
-∀m ≥ n 0 (x m B and y m B): Then, by (IV), there are infinitely many odd numbers k such that y k C. Thus, by induction hypothesis, there exists n 1 ∈ ω such that ∀m ≥ n 1 (y m C).Then, we define n := max{n 0 , n 1 } + 1.
Let m ≥ n and z ∈ W be such that x m Rz and z B. We show that there exists v ∈ W such that zS xm v and v C. Let i be an index of z.If i is odd, then zS xm z and z C by (III) and (IV).Assume that i is even.We distinguish the following two cases.Then IL − (J1, J5) is valid in F. Let be a satisfaction relation on F such that v q and for each n ∈ ω, n q.Claim 7.For any formula A with v(A) ⊆ {q}, there exists n ∈ ω such that ∀m ≥ n(m A) or ∀m ≥ n(m A).
Proof.We prove by induction on the construction of A. We only prove the case of A ≡ B C. We distinguish the following three cases.(I) For any even number k, if k B, then there exists j ≤ k such that j C or v C.

• (Case
(II) For any odd number k, if k B, then there exists j ≤ k such that j C.
By induction hypothesis, there exists an n 0 ∈ ω such that ∀m ≥ n 0 (m B) or ∀m ≥ n 0 (m B).We may assume that n 0 is an odd number.We distinguish the following two cases.
-∀m ≥ n 0 (m B): Let m ≥ n 0 + 1 and k be any element in W satisfying mRk and k B. Since n 0 is odd and n 0 B, there exists j 0 ≤ n 0 such that j 0 C by (II).We distinguish the following three cases.* k is odd: By (II), there exists j ≤ k such that j C. Then kS m j and j C. * k is even and k ≥ n 0 : Since k ≥ j 0 , we have kS m j 0 and j 0 C. * k is even and k < n 0 : By (I), there exists j ≤ k such that j C or v C. Since k < n 0 ≤ m − 1, we obtain k < m − 1.Hence, kS m j and kS m v.
In any case, there exists w ∈ W such that kS m w and w C. Therefore, m B C.
-∀m ≥ n 0 (m B): Let m ≥ n 0 + 1 and k be any element in W satisfying mRk and k B. Then k < n 0 because k B. We distinguish the following two cases.* k is odd: Since there exists j ≤ k such that j C by (II), kS m j and j C. * k is even: By (I), there exists j ≤ k such that j C or v C.
Since k < n 0 ≤ m − 1, we obtain k < m − 1 and hence kS m j and kS m v.
In any case, there exists w ∈ W such that kS m w and w C. Therefore, m B C.
We suppose, towards a contradiction, that there exists a formula A such that v(A) ⊆ {q} and IL − (J1, J5) A ↔ A q. Since IL − (J1, J5) is valid in F, A ↔ A q is also valid in F. Then the following claim holds.Claim 8.For any n ∈ ω, n is even if and only if n A.
Proof.We prove by induction on n.
For n = 0, since obviously 0 A q, we have 0 A. Suppose n > 0 and the claim holds for any natural number less than n.
(⇐): Assume that n is odd.Then nRn−1 and since n−1 is even, n−1 A by induction hypothesis.Let w be the any element in W which satisfies n − 1S n w.By the definition of S n , w ≤ n − 1 and hence w q.Therefore n A q, and thus n A.
(⇒): Assume that n is even.Let m be the any element in W which satisfies nRm and m A. By induction hypothesis, m is even and hence m < n − 1.Then mS n v and v q.Therefore n A q and hence, n A.
As in Corollary 6.2, we obtain the following corollary.Corollary 6.4.Let L be any logic such that IL − ⊆ L ⊆ IL − (J1, J5).Then L has neither FPP nor CIP.
We draw the relations S 3 and S 4 .As in the proof of Theorem 6.3, in the case of xRy for x, y < n, xS n y holds, and the corresponding broken lines are omitted in the figure .Then IL − (J1, J4 + , J5) is valid in F. Let be an arbitrary satisfaction relation on F. Proof.This is proved by induction on the construction of A. We prove only the case of A ≡ B C. We distinguish the following three cases.In our proofs of Theorem 4.1, Theorem 5.2 and Theorem 5.9, the use of the axiom scheme J5 seems inevitable.In fact, CL (= IL − (J1, J2 + )) fails to have FPP.Thus we propose a question whether J5 is necessary or not for FPP and FPP.For this question, we keep in mind the fact that an extension L of K4 proves the axiom scheme L3 if L has FPP.Problem 7.2.
1.For every extension L of IL − (J2 + ), if L has FPP, then does L prove J5? 2. For every extension L of IL − (J4), if L has FPP, then does L prove J5?

Theorem 4 . 1 .
The logic IL − (J2 + , J5) has CIP.Our proof of Theorem 4.1 is based on a semantical proof of CIP for IL due to Areces, Hoogland and de Jongh[1].

Figure 3 :
Figure 3: A counter model of FPP for CL 1): There exists an even number k such that x k B, x k C and x k+1 C. Let m ≥ k + 2.Then, x m Rx k and x k B. For any v ∈ W which satisfies x k S xm v, either v = x k or v = x k+1 by the definition of S xm .Thus, v C. Therefore, we obtain x m B C. Since y m Rx k+1 , we also obtain y m B C in a similar way.• (Case 2): There exists an even number k such that y k B, y k C and x k+1 C. It is proved that k + 2 witnesses the claim as in Case 1. • (Case 3): There exists an odd number k such that x k B and x k C. Let m ≥ k + 1.Then, x m Rx k and x k B. For any v ∈ W satisfying x k S xm v, v = x k by the definition of S xm .Thus, v C. Therefore, we obtain x m B C. Since y m Rx k , y m B C is also proved.• (Case 4): There exists an odd number k such that y k B and y k C. It is proved that k + 1 witnesses the claim as in Case 3. • (Case 5): Otherwise, all of the following conditions are satisfied.(I) For any even number k, if x k B, then either x k C or x k+1 C. (II) For any even number k, if y k B, then either y k C or x k+1 C. (III) For any odd number k, if x k B, then x k C. (IV) For any odd number k, if y k B, then y k C.

-n − 1 ≤
i < m: We obtain z ¬B ∨ C by the definition of n.Since z B, z C. By the definition of S xm , zS xm z. i < n − 1: Then i < m − 1. Therefore zS xm z and zS xm x i+1 .Furthermore, by (I) and (II), we obtain z C or x i+1 C. In any case, there exists v ∈ W such that zS xm v and v C. Therefore, we obtain x m B C. Similarly, we have y m B C. For instance, the relations S 3 and S 4 are shown in the following figure.In the case of xRy for x, y < n, xS n y holds, and the corresponding broken lines are omitted in the figure.

Figure 4 :
Figure 4: A counter model of FPP for IL − (J1, J5) 1): There exists an even number k such that k B, for all j ≤ k, j C and v C: Let m ≥ k + 1.Then mRk and k B. For any w ∈ W which satisfies kS m w, since either w ≤ k or w = v, we obtain w C. Therefore, m B C. • (Case 2): There exists an odd number k such that k B and for all j ≤ k, j C: Let m ≥ k + 1.Then mRk and k B. For any w ∈ W which satisfies kS m w, w C because w ≤ k.Therefore, m B C.• (Case 3): Otherwise: Then, the following conditions (I) and (II) are fulfilled.

Claim 9 .
For any formula A with v(A) = ∅, there exists an n ∈ W such that ∀m ≥ n(m A) or ∀m ≥ n(m A).

•
(Case 1): There exists n > 0 such that n B and for all k ≤ n, k C. Let m ≥ n + 1.Then mRn and n B. Also, for any k ∈ W , if nS m k, then k ≤ n because n = 0. Therefore k C. Thus, m B C.

1 .
J1 is valid in F if and only if for any w, x ∈ W , if wRx, then xS w x. 2. J2 + is valid in F if and only if J4 + is valid in F and for any w ∈ W , S w is transitive.3. J4 + is valid in F if and only if for any w ∈ W , S w is a binary relation on ↑ (w). 4. J5 is valid in F if and only if for any w, x, y ∈ W , wRx and xRy imply xS w y.