Modal completeness of sublogics of the interpretability logic IL
MModal completeness of sublogics of theinterpretability logic IL Taishi Kurahashi and Yuya Okawa
Abstract
We study modal completeness and incompleteness of several sublogicsof the interpretability logic IL . We introduce the sublogic IL − , and provethat IL − is sound and complete with respect to Veltman prestructureswhich are introduced by Visser. Moreover, we prove the modal com-pleteness of twelve logics between IL − and IL with respect to Veltmanprestructures. On the other hand, we prove that eight natural sublogicsof IL are modally incomplete. Finally, we prove that these incompletelogics are complete with respect to generalized Veltman prestructures. Asa consequence of these investigations, we obtain that the twenty logicsstudied in this paper are all decidable. The notion of formalized provability is well-studied in the framework of modallogic. The provability logic of Peano Arithmetic PA is the set of all modalformulas that are verifiable in PA when the modal operator (cid:3) is interpreted asthe provability predicate Pr PA ( x ) of PA . Solovay’s arithmetical completenesstheorem [11] states that the provability logic of PA is exactly axiomatized bythe modal logic GL that is obtained from the smallest normal modal logic K by adding the axiom scheme (cid:3) ( (cid:3) A → A ) → (cid:3) A . Segerberg [9] proved that thelogic GL is sound and complete with respect to the class of all transitive andconversely well-founded finite Kripke frames.The interpretability logic IL is the base logic for modal logical investigationsof the notion of relative interpretability. The language of IL is that of GL withthe additional binary modal operator (cid:66) . The intended meaning of the formula A (cid:66) B is “ PA + B is relatively interpretable in PA + A ”. The inference rules of IL are the same as that of GL , and the axioms of IL are that of GL togetherwith the following axioms: J1 (cid:3) ( A → B ) → A (cid:66) B ; J2 ( A (cid:66) B ) ∧ ( B (cid:66) C ) → A (cid:66) C ; J3 ( A (cid:66) C ) ∧ ( B (cid:66) C ) → ( A ∨ B ) (cid:66) C ;1 a r X i v : . [ m a t h . L O ] M a y A (cid:66) B → ( ♦ A → ♦ B ); J5 ♦ A (cid:66) A .The logic IL is not arithmetically complete by itself. The logic ILM isobtained from IL by adding Montagna’s Principle A (cid:66) B → ( A ∧ (cid:3) C ) (cid:66) ( B (cid:66)(cid:3) C ), then Berarducci [1] and Shavrukov [10] independently proved that ILM isarithmetically sound and complete with respect to arithmetical interpretationsfor PA . Also let ILP be the logic IL with the Persistence Principle A (cid:66) B → (cid:3) ( A (cid:66) B ). Visser [14] proved the arithmetical completeness theorem of the logic ILP with respect to arithmetical interpretations for suitable finitely axiomatizedfragments of PA .These logics have Kripkean semantics. A triple (cid:104) W, R, { S x } x ∈ W (cid:105) is said to bean IL -frame or a Veltman frame if (cid:104) W, R (cid:105) is a Kripke frame of GL and for each x ∈ W , S x is a transitive and reflexive binary relation on ↑ ( x ) := { y ∈ W : xRy } satisfying the following property: ( ∗ ) ∀ y, z ∈ W ( xRy & yRz ⇒ yS x z ). DeJongh and Veltman [3] proved that IL is sound and complete with respect to allfinite IL -frames. Also they proved that the logics ILM and
ILP are sound andcomplete with respect to corresponding classes of finite IL -frames, respectively.The logic IL and its extensions are not only arithmetically significant. Itis known that for extensions of PA , relative interpretability is equivalent toΠ -conservativity, and this equivalence is provable in PA (see [7]). Thereforethe logic ILM is also the logic of Π -conservativity for PA (see also [5]). Onthe other hand, when we consider the logics of Γ-conservativity for Γ (cid:54) = Π ,the principle J5 is no longer arithmetically valid. Ignatiev [6] introduced thelogic of conservativity CL which is obtained from IL by removing J5 , and heproved that the extensions SbCLM and
SCL of CL are exactly the logic ofΠ -conservativity and the logic of Γ-conservativity for Γ ∈ { Σ n , Π n : n ≥ } ,respectively.Ignatiev also proved that CL is complete with respect to Kripkean semantics.We say a triple (cid:104) W, R, { S x } x ∈ W (cid:105) is a CL -frame if it is an IL -frame without theproperty ( ∗ ). Then CL is sound and complete with respect to the class ofall finite CL -frames. The correspondence between J5 and the property ( ∗ ) isexplained in the framework of IL − -frames. A triple (cid:104) W, R, { S x } x ∈ W (cid:105) is calledan IL − -frame or a Veltman prestructure if (cid:104) W, R (cid:105) is a frame of GL and for each x ∈ W , S x is a binary relation on W with ∀ y, z ∈ W ( yS x z ⇒ xRy ). Then Visser[13] stated that for any IL − -frame, the validity of the scheme J5 is equivalentto the property ( ∗ ).Visser also showed, for example, that for any IL − -frame, the validity of J4 is equivalent to the property ∀ x, y, z ∈ W ( yS x z ⇒ xRz ). However, systematicstudy of sublogics of IL through IL − -frames has not been done so far. In thispaper, we do this study, and prove the modal completeness and incompletenessof several sublogics of IL .In Section 2, we introduce the logic IL − that is valid in all IL − -frames. Weintroduce the notion of IL − set -frames that serves a wider class of models than IL − -frames. Then we show that IL − is also valid in all IL − set -frames. In Section2, we investigate several axiom schemata and extensions of IL − . Section 4 isdevoted to proving lemmas used to prove our modal completeness theorems. Ourmodal completeness theorem with respect to IL − -frames is proved in Section5. In Section 6, we prove several natural sublogics of IL are incomplete withrespect to IL − -frames. Finally, in Section 7, we prove these incomplete logicsare complete with respect to IL − set -frames. − In this section, we introduce and investigate the logic IL − . The language of IL − consists of countably many propositional variables p, q, r, . . . , logical constants (cid:62) , ⊥ , connectives ¬ , ∧ , ∨ , → and modal operators (cid:3) , (cid:66) . We show that everytheorem of IL − is valid in all IL − -frames. In fact, we will prove in Section 5 that IL − is sound and complete with respect to the class of all (finite) IL − -frames.The logic IL − is the basis for our logics discussed in this paper.First, we introduce the logic IL − . Definition 2.1.
The axiom schemata of the logic IL − are as follows: L1 All tautologies in the language of IL − ; L2 (cid:3) ( A → B ) → ( (cid:3) A → (cid:3) B ); L3 (cid:3) ( (cid:3) A → A ) → (cid:3) A ; J3 ( A (cid:66) C ) ∧ ( B (cid:66) C ) → ( A ∨ B ) (cid:66) C ; J6 (cid:3) A ↔ ( ¬ A ) (cid:66) ⊥ .The inference rules of IL − are Modus Ponens A A → BB , Necessitation A (cid:3) A , R1 and R2 . Here the rules R1 and R2 are defined as follows: R1 A → BC (cid:66) A → C (cid:66) B ; R2 A → BB (cid:66) C → A (cid:66) C .The logic GL consists of the axiom schemata L1 , L2 and L3 , and of theinference rules Modus Ponens and Necessitation (in the language without (cid:66) ).Hence IL − is an extension of GL .We introduce IL − -frames that are originally introduced by Visser [13] asVeltman prestructures. Definition 2.2.
We say that a triple (cid:104)
W, R, { S x } x ∈ W (cid:105) is an IL − -frame if itsatisfies the following conditions:1. W is a non-empty set;2. R is a transitive and conversely well-founded binary relation on W ;3. For each x ∈ W , S x is a binary relation on W satisfying ∀ y, z ∈ W ( yS x z ⇒ xRy ).For each x ∈ W , let ↑ ( x ) := { y ∈ W : xRy } . In this notation, the third clauseabove states that S x is a relation on ↑ ( x ) × W .A quadruple (cid:104) W, R, { S x } x ∈ W , (cid:13) (cid:105) is called an IL − -model if (cid:104) W, R, { S x } x ∈ W (cid:105) is an IL − -frame and (cid:13) is a binary relation between W and the set of all formulassatisfying the usual conditions for satisfaction with the following conditions: • x (cid:13) (cid:3) A ⇐⇒ ∀ y ∈ W ( xRy ⇒ y (cid:13) A ). • x (cid:13) A (cid:66) B ⇐⇒ ∀ y ∈ W ( y (cid:13) A ⇒ ∃ z ∈ W ( yS x z & z (cid:13) B )).A formula A is said to be valid in an IL − -frame (cid:104) W, R, { S x } x ∈ W (cid:105) if for allsatisfaction relations (cid:13) on the frame and all x ∈ W , x (cid:13) A .We prove that IL − is sound with respect to the class of all IL − -frames. Proposition 2.3.
Every theorem of IL − is valid in all IL − -frames.Proof. We prove by induction on the length of proofs in IL − . Since the modallogic GL is sound with respect to the class of all transitive and conversely well-founded Kripke frames (see [2]), all axioms of GL in the language of IL − arevalid in all IL − -frames. That is, L1 , L2 and L3 are valid in all IL − -frames.Then it suffices to prove that J3 and J6 are valid in all IL − -frames, and therules R1 and R2 preserve the validity.Let x ∈ W be any element and (cid:13) be any satisfaction relation on F . J3 : Suppose x (cid:13) ( A (cid:66) C ) ∧ ( B (cid:66) C ). Let y ∈ W be any element with xRy and y (cid:13) A ∨ B . In either case of y (cid:13) A and y (cid:13) B , there exists z ∈ W suchthat yS x z and z (cid:13) C . Thus we obtain x (cid:13) ( A ∨ B ) (cid:66) C . J6 : ( → ): Suppose x (cid:13) (cid:3) A . Then there is no y ∈ W such that xRy and y (cid:13) ¬ A . Hence x (cid:13) ( ¬ A ) (cid:66) ⊥ .( ← ): Suppose x (cid:13) ( ¬ A ) (cid:66) ⊥ . If there were y ∈ W with xRy and y (cid:13) ¬ A ,then there would be some z ∈ W such that z (cid:13) ⊥ , a contradiction. Thus if xRy , then y (cid:13) A , and this means x (cid:13) (cid:3) A . R1 : Assume A → B is valid in F . Suppose x (cid:13) C (cid:66) A and let y ∈ W besuch that xRy and y (cid:13) C . Then there exists z ∈ W such that yS x z and z (cid:13) A .By the assumption, z (cid:13) B . Then we obtain x (cid:13) C (cid:66) B . R2 : Assume A → B is valid in F . Suppose x (cid:13) B (cid:66) C and let y ∈ W besuch that xRy and y (cid:13) A . By the assumption, y (cid:13) B , and hence there exists z ∈ W such that yS x z and z (cid:13) C . Thus we have x (cid:13) A (cid:66) C .By the rules R1 and R2 , we immediately obtain the following proposition. Proposition 2.4.
Let L be a logic with the inference rules R1 and R2 . If L (cid:96) A ↔ A and L (cid:96) B ↔ B , then L (cid:96) A (cid:66) B ↔ A (cid:66) B . In this paper, we freely use Proposition 2.4 without any mention. In IL − ,the inference rule R2 is strengthened as follows.4 roposition 2.5. IL − (cid:96) (cid:3) ¬ A → A (cid:66) B .2. IL − (cid:96) (cid:3) ( A → B ) → ( B (cid:66) C → A (cid:66) C ) .Proof.
1. Since IL − (cid:96) ⊥ → B , we have IL − (cid:96) A (cid:66) ⊥ → A (cid:66) B by the rule R1 .By the axiom J6 , we obtain IL − (cid:96) (cid:3) ¬ A → A (cid:66) B .2. Since IL − (cid:96) (cid:3) ( A → B ) → (cid:3) ¬ ( A ∧ ¬ B ), we have IL − (cid:96) (cid:3) ( A → B ) → ( A ∧¬ B ) (cid:66) C by 1. Then IL − (cid:96) (cid:3) ( A → B ) ∧ ( B (cid:66) C ) → (( A ∧¬ B ) ∨ B ) (cid:66) C by theaxiom J3 . Since IL − (cid:96) A → ( A ∧¬ B ) ∨ B , we have IL − (cid:96) (( A ∧¬ B ) ∨ B ) (cid:66) C → A (cid:66) C by the rule R2 . Therefore we conclude IL − (cid:96) (cid:3) ( A → B ) ∧ ( B (cid:66) C ) → A (cid:66) C .Thus IL − is deductively equivalent to the system obtained from IL − byreplacing the rule R2 by the axiom scheme (cid:3) ( A → B ) → ( B (cid:66) C → A (cid:66) C ).In Section 5, we will prove that several extensions of IL − are complete withrespect to corresponding classes of IL − -frames. On the other hand, we will alsoprove that several logics are not complete. To prove this incompleteness, we usethe notion of IL − set -frames that is a general notion of IL -frames or generalizedVeltman frames introduced by Verbrugge [12] (see also [15, 8]). Definition 2.6.
A tuple (cid:104)
W, R, { S x } x ∈ W (cid:105) is called an IL − set -frame if it satisfiesthe following conditions:1. W is a non-empty set;2. R is a transitive and conversely well-founded binary relation on W ;3. For each x ∈ W , S x is a relation on W × ( P ( W ) \ {∅} ) such that ∀ y ∈ W, ∀ V ∈ P ( W ) \ {∅} ( yS x V ⇒ xRy );4. (Monotonicity) ∀ x, y ∈ W, ∀ V, U ∈ P ( W ) \ {∅} ( yS x V & V ⊆ U ⇒ yS x U ).As in the definition of IL − -frames, we can define IL − set -models (cid:104) W, R, { S x } x ∈ W , (cid:13) (cid:105) with the following clause: • x (cid:13) A (cid:66) B ⇐⇒ ∀ y ∈ W ( xRy & y (cid:13) A ⇒ ∃ V ∈ P ( W ) \ {∅} ( yS x V & ∀ z ∈ V ( z (cid:13) B ))).Let M = (cid:104) W, R, { S x } x ∈ W , (cid:13) (cid:105) be an IL − -model. For each x ∈ W , we definethe relation S (cid:48) x ⊆ W × ( P ( W ) \ {∅} ) by yS (cid:48) x V : ⇐⇒ ∃ z ∈ V ( yS x z ). Then it isshown that (cid:104) W, R, { S (cid:48) x } x ∈ W (cid:105) is an IL − set -frame. Let (cid:13) (cid:48) be the unique satisfactionrelation on this IL − set -frame satisfying that for any x ∈ W and any propositionalvariable p , x (cid:13) (cid:48) p if and only if x (cid:13) p . Then (cid:104) W, R, { S (cid:48) x } x ∈ W , (cid:13) (cid:48) (cid:105) is an IL − set -model, and for any x ∈ W and any formula A , x (cid:13) A if and only if x (cid:13) (cid:48) A .Therefore, in this sense, every IL − -frame (resp. model) can be recognized as an IL − set -frame (resp. model). We strengthen Proposition 2.3. Proposition 2.7.
Every theorem of IL − is valid in all IL − set -frames. roof. Let F = (cid:104) W, R, { S x } x ∈ W , (cid:13) (cid:105) be an IL − set -model, x ∈ W be any elementand (cid:13) be any satisfaction relation on F . J3 : Suppose x (cid:13) ( A (cid:66) C ) ∧ ( B (cid:66) C ). Let y ∈ W be any element suchthat xRy and y (cid:13) A ∨ B . In either case that y (cid:13) A and y (cid:13) B , there exists V ∈ P ( W ) \{∅} such that yS x V and ∀ z ∈ V ( z (cid:13) C ). Therefore x (cid:13) ( A ∨ B ) (cid:66) C . J6 : This follows from the following equivalences: x (cid:13) (cid:3) A ⇐⇒ ∀ y ( xRy ⇒ y (cid:13) A ) , ⇐⇒ ∀ y ( xRy & y (cid:13) ¬ A ⇒ ∃ V ( yS x V & ∀ z ∈ V ( z (cid:13) ⊥ ))) , ⇐⇒ x (cid:13) ( ¬ A ) (cid:66) ⊥ . R1 : Assume that A → B is valid in F . Suppose x (cid:13) B (cid:66) C and let y ∈ W be any element with xRy and y (cid:13) A . Then y (cid:13) B by the assumption, andhence there exists V ∈ P ( W ) \ {∅} such that yS x V and ∀ z ∈ V ( z (cid:13) C ). Thuswe have x (cid:13) A (cid:66) C . R2 : Assume that A → B is valid in F . Suppose x (cid:13) C (cid:66) A . Let y ∈ W besuch that xRy and y (cid:13) C . Then there exists V ∈ P ( W ) \ {∅} such that yS x V and ∀ z ∈ V ( z (cid:13) A ). For each z ∈ V , z (cid:13) B by the assumption. Therefore weconclude x (cid:13) C (cid:66) B . − In this section, we investigate several additional axiom schemata and severalextensions of IL − . Let Σ , . . . , Σ n be axiom schemata. Then IL − (Σ , . . . , Σ n )is the logic IL − together with the axiom schemata Σ , . . . , Σ n . Let L be anextension of IL − . We say that L is complete with respect to finite IL − -frames (resp. IL − set -frames) if for any formula A , L (cid:96) A if and only if A is valid in allfinite IL − -frames (resp. IL − set -frames) where all axioms of L are valid. In this subsection, we investigate the axiom scheme J1 . J1 (cid:3) ( A → B ) → A (cid:66) B .First, we show that the following axiom scheme J1 (cid:48) is equivalent to J1 over IL − . J1 (cid:48) A (cid:66) A . Proposition 3.1.
The logics IL − ( J1 ) and IL − ( J1 (cid:48) ) are deductively equivalent.Proof. IL − ( J1 ) (cid:96) J1 (cid:48) : This is because IL − (cid:96) (cid:3) ( A → A ) and IL − ( J1 ) (cid:96) (cid:3) ( A → A ) → A (cid:66) A . IL − ( J1 (cid:48) ) (cid:96) J1 : By Proposition 2.5.2, IL − (cid:96) (cid:3) ( A → B ) → ( B (cid:66) B → A (cid:66) B ).Since IL − ( J1 (cid:48) ) (cid:96) B (cid:66) B , we obtain the desired result.6herefore, in this paper, we sometimes identify the axiom schemata J1 and J1 (cid:48) .The following proposition is due to Visser. Proposition 3.2 (Visser [13]) . Let F = (cid:104) W, R, { S x } x ∈ W (cid:105) be any IL − -frame.Then the following are equivalent:1. J1 is valid in F .2. ∀ x, y ∈ W ( xRy ⇒ yS x y ) . We prove a similar equivalence concerning IL − set -frames. Proposition 3.3.
Let F = (cid:104) W, R, { S x } x ∈ W (cid:105) be any IL − set -frame. Then thefollowing are equivalent:1. J1 is valid in F .2. ∀ x, y ∈ W ( xRy ⇒ yS x { y } ) .Proof. (1 ⇒ J1 is valid in F . Suppose xRy . Let (cid:13) be asatisfaction relation on F satisfying for any u ∈ W , u (cid:13) p if and only if u = y for some fixed propositional variable p . Then xRy and y (cid:13) p . Since x (cid:13) p (cid:66) p ,there exists V ∈ P ( W ) \ {∅} such that yS x V and ∀ z ∈ V ( z (cid:13) p ). By thedefinition of (cid:13) , V = { y } because V is non-empty. We obtain yS x { y } .(2 ⇒ ∀ x, y ∈ W ( xRy ⇒ yS x { y } ). Let y ∈ W be such that xRy and y (cid:13) A . Then yS x { y } and ∀ z ∈ { y } ( z (cid:13) A ). Thus we conclude x (cid:13) A (cid:66) A . This subsection is devoted to investigating the axiom scheme J4 . J4 ( A (cid:66) B ) → ( ♦ A → ♦ B ).First, we prove that J4 is equivalent to the following axiom scheme J4 (cid:48) over IL − . The principle J4 (cid:48) is introduced in Visser [13]. J4 (cid:48) ( A (cid:66) B ) → ( B (cid:66) ⊥ → A (cid:66) ⊥ ). Proposition 3.4.
The logics IL − ( J4 ) and IL − ( J4 (cid:48) ) are deductively equivalent.Proof. This is because IL − (cid:96) ( ♦ A → ♦ B ) ↔ ( B (cid:66) ⊥ → A (cid:66) ⊥ ) by J6 .Since J4 (cid:48) is a particular instance of the axiom scheme J2 , we obtain thefollowing corollary. Corollary 3.5. IL − ( J2 ) (cid:96) J4 . J4 does not behave well by itself in the sense of modalcompleteness. In fact, we will prove in Section 6 that for instance, IL − ( J4 )is not complete with respect to corresponding class of IL − -frames. Thus weintroduce a well-behaved axiom scheme J4 + whose corresponding class of IL − -frames is same as that of J4 . The principle J4 + is originally introduced inVisser [13]. We also introduce the schemata J4 (cid:48) + and J4 (cid:48)(cid:48) + as follows: J4 + (cid:3) ( A → B ) → ( C (cid:66) A → C (cid:66) B ). J4 (cid:48) + (cid:3) A → ( C (cid:66) ( A → B ) → C (cid:66) B ). J4 (cid:48)(cid:48) + (cid:3) A → ( C (cid:66) B → C (cid:66) ( A ∧ B )). Proposition 3.6.
The logics IL − ( J4 + ) , IL − ( J4 (cid:48) + ) and IL − ( J4 (cid:48)(cid:48) + ) are deduc-tively equivalent.Proof. IL − ( J4 + ) (cid:96) J4 (cid:48) + : Since IL − (cid:96) A → (( A → B ) → B ), IL − (cid:96) (cid:3) A → (cid:3) (( A → B ) → B ). Then we have IL − ( J4 + ) (cid:96) (cid:3) A → ( C (cid:66) ( A → B ) → C (cid:66) B ). IL − ( J4 (cid:48) + ) (cid:96) J4 (cid:48)(cid:48) + : Since IL − (cid:96) B → ( A → A ∧ B ), IL − (cid:96) C (cid:66) B → C (cid:66) ( A → A ∧ B ) by the rule R1 . Then IL − ( J4 (cid:48) + ) (cid:96) (cid:3) A → ( C (cid:66) B → C (cid:66) ( A ∧ B )). IL − ( J4 (cid:48)(cid:48) + ) (cid:96) J4 + : By the axiom J4 (cid:48)(cid:48) + , we have IL − ( J4 (cid:48)(cid:48) + ) (cid:96) (cid:3) ( A → B ) → ( C (cid:66) A → C (cid:66) (( A → B ) ∧ A )). Since IL − (cid:96) ( A → B ) ∧ A → B , we have IL − (cid:96) C (cid:66) (( A → B ) ∧ A ) → C (cid:66) B by the rule R1 . Thus IL − ( J4 (cid:48)(cid:48) + ) (cid:96) (cid:3) ( A → B ) → ( C (cid:66) A → C (cid:66) B ).The axiom scheme J4 + is a strengthening of the inference rule R1 , andhence in extensions of IL − ( J4 + ), the inference rule R1 is redundant.We show that J4 + implies J4 over IL − . Proposition 3.7. IL − ( J4 + ) (cid:96) J4 .Proof. Since IL − (cid:96) B (cid:66) ⊥ → (cid:3) ¬ B by J6 , IL − (cid:96) B (cid:66) ⊥ → (cid:3) ( B → ⊥ ). Thenby J4 + , we have IL − ( J4 + ) (cid:96) A (cid:66) B → ( B (cid:66) ⊥ → A (cid:66) ⊥ ). By Proposition 3.4,we conclude IL − ( J4 + ) (cid:96) J4 .We prove that J4 and J4 + have the same frame condition with respect to IL − -frames. This is stated in Visser [13]. Proposition 3.8 (Visser [13]) . Let F = (cid:104) W, R, { S x } x ∈ W (cid:105) be any IL − -frame.Then the following are equivalent:1. J4 + is valid in F .2. J4 is valid in F .3. ∀ x, y, z ∈ W ( yS x z ⇒ xRz ) . On the other hand, J4 and J4 + can be distinguished by considering IL − set -frames. That is, these logics have different frame conditions with respect to IL − set -frames. 8 roposition 3.9. Let F = (cid:104) W, R, { S x } x ∈ W (cid:105) be any IL − set -frame. Then thefollowing are equivalent:1. J4 is valid in F .2. ∀ x, y ∈ W, ∀ V ∈ P ( W ) \ {∅} ( yS x V ⇒ ∃ z ∈ V ( xRz )) .Proof. (1 ⇒ J4 is valid in F , and suppose yS x V . Let (cid:13) be asatisfaction relation on F such that for any u ∈ W , u (cid:13) p if and only if u = y ,and u (cid:13) q if and only if u ∈ V for some fixed propositional variables p and q .Then x (cid:13) p (cid:66) q because V is non-empty. Since xRy and y (cid:13) p , we have x (cid:13) ♦ p .Then by the validity of J4 , x (cid:13) ♦ q . Hence there exists z ∈ W such that xRz and z (cid:13) q . By the definition of (cid:13) , we obtain z ∈ V .(2 ⇒ ∀ x, y ∈ W, ∀ V ∈ P ( W ) \ {∅} ( yS x V ⇒ ∃ z ∈ V ( xRz )).Suppose x (cid:13) ( A (cid:66) B ) ∧ ♦ A . Then there exists y ∈ W such that xRy and y (cid:13) A , and also there exists a non-empty V ∈ P ( W ) such that yS x V and ∀ z ∈ V ( z (cid:13) B ). By the assumption, xRz for some z ∈ V . Hence x (cid:13) ♦ B . Thisshows that J4 is valid in F . Proposition 3.10.
Let F = (cid:104) W, R, { S x } x ∈ W (cid:105) be any IL − set -frame. Then thefollowing are equivalent:1. J4 + is valid in F .2. ∀ x, y ∈ W, ∀ V ∈ P ( W ) \ {∅} ( yS x V ⇒ yS x ( V ∩ ↑ ( x ))) .Proof. (1 ⇒ J4 + is valid in F . Suppose yS x V . Let (cid:13) be asatisfaction relation on F such that for any u ∈ W , u (cid:13) p if and only if u = y , u (cid:13) q if and only if u ∈ V , and u (cid:13) r if and only if ( u ∈ V and xRu ), for somefixed propositional variables p, q and r . Then x (cid:13) p (cid:66) q because V is non-empty.Let y ∈ W be any element such that xRy and y (cid:13) q , then y ∈ V and xRy .This means y (cid:13) r . Therefore x (cid:13) (cid:3) ( q → r ). By the validity of J4 + , we obtain x (cid:13) p (cid:66) r . Since xRy and y (cid:13) p , there exists a non-empty U ∈ P ( W ) such that yS x U and ∀ z ∈ U ( z (cid:13) r ). By the definition of (cid:13) , for each z ∈ U , z ∈ V and xRz . That is, U ⊆ V ∩ ↑ ( x ). By Monotonicity, we conclude yS x ( V ∩ ↑ ( x )).(2 ⇒ ∀ x, y ∈ W, ∀ V ∈ P ( W ) \ {∅} ( yS x V ⇒ yS x ( V ∩ ↑ ( x ))).Suppose x (cid:13) ( A (cid:66) B ) ∧ (cid:3) ( B → C ). Let y ∈ W be such that xRy and y (cid:13) A ,then there exists a non-empty V ∈ P ( W ) such that yS x V and ∀ z ∈ V ( z (cid:13) B ).By the assumption, yS x ( V ∩ ↑ ( x )). In particular, for each z ∈ V ∩ ↑ ( x ), z (cid:13) B and z (cid:13) B → C , and hence z (cid:13) C . We have shown x (cid:13) A (cid:66) C . Therefore J4 + is valid in F .By Proposition 3.10, when we consider logics containing J4 + , then for each IL − set -frame (cid:104) W, R, { S x } x ∈ W (cid:105) , we may assume that for every x ∈ W , S x is arelation on ↑ ( x ) × ( P ( ↑ ( x )) \ {∅} ). This is required in the original definitionof IL set -frames (see [8, 15]). 9 .3 The axiom scheme J2 In this subsection, we discuss the axiom scheme J2 . J2 ( A (cid:66) B ) ∧ ( B (cid:66) C ) → A (cid:66) C .As in the case of the axiom J4 , we introduce the following new axiomschemata J2 + and J2 (cid:48) + which are stronger than J2 . J2 + ( A (cid:66) ( B ∨ C )) ∧ ( B (cid:66) C ) → A (cid:66) C . J2 (cid:48) + ( A (cid:66) B ) ∧ (( B ∧ ¬ C ) (cid:66) C ) → A (cid:66) C . Proposition 3.11.
The logics IL − ( J2 + ) and IL − ( J2 (cid:48) + ) are deductively equiv-alent.Proof. IL − ( J2 + ) (cid:96) J2 (cid:48) + : Since IL − (cid:96) B → ( B ∧ ¬ C ) ∨ C , we have IL − (cid:96) A (cid:66) B → A (cid:66) (( B ∧ ¬ C ) ∨ C ) by the rule R1 . Then IL − (cid:96) ( A (cid:66) B ) ∧ (( B ∧¬ C ) (cid:66) C ) → ( A (cid:66) (( B ∧ ¬ C ) ∨ C )) ∧ (( B ∧ ¬ C ) (cid:66) C ). Thus IL − ( J2 + ) (cid:96) ( A (cid:66) B ) ∧ (( B ∧ ¬ C ) (cid:66) C ) → A (cid:66) C . IL − ( J2 (cid:48) + ) (cid:96) J2 + : Since IL − (cid:96) ( B ∨ C ) ∧ ¬ C → B , IL − (cid:96) B (cid:66) C → (( B ∨ C ) ∧ ¬ C ) (cid:66) C by the rule R2 . Then IL − (cid:96) ( A (cid:66) ( B ∨ C )) ∧ ( B (cid:66) C ) → ( A (cid:66) ( B ∨ C )) ∧ (( B ∨ C ) ∧ ¬ C ) (cid:66) C . Therefore we conclude IL − ( J2 (cid:48) + ) (cid:96) ( A ∧ ( B ∨ C )) ∧ ( B (cid:66) C ) → A (cid:66) C .The axiom scheme J2 + is slightly stronger than J2 . In fact, the followingproposition shows that J2 and J2 + are equivalent over the logic IL − ( J1 ). Proposition 3.12. IL − ( J2 + ) (cid:96) J2 .2. IL − ( J1 , J2 ) (cid:96) J2 + .Proof.
1. This is because IL − (cid:96) A (cid:66) B → A (cid:66) ( B ∨ C ).2. Since IL − ( J1 ) (cid:96) B (cid:66) C → ( B ∨ C ) (cid:66) C by J1 and J3 , we have IL − ( J1 , J2 ) (cid:96) ( A (cid:66) ( B ∨ C )) ∧ ( B (cid:66) C ) → A (cid:66) C .We proved in Corollary 3.5 that IL − ( J2 ) proves J4 . Analogously, we provethat J2 + is stronger than J4 + over IL − . Proposition 3.13. IL − ( J2 + ) (cid:96) J4 + .Proof. Since IL − (cid:96) (cid:3) ( A → B ) → (cid:3) ¬ ( A ∧ ¬ B ), we have IL − (cid:96) (cid:3) ( A → B ) → ( A ∧ ¬ B ) (cid:66) B by Proposition 2.5.1. Then IL − ( J2 (cid:48) + ) (cid:96) (cid:3) ( A → B ) → ( C (cid:66) A → C (cid:66) B ). By Proposition 3.11, we obtain IL − ( J2 + ) (cid:96) J4 + .The following corollary is straightforward from Proposition 3.12.2 and Propo-sition 3.13. Corollary 3.14. IL − ( J1 , J2 ) (cid:96) J4 + .
10e prove that J2 and J2 + have the same frame condition. Proposition 3.15.
Let F = (cid:104) W, R, { S x } x ∈ W (cid:105) be any IL − -frame. Then thefollowing are equivalent:1. J2 + is valid in F .2. J2 is valid in F .3. J4 is valid in F and for any x ∈ W , S x is transitive.Proof. (1 ⇒ ⇒ ⇒ J4 is valid in F and for any x ∈ W , S x is transitive.Suppose x (cid:13) ( A (cid:66) ( B ∨ C )) ∧ ( B (cid:66) C ). Let y ∈ W be any element such that xRy and y (cid:13) A . Then there exists z ∈ W such that yS x z and z (cid:13) B ∨ C . Weshall show that there exists u ∈ W such that yS x u and u (cid:13) C . If z (cid:13) C , thenthis is done. If z (cid:49) C , then z (cid:13) B . Since xRz , by our supposition, there exists u ∈ W such that zS x u and u (cid:13) C . By the transitivity of S x , we obtain yS x u .Therefore we conclude x (cid:13) A (cid:66) C . That is to say, J2 + is valid in F .We prove that J2 and J2 + have different frame conditions with respect to IL − set -frames. Proposition 3.16.
Let F = (cid:104) W, R, { S x } x ∈ W (cid:105) be an IL − set -frame. Then thefollowing are equivalent:1. J2 is valid in F .2. J4 is valid in F and ∀ x, y ∈ W, ∀ V ∈ P ( W ) \ {∅} ( yS x V & ∀ z ∈ V ∩ ↑ ( x )( zS x U z ) ⇒ yS x ( (cid:83) z ∈ V ∩↑ ( x ) U z )) .Proof. (1 ⇒ J2 is valid in F . Then by Corollary 3.5, J4 isvalid in F . Suppose yS x V and ∀ z ∈ V ∩ ↑ ( x )( zS x U z ). Let (cid:13) be a satisfactionrelation on F such that for any u ∈ W , u (cid:13) p if and only if u = y , u (cid:13) q if andonly if u ∈ V , and u (cid:13) r if and only if ∃ z ∈ V ∩ ↑ ( x )( u ∈ U z ). Then x (cid:13) p (cid:66) q and x (cid:13) q (cid:66) r . By the validity of J2 , x (cid:13) p (cid:66) r . Since xRy and y (cid:13) p , thereexists a non-empty U ∈ P ( W ) such that yS x U and ∀ w ∈ U ( w (cid:13) r ). By thedefinition of (cid:13) , U ⊆ (cid:83) z ∈ V ∩↑ ( x ) U z . By Monotonicity, yS x ( (cid:83) z ∈ V ∩↑ ( x ) U z ).(2 ⇒ J4 is valid in F and ∀ x, y ∈ W, ∀ V ∈ P ( W ) \{ } ( yS x V & ∀ z ∈ V ∩ ↑ ( x )( zS x U z ) ⇒ yS x ( (cid:83) z ∈ V ∩↑ ( x ) U z )). Suppose x (cid:13) ( A (cid:66) B ) ∧ ( B (cid:66) C ). Let y ∈ W be any element with xRy and y (cid:13) A . Thenthere exists a non-empty V ∈ P ( W ) such that yS x V and ∀ z ∈ V ( z (cid:13) B ).Since J4 is valid in F , we have V ∩ ↑ ( x ) (cid:54) = ∅ . Then for each z ∈ V ∩ ↑ ( x ),there exists U z ∈ P ( W ) \ {∅} such that zS x U z and ∀ w ∈ U z ( w (cid:13) C ). Bythe assumption, yS x ( (cid:83) z ∈ V ∩↑ ( x ) U z ) because the set (cid:83) z ∈ V ∩↑ ( x ) U z is non-empty.Also ∀ w ∈ (cid:83) z ∈ V ∩↑ ( x ) U z ( w (cid:13) C ). We have shown w (cid:13) A (cid:66) C . Hence J2 is validin F . 11he condition ∀ x, y ∈ W, ∀ V ∈ P ( W ) \{∅} ( yS x V & ∀ z ∈ V ∩ ↑ ( x )( zS x U z ) ⇒ yS x ( (cid:83) z ∈ V ∩↑ ( x ) U z )) stated in Proposition 3.16 is required in the original defini-tion of IL set -frames. Proposition 3.17.
Let F = (cid:104) W, R, { S x } x ∈ W (cid:105) be any IL − set -frame. Then thefollowing are equivalent:1. J2 + is valid in F .2. J4 is valid in F and ∀ x, y ∈ W, ∀ V , V ∈ P ( W ) \ {∅} ( yS x ( V ∪ V ) & ∀ z ∈ V ∩ ↑ ( x )( zS x U z ) ⇒ yS x ( (cid:83) z ∈ V ∩↑ ( x ) U z ∪ V )) .Proof. (1 ⇒ J2 + is valid in F . Since IL − ( J2 + ) (cid:96) J4 , J4 is alsovalid in F . Suppose yS x ( V ∪ V ) and ∀ z ∈ V ∩ ↑ ( x )( zS x U z ). Let (cid:13) be asatisfaction relation on F such that for any u ∈ W , u (cid:13) p if and only if u = y , u (cid:13) q if and only if u ∈ V , and u (cid:13) r if and only if ( ∃ z ∈ V ∩ ↑ ( x )( u ∈ U z )or u ∈ V ). Then x (cid:13) p (cid:66) ( q ∨ r ) and x (cid:13) q (cid:66) r . By the validity of J2 + , x (cid:13) p (cid:66) r . Since xRy and y (cid:13) p , there exists a non-empty U ∈ P ( W ) \ {∅} such that yS x U and ∀ w ∈ U ( w (cid:13) r ). Then by the definition of (cid:13) , we have U ⊆ (cid:83) z ∈ V ∩↑ ( x ) U z ∪ V . By Monotonicity, yS x ( (cid:83) z ∈ V ∩↑ ( x ) U z ∪ V ).(2 ⇒ J4 is valid in F and ∀ x, y ∈ W, ∀ V , V ∈ P ( W ) \{∅} ( yS x ( V ∪ V ) & ∀ z ∈ V ∩ ↑ ( x )( zS x U z ) ⇒ yS x ( (cid:83) z ∈ V ∩↑ ( x ) U z ∪ V )). Let x (cid:13) ( A (cid:66) ( B ∨ C )) ∧ ( B (cid:66) C ). Let y ∈ W be such that xRy and y (cid:13) A , thenthere exists a non-empty V ∈ P ( W ) such that yS x V and ∀ z ∈ V ( z (cid:13) B ∨ C ).Since J4 is valid, we have V ∩ ↑ ( x ) (cid:54) = ∅ . Let V := { z ∈ V : z (cid:13) B } and V := { z ∈ V : z (cid:13) C } , then V = V ∪ V . In particular, for each z ∈ V ∩ ↑ ( x ),there exists a non-empty U z ∈ P ( W ) such that zS x U z and ∀ w ∈ U z ( w (cid:13) C ).By the assumption, we have yS x ( (cid:83) z ∈ V ∩↑ ( x ) U z ∪ V ) because (cid:83) z ∈ V ∩↑ ( x ) U z ∪ V is non-empty. Since ∀ w ∈ (cid:83) z ∈ V ∩↑ ( x ) U z ∪ V ( w (cid:13) C ), we obtain w (cid:13) A (cid:66) C .Therefore J2 + is valid in F . We investigate J5 . J5 ♦ A (cid:66) A .The following proposition is stated in Visser [13]. Proposition 3.18 (Visser [13]) . Let F = (cid:104) W, R, { S x } x ∈ W (cid:105) be any IL − -frame.The following are equivalent:1. J5 is valid in F .2. ∀ x, y, z ∈ W ( xRy & yRz ⇒ yS x z ) . Proposition 3.19.
Let F = (cid:104) W, R, { S x } x ∈ W (cid:105) be any IL − set -frame. Then thefollowing are equivalent:1. J5 is valid in F . . ∀ x, y, z ∈ W ( xRy & yRz ⇒ yS x { z } ) .Proof. (1 ⇒ J5 is valid in F . Suppose xRy and yRz . Let (cid:13) be a satisfaction relation on F such that for any u ∈ W , u (cid:13) p if and only if u = z for some fixed propositional variable p . Then yRz and z (cid:13) p , and hence xRy and y (cid:13) ♦ p . Since x (cid:13) ♦ p (cid:66) p , there exists a non-empty set V ∈ P ( W )such that yS x V and ∀ w ∈ V ( w (cid:13) p ). By the definition of (cid:13) , we have V = { z } .Therefore yS x { z } .(2 ⇒ ∀ x, y, z ∈ W ( xRy & yRz ⇒ yS x { z } ). Let y ∈ W be anyelement such that xRy and y (cid:13) ♦ A . Then there exists z ∈ W such that yRz and z (cid:13) A . By the assumption, yS x { z } . Since ∀ w ∈ { z } ( w (cid:13) A ), we obtain x (cid:13) ♦ A (cid:66) A . That is, J5 is valid in F .The condition stated in the second clause in Proposition 3.19 is required inthe original definition of IL set -frames. In this subsection, we show that the logics CL and IL are exactly IL − ( J1 , J2 )and IL − ( J1 , J2 , J5 ), respectively. Since IL − ( J1 , J2 ) proves J2 + and J4 + byProposition 3.12 and Corollary 3.14, our logics studied in this paper are actuallysublogics of IL . The logic CL is GL plus J1 , J2 , J3 and J4 . Also the logic IL is CL plus J5 . Proposition 3.20. CL (cid:96) (cid:3) A ↔ ( ¬ A ) (cid:66) ⊥ .2. CL (cid:96) (cid:3) ( A → B ) → ( C (cid:66) A → C (cid:66) B ) .3. CL (cid:96) (cid:3) ( A → B ) → ( B (cid:66) C → A (cid:66) C ) .Proof.
1. ( → ): Since CL (cid:96) (cid:3) A → (cid:3) ( ¬ A → ⊥ ), CL (cid:96) (cid:3) A → ( ¬ A ) (cid:66) ⊥ by J1 .( ← ): By J4 , CL (cid:96) ( ¬ A ) (cid:66) ⊥ → ( ♦ ¬ A → ♦ ⊥ ). Since CL (cid:96) ¬ ♦ ⊥ , CL (cid:96) ( ¬ A ) (cid:66) ⊥ → ¬ ♦ ¬ A . That is, CL (cid:96) ( ¬ A ) (cid:66) ⊥ → (cid:3) A .2. This is because CL (cid:96) (cid:3) ( A → B ) → A (cid:66) B by J1 and CL (cid:96) ( C (cid:66) A ) ∧ ( A (cid:66) B ) → C (cid:66) B by J2 .3. This is because CL (cid:96) (cid:3) ( A → B ) → A (cid:66) B by J1 and CL (cid:96) ( A (cid:66) B ) ∧ ( B (cid:66) C ) → A (cid:66) C by J2 . Proposition 3.21.
The logics CL and IL − ( J1 , J2 ) are deductively equivalent.Proof. CL (cid:96) IL − ( J1 , J2 ): This follows from Proposition 3.20. IL − ( J1 , J2 ) (cid:96) CL : This is because IL − ( J2 ) (cid:96) J4 by Corollary 3.5. Corollary 3.22.
The logics IL and IL − ( J1 , J2 , J5 ) are deductively equivalent. Then de Jongh and Veltman’s and Ignatiev’s theorems are restated as fol-lows: 13 heorem 3.23 (de Jongh and Veltman [3]) . For any formula A , the followingare equivalent:1. IL − ( J1 , J2 , J5 ) (cid:96) A .2. A is valid in all finite IL − -frames where all axioms of IL − ( J1 , J2 , J5 ) arevalid. Theorem 3.24 (Ignatiev [6]) . For any formula A , the following are equivalent:1. IL − ( J1 , J2 ) (cid:96) A .2. A is valid in all finite IL − -frames where all axioms of IL − ( J1 , J2 ) arevalid. In the following, we identify CL with IL − ( J1 , J2 ), and IL with IL − ( J1 , J2 , J5 ). In this section, we prepare some definitions and lemmas for our proofs of themodal completeness theorems of several logics. In this section, let L be anyconsistent logic containing IL − . For a set Φ of formulas, define Φ (cid:66) := { B :there exists a formula C such that either B (cid:66) C ∈ Φ or C (cid:66) B ∈ Φ } . Foreach formula A , let ∼ A : ≡ (cid:40) B if A is of the form ¬ B ¬ A otherwise . We say a finite setΓ of formulas is L -consistent if L (cid:48) (cid:86) Γ → ⊥ , where (cid:86) Γ is a conjunction ofall elements of Γ. Also we say Γ ⊆ Φ is Φ -maximally L -consistent if Γ is L -consistent and for any A ∈ Φ, either A ∈ Γ or ∼ A ∈ Γ. Notice that if Γ isΦ-maximally L -consistent and L (cid:96) (cid:86) Γ → A for A ∈ Φ, then A ∈ Γ. Definition 4.1.
A set Φ of formulas is said to be adequate if it satisfies thefollowing conditions:1. Φ is closed under taking subformulas and applying ∼ ;2. ⊥ ∈ Φ (cid:66) ;3. If B, C ∈ Φ (cid:66) , then B (cid:66) C ∈ Φ;4. If B ∈ Φ (cid:66) , then (cid:3) ∼ B ∈ Φ;5. If
B, C , . . . , C m , D , . . . , D n ∈ Φ (cid:66) , then (cid:3) (cid:16) B → (cid:87) mi =1 C i ∨ (cid:87) nj =1 ♦ D j (cid:17) ∈ Φ.Then the following proposition clearly holds.
Proposition 4.2.
Every finite set of formulas is contained in some finite ade-quate set. K L := { Γ ⊆ Φ : Γ is Φ-maximally L -consistent } . Then K L is also a finite set. Definition 4.3.
Let Γ , ∆ ∈ K L and C ∈ Φ (cid:66) .1. Γ ≺ ∆ : ⇐⇒
1. for any (cid:3) B ∈ Φ, if (cid:3) B ∈ Γ, then B, (cid:3) B ∈ ∆ and 2.there exists (cid:3) B ∈ Φ such that (cid:3)
B / ∈ Γ and (cid:3) B ∈ ∆.2. Γ ≺ C ∆ : ⇐⇒ Γ ≺ ∆ and for any B ∈ Φ, if B (cid:66) C ∈ Γ, then ∼ B ∈ ∆.3. Γ ≺ ∗ C ∆ : ⇐⇒ Γ ≺ ∆ and for any B ∈ Φ, if B (cid:66) C ∈ Γ, then ∼ B, (cid:3) ∼ B ∈ ∆.The relation ≺ ∗ C was introduced by de Jongh and Veltman [3], and Γ ≺ ∗ C ∆is read as “∆ is a C -critical successor of Γ” . The relation ≺ C was introducedby Ingatiev [6]. Obviously, Γ ≺ ∗ C ∆ implies Γ ≺ C ∆. Lemma 4.4.
For Γ , ∆ ∈ K L , if Γ ≺ ∆ , then Γ ≺ ∗⊥ ∆ .Proof. Suppose Γ ≺ ∆. If B (cid:66) ⊥ ∈ Γ, then (cid:3) ∼ B ∈ Γ by J6 . Then ∼ B, (cid:3) ∼ B ∈ ∆. This means Γ ≺ ∗⊥ ∆. Lemma 4.5.
Let Γ , ∆ , Θ ∈ K L and C ∈ Φ (cid:66) . If Γ ≺ ∗ C ∆ and ∆ ≺ Θ , then Γ ≺ ∗ C Θ .Proof. Suppose Γ ≺ ∗ C ∆ and ∆ ≺ Θ. If B (cid:66) C ∈ Γ, then (cid:3) ∼ B ∈ ∆. Then ∼ B, (cid:3) ∼ B ∈ Θ. Therefore Γ ≺ ∗ C Θ. Lemma 4.6.
Let Γ ∈ K L and D, E ∈ Φ (cid:66) . If D (cid:66) E / ∈ Γ , then there exists ∆ ∈ K L such that D ∈ ∆ and Γ ≺ E ∆ . Moreover:a. If L contains J5 , then (cid:3) ∼ E ∈ ∆ also holds.b. If L contains J2 and J5 , then Γ ≺ ∗ E ∆ and (cid:3) ∼ E ∈ ∆ also hold.Proof. Suppose D (cid:66) E / ∈ Γ. Let X := { G : G (cid:66) E ∈ Γ } . Then (cid:3) ( D → (cid:87) X ) ∈ Φ.By J3 , we have IL − (cid:96) (cid:86) Γ → (cid:87) X (cid:66) E . • Suppose, for the contradiction, that (cid:3) ( D → (cid:87) X ) ∈ Γ. Then IL − (cid:96) (cid:86) Γ → ( (cid:87) X (cid:66) E → D (cid:66) E ) by Proposition 2.5.2. Hence IL − (cid:96) (cid:86) Γ → D (cid:66) E , and thus D (cid:66) E ∈ Γ. This contradicts our supposition. Therefore (cid:3) ( D → (cid:87) X ) / ∈ Γ.Let Y := { B, (cid:3) B : (cid:3) B ∈ Γ } ∪ { D, (cid:3) ( D → (cid:95) X ) } ∪ {∼ G : G ∈ X } , For every set S of formulas, more general notion of assuring successor ≺ S was introducedand investigated in Goris et al. [4]. Then ≺ ∗ C is exactly ≺ {¬ C } . However, in this paper, ≺ C and ≺ ∗ C are sufficient for our purpose. Y ⊆ Φ. Suppose that the set Y were L -inconsistent. Then for some (cid:3) B , . . . , (cid:3) B k ∈ Γ, L (cid:96) k (cid:94) i =1 ( B i ∧ (cid:3) B i ) → ( (cid:3) ( D → (cid:95) X ) → ( D → (cid:95) X )) ,L (cid:96) k (cid:94) i =1 (cid:3) B i → (cid:3) ( (cid:3) ( D → (cid:95) X ) → ( D → (cid:95) X )) ,L (cid:96) (cid:94) Γ → (cid:3) ( D → (cid:95) X ) . Thus (cid:3) ( D → (cid:87) X ) ∈ Γ, and this is a contradiction. We have shown that Y is L -consistent.Let ∆ ∈ K L be such that Y ⊆ ∆. Then D ∈ ∆. Since (cid:3) ( D → (cid:87) X ) ∈ ∆ \ Γ, Γ ≺ ∆. Moreover, if G (cid:66) E ∈ Γ, then G ∈ X , and hence ∼ G ∈ ∆.This means Γ ≺ E ∆. • a. Assume that L contains J5 . Let X := X ∪ { ♦ E } . Then (cid:3) ( D → (cid:87) X ) ∈ Φ. If (cid:3) ( D → (cid:87) X ) ∈ Γ, then IL − (cid:96) (cid:86) Γ → ( (cid:87) X (cid:66) E → D (cid:66) E ). Since IL − (cid:96) (cid:86) Γ → (cid:87) X (cid:66) E and L (cid:96) ♦ E (cid:66) E by J5 , we obtain L (cid:96) (cid:86) Γ → (cid:87) X (cid:66) E . Thus L (cid:96) (cid:86) Γ → D (cid:66) E and D (cid:66) E ∈ Γ. This is acontradiction. Therefore (cid:3) ( D → (cid:87) X ) / ∈ Γ.Let Y := { B, (cid:3) B : (cid:3) B ∈ Γ }∪{ D, (cid:3) ( D → (cid:95) X ) }∪{∼ G : G ∈ X }∪{ (cid:3) ∼ E } . Then it can be proved that Y is also an L -consistent subset of Φ asabove. Let ∆ ∈ K L be such that Y ⊆ ∆. Then ∆ satisfies the requiredconditions. • b. Assume that L contains J2 and J5 . Let X := X ∪ { ♦ G : G ∈ X } ∪ { ♦ E } . Then (cid:3) ( D → (cid:87) X ) ∈ Φ. For each G ∈ X , we have L (cid:96) (cid:86) Γ → ( ♦ G (cid:66) G ) ∧ ( G (cid:66) E ) by J5 . Then by J2 , L (cid:96) (cid:86) Γ → ♦ G (cid:66) E .Therefore we obtain L (cid:96) (cid:86) Γ → (cid:86) G ∈ X ( ♦ G (cid:66) E ). Since we also have IL − (cid:96) (cid:86) Γ → (cid:87) X (cid:66) E and L (cid:96) ♦ E (cid:66) E , we get L (cid:96) (cid:86) Γ → (cid:87) X (cid:66) E .This implies (cid:3) ( D → (cid:87) X ) / ∈ Γ.Let Y := { B, (cid:3) B : (cid:3) B ∈ Γ }∪{ D, (cid:3) ( D → (cid:95) X ) }∪{∼ G, (cid:3) ∼ G : G ∈ X }∪{ (cid:3) ∼ E } . Then Y is also an L -consistent subset of Φ, and any ∆ ∈ K L with Y ⊆ ∆is a desired set. Lemma 4.7.
Let Γ , ∆ ∈ K L and D, E, F ∈ Φ (cid:66) . If D (cid:66) E ∈ Γ , Γ ≺ F ∆ and D ∈ ∆ , then there exists Θ ∈ K L such that E ∈ Θ and ∼ F ∈ Θ . Moreover: . If L contains J4 + , then Γ ≺ Θ also holds.b. If L contains J2 + , then Γ ≺ F Θ also holds.c. If L contains J2 + and J5 , then Γ ≺ ∗ F Θ and (cid:3) ∼ F ∈ Θ also hold.Proof. Suppose D (cid:66) E ∈ Γ, Γ ≺ F ∆ and D ∈ ∆. • Suppose, towards a contradiction, that the set { E, ∼ F } is L -inconsistent.Then L (cid:96) E → F . By the rule R1 , we have L (cid:96) D (cid:66) E → D (cid:66) F , andhence D (cid:66) F ∈ Γ. Since Γ ≺ F ∆, we have ∼ D ∈ ∆. This contradicts the L -consistency of ∆. Therefore { E, ∼ F } is L -consistent. Let Θ ∈ K L besuch that { E, ∼ F } ⊆ Θ, and then Θ satisfies the required conditions. • a. Assume that L contains J4 + . If (cid:3) ( E → F ) ∈ Γ, then by J4 + , L (cid:96) (cid:86) Γ → ( D (cid:66) E → D (cid:66) F ). Since D (cid:66) E ∈ Γ, we obtain D (cid:66) F ∈ Γ.Then ∼ D ∈ ∆ because Γ ≺ F ∆, and this is a contradiction. Therefore (cid:3) ( E → F ) / ∈ Γ.Suppose Y := { B, (cid:3) B : (cid:3) B ∈ Γ } ∪ { E, ∼ F, (cid:3) ( E → F ) } were L -inconsistent. Then there would be (cid:3) B , . . . , (cid:3) B k ∈ Γ such that L (cid:96) k (cid:94) i =1 ( B i ∧ (cid:3) B i ) → ( (cid:3) ( E → F ) → ( E → F )) ,L (cid:96) k (cid:94) i =1 (cid:3) B i → (cid:3) ( (cid:3) ( E → F ) → ( E → F )) ,L (cid:96) (cid:94) Γ → (cid:3) ( E → F ) . Then (cid:3) ( E → F ) ∈ Γ, and this is a contradiction. Thus Y is L -consistent.Let Θ ∈ K L be such that Y ⊆ Θ. Then (cid:3) ( E → F ) ∈ Θ \ Γ, and hencewe conclude Γ ≺ Θ. • b. Assume that L contains J2 + . Let X := { G : G (cid:66) F ∈ Γ } . If (cid:3) ( E → (cid:87) X ∨ F ) ∈ Γ, then L (cid:96) (cid:86) Γ → (cid:3) ( E ∧ ¬ F → (cid:87) X ), and hence L (cid:96) (cid:86) Γ → ( (cid:87) X (cid:66) F → ( E ∧¬ F ) (cid:66) F ) by Proposition 2.5.2. Since L (cid:96) (cid:86) Γ → (cid:87) X (cid:66) F ,we have L (cid:96) (cid:86) Γ → ( E ∧ ¬ F ) (cid:66) F . Since D (cid:66) E ∈ Γ, L (cid:96) (cid:86) Γ → D (cid:66) F by J2 (cid:48) + . Thus D (cid:66) F ∈ Γ. Then ∼ F ∈ ∆ because Γ ≺ F ∆, and this is acontradiction. Hence (cid:3) ( E → (cid:87) X ∨ F ) / ∈ Γ.Let Y := { B, (cid:3) B : (cid:3) B ∈ Γ }∪{ E, ∼ F, (cid:3) ( E → (cid:95) X ∨ F ) }∪{∼ G : G ∈ X } , then Y is L -consistent. Let Θ ∈ K L be such that Y ⊆ Θ. Then (cid:3) ( E → (cid:87) X ∨ F ) ∈ Θ \ Γ, and hence Γ ≺ Θ. Moreover, Γ ≺ F Θ. • c. Assume that L contains J2 + and J5 . Let X := { G : G (cid:66) F ∈ Γ } and X := X ∪ { ♦ G : G ∈ X } ∪ { ♦ F } . Then (cid:3) ( E → (cid:87) X ∨ F ) ∈ Φ. For each17 ∈ X , L (cid:96) (cid:86) Γ → ( ♦ G (cid:66) G ) ∧ ( G (cid:66) F ) by J5 . Then L (cid:96) (cid:86) Γ → ♦ G (cid:66) F by J2 . Since L (cid:96) (cid:86) Γ → (cid:87) X (cid:66) F and L (cid:96) ♦ F (cid:66) F , we obtain L (cid:96) (cid:86) Γ → (cid:87) X (cid:66) F .Suppose, towards a contradiction, (cid:3) ( E → (cid:87) X ∨ F ) ∈ Γ. Then L (cid:96) (cid:86) Γ → (cid:3) ( E ∧ ¬ F → (cid:87) X ), and thus L (cid:96) (cid:86) Γ → ( (cid:87) X (cid:66) F → ( E ∧¬ F ) (cid:66) F ) by Proposition 2.5.2. Hence L (cid:96) (cid:86) Γ → ( E ∧ ¬ F ) (cid:66) F . Since D (cid:66) E ∈ Γ, by J2 (cid:48) + , we have L (cid:96) (cid:86) Γ → D (cid:66) F . Thus D (cid:66) F ∈ Γ. Then ∼ F ∈ ∆ because Γ ≺ F ∆. This is a contradiction. Hence we obtain (cid:3) ( E → (cid:87) X ∨ F ) / ∈ Γ.Let Y := { B, (cid:3) B : (cid:3) B ∈ Γ }∪{ E, ∼ F, (cid:3) ( E → (cid:95) X ∨ F ) }∪{∼ G, (cid:3) ∼ G : G ∈ X }∪{ (cid:3) ∼ F } , then we can prove that Y is L -inconsistent. Let Θ ∈ K L be such that Y ⊆ Θ. Then (cid:3) ( E → (cid:87) X ∨ F ) ∈ Θ \ Γ, and hence Γ ≺ Θ. Moreover,Γ ≺ ∗ F Θ. Lemma 4.8.
Assume that L contains J4 . Let Γ , ∆ ∈ K L and D, E ∈ Φ (cid:66) . If D (cid:66) E ∈ Γ , Γ ≺ ∆ and D ∈ ∆ , then there exists Θ ∈ K L such that Γ ≺ Θ and E ∈ Θ .Proof. Since D (cid:66) E ∈ Γ, L (cid:96) (cid:86) Γ → ( ♦ D → ♦ E ) by J4 . If (cid:3) ∼ E ∈ Γ, then (cid:3) ∼ D ∈ Γ. Since Γ ≺ ∆, we have ∼ D ∈ ∆, a contradiction. Thus (cid:3) ∼ E / ∈ Γ.Let Y := { B, (cid:3) B : (cid:3) B ∈ Γ } ∪ { E, (cid:3) ∼ E } , then it is proved that Y is L -consistent. Thus for some Θ ∈ K L , Y ⊆ Θ. Since (cid:3) ∼ E ∈ Θ \ Γ, we obtainΓ ≺ Θ. Lemma 4.9.
Assume that L contains J2 . Let Γ , ∆ ∈ K L and D, E, F ∈ Φ (cid:66) .If D (cid:66) E ∈ Γ , Γ ≺ F ∆ and D ∈ ∆ , then there exists Θ ∈ K L such that Γ ≺ F Θ and E ∈ Θ . Moreover:a. If L contains J5 , then Γ ≺ ∗ F Θ and (cid:3) ∼ F ∈ Θ also hold.Proof. Let X := { G : G (cid:66) F ∈ Γ } . Suppose, towards a contradiction, that (cid:3) ( E → (cid:87) X ) ∈ Γ. Then by Proposition 2.5.2, IL − (cid:96) (cid:86) Γ → ( (cid:87) X (cid:66) F → E (cid:66) F ). Since IL − (cid:96) (cid:86) Γ → (cid:87) X (cid:66) F , IL − (cid:96) (cid:86) Γ → E (cid:66) F . Also since D (cid:66) E ∈ Γ, we obtain L (cid:96) (cid:86) Γ → D (cid:66) F by J2 . Thus D (cid:66) F ∈ Γ. SinceΓ ≺ F ∆, ∼ D ∈ ∆ and hence this contradicts the L -consistency of ∆. Therefore (cid:3) ( E → (cid:87) X ) / ∈ Γ.Let Y := { B, (cid:3) B : (cid:3) B ∈ Γ } ∪ { E, (cid:3) ( E → (cid:87) X ) } ∪ {∼ G : G ∈ X } , then Y is L -consistent. Let Θ ∈ K L be such that Y ⊆ Θ. Then Θ is a desired seta. Assume that L contains J5 . Let X := X ∪ { ♦ G : G ∈ X } ∪ { ♦ F } . As inthe proof of Lemma 4.6.b, it can be shown that (cid:3) ( E → (cid:87) X ) is not in Γ. Let Y := { B, (cid:3) B : (cid:3) B ∈ Γ } ∪ { E, (cid:3) ( E → (cid:87) X ) } ∪ {∼ G, (cid:3) ∼ G : G ∈ X } ∪ { (cid:3) ∼ F } . Then Y is L -consistent. Let Θ ∈ K L be such that Y ⊆ Θ.Table 1 summarizes several conclusions of Lemmas 4.7, 4.8 and 4.9.18 ∈ Θ ∼ F ∈ Θ Γ ≺ Θ Γ ≺ F Θ Γ ≺ ∗ F Θ& (cid:3) ∼ F ∈ Θ (cid:88) (cid:88) Lemma 4.7 J4 (cid:88) (cid:88) Lemma 4.8 J4 + (cid:88) (cid:88) (cid:88) Lemma 4.7.a J2 (cid:88) (cid:88) Lemma 4.9 J2 , J5 (cid:88) (cid:88) Lemma 4.9.a J2 + (cid:88) (cid:88) (cid:88) Lemma 4.8.b J2 + , J5 (cid:88) (cid:88) (cid:88) Lemma 4.8.cTable 1: Conclusions of Lemmas 4.7, 4.8 and 4.9 − -frames In this section, we prove modal completeness theorems with respect to IL − -frames for twelve sublogics of IL . First, we prove the completeness theoremfor logics in Figure 1 other than IL − ( J2 + , J5 ) and IL . Secondly, we prove thecompleteness theorem for logics IL − ( J2 + , J5 ) and IL . Our proof technique ofthe second completeness theorem is essentially same as in the proof of de Jonghand Veltman [3]. On the other hand, our proof of the first completeness theoremadmits a simpler technique than that of the second theorem. More precisely, inthe proof of the second theorem, the universe a countermodel is defined as a setof tuples (cid:104) Γ , τ (cid:105) where Γ is a Φ-maximal L -consistent subset of a finite adequateset Φ and τ is a finite sequence of formulas in Φ. On the other hand, in ourproof of the first theorem, we simply consider tuples (cid:104) Γ , B (cid:105) where B is a formulain Φ to define a countermodel. As a consequence, our proof of the completenesstheorem of the logic CL is simpler than Ignatiev’s proof in [6]. IL − IL − ( J5 ) IL − ( J1 ) IL − ( J4 + ) IL − ( J1 , J5 ) IL − ( J4 + , J5 ) IL − ( J1 , J4 + ) IL − ( J2 + ) IL − ( J1 , J4 + , J5 ) IL − ( J2 + , J5 ) CL IL
Figure 1: Sublogics of IL complete with respect to IL − -framesFirst, we prove the completeness theorem for logics other than IL − ( J2 + , J5 )and IL . 19 heorem 5.1. Let L be one of the logics IL − , IL − ( J4 + ) , IL − ( J1 ) , IL − ( J5 ) , IL − ( J2 + ) , IL − ( J1 , J4 + ) , IL − ( J4 + , J5 ) , IL − ( J1 , J5 ) , CL and IL − ( J1 , J4 + , J5 ) .Then for any formula A , the following are equivalent:1. L (cid:96) A .2. A is valid in all (finite) IL − -frames where all axioms of L are valid.Proof. (1 ⇒ ⇒ L (cid:48) A . Let Φ be a finite adequate set of formulas with ∼ A ∈ Φ. The existence of such a set Φ is guaranteed by Proposition 4.2. Bythe supposition, there exists Γ ∈ K L such that ∼ A ∈ Γ .Let M = (cid:104) W, R, { S x } x ∈ W , (cid:13) (cid:105) be a model satisfying the following clauses:1. W = {(cid:104) Γ , B (cid:105) : Γ ∈ K L and B ∈ Φ (cid:66) } ;2. (cid:104) Γ , B (cid:105) R (cid:104) ∆ , C (cid:105) ⇐⇒ Γ ≺ ∆;3. (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) (cid:104) Θ , D (cid:105) ⇐⇒ (cid:104) Γ , B (cid:105) R (cid:104) ∆ , C (cid:105) and the condition C L which isdefined below holds;4. (cid:104) Γ , B (cid:105) (cid:13) p ⇐⇒ p ∈ Γ.The condition C L is different for each L , and defined as follows: • L ∈ { IL − , IL − ( J1 ) } : If Γ ≺ C ∆, then ∼ C ∈ Θ. • L ∈ { IL − ( J4 + ) , IL − ( J1 , J4 + ) } : (cid:104) Γ , B (cid:105) R (cid:104) Θ , D (cid:105) and if Γ ≺ C ∆, then ∼ C ∈ Θ. • L ∈ { IL − ( J2 + ) , CL } : (cid:104) Γ , B (cid:105) R (cid:104) Θ , D (cid:105) and if Γ ≺ C ∆, then D ≡ C , Γ ≺ C Θ and ∼ C ∈ Θ. • L ∈ { IL − ( J5 ) , IL − ( J1 , J5 ) } : If Γ ≺ C ∆ and (cid:3) ∼ C ∈ ∆, then ∼ C ∈ Θ. • L ∈ { IL − ( J4 + , J5 ) , IL − ( J1 , J4 + , J5 ) } : (cid:104) Γ , B (cid:105) R (cid:104) Θ , D (cid:105) and if Γ ≺ C ∆and (cid:3) ∼ C ∈ ∆, then ∼ C ∈ Θ.Here D ≡ C means that formulas D and C are identical. Then W is finiteand R is a transitive and conversely well-founded binary relation on W . Thus (cid:104) W, R, { S x } x ∈ W (cid:105) is an IL − -frame. Lemma 5.2.
Every axiom of L is valid in the frame F = (cid:104) W, R, { S x } x ∈ W (cid:105) of M .Proof. We distinguish the following several cases: • L = IL − ( J1 ): Suppose (cid:104) Γ , B (cid:105) R (cid:104) ∆ , C (cid:105) . If Γ ≺ C ∆, then ∼ C ∈ ∆ because C (cid:66) C ∈ Γ. Thus (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) (cid:104) ∆ , C (cid:105) by the definition of C L . Therefore J1 is valid in F by Proposition 3.2.20 L = IL − ( J4 + ): If (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) (cid:104) Θ , D (cid:105) , then (cid:104) Γ , B (cid:105) R (cid:104) Θ , D (cid:105) . By Proposi-tion 3.8, J4 + is valid in F . • L = IL − ( J2 + ): As in the case of IL − ( J4 + ), J4 + is valid in F . Suppose (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) (cid:104) ∆ , C (cid:105) and (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) (cid:104) ∆ , C (cid:105) . Then (cid:104) Γ , B (cid:105) R (cid:104) ∆ , C (cid:105) .If Γ ≺ C ∆ , then C ≡ C and Γ ≺ C ∆ . Since (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) (cid:104) ∆ , C (cid:105) and Γ ≺ C ∆ , we have C ≡ C , Γ ≺ C ∆ and ∼ C ∈ ∆ . Thus we ob-tain (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) (cid:104) ∆ , C (cid:105) . Therefore J2 + is valid in F by Proposition3.15. • L = IL − ( J5 ): Suppose (cid:104) Γ , B (cid:105) R (cid:104) ∆ , C (cid:105) and (cid:104) ∆ , C (cid:105) R (cid:104) Θ , D (cid:105) . If Γ ≺ C ∆and (cid:3) ∼ C ∈ ∆, then ∼ C ∈ Θ because ∆ ≺ Θ. Thus (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) (cid:104) Θ , D (cid:105) holds. Then by Proposition 3.18, J5 is valid in F . • For other cases, the lemma is proved in a similar way as above.
Lemma 5.3 (Truth Lemma) . For any formula C ∈ Φ and any (cid:104) Γ , B (cid:105) ∈ W , C ∈ Γ if and only if (cid:104) Γ , B (cid:105) (cid:13) C .Proof. We prove by induction on the construction of C . We only give a proofof the case C ≡ ( D (cid:66) E ).( ⇒ ): Assume D (cid:66) E ∈ Γ. Let (cid:104) ∆ , F (cid:105) be any element of W such that (cid:104) Γ , B (cid:105) R (cid:104) ∆ , F (cid:105) and (cid:104) ∆ , F (cid:105) (cid:13) D . Then by induction hypothesis, D ∈ ∆. Wedistinguish the following two cases. • If Γ ≺ F ∆, then by Lemma 4.7, there exists Θ ∈ K L such that E ∈ Θ and ∼ F ∈ Θ. Moreover, if L (cid:96) J4 + , Γ ≺ Θ holds. Also if L (cid:96) J2 + , Γ ≺ F Θholds. • If Γ ⊀ F ∆, by Lemma 4.7, there exists Θ ∈ K L such that E ∈ Θ becauseΓ ≺ ⊥ ∆. Moreover, if L (cid:96) J4 + , Γ ≺ Θ holds.In either case, we have (cid:104) Θ , F (cid:105) ∈ W . Also E ∈ Θ and (cid:104) ∆ , F (cid:105) S (cid:104) Γ ,B (cid:105) (cid:104) Θ , F (cid:105) .Then by induction hypothesis, (cid:104) Θ , F (cid:105) (cid:13) E . Therefore we conclude (cid:104) Γ , B (cid:105) (cid:13) D (cid:66) E .( ⇐ ): Assume D (cid:66) E / ∈ Γ. By Lemma 4.6, there exists ∆ ∈ K L such that D ∈ ∆ and Γ ≺ E ∆. Moreover, if L contains J5 , then (cid:3) ∼ E ∈ ∆ also holds.Since (cid:104) ∆ , E (cid:105) ∈ W , (cid:104) ∆ , E (cid:105) (cid:13) D by induction hypothesis. Let (cid:104) Θ , F (cid:105) be anyelement of W with (cid:104) ∆ , E (cid:105) S (cid:104) Γ ,B (cid:105) (cid:104) Θ , F (cid:105) . By the definitions of the relation S andthe condition C L , we have ∼ E ∈ Θ in all cases of L By induction hypothesis, (cid:104) Θ , F (cid:105) (cid:49) E . Therefore we obtain (cid:104) Γ , B (cid:105) (cid:49) D (cid:66) E .Since ⊥ ∈ Φ (cid:66) , we have (cid:104) Γ , ⊥(cid:105) ∈ W . Since A / ∈ Γ , (cid:104) Γ , ⊥(cid:105) (cid:49) A by TruthLemma. Therefore A is not valid in M .21ur proof of Theorem 5.1 cannot be applied to logics containing both J2 and J5 . For example, for L = IL − ( J2 + , J5 ), the condition C L which is usedto define the relation S might be as follows: (cid:104) Γ , B (cid:105) R (cid:104) Θ , D (cid:105) and if Γ ≺ C ∆ and (cid:3) ∼ C ∈ ∆, then D ≡ C , Γ ≺ C Θ and ∼ C ∈ Θ. Then J5 is no longer validin the resulting frame (cid:104) W, R, { S x } x ∈ W (cid:105) . To avoid this obstacle, as mentionedabove, for the modal completeness of such logics, we consider tuples (cid:104) W, τ (cid:105) asmembers of the universe of our countermodel, where τ is a finite sequence offormulas.For finite sequences τ and σ of formulas, τ ⊆ σ denotes that τ is an initialsegment of σ . Also τ (cid:40) σ denotes that τ is a proper initial segment of σ , thatis, τ ⊆ σ and | τ | < | σ | , where | τ | is the length of τ . Let τ ∗ (cid:104) B (cid:105) be the sequenceobtained from τ by concatenating B as the last element. Theorem 5.4.
Let L be one of the logics IL − ( J2 + , J5 ) and IL . Then for anyformula A , the following are equivalent:1. L (cid:96) A .2. A is valid in all (finite) IL − -frames where all axioms of L are valid.Proof. (1 ⇒ ⇒ L (cid:48) A . Let Φ be any finite adequate set with ∼ A ∈ Φ.Let Γ ∈ K L be such that ∼ A ∈ Γ .For each Γ ∈ K L , we define the rank of Γ (write rank(Γ)) as follows:rank(Γ) := sup { rank(∆) + 1 : Γ ≺ ∆ } , where sup ∅ = 0. This is well-definedbecause ≺ is conversely well-founded.Let M = (cid:104) W, R, { S x } x ∈ W , (cid:13) (cid:105) be a model satisfying the following clauses:1. W = {(cid:104) Γ , τ (cid:105) : Γ ∈ K L and τ is a finite sequence of elements of Φ (cid:66) withrank(Γ) + | τ | ≤ rank(Γ ) } ;2. (cid:104) Γ , τ (cid:105) R (cid:104) ∆ , σ (cid:105) ⇐⇒ Γ ≺ ∆ and τ (cid:40) σ ;3. (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) (cid:104) Θ , ρ (cid:105) ⇐⇒ (cid:104) Γ , τ (cid:105) R (cid:104) ∆ , σ (cid:105) , (cid:104) Γ , τ (cid:105) R (cid:104) Θ , ρ (cid:105) and if τ ∗ (cid:104) C (cid:105) ⊆ σ ,Γ ≺ ∗ C ∆ and (cid:3) ∼ C ∈ ∆, then τ ∗ (cid:104) C (cid:105) ⊆ ρ , Γ ≺ ∗ C Θ and ∼ C, (cid:3) ∼ C ∈ Θ;4. (cid:104) Γ , τ (cid:105) (cid:13) p ⇐⇒ p ∈ Γ.Then W is finite because of the condition rank(Γ) + | τ | ≤ rank(Γ ). Also (cid:104) W, R, { S x } x ∈ W (cid:105) is an IL − -frame. Lemma 5.5.
Every axiom of L is valid in the frame F = (cid:104) W, R, { S x } x ∈ W (cid:105) of M .Proof. J2 + : By the definition of S , J4 + is obviously valid in F . Assume (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) (cid:104) ∆ , σ (cid:105) and (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) (cid:104) ∆ , σ (cid:105) . Suppose τ ∗ (cid:104) C (cid:105) ⊆ σ , Γ ≺ ∗ C ∆ and (cid:3) ∼ C ∈ ∆ . Then τ ∗ (cid:104) C (cid:105) ⊆ σ , Γ ≺ ∗ C ∆ and (cid:3) ∼ C ∈ ∆ because (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) (cid:104) ∆ , σ (cid:105) . Then also τ ∗ (cid:104) C (cid:105) ⊆ σ , Γ ≺ ∗ C ∆ and ∼ C, (cid:3) ∼ C ∈ ∆ because (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) (cid:104) ∆ , σ (cid:105) . Thus we obtain (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) (cid:104) ∆ , σ (cid:105) . There-fore J2 + is valid in F by Proposition 3.17.22 : Assume that (cid:104) Γ , τ (cid:105) R (cid:104) ∆ , σ (cid:105) and (cid:104) ∆ , σ (cid:105) R (cid:104) Θ , ρ (cid:105) . Suppose τ ∗ (cid:104) C (cid:105) ⊆ σ ,Γ ≺ ∗ C ∆ and (cid:3) ∼ C ∈ ∆. Since σ (cid:40) ρ , we have τ ∗ (cid:104) C (cid:105) ⊆ ρ . Since Γ ≺ ∗ C ∆ and∆ ≺ Θ, Γ ≺ ∗ C Θ by Lemma 4.5. Also we have ∼ C, (cid:3) ∼ C ∈ Θ because ∆ ≺ Θ.Therefore we obtain (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) (cid:104) Θ , ρ (cid:105) . By Proposition 3.18, J5 is valid in F .At last, we assume L = IL and show that J1 is valid in F . Suppose (cid:104) Γ , τ (cid:105) R (cid:104) ∆ , σ (cid:105) , Γ ≺ ∗ C ∆ and (cid:3) ∼ C ∈ ∆. Since C (cid:66) C ∈ Γ, ∼ C ∈ ∆. Thus wehave (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) (cid:104) ∆ , σ (cid:105) . By Proposition 3.2, J1 is valid in F . Lemma 5.6 (Truth Lemma) . For any formula C ∈ Φ and any (cid:104) Γ , B (cid:105) ∈ W , C ∈ Γ if and only if (cid:104) Γ , B (cid:105) (cid:13) C .Proof. This is proved by induction on the construction of C , and we prove onlyfor C ≡ ( D (cid:66) E ).( ⇒ ): Assume D (cid:66) E ∈ Γ. Let (cid:104) ∆ , σ (cid:105) be any element of W such that (cid:104) Γ , τ (cid:105) R (cid:104) ∆ , σ (cid:105) and (cid:104) ∆ , σ (cid:105) (cid:13) D . Then by induction hypothesis, D ∈ ∆. Wedistinguish the following two cases. • If τ ∗ (cid:104) F (cid:105) ⊆ σ , Γ ≺ ∗ F ∆ and (cid:3) ∼ F ∈ ∆ for some F , then by Lemma 4.7,there exists Θ ∈ K L such that E ∈ Θ, Γ ≺ ∗ F Θ and ∼ F, (cid:3) ∼ F ∈ Θ. • If not, by Lemma 4.7, there exists Θ ∈ K L such that E ∈ Θ and Γ ≺ Θbecause Γ ≺ ⊥ ∆.In either case, we have rank(Θ) + 1 ≤ rank(Γ). Let ρ := τ ∗ (cid:104) F (cid:105) . Then | ρ | = | τ | + 1, and hence we obtainrank(Θ) + | ρ | = rank(Θ) + 1 + | τ | ≤ rank(Γ) + | τ | ≤ rank(Γ ) . It follows (cid:104) Θ , ρ (cid:105) ∈ W . By the definition of S , we have (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) (cid:104) Θ , ρ (cid:105) . Alsoby induction hypothesis, (cid:104) Θ , ρ (cid:105) (cid:13) E . Therefore we conclude (cid:104) Γ , τ (cid:105) (cid:13) D (cid:66) E .( ⇐ ): Assume D (cid:66) E / ∈ Γ. By Lemma 4.6, there exists ∆ ∈ K L such that D ∈ ∆, Γ ≺ ∗ E ∆ and (cid:3) ∼ E ∈ ∆. Let σ := τ ∗ (cid:104) E (cid:105) , then it is proved that (cid:104) ∆ , σ (cid:105) is an element of W as above. Then (cid:104) ∆ , σ (cid:105) (cid:13) D by induction hypothesis.Let (cid:104) Θ , ρ (cid:105) be any element of W with (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) (cid:104) Θ , ρ (cid:105) . Since τ ∗ (cid:104) E (cid:105) = σ ,Γ ≺ ∗ E ∆ and (cid:3) ∼ E ∈ ∆, we have ∼ E ∈ Θ by the definition of S By inductionhypothesis, (cid:104) Θ , ρ (cid:105) (cid:49) E . Therefore we conclude (cid:104) Γ , τ (cid:105) (cid:49) D (cid:66) E .Let (cid:15) be the empty sequence. Then rank(Γ ) + | (cid:15) | = rank(Γ ), and hence (cid:104) Γ , (cid:15) (cid:105) ∈ W . Since A / ∈ Γ , (cid:104) Γ , (cid:15) (cid:105) (cid:49) A by Truth Lemma. Therefore A is notvalid in M .As a corollary to Theorems 5.1 and 5.4, we have the decidability of theselogics. Corollary 5.7.
Every logic shown in Figure 1 is decidable.
Since every IL − -frame can be transformed into an IL − set -frame, we obtainthe following corollary. 23 orollary 5.8. Let L be one of twelve logics in Figure 1 and let A be anyformula. Then the following are equivalent:1. L (cid:96) A .2. A is valid in all (finite) IL − set -frames in which all axioms of L are valid. − -frames In this section, we prove the modal incompleteness of eight logics shown inFigure 2 with respect to IL − -frames. IL − ( J4 ) IL − ( J1 , J4 ) IL − ( J4 , J5 ) IL − ( J2 ) IL − ( J1 , J4 , J5 ) IL − ( J2 , J5 ) IL − ( J2 , J4 + ) IL − ( J2 , J4 + , J5 )Figure 2: Sublogics of IL incomplete with respect to IL − -framesFirst, we prove the incompleteness of the logics IL − ( J2 ), IL − ( J2 , J4 + ), IL − ( J2 , J5 ) and IL − ( J2 , J4 + , J5 ). Proposition 6.1. IL − ( J2 , J4 + , J5 ) (cid:48) J2 + .Proof. Let F = (cid:104) W, R, { S x } x ∈ W (cid:105) be the IL − set -frame defined as follows:1. W := { x, y , y , y } ;2. R := { ( x, y ) , ( x, y ) , ( x, y ) } ;3. y S x V : ⇐⇒ V ⊇ { y , y } ; y S x V : ⇐⇒ V ⊇ { y } ; y S x V : ⇐⇒ V ⊇ { y , y , y } . xy y y By Monotonicity of S x , F is actually an IL − set -frame. First, we prove that J2 , J4 + and J5 are valid in F . 24 J4 + : If yS x V , then V ∩ ↑ ( x ) = V \ { x } . By the definition of S x , we have yS x ( V \ { x } ). Thus yS x ( V ∩ ↑ ( x )). By Proposition 3.10, J4 + is valid in F . • J2 : Since IL − ( J4 + ) (cid:96) J4 , J4 is also valid in F . Suppose yS x V and ∀ z ∈ V ∩ ↑ ( x )( zS x U z ). Then y ∈ V if y is either y , y or y . Also since y ∈ V ∩ ↑ ( x ), there exists U y ⊆ W such that y S x U y . By the definitionof S x , U y ⊇ { y , y , y } . Thus (cid:83) z ∈ V ∩↑ ( x ) U z ⊇ { y , y , y } . Then we have yS x ( (cid:83) z ∈ V ∩↑ ( x ) U z ) if y is either y , y or y . Therefore J2 is valid in F by Proposition 3.16. • J5 : Since there are no y, z ∈ W such that xRy and yRz , by Proposition3.19, J5 is trivially valid in F .It suffices to show that J2 + is not valid in F . Let V = { y } and V = { y } ,then y S x ( V ∪ V ). Also let U y = { y } , then ∀ z ∈ V ∩ ↑ ( x )( zS x U z ). Onthe other hand, since (cid:83) z ∈ V ∩↑ ( x ) U z ∪ V = U y ∪ V = { y } ∪ { y } = { y } , y S x ( (cid:83) z ∈ V ∩↑ ( x ) U z ∪ V ) does not hold. Therefore J2 + is not valid in F byProposition 3.17. Corollary 6.2.
Let L be any logic with IL − ( J2 ) ⊆ L ⊆ IL − ( J2 , J4 + , J5 ) .Then L is not complete with respect to IL − -frames.Proof. Let F be any IL − -frame in which all axioms of L are valid. Then J2 is valid in F , and hence J2 + is also valid in F by Proposition 3.15. However,by Proposition 6.1, L (cid:48) J2 + . Therefore L is not complete with respect to IL − -frames.Secondly, we prove the incompleteness of the logics IL − ( J4 ), IL − ( J1 , J4 ), IL − ( J4 , J5 ) and IL − ( J1 , J4 , J5 ). Proposition 6.3. IL − ( J1 , J4 , J5 ) (cid:48) J4 + .Proof. We define the IL − set -frame F = (cid:104) W, R, { S x } x ∈ W (cid:105) as follows:1. W := { x, y , y , y } ;2. R := { ( x, y ) , ( x, y ) } ;3. y S x V : ⇐⇒ V ⊇ { y } or V ⊇ { y , y } ; y S x V : ⇐⇒ V ⊇ { y } . xy y y Indeed, F is an IL − set -frame. We show J1 , J4 and J5 are valid in F .25 J1 : Since y S x { y } and y S x { y } , J1 is valid by Proposition 3.3. • J4 : Suppose yS x V . Then whatever y is, either y ∈ V or y ∈ V . Thusthere exists z ∈ V such that xRz . Hence J4 is valid in F by Proposition3.9. • J5 : As in the proof of Proposition 6.1, J5 is trivially valid in F .Then we show that J4 + is not valid in F . Let V = { y , y } , then y S x V .On the other hand, since V ∩ ↑ ( x ) = { y } , y S x ( V ∩ ↑ ( x )) does not hold.Therefore J4 + is not valid in F by Proposition 3.10. Corollary 6.4.
Let L be any logic with IL − ( J4 ) ⊆ L ⊆ IL − ( J1 , J4 , J5 ) . Then L is incomplete with respect to IL − -frames. − set -frames In this section, we prove eight logics shown in Figure 2 are complete with respectto IL − set -frames. As in Section 5, at first we prove the completeness theorem oflogics other than IL − ( J2 , J5 ) and IL − ( J2 , J4 + , J5 ). Theorem 7.1.
Let L be one of IL − ( J4 ) , IL − ( J1 , J4 ) , IL − ( J4 , J5 ) , IL − ( J1 , J4 , J5 ) , IL − ( J2 ) and IL − ( J2 , J4 + ) . Then for any formula A , the following are equiv-alent:1. L (cid:96) A .2. A is valid in all (finite) IL − set -frames where all axioms of L are valid.Proof. (1 ⇒ ⇒ L (cid:48) A . Let Φ be any finite adequate set of formulascontaining {∼ A } . Let Γ ∈ K L be such that ∼ A ∈ Γ .We define a model M = (cid:104) W, R, { S x } x ∈ W , (cid:13) (cid:105) as follows:1. W = {(cid:104) Γ , B (cid:105) : Γ ∈ K L and B ∈ Φ (cid:66) } ;2. (cid:104) Γ , B (cid:105) R (cid:104) ∆ , C (cid:105) : ⇐⇒ Γ ≺ ∆;3. (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) V : ⇐⇒ (a) (cid:104) Γ , B (cid:105) R (cid:104) ∆ , C (cid:105) ;(b) For some (cid:104) Θ , D (cid:105) ∈ V , (cid:104) Γ , B (cid:105) R (cid:104) Θ , D (cid:105) ;(c) The condition C L holds.4. (cid:104) Γ , B (cid:105) (cid:13) p : ⇐⇒ p ∈ Γ.The condition C L is different for each L , and defined as follows: • L ∈ { IL − ( J4 ) , IL − ( J1 , J4 ) } : If Γ ≺ C ∆, then there exists (cid:104) Λ , G (cid:105) ∈ V such that ∼ C ∈ Λ. 26 L ∈ { IL − ( J4 , J5 ) , IL − ( J1 , J4 , J5 ) } : If Γ ≺ C ∆ and (cid:3) ∼ C ∈ ∆, thenthere exists (cid:104) Λ , G (cid:105) ∈ V such that ∼ C ∈ Λ. • L = IL − ( J2 ): If Γ ≺ C ∆, then there exist (cid:104) Λ , G (cid:105) , (cid:104) Λ , C (cid:105) ∈ V such that ∼ C ∈ Λ and Γ ≺ C Λ . • L = IL − ( J2 , J4 + ): If Γ ≺ C ∆, then there exist (cid:104) Λ , G (cid:105) , (cid:104) Λ , C (cid:105) ∈ V suchthat Γ ≺ Λ , ∼ C ∈ Λ and Γ ≺ C Λ .The set W is finite and the relation R is transitive and conversely well-founded. Moreover, by Monotonicity of S x , F = (cid:104) W, R, { S x } x ∈ W (cid:105) is an IL − set -frame. Lemma 7.2.
Every axiom of L is valid in F .Proof. If (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) V , then for some (cid:104) Θ , D (cid:105) ∈ V , (cid:104) Γ , B (cid:105) R (cid:104) Θ , D (cid:105) . Thus J4 isvalid in F by Proposition 3.9.We distinguish the following five cases: • L = IL − ( J1 , J4 ): Suppose (cid:104) Γ , B (cid:105) R (cid:104) ∆ , C (cid:105) . If Γ ≺ C ∆, then ∼ C ∈ ∆because C (cid:66) C ∈ Γ. Hence (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) {(cid:104) ∆ , C (cid:105)} . We conclude that J1 isvalid in F by Proposition 3.3. • L = IL − ( J2 ): Assume that (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) V and for any (cid:104) ∆ (cid:48) , C (cid:48) (cid:105) ∈ V ∩ ↑ ( (cid:104) Γ , B (cid:105) ), (cid:104) ∆ (cid:48) , C (cid:48) (cid:105) S (cid:104) Γ ,B (cid:105) U (cid:104) ∆ (cid:48) ,C (cid:48) (cid:105) . We distinguish the following two cases: – If Γ ≺ C ∆, then for some (cid:104) Λ , C (cid:105) ∈ V , we have Γ ≺ C Λ . Then (cid:104) Λ , C (cid:105) ∈ V ∩ ↑ ( (cid:104) Γ , B (cid:105) ). Since (cid:104) Λ , C (cid:105) S (cid:104) Γ ,B (cid:105) U (cid:104) Λ ,C (cid:105) and Γ ≺ C Λ ,there exist (cid:104) Λ (cid:48) , G (cid:105) , (cid:104) Λ (cid:48) , C (cid:105) ∈ U (cid:104) Λ ,C (cid:105) such that ∼ C ∈ Λ (cid:48) and Γ ≺ C Λ (cid:48) by the definition of S . Therefore (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) U (cid:104) Λ ,C (cid:105) . – If Γ ⊀ C ∆, then for some (cid:104) Θ , D (cid:105) ∈ V , (cid:104) Γ , B (cid:105) R (cid:104) Θ , D (cid:105) , and hence (cid:104) Θ , D (cid:105) S (cid:104) Γ ,B (cid:105) U (cid:104) Θ ,D (cid:105) . Then there exists (cid:104) Θ (cid:48) , D (cid:48) (cid:105) ∈ U (cid:104) Θ ,D (cid:105) such that (cid:104) Γ , B (cid:105) R (cid:104) Θ (cid:48) , D (cid:48) (cid:105) . Therefore, we have (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) U (cid:104) Θ ,D (cid:105) .In either case, we obtain (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) ( (cid:83) (cid:104) ∆ (cid:48) ,C (cid:48) (cid:105)∈ V ∩↑ ( (cid:104) Γ ,B (cid:105) ) U (cid:104) ∆ (cid:48) ,C (cid:48) (cid:105) ) byMonotonicity. Thus we conclude that J2 is valid in F by Proposition3.13. • L = IL − ( J2 , J4 + ): As in the case of IL − ( J2 ), J2 is valid in F .Suppose (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) V . We distinguish the following two cases: – If Γ ≺ C ∆, then there exist (cid:104) Λ , G (cid:105) , (cid:104) Λ , C (cid:105) ∈ V such that Γ ≺ Λ , ∼ C ∈ Λ and Γ ≺ C Λ . Let V (cid:48) := {(cid:104) Λ , G (cid:105) , (cid:104) Λ , C (cid:105)} . – If Γ ⊀ C ∆, then for some (cid:104) Θ , D (cid:105) ∈ V with (cid:104) Γ , B (cid:105) R (cid:104) Θ , D (cid:105) , let V (cid:48) := {(cid:104) Θ , D (cid:105)} .In either case, we have (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) V (cid:48) . Also we have V (cid:48) ⊆ V ∩ ↑ ( (cid:104) Γ , B (cid:105) ).By Monotonicity, (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) ( V ∩ ↑ ( (cid:104) Γ , B (cid:105) )). Therefore J4 + is valid in F by Proposition 3.10. 27 L = IL − ( J4 , J5 ): Suppose (cid:104) Γ , B (cid:105) R (cid:104) ∆ , C (cid:105) and (cid:104) ∆ , C (cid:105) R (cid:104) Θ , D (cid:105) . Let V := {(cid:104) Θ , D (cid:105)} , then (cid:104) Θ , D (cid:105) ∈ V ∩ ↑ ( (cid:104) Γ , B (cid:105) ). If Γ ≺ C ∆ and (cid:3) ∼ C ∈ ∆, then ∼ C ∈ Θ because ∆ ≺ Θ. Thus (cid:104) ∆ , C (cid:105) S (cid:104) Γ ,B (cid:105) V . By Proposition 3.19, J5 is valid in F . • L = IL − ( J1 , J4 , J5 ): As in the cases of IL − ( J1 , J4 ) and IL − ( J4 , J5 ),the axiom schemata J1 , J4 and J5 are valid in F . Lemma 7.3 (Truth Lemma) . For any C ∈ Φ and any (cid:104) Γ , B (cid:105) ∈ W , C ∈ Γ ifand only if (cid:104) Γ , B (cid:105) (cid:13) C .Proof. We prove by induction on C , and we only give a proof of the case C ≡ ( D (cid:66) E ).( ⇒ ): Assume D (cid:66) E ∈ Γ. Let (cid:104) ∆ , F (cid:105) be any element of W such that (cid:104) Γ , B (cid:105) R (cid:104) ∆ , F (cid:105) and (cid:104) ∆ , F (cid:105) (cid:13) D . By induction hypothesis, D ∈ ∆. Since L contains J4 , by Lemma 4.8, there exists Θ ∈ K L such that Γ ≺ Θ and E ∈ Θ. • If Γ ≺ F ∆, then by Lemma 4.7, there exists Λ ∈ K L such that E ∈ Λand ∼ F ∈ Λ. In particular, if L = IL − ( J2 , J4 + ), then Γ ≺ Λ holds.Moreover, if L ∈ { IL − ( J2 ) , IL − ( J2 , J4 + ) } , then we may assume Γ ≺ F Θby Lemma 4.9. Let V := {(cid:104) Θ , F (cid:105) , (cid:104) Λ , F (cid:105)} . • If Γ ⊀ F ∆, then let V := {(cid:104) Θ , F (cid:105)} .In either case, (cid:104) ∆ , F (cid:105) S (cid:104) Γ ,B (cid:105) V . By induction hypothesis, (cid:104) Θ , F (cid:105) (cid:13) E and (cid:104) Λ , F (cid:105) (cid:13) E . We conclude (cid:104) Γ , B (cid:105) (cid:13) D (cid:66) E .( ⇐ ): Assume D (cid:66) E / ∈ Γ. Then by Lemma 4.6, there exists ∆ ∈ K L such that D ∈ ∆ and Γ ≺ E ∆. Moreover if L contains J5 , then (cid:3) ∼ E ∈ ∆ also holds.We have (cid:104) ∆ , E (cid:105) (cid:13) D by induction hypothesis. Let V be any non-empty subsetof W such that (cid:104) ∆ , E (cid:105) S (cid:104) Γ ,B (cid:105) V . By the definition of S , there exists (cid:104) Λ , G (cid:105) ∈ V such that ∼ E ∈ Λ. Then by induction hypothesis, (cid:104) Λ , G (cid:105) (cid:49) E . Thus we obtain (cid:104) Γ , B (cid:105) (cid:49) D (cid:66) E .Since (cid:104) Γ , ⊥(cid:105) ∈ W and A / ∈ Γ , it follows from Truth Lemma that (cid:104) Γ , ⊥(cid:105) (cid:49) A . Thus A is not valid in M .At last, we prove the completeness of the logics IL − ( J2 , J5 ) and IL − ( J2 , J4 + , J5 )with respect to IL − set -frames. Theorem 7.4.
Let L be one of IL − ( J2 , J5 ) and IL − ( J2 , J4 + , J5 ) . Then forany formula A , the following are equivalent:1. L (cid:96) A .2. A is valid in all (finite) IL − set -frames where all axioms of L are valid. roof. (1 ⇒ ⇒ L (cid:48) A . Let Φ be any finite adequate set containing ∼ A .Let Γ ∈ K L be such that ∼ A ∈ Γ . For each Γ ∈ K L , rank(Γ) is defined as inthe proof of Theorem 5.4.We define the model M := (cid:104) W, R, { S x } x ∈ W , (cid:13) (cid:105) as follows:1. W = {(cid:104) Γ , τ (cid:105) : Γ ∈ K L and τ is a finite sequence of elements of Φ (cid:66) withrank(Γ) + | τ | ≤ rank(Γ ) } ;2. (cid:104) Γ , τ (cid:105) R (cid:104) ∆ , σ (cid:105) : ⇐⇒ Γ ≺ ∆ and τ (cid:40) σ ;3. (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) (cid:104) Θ , ρ (cid:105) : ⇐⇒ (a) (cid:104) Γ , τ (cid:105) R (cid:104) ∆ , σ (cid:105) ;(b) For some (cid:104) Θ , ρ (cid:105) ∈ V , (cid:104) Γ , τ (cid:105) R (cid:104) Θ , ρ (cid:105) ;(c) If τ ∗ (cid:104) C (cid:105) ⊆ σ , Γ ≺ ∗ C ∆ and (cid:3) ∼ C ∈ ∆, then the condition C L holds.4. (cid:104) Γ , τ (cid:105) (cid:13) p : ⇐⇒ p ∈ Γ.The condition C L is different for each L , and defined as follows: • IL − ( J2 , J5 ): There exist (cid:104) Λ , ρ (cid:105) , (cid:104) Λ , ρ (cid:105) ∈ V such that τ ∗ (cid:104) C (cid:105) ⊆ ρ , ∼ C ∈ Λ , Γ ≺ ∗ C Λ and (cid:3) ∼ C ∈ Λ . • IL − ( J2 , J4 + , J5 ): There exist (cid:104) Λ , ρ (cid:105) , (cid:104) Λ , ρ (cid:105) ∈ V such that τ ∗ (cid:104) C (cid:105) ⊆ ρ , τ ∗ (cid:104) C (cid:105) ⊆ ρ , Γ ≺ Λ , ∼ C ∈ Λ , Γ ≺ ∗ C Λ and (cid:3) ∼ C ∈ Λ .Then (cid:104) W, R, { S x } x ∈ W (cid:105) is a finite IL − set -frame. Lemma 7.5.
Every axiom of L is valid in the frame F = (cid:104) W, R, { S x } x ∈ W (cid:105) of M .Proof. J2 : As in the proof of Theorem 5.4, J4 is valid in F . Suppose (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) V and for any (cid:104) ∆ (cid:48) , σ (cid:48) (cid:105) ∈ V ∩ ↑ ( (cid:104) Γ , τ (cid:105) ), (cid:104) ∆ (cid:48) , σ (cid:48) (cid:105) S (cid:104) Γ ,τ (cid:105) U (cid:104) ∆ (cid:48) ,σ (cid:48) (cid:105) . • If τ ∗ (cid:104) C (cid:105) ⊆ σ , Γ ≺ ∗ C ∆ and (cid:3) ∼ C ∈ ∆, then there exists (cid:104) Λ , ρ (cid:105) ∈ V such that τ ∗ (cid:104) C (cid:105) ⊆ ρ , Γ ≺ ∗ C Λ and (cid:3) ∼ C ∈ Λ . Since (cid:104) Λ , ρ (cid:105) ∈ V ∩ ↑ ( (cid:104) Γ , τ (cid:105) ), we have (cid:104) Λ , ρ (cid:105) S (cid:104) Γ ,τ (cid:105) U (cid:104) Λ ,ρ (cid:105) . Since τ ∗ (cid:104) C (cid:105) ⊆ ρ , Γ ≺ ∗ C Λ and (cid:3) ∼ C ∈ Λ , by the definition of S , the set U (cid:104) Λ ,ρ (cid:105) satisfies the condition C L . Thus we obtain (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) U (cid:104) Λ ,ρ (cid:105) . • If not, then let (cid:104) Θ , ρ (cid:105) ∈ V be such that (cid:104) Γ , τ (cid:105) R (cid:104) Θ , ρ (cid:105) . Since (cid:104) Θ , ρ (cid:105) ∈ V ∩ ↑ ( (cid:104) Γ , τ (cid:105) ), we have (cid:104) Θ , ρ (cid:105) S (cid:104) Γ ,τ (cid:105) U (cid:104) Θ ,ρ (cid:105) . In particular, (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) U (cid:104) Θ ,ρ (cid:105) .In either case, by Monotonicity, (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) ( (cid:83) (cid:104) ∆ (cid:48) ,σ (cid:48) (cid:105)∈ V ∩↑ ( (cid:104) Γ ,τ (cid:105) ) U (cid:104) ∆ (cid:48) ,σ (cid:48) (cid:105) ).Therefore J2 is valid in F by Proposition 3.16. J5 : Suppose (cid:104) Γ , τ (cid:105) R (cid:104) ∆ , σ (cid:105) and (cid:104) ∆ , σ (cid:105) R (cid:104) Θ , ρ (cid:105) . If there exists C such that τ ∗ (cid:104) C (cid:105) ⊆ σ , Γ ≺ ∗ C ∆ and (cid:3) ∼ C ∈ ∆, then τ ∗ (cid:104) C (cid:105) ⊆ ρ because σ (cid:40) ρ . SinceΓ ≺ ∗ C ∆ and ∆ ≺ Θ, we have Γ ≺ ∗ C Θ by Lemma 4.5. Also ∼ C, (cid:3) ∼ C ∈ Θbecause ∆ ≺ Θ. Therefore we obtain (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) {(cid:104) Θ , ρ (cid:105)} . By Proposition 3.19, J5 is valid in F . 29t last, when L = IL − ( J2 , J4 + , J5 ), we prove that J4 + is valid in F .Suppose (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) V . Then there exists (cid:104) Θ , ρ (cid:105) ∈ V such that (cid:104) Γ , τ (cid:105) R (cid:104) Θ , ρ (cid:105) ,and hence (cid:104) Θ , ρ (cid:105) ∈ V ∩ ↑ ( (cid:104) Γ , τ (cid:105) ). If τ ∗ (cid:104) C (cid:105) ⊆ ρ , Γ ≺ ∗ C ∆ and (cid:3) ∼ C ∈ ∆for some C , then there exist (cid:104) Λ , ρ (cid:105) , (cid:104) Λ , ρ (cid:105) ∈ V such that τ ∗ (cid:104) C (cid:105) ⊆ ρ , τ ∗ (cid:104) C (cid:105) ⊆ ρ , Γ ≺ Λ , ∼ C ∈ Λ , Γ ≺ ∗ C Λ and (cid:3) ∼ C ∈ Λ . In particular, (cid:104) Λ , ρ (cid:105) , (cid:104) Λ , ρ (cid:105) ∈ V ∩ ↑ ( (cid:104) Γ , τ (cid:105) ). Thus we obtain (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) ( V ∩ ↑ ( (cid:104) Γ , τ (cid:105) ).By Proposition 3.10, J4 + is valid in F . Lemma 7.6 (Truth Lemma) . For any formula C ∈ Φ and any (cid:104) Γ , τ (cid:105) ∈ W , C ∈ Γ if and only if (cid:104) Γ , τ (cid:105) (cid:13) C .Proof. We prove the lemma by induction on the construction of C , and we givea proof only for C ≡ ( D (cid:66) E ).( ⇒ ): Assume D (cid:66) E ∈ Γ. Let (cid:104) ∆ , σ (cid:105) be any element of W such that (cid:104) Γ , τ (cid:105) R (cid:104) ∆ , σ (cid:105) and (cid:104) ∆ , σ (cid:105) (cid:13) D . Then by induction hypothesis, D ∈ ∆. Wedistinguish the following two cases. • If τ ∗ (cid:104) F (cid:105) ⊆ σ , Γ ≺ ∗ F ∆ and (cid:3) ∼ F ∈ ∆, then there exists Λ ∈ K L suchthat E, ∼ F ∈ Λ by Lemma 4.7. Moreover, if L = IL − ( J2 , J4 + , J5 ), Γ ≺ Λ also holds. By Lemma 4.9, there exists Λ ∈ K L such that Γ ≺ ∗ F Λ and E, (cid:3) ∼ F ∈ Λ . Let ρ := τ ∗ (cid:104) F (cid:105) , then we can show (cid:104) Λ , ρ (cid:105) , (cid:104) Λ , ρ (cid:105) ∈ W as in the proof of Theorem 5.4. Let V := {(cid:104) Λ , ρ (cid:105) , (cid:104) Λ , ρ (cid:105)} , then (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) V by the definition of S . By induction hypothesis, (cid:104) Λ , ρ (cid:105) (cid:13) E and (cid:104) Λ , ρ (cid:105) (cid:13) E . We conclude (cid:104) Γ , τ (cid:105) (cid:13) D (cid:66) E . • If not, then there exists Θ ∈ K L such that Γ ≺ Θ and E ∈ Θ by Lemma4.8. Let ρ := τ ∗ (cid:104) E (cid:105) , then (cid:104) Θ , ρ (cid:105) ∈ W . By induction hypothesis, (cid:104) Θ , ρ (cid:105) (cid:13) E . Let V := {(cid:104) Θ , ρ (cid:105)} , then (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) V . We conclude (cid:104) Γ , τ (cid:105) (cid:13) D (cid:66) E .( ⇐ ): Assume D (cid:66) E / ∈ Γ. By Lemma 4.6, there exists ∆ ∈ K L such that D ∈ ∆, Γ ≺ ∗ E ∆ and (cid:3) ∼ E ∈ ∆. Let σ := τ ∗ (cid:104) E (cid:105) , then (cid:104) ∆ , σ (cid:105) ∈ W . We have (cid:104) ∆ , σ (cid:105) (cid:13) D by induction hypothesis.Let V be any non-empty subset of W with (cid:104) ∆ , σ (cid:105) S (cid:104) Γ ,τ (cid:105) V . Since τ ∗ (cid:104) E (cid:105) = σ ,Γ ≺ ∗ E ∆ and (cid:3) ∼ E ∈ ∆, there exists (cid:104) Λ , ρ (cid:105) ∈ V such that ∼ E ∈ Λ by thedefinition of S By induction hypothesis, (cid:104) Λ , ρ (cid:105) (cid:49) E . Therefore we conclude (cid:104) Γ , τ (cid:105) (cid:49) D (cid:66) E .For the empty sequence (cid:15) , (cid:104) Γ , (cid:15) (cid:105) ∈ W . By Truth Lemma, we obtain (cid:104) Γ , (cid:15) (cid:105) (cid:49) A because A / ∈ Γ . Therefore A is not valid in M . Corollary 7.7.
Every logic shown in Figure 2 is decidable.
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