Digital Object Identifier (DOI):10.1007/s00153-002-0151-1

Arch. Math. Logic 42, 261–277 (2003) Mathematical Logic

V.V. Rybakov

Barwise’s information frames and modal logics

Received: 7 May 2000Published online: 10 October 2002 – © Springer-Verlag 2002

Abstract. The paper studies Barwise’s information frames and answers the John Barwisequestion: to find axiomatizations for the modal logics generated by information frames. Wefind axiomatic systems for (i) the modal logic of all complete information frames, (ii) thelogic of all sound and complete information frames, (iii) the logic of all hereditary and com-plete information frames, (iv) the logic of all complete, sound and hereditary informationframes, and (v) the logic of all consistent and complete information frames. The notion ofweak modal logics is also proposed, and it is shown that the weak modal logics generated byall information frames and by all hereditary information frames are K and K4 respectively.To develop general theory, we prove that (i) any Kripke complete modal logic is the modallogic of a certain class of information frames and that (ii) the modal logic generated by anygiven class of complete, rarefied and fully classified information frames is Kripke complete.This paper is dedicated to the memory of talented mathematician John Barwise.

1. Introduction, motivation

It is not an exaggeration nowadays to say that the scientific disciplines dealingwith information are ubiquitous. The research concerning the notion of informa-tion ranges from pure mathematics, informatics and computer science, artificialintelligence and knowledge representation, through technical sciences to philos-ophy, linguistics and many other subjects of the human sciences. In the presentpaper 1 we explore informational frames introduced by John Barwise [1]. Theseframes are special multi-based models, or algebraic systems, with the relations

s which express the context’s depending consequence relations on classificationtypes (i.e. knowledge about consequences). They are aimed to handle informationabout situations which occur the base-set in these frames.

For an introduction to information frames the reader can refer to Barwise [1],where a comprehensive motivation is given. Stated briefly, in [1] a theory of pos-sible states is offered with the aim to describe relations between situations. Modaloperations appear as a certain mathematical tool aimed to clarify which situations

V.V. Rybakov: Department of Computing and Mathematics, Manchester MetropolitanUniversity, All Saints, Manchester M15 6BH, U.K. e-mail: V.Rybakov@mmu.ac.uk

Key words or phrases: Knowledge presentation – Information – Information flow – Infor-mation frames – Modal logic-Kripke model

1 This research is supported by the Russian Science Foundation (RFFI) and by theDepartment of Computer Science and Informatics of Bern University, Switzerland, 1999.

262 V.V. Rybakov

are possible and which are not. The point is that the possibility of states introducesan effect which has a second order nature and cannot be expressed by first orderformulas. This approach has some roots in Fred Dretske’s semantic theory of in-formation [5] and the research of Stalnaker [9] and Shannon [8], as well as someparts of [2], [3] and [6] though these do not refer to directly.

The modal logics generated by information frames are defined in a semanticway. As an intriguing open question, Barwise [1] set the question to find axiomati-zations for modal logics generated by information frames. This question was initialpoint of the research presented here. The aim of our paper is to answer this questionand to establish a general mathematical theory concerning information frames. Wefind axiomatic systems for modal logics of (i) all complete information frames, (ii)all sound and complete information frames, (iii) all hereditary and complete in-formation frames, (iv) all complete, sound and hereditary information frames, and(v) all consistent and complete information frames. Also we consider weak modallogics generated by information frames and show that the weak modal logics gen-erated by all information frames and by all hereditary information frames are K

and K4 respectively. Towards general theory on relations between Kripke completemodal logics and modal logics generated by information frames, it is shown (The-orem 3.17) that any Kripke complete modal logic is generated by a certain class ofinformation frames. To make a step for the converse, in the final part of the paper,we show that, for rarefied information frames, the context’s logical consequence

relation s can be presented in some way by usual Kripke frames. This allowsus, partly, to answer the question: which logics generated by information framesare Kripke complete. We show that the modal logic generated by any given class ofcomplete, rarefied and fully classified information frames is Kripke complete. Allpreliminary information is given, and the paper, in this respect, is self-contained.

2. Denotation, preliminary information

For common knowledge concerning modal logics and their semantics (i.e. Kripkemodels and modal algebras), we refer to the literature (among modern sources, cf.[10] or [4], for instance). Information frames and modal logics generated by theseframes were introduced in Barwise [2]. The exposition of our results is based onthis paper, and we recall all necessary definitions, notation and facts from [2].

A classification A is a triple 〈S(A), �(A), |=A〉, where S(A) is a set of ob-jects called situations, the set �(A) is a set of objects called situation types, and|=A⊆ S(A) × �(A). The notation s |=A σ is read s is of type σ or σ holds s, orstill σ is valid in s. In writings, as above, we will omit A if the context is clearand then we write merely A := 〈S, �, |=〉 . A sequent I is a pair 〈�, �〉, where� ∪ � ⊆ �. For any s ∈ S, I holds s (denotations � |=s �, or s |= I) iff∀σ ∈ �(s |= σ) ⇒ ∃σ ∈ �(s |= σ). A sequent I is an information about a set ofsituations S1 ⊆ S iff ∀s ∈ S1(s |= I), we write then � |=S1 � and S1 |= I.

Lemma 2.1 ([1] Proposition 1). Let A be a classification and S1 be a set of situa-tions from S. Then the relation |=S1 satisfies the following conditions:

• (i) Identity: � |=S1 �,

Barwise’s information frames and modal Logics 263

• (ii) Weakening: (� |=S1 �⇒�, �1 |=S1 �, �1),• (iii) Global Cut: If �, �1 |=S1 �, �1 for each partition 〈�1, �1〉 of some set

�1 ⊆ � then � |=S1 �.

An information context C(A) := 〈A, c, N〉 consists of a certain classifica-

tion A = 〈S, �, |=〉 together with c ⊆ �×� and a set N ⊆ S (objects of N are

called normal situations), where c satisfies conditions (i) – (iii) of Lemma 2.1

and each sequent in c is an information about the set N . Thus we can refer to

information contexts as tuples C := 〈S, �, |=, c, N〉 satisfying the conditionsabove.

An information context is (1) Sound if every s ∈ S is normal, (2) Complete

if for all sequents I := 〈�, �〉, ��� c� ⇒ (∃s ∈ N)(� |=s �), (3) Consistent

if ∃I := 〈�, �〉(��� c�) (or equivalently ∅�� c∅, cf. weakening). Let C be aninformation context. A state is a sequent I which is a partition of the set of situationtypes �. We denote by St (�) the set of all states. For any s ∈ S, the state of s is thestate st (s) := 〈�s, �s〉, where �s := {σ ∈ � | s |= σ } and �s := � − �s . A state

ω ∈ St (�) is realized if ∃s ∈ S(st (s) = ω).A state ω := 〈�, �〉 is c-impossible

if � c�. Otherwise ω is c-possible.An information frame IF (A) is the following structure defined over a clas-

sification A := 〈S, �, |=〉: IF (A) := 〈A, {Cs | s ∈ S}〉, where all structures

Cs := 〈S, �, |=, s, Ns〉 are certain information contexts over S (any is the con-text of the object s), and Ns ⊆ S, such that if s1, s2 ∈ S and st (s1) = st (s2)

then s1 = s2 . We say IF (A) is complete, sound or consistent if any Cs fors ∈ S(A) does.

A classificationA := 〈S, �, |=〉 is Boolean if there are fixed operations∧, ∨, →and ¬ defined over classification types from � which satisfy usual Boolean condi-tions:

s |= ω1 ∧ ω2⇔s |= ω1&s |= ω2; s |= ω1 ∨ ω2⇔s |= ω1 ∨ s |= ω2

s |= ω1 → ω2⇔s |= ω1 ∨ s |= ω2; s |= ¬ω⇔s |= ω.

It is easy to see that each classification give rise to an associated Boolean classifica-tion B(A) if we extend classification situation types to their Boolean combinationsdefining new |= in accordance with the conditions above.

Given a classification A := 〈S, � |=〉. For any propositional letter p, we canchoose a valuation V of p in A to be a subset of St (�). Then we can extend V toall Boolean formulas constructed over valuated by V letters in usual way. We needto extend V to modal formulas which employ � and � also. For this we involve inthe information frames but not only classifications.

Definition 2.2. Let IF (A) be an information frame.1. A state ω ∈ St (�) is accessible from a state ω1 (denotation ω1 ≤A ω) iff

∃s ∈ S with st (s) = ω1 such that the state ω is s-possible, i.e. ω = 〈�, �〉 and

��� s�.

264 V.V. Rybakov

For any p ⊆ St (�), we set

�p := {ω | ω ∈ St (�) ∧ ∃ω1 ∈ St (�)(ω ≤A ω1 ∧ ω1 ∈ p)}.�p := {ω | ω ∈ St (�) ∧ ∀ω1 ∈ St (�)(ω ≤A ω1 ⇒ ω1 ∈ p)}.

Note that these definitions are consistent w.r.t. mutual presentation of modal op-erations by using ¬ because, for situations s1, s2 with st (s1) = st (s2), always

s1 = s2 holds. And now we can extend valuations on information frames tomodal formulas by V (�A) = �(V (A)) and V (�A) = �(V (A)).

Lemma 2.3. (cf. [2], Proposition 3.) For any information frame IF (A) and anysituation s from A and any p ⊆ St (�),

1. St (s) ∈ �p iff ∃ω ∈ St (�) such that ω is s-possible and ω ∈ p.2. St (s) ∈ �p iff ∀ω ∈ St (�) if ω is s-possible then ω ∈ p.

Given an information frame IF (A) and a valuation V of all propositional let-ters of a modal formula A in the classification A, we say A is valid in IF (A) underV and write IF (A) |=V A iff ∀s ∈ S(st (s) ∈ V (A)). If this holds for any V wewrite IF (A) |= A.

3. Modal logics and information frames

Let K be a class of information frames. The modal logic L(K) of K is the set ofall propositional modal formulas which are valid in all information frames from Kunder all valuations. To show this definition is consistent we need, first, to verifythat L(K) is really a logic, i.e. to verify that this set is closed w.r.t. substitutions offormulas in place of variables.

Lemma 3.1. For any modal formula A(p1, ..., pn) ∈ L(K) and any tuple offormulas B1, ..., Bn, A(B1, ..., Bn) ∈ L(K).

Proof. Indeed, given an IF (A) from K and a valuation V of all propositional let-ters from A(B1, ..., Bn) in the classification A, we take the new valuation V1 forp1, ..., pn in A assuming V1(pi) := V (Bi). Then we have IF (A) |=V1 A becauseA ∈ L(K). But then IF (A) |=V A(B1, ..., Bn). ��

For a modal formula A, notation �PC A means that we can derive A usingmodus ponens from schemes of axioms of the classical propositional calculus PC

for the case of modal formulas.

Lemma 3.2. For any information frame IF (A), A := 〈S, � |=〉, and any valua-tion V of propositional letters of any modal formula A in St (�), if �PC A then∀ω ∈ St (�)(ω ∈ V (A)).

The proof is straightforward: first to verify that all axioms of PC satisfy thecondition of our lemma and then to show that using of modus ponens preserves thiscondition.

So, as we meant, modus ponens preserves validity and hence L(K) is a logicbased on classical propositional calculus. For to be a modal logic this set mustsatisfy certain minimal requirements. Concerning the modal axiom of the minimalnormal modal logic K, the following holds

Barwise’s information frames and modal Logics 265

Lemma 3.3. ([2], Proposition 4) For any information frame IF (A) and anyvaluation V of letters p and q in A, IF (A) |=V �(p → q) → (�p → �q).

So, logics L(K) include axioms of the minimal normal modal logic K. But it iseasy to show that the normalization rule x/�x does not always preserve the validityof formulas on information frames. Nevertheless for good classes of informationframes the necessitating rule works:

Lemma 3.4. (cf. [1], Proposition 6) For any given complete information frameIF (A), the rule x/�x preserves the validity of modal formulas in IF (A) (w.r.t.any given valuation).

Proof. Assume that a formula A is valid in IF (A) for a given V . We have to showthat ∀s ∈ S(st (s) ∈ V (�p)). Assume that a state ω ∈ St (�) is s-possible. SinceIF (A) is complete any Cs1 for s1 ∈ S is complete, in particular Cs does. Thusthere is an s-normal situation s1 with st (s1) = ω. Then by the validity of A itselfst (s1) = ω ∈ V (A). Hence ω ∈ V (�A) as required. ��

Thus, for any class K of complete information frames, L(K) is a normal modallogic. And any L(K) is a modal logic but maybe non-normal. We can easy showthat, for any K, the logic L(K) differs from the minimal non-normal modal logic.Indeed,

Lemma 3.5. For any classification A := 〈S, � |=〉, any IF (A) and any valuationV of propositional letters of any modal formula A in St (�), if ∀ω ∈ St (�)(ω ∈V (A)) then ∀ω ∈ St (�)(ω ∈ V (�A)).

It is evident: assume ω, ω1 ∈ St (�) and assume ω ≤A ω1. Then ω1 ∈ V (A)

and consequently we get ω ∈ V (�A).Thus from these two lemmas we immediately obtain

Corollary 3.6. For any logic L(K) and for any modal formula A, if �PC A thenfor any k ∈ N , �kA ∈ L(K).

Now we slightly generalize Lemma 3.3:

Lemma 3.7. For any classification A := 〈S, � |=〉, and any information frameIF (A) and any valuation V of propositional letters of the formula

AxK := �(p → q) → (�p → �q)

in St (�), ∀ω ∈ St (�)(ω ∈ V (�(p → q) → (�p → �q)))

Proof. Let ω ∈ St (�) and ω ∈ V (�(p → q)) = �V (p → q) and ω ∈ V (�p) =�V (p). Suppose ω ≤A ω1 for some ω1 ∈ St (�). Then using our assumptionsabove, it follows ω1 ∈ V (p) and ω1 ∈ V (p → q) and consequently ω1 ∈ V (q).Thus ω ∈ �V (q). ��

Using this lemma and Lemma 3.5 we derive

266 V.V. Rybakov

Corollary 3.8. For any classification A := 〈S, � |=〉, any IF (A) and any valu-ation V of propositional letters of AxK in St (�), ∀ω ∈ St (�)(ω ∈ V (�nAxK))

for any n. For any logic L(K) and for any n ∈ N , �nAxK ∈ L(K).

Thus all modal logics L(K) have formulas �nAxK and �nA among theoremsfor any n and any PC-provable A. So no one of these logics coincides with theminimal non-normal modal logic. Now, after these simple observations, we turnto Barwise’s question from [1] concerning finding of axiomatizations for modallogics generated by information frames. We begin with the logic L(Kc), where Kc

is the set of all complete frames.Consider the minimal normal modal logic K whose axioms are AxK and A,

where �PC A, and inference rules are modus ponens and the normalization(necessitation) rule x/�x. By Corollaries 3.6 and 3.8 K ⊆ L(Kc). We claim thatthe axiomatization of K is the axiomatization for L(Kc):

Theorem 3.9. The modal logic L(Kc) coincides with K.

Proof. The inclusion K ⊆ L(Kc) is shown above. Assume that A(p1, ..., pn) isa modal propositional formula built up out of propositional letters p1, ..., pn andsuppose that A ∈ K. Since, as well known, K is Kripke-complete (cf. [4] or [10]for instance), there is a Kripke model M := 〈W, R, V 〉 disproving A. That is forsome a ∈ W

a�� V A. (1)

To continue our proof we need to recall the notion of the information frame gener-ated by any given Kripke frame introduced by J.Barwise in [1].

Given an arbitrary Kripke frame F := 〈W, R〉. As the classification A(F) gen-erated by F we take 〈W, 2W, ∈〉, where 2W is the Boolean algebra of all subsets ofW . Thus we can assume that this classification is Boolean.

For any s ∈ W the context Cs consists of the set of all situations W together

with the set of normal situations Ns := {a | a ∈ W, sRa} and the relation s

defined as follows:

∀〈�, �〉 ∈ 22W × 22W

(� s� ⇔ ∀a ∈ W(sRa ⇒ (a ∈⋂

� ⇒ a ∈⋃

�))).

We setIFF := 〈W, 2W, ∈, {〈W, 2W, ∈, s, Ns〉 | s ∈ W }〉

and call IFF information frame associated with F . It can be simply verified thatIFF satisfies all requirements for the internal contexts. Not trivial parts are only

the cases of Global Cut and that s is an information about the set of all normalstates. We consider these two cases below.

Global Cut: Assume �, �1 s�, �1 for all partitions 〈�1, �1〉 of a set �1 ⊆2W . Take arbitrary s1 ∈ W , where sRs1 and assume s1 ∈ ⋂

�. Consider �1 :={X | X ⊆ W, X ∈ �1, s1 ∈ X } and � := �1 −�1. Then 〈�1, �1〉 is a partition of

�1 and by our assumption above �, �1 s�, �1 holds and because s1 ∈ ⋂�, �1

Barwise’s information frames and modal Logics 267

we conclude s1 ∈ ⋃�, �1. However s1 ∈ ⋃

�1 therefore s1 ∈ ⋂�. Thus Global

Cut holds.Now assume a is a normal situation. Then a ∈ Ns which means sRa. Let 〈�, �〉

be a sequent and � s�. Assume now a ∈ X for any X ∈ �. Then a ∈ ⋂� and

using � s� and sRa we conclude a ∈ ⋂� which means ∃X ∈ �(a ∈ X ).

Thus the sequent 〈�, �〉 is an information about a (which, to recall the definition,means � |={a} �).

To verify that s1 = s2 for s1 and s2 with st (s1) = st (s2) consider lasttwo states. st (si) := 〈{X | X ⊆ W, si ∈ X }, {Y | Y ⊆ W, si ∈ Y}〉. Therefore ifwe denote st (si) := 〈�si , �si 〉 we get si = ⋃

�si . Thus since st (s1) = st (s2) wederive s1 = s2. Hence the information frame IFF is well defined.

Given an arbitrary Kripke model M := 〈W, R, V 〉. Consider the associatedinformation frame IFM for the frame of M with the valuation V1 of propositionalletters from the domain of V given by

V1(pi) := {st (s) | s ∈ W, (M, s) V pi.} (2)

Lemma 3.10. The following hold:(i) The definition of V1 is consistent: any st (s) is a partition of 2W ;(ii) The information frame IFM is complete;(iii) For any s ∈ W and any modal formula B built up out of variables from the

domain P of V ,

(M, s) V B ⇔ st (s) ∈ V1(B).

Proof. For any s ∈ W the state st (s) by its definition is 〈�s, �s〉, where

�s := {X | X ⊆ W, s ∈ X } and �s := {Y | Y ⊆ W, s ∈ Y}.(i): Clearly that st (s) is a partition of 2W : observe �s = {X | X ⊆ W, s ∈ X },

�s := {Y | Y ⊆ W, s ∈ Y} and, obviously, it is a partition of 2W . So the valuationV1 is well defined.

(ii): To show that IFM is complete assume, for a certain sequent I := 〈�, �〉,��� s� holds. Then for some a, sRa and a ∈ ⋂

� but a ∈ ⋃�. Moreover a ∈ Ns

and � |=a �. Thus our information frame is complete.(iii): Induction on the length of B. Base: for B := pi by the definition of V1 we

have (M, s) V pi⇔st (s) ∈ V1(pi) as we needed. Inductive steps for Booleanconnectives are routine and are evident. Assume B := �E. By the definition of theoperation � (to evaluate modal formulas) within information frames, we have

�V1(E) := {ω | ω ∈ St (2W)&∀ω1 ∈ St (2W)(ω ≤A ω1 ⇒ ω1 ∈ V1(E))} (3)

Fix for short ω := 〈�, �〉 and ω1 := 〈�1, �1〉 for all ω and ω1 mentionedabove. By decomposing the relation ω ≤A ω1 in accordance with the definition of≤A, from (3) we derive

�V1(E) := {ω | ω ∈ St (2W)&∀ω1 ∈ St (2W)

268 V.V. Rybakov

[∃s0 ∈ W(st (s0) = ω) & (ω1 = 〈�1, �1〉) & (4)

(�1 �� s0�1)] ⇒ ω1 ∈ V1(E))}.Any relation �1 �� s0�1 above is equivalent to

∃s1[(s0Rs1)&(s1 ∈⋂

�1) & (s1 ∈⋃

�1)]. (5)

First suppose st (s) ∈ V1(�E) = �V1(E). Assume sRs1 for some s1. Con-

sider st (s1) := 〈�s1 , �s1〉. Then �s1 �� s�s1 , indeed, to verify this using (5) notethat s1 ∈ ⋂

�s1 = {s1} but s1 ∈ ⋃�1. Therefore applying (4) and st (s) ∈

�V1(E) we conclude st (s1) ∈ V1(E) and involving inductive hypotheses we derive

(M, s1) V E. And, since this holds for all s1 mentioned above, (M, s) V �E.

Conversely, assume now that (M, s) V �E. Then for all s1 such that sRs1,

(M, s1) V E holds. And by inductive hypothesis it follows

∀s1[(sRs1) ⇒ (st (s1) ∈ V1(E))]. (6)

To show that st (s) ∈ �V1(E) assume that ω1 := 〈�1, �1〉 is a partition of 2W and

�1 �� s�1. Using (5) we derive ∃s1[(sRs1)&(s1 ∈ ⋂�1)&(s1 ∈ ⋃

�1)]. But ω1is a partition, therefore st (s1) = ω1, and applying (6) we conclude ω1 ∈ V1(E).These observations by (4) imply st (s) ∈ �V1(E) which concludes the proof of ourlemma. ��

To continue the proof of our theorem, note that our given Kripke model M :=〈W, R, V 〉 disproves A. That is (see (1)) for some a ∈ W , (M, a)�� V A. Thereforeby Lemma 3.10 st (a) ∈ V1(A), hence IFM |=V1 A and still the information frameIFM is complete. Thus A ∈ L(Kc) and our theorem is proved. ��

Using this theorem as a bases of our further research we want to look at modallogics generated by distinct classes of information frames. First we turn to the logicof all complete and sound information frames. Let Ks,c be the class of all soundand complete information frames.

Lemma 3.11 ([1], Proposition 5). For any sound information frame IF (A) and

any valuation V for the letter p in A, IF (A) V �p → p (and IF (A) V p →�p dually holds).

Proof. Assume s is a situation from A and st (s) ∈ �V (p). We have

�V (p) := {ω | ω ∈ St (�(A)) & ∀ω1 ∈ St (�(A))(ω ≤A ω1 ⇒ ω1 ∈ V (p))}.and to show that st (s) ∈ V (p) it is sufficient to verify that st (s) ≤A st (s). The last

relation is equivalent to �s �� s�s . But IF (A) is sound, consequently s is normal

in Cs , that is s ∈ A and s ∈ Ns and, for any sequent 〈�, �〉, if � s� then � |= �

(because the sequent 〈�, �〉 must be the information about all normal situations, in

particular for s). Because the last fails for the sequent 〈�s, �s〉 it follows �s �� s�s ,and st (s) ∈ V (p). ��

Barwise’s information frames and modal Logics 269

Thus the formula �p → p is valid in all sound information frames and hence�p → p ∈ L(Ks,c). Consider the von Wright modal logic T := K ⊕ (�p → p).Recall that it has as inference rules modus ponens and the necessitation rule.

Theorem 3.12. von Wright modal logic T and the modal logic of all sound andcomplete information frames L(Ks,c) coincide.

Proof. The inclusion T ⊆ L(Ks,c) immediately follows from Lemmas 3.11, 3.4and other previous results. For the converse, assume A(p1, ..., pn) is a modal prop-ositional formula (built up out of propositional letters p1, ..., pn) and A ∈ T holds.T is Kripke-complete (cf. [4] or [10] for instance) w.r.t. reflexive Kripke models,i.e. there is a reflexive model M := 〈W, R, V 〉 disproving A. We take the asso-ciated information frame IFM generated by M with the valuation V1 as abovein Lemma 3.10. By this lemma the information frame IFM is complete and V1disproves the formula A in IFM. Still, as M is reflexive, the information frameIFM is sound. So, it follows A ∈ L(Ks,c). ��

Next reasonable step is to explore which conditions on information frames couldyield the validity of the modal formula �p → ��p, which one is responsible fortransitivity in Kripke models.

Definition 3.13. We say an information frame IF (A) is hereditary if, for any sit-uations s1, s2 ∈ S(A) and any state ω ∈ St (�(A)) if st (s1) ≤A st (s2) andst (s2) ≤A ω then st (s1) ≤A ω.

Lemma 3.14. If an information frame IF (A) is hereditary then, for any valuationV of a letter p in St (�) of A := 〈S, �, |=〉, IF (A) |=V �p → ��p holds.

Proof. Assume st (s) ∈ �V (p) but st (s) ∈ ��V (p). Then there is a state ω1 :=〈�1, �1〉 with st (s) ≤A ω1 and ω1 ∈ �V (p). Therefore ∃ω2 := 〈�2, �2〉 ∈ St (�)

such that ω1 ≤A ω2 and ω2 ∈ V (p). Thus there is s1 ∈ S with st (s1) = ω1 and

�2 �� s1�2. Thus st (s) ≤A st (s1) ≤A ω2. But the information frame IF (A) ishereditary. Therefore st (s) ≤ ω2 and since st (s) ∈ �V (p), it follows ω2 ∈ V (p)

– a contradiction. ��Theorem 3.15. The modal logic L(Kh,c) of all hereditary and complete informa-tion frames is the normal modal logic K4 := K ⊕ �p → ��p.

Proof. By previously shown facts and Lemma 3.14 it follows that L(Kh,c) ⊆ K4.Assume A ∈ K4. It is well known that K4 is Kripke complete w.r.t transitiveframes (cf. [10] or [4], for instance). Therefore there exists a Kripke model M :=〈W, R, V 〉 based on a transitive frame with a valuation V disproving A. Take theinformation frame IFM generated by M and the valuation V1 as in (2) beforeLemma 3.10.

By this lemma the information frame IFM is always complete and is, in ourcase, hereditary. Indeed, let s1, s2 ∈ W , ω ∈ St (2W), st (s1) ≤A st (s2), and

st (s2) ≤A ω. Hence ω := 〈�, �〉 and ��� s2�. This means for some a ∈ W s2Ra

270 V.V. Rybakov

and a ∈ ⋂� and a ∈ ⋃

�. That st (s1) ≤A st (s2) implies for some b s1Rb andb ∈ ⋂

�st(s2) and b ∈ ⋃�st(s2) where st (s2) = 〈�st(s2), �st(s2)〉. But then b = s2.

In sum we got s1Rs2 and s2Ra. Our model M is transitive. Therefore s1Ra and

we know that a ∈ ⋂� and a ∈ ⋃

�. Hence ��� s1� and st (s1) ≤A ω what iswhat we needed. So, IFM is hereditary. Since this point we continue the proof asit was done for Theorem 3.9 using Lemma 3.10 following in detail this proof. ��

Let L(Ks,h,c) be the modal logic of all sound, hereditary and complete infor-mation frames.

Theorem 3.16. The modal logic L(Ks,h,c) of all sound, hereditary and completeinformation frames is the normal modal logic S4 := K4 ⊕ �p → ��p.

Proof. Indeed, L(Ks,h,c) ⊆ S4, which follows directly from Lemmas 3.11 and3.14. If A ∈ S4 then, using that S4 is Kripke complete w.r.t. reflexive and transitivemodels (cf, [4] or [10], for instance) it follows there is a transitive and reflexiveKripke model disproving A. We take the associated information frame IFM andthe valuation V1 as in (2) above Lemma 3.10. By this lemma the information frameIFM disproves A by V1.And we can show exactly as in Theorems 3.15 and 3.12 thatthis information frame is hereditary and sound, and we know that this informationframe is always complete (cf. Lemma 3.10). ��

Thus, from our research, it follows that the basic modal logics can be representedas modal logics of specific classes of complete information frames. Up to now wehave restricted ourselves in our consideration to modal logics which are not toostrong – basic ones – which occur in the lower part of the lattice of all normal modallogics. Certainly it would be relevant to look at stronger modal logics from theinformation frames viewpoint. First reasonable general question is: which normalmodal logics can be represented as modal logics of information frames? A goodsufficient condition is:

Theorem 3.17. For any Kripke complete normal modal logic L there is a class K ofcomplete information frames such that L(K) = L, i.e. any Kripke complete normalmodal logic is complete w.r.t. a specific class of complete information frames.

Proof. Indeed, if L is Kripke complete then there is a class KF of Kripke framessuch that

L = {A | A ∈ For, ∀F ∈ KF(F A)}.Take the class IF (KF) of all information frames associated with frames from KF

which are defined above in Lemma 3.10. By Lemma 3.10 L(IF (KF)) ⊆ L. As-sume that there is a formula A ∈ L such that A ∈ L(IF (KF)). Then there is aframe IFF from IF (KF) and a valuation V1 of A in this frame disproving A. Thatis, there is an element a ∈ F such that st (a) ∈ V1(A). Take the valuation V in theframe F by setting

V (pi) := {s | s ∈ F, st (s) ∈ V1(pi)}.

Barwise’s information frames and modal Logics 271

Then our both valuations V1 and V will satisfy the condition (3.5) precedingLemma 3.10. Therefore Lemma 3.10 is applicable to both of them, and by thislemma it follows that for all s ∈ F

s V A ⇔ st (s) ∈ V1(A).

But we had that st (a) ∈ V1(a). Therefore it follows a�� V A and F �� V A whichin sum yields A ∈ L. ��

Thus all Kripke complete normal modal logics are representable as modal log-ics of certain classes of complete information frames, i.e. a comprehensive answeris given.

As we noted above the generalization (necessitation) rule x/�x does not pre-serve validity of formulas in information frames. Now we briefly pause to discusswhat is a reason for, and how we could fix it by altering the notion of the validity.

For a given information frame IF (A) with the classification A and situationtypes � we defined a valuation V of propositional letters by states from St (�)

(note, not by states of situations st (t) with s ∈ S(A)). But the definition of va-lidity of a formula A in IF (A) under V was given as follows: IF (A) |=V A iff∀s ∈ S(st (s) ∈ V (A)). Thus we restrict ourselves by looking only at the validityof A in states which are states of situations (i.e. are realized) but not in all states.And an immediate hypothesis comes that this dissonance is the reason for the fail-ure of the necessitating rule (note however that our previous definition of validity,with the restriction to realized states, was taken wisely, because non-realized statescould exist and we could be particularly interested to avoid non-realized situations).Below we confirm our hypothesis. So, we alter slightly the definition of the validityand fix the denotation

IF (A) |=s,V A ⇔ ∀ω ∈ St (�)(ω ∈ V (A)).

Evidently |=s,V is stronger than |=V and we refer to |=s,V as strong validity and to|=V as weak validity in information frames.

Definition 3.18. For a given class K of information frames, the weak modal logicof K is the set

Lw(K) := {A | ∀IF (A) ∈ K, ∀V (IF (A) |=s,V A)}.Clearly, we always have Lw(K) ⊆ L(K), therefore we refer to the logics Lw(K)

as weak logics. The same argument as for L(K) shows that the weak modal logicsare in fact modal logics, i.e. they contain all the necessary schemes of axioms ofthe logic K and are closed w.r.t modus ponens.

Lemma 3.19. Let IF (A) be an arbitrary information frame, let A be a formula A

and V be a valuation of A in IF (A). If IF (A) |=s,V A then IF (A) |=s,V �A.

Proof. Let IF (A) |=s,V A. Then ∀ω ∈ St (�(A)), ω ∈ V (A). We must show that∀ω ∈ St (�(A))(ω ∈ V (�A)) holds. Assume that a certain state ω1 ∈ St (�(A))

is happen to be ω-possible. Then by the strong validity of the formula A itself itfollows ω1 ∈ V (A). Hence we get ω ∈ V (�A) what we needed. ��

272 V.V. Rybakov

Theorem 3.20. The weak modal logic Lw(IF ) of all information frames is theminimal normal modal logic K.

Proof. We know already that K ⊆ Lw(IF ); for the converse use Lemma 3.10.��

We cannot claim that, for all sound information frames IF (A), the relationIF (A) |=s,V �p → p holds for all V . Because, in sound frames certain non-realized states ω still could be and ω ∈ �V (⊥) for all V . Therefore we cannotoffer immediately axiomatization for the weak modal logic of the class Ks of allsound information frames, and cannot say that Lw(Ks) = T . However, for the classKh of all hereditary information frames, our techniques works and we can offeraxiomatic system for the weak logic Lw(Kh).

Lemma 3.21. For any hereditary information frame IF (A) and for any valuationV of a letter p in St (�) of A := 〈S, �, |=〉, IF (A) |=s,V �p → ��p holds.

Proof. Suppose ω ∈ St (�) and ω ∈ �V (p) but ω ∈ ��V (p). Then there is astate ω1 := 〈�1, �1〉 with ω ≤A ω1 and ω1 ∈ �V (p). In particular, there is asituation s with st (s) = ω and st (s) ≤A ω1. Then ∃ω2 := 〈�2, �2〉 ∈ St (�) suchthat ω1 ≤A ω2 and ω2 ∈ V (p). Again, then there is a state s1 with st (s1) = ω1 and

st (s1) ≤ ω2. Therefore it follows that �2 �� s1�2. Thus st (s) ≤A st (s1) ≤A ω2and, since the information frame IF (A) is hereditary, it follows st (s) ≤ ω2. Butbecause st (s) ∈ �V (p), it follows ω2 ∈ V (p) which contradicts ω2 ∈ V (p). ��

Theorem 3.22. The weak modal logic Lw(Kh) of the family of all hereditaryinformation frames is the normal modal logic

K4 := K ⊕ �p → ��p.

Proof. Using Lemma 3.21 it follows Lw(Kh) ⊆ K4. Suppose A ∈ K4. Since K4is Kripke complete w.r.t transitive frames (cf. [10] or [4], for instance), there existsa Kripke model M := 〈W, R, V 〉 based on a transitive frame with a valuation V

with M�� A. Take the information frame IFM generated by M and the valuationV1 as in (2) before Lemma 3.10. By this lemma V1 disproves A in IFM. Exactlyby the same argument as in Theorem 3.15 we can show that IFM is hereditarywhich completes the proof. ��

Thus there is a skew concerning typical expectations on behavior of the modallogics K4 and T for the cases of weak and usual modal logics of informationframes. K4 and T do not play the same role for weak modal logics as they do forusual modal logics of information frames.

Now we explore how to handle consistent frames through modal framework.

Lemma 3.23. Any information frame IF (A) is consistent iff, for any valuation V ,IF (A) |=V �p → �p.

Barwise’s information frames and modal Logics 273

Proof. Assume IF (A) is not consistent. Take any V for p. Consider st (s), forsome s ∈ S. Then there are no s-possible states and then st (s) ∈ �V (p) butst (s) ∈ �V (p), i.e. st (s) ∈ V (�p → �p). Thus IF (A) |=V �p → �p.

Conversely, let st (s) ∈ �V (p) and st (s) ∈ �V (p). Then s-possible states do

not exist and, for any state 〈�, � 〉, � s� holds. Using this fact by Global Cut

we derive ∅ s∅, i.e. F(A) is inconsistent. ��

We cannot claim immediately that the failure of von Wright formula �p → �p

by |=s,V would imply inconsistency since situations which are not-realized couldexist. And for any non-realized situation ω always ω ∈ �V (p) and ω ∈ �V (p)

for any p.

Theorem 3.24. The modal logic L(Kcon,c) of all consistent and complete informa-tion frames is the normal modal logic D := K ⊕ �p → �p.

Proof. From previous information and Lemma 3.23 we know that L(Kcon,s) ⊆ D.If A ∈ D then we employ again Kripke completeness of D. It is well known that D isKripke complete w.r.t serial frames (recall a frame F is serial if F |= ∀x∃y(xRy)),(cf. [4] page 143, for instance). Therefore there is a Kripke model M := 〈W, R, V 〉with a valuation V based on a serial frame disproving A. Take the informationframe IFM associated with the frame of M and the valuation V1 as in (3.5) beforeLemma 3.10.

The information frame IFM is consistent. Indeed, for any s ∈ M there iss1 ∈ M such that sRs1 because M is serial. Take ω1 := st (s1) = 〈�s1 , �s1〉, Then

s1 ∈ ⋂�s1 and s1 ∈ ⋃

�s1 and consequently �s1 �� s�s1 so IFM is consistent.But by Lemma 3.10 the frame IFM is complete and disproves A by V1. ��

Using the information frames IFF generated by given Kripke frames F , weshowed the completeness of the basic normal modal logics w.r.t. classes of specialinformation frames.And we verified that IFF and F itself generate the same modallogics (cf. Lemma 3.10 and proof of Theorem 3.17). Now we wont to explorewhether it is possible to make the similar for the converse: by any given informa-tion frame IF to construct a Kripke frame KFIF such that the information frameIFKFIF

associated with KFKF (i) would be based on the same set of situations,(ii) would have similar classification types and the same information for interiorcontexts as IF itself, so would be so close to IF as possible.

Suppose a classification A and a certain based on A information frame

IF (A) := 〈〈S, �, |=〉, {〈S, �, |=, s, Ns〉 | s ∈ S}〉

be given.

Definition 3.25. We say the classification 〈S, �, |=〉 is rarefied if, for any ω, ω1 ∈�, if ω = ω1 then there is a situation s ∈ S such that ¬(s |= ω⇔s |= ω1). Aninformation frame is rarefied if its classification is rarefied.

274 V.V. Rybakov

Evident examples of rarefied information frames are information frames overclassifications 〈S, �, ∈〉, where � ⊆ 2S . The informal meaning of rarefying is: weeliminate duplications from classification types. The lemma below shows that thementioned examples are only the case for rarefied information frames.

Lemma 3.26. For any rarefied information frame

IF (A) := 〈〈S, �, |=〉, {〈S, �, |=, s, Ns〉 | s ∈ S}〉

there is an information frame

IF (A)1 := 〈〈S, �(2S), ∈〉, {〈S, �(2S), ∈, s,1, Ns〉 | s ∈ S}〉,

where �(2S) ⊆ 2S , which is isomorphic to IFA.

Proof. For any ω ∈ � take the set Xω := {s | s ∈ S, s |= ω}. Consider theset �(2S) := {Xω | ω ∈ �}. Then s |= ω⇔s ∈ Xω and the classifications〈S, �, |=〉 and 〈S, �(2S), ∈〉 are isomorphic since the classification 〈S, �, |=〉 israrefied, i.e. Xω = Xω1 for ω = ω1. Take the information frame IF (A)1 as in theformulation of our lemma, where, for any sequent 〈�, �〉, where �, � ⊆ �, and�1 := {Xω | ω ∈ �}, �1 := {Xω | ω ∈ �}

�1 s,1�1 ⇔def � s�.

Note that the relation s,1 is well defined. Indeed, for any sequent 〈�2, �2〉, where�2 ⊆ �(2S) and �2 ⊆ �(2S), there is a unique sequent 〈�, �〉, where �, � ⊆ �

such that �1 = �2 and �1 = �2 since IF (A) is rarefied. From this observationwe immediately derive that frames IF (A) and IF (A)1 are isomorphic. ��

Thus, considering rarefied information frames IF (A), we may assume thatthese frames all are of kind:

IF (A) := 〈〈S, �(2S), ∈〉, {〈S, �(2S), ∈, s, Ns〉 | s ∈ S}. (7)

Given a rarefied information frame IF (A) as above, we introduce the Kripkeframe KFIF(A) associated with IF (A) as follows: KFIF(A) := 〈S, R〉, where∀a, b ∈ S(aRb⇔b ∈ Na). Next, we take the information frame

IF (KFIF(A)) := 〈〈S, 2S, ∈〉, {〈S, 2S, ∈, s,1, Ns,1 | s ∈ S}〉

associated with KFIF(A). Note that Ns = Ns,1 for all s by the definition.

Lemma 3.27. Suppose IF (A) is complete. Then, for any sequent 〈�, �〉, where�, � ⊆ St (�(2s)) and any s ∈ S

� s� ⇔� s,1�.

Barwise’s information frames and modal Logics 275

Proof. Assume ��� s,1�. Then there is an a such that sRa and a ∈ ⋂� but

a ∈ ⋃�. Observe a ∈ Ns and a ∈ X for all X ∈ � but a ∈ X for all X ∈ �.

Then it follows that ��� s� because s is the information about all s-normalsituations.

Conversely, assume that ��� s�. Since IF (A) is complete there is an s-normalsituation a (a ∈ Ns) which disproves this sequent, i.e. a ∈ ⋂

� and a ∈ �. Still,

because a ∈ Ns it follows sRa. And by the definition of s,1 we immediately

derive that ��� s,1�. ��Thus, partly, for the case of complete and rarefied information frames, we can

represent their contexts information relation s by information frames generatedby normal Kripke frames. To make one step more in order to specify which modallogics complete w.r.t information frames are Kripke complete, we consider fullyclassified information frames. Taking in accordance (7) we say:

Definition 3.28. A rarefied information frame IF (A) is said to be fully classifiedif �(2S) = 2S .

Theorem 3.29. Any normal modal logic L complete w.r.t a class of complete,rarefied and fully classified information frames is Kripke complete.

Proof. Indeed, L := L(K) holds for a class K of complete, rarefied and fully clas-sified information frames. Consider the class of Kripke frames K1 consisting offrames associated to information frames from K. Take the frame KFIF(A) for anyIF (A) ∈ K. Consider the structure of these frames;

IF (A) := 〈〈S, 2S, ∈〉, {〈S, 2S, ∈, s, Ns〉 | s ∈ S}〉

KFIF(A) := 〈S, R〉, ∀a, b ∈ S(aRb⇔b ∈ Na).

Take the information frame IF (KFIF(A)) associated with KFIF(A),

IF (KFIF(A)) := 〈〈S, 2S, ∈〉, {〈S, 2S, ∈, s,1, Ns | s ∈ S}〉.Consider any valuation V of a modal formula A in St (2S). Denote VIF (A) the valueof A under V in IF (A) and VIFKF (A) the value of A under V in IF (KFIF(A)).Then for any ω ∈ St (2S)

ω ∈ VIF (A) ⇔ ω ∈ VIFKF (A). (8)

To show this we merely use induction on length of A. The base of the inductionand steps for Boolean operations are evident, and the step for � follows fromLemma 3.27. Next relation we need is

∀A, IF (A) |= A ⇔ KFIF(A) A. (9)

Indeed, if A is disproved in IF (A) by a valuation V then, for some s ∈ S, st (s) ∈VIF (A). By (8) it follows st (s) ∈ VIFKF (A). Take the valuation V1 on KFIF(A)

276 V.V. Rybakov

setting V1(p) := {s | st (s) ∈ V (p)}. Then these V and V1 (with converse renam-ing) satisfy the condition (2) above Lemma 3.10. Therefore applying Lemma 3.10

we derive s�� V1A. Thus KFIF(A)�� A.

Conversely assume KFIF(A)�� A. Then for some s ∈ S, s�� V A for a valua-tion V . Take the valuation V1 in St (2S) as follows: V1(p) := {st (s) | s ∈ V (p)}.Then again the condition (2) above Lemma 3.10 holds and applying this lemma itfollows st (s) ∈ V1(A), i.e. st (s) ∈ V1,IFKF and by using (8) we can conclude thatst (s) ∈ V1,IF . Thus IF (A) |= A and relation (9) holds. Using (9) we immediatelyderive L = L(K1). ��

Conclusion. In this paper we investigated logics generated by Barwise’s infor-mation frames. An attempt was made for understanding what these logics are andhow they could be depicted in terms of pure modal logics framework. Our pri-mary goal was finding axiomatizations for modal logics generated by classes ofinformation frames, describing modal logics which have representation as logicson information frames. Undoubtedly Barwise’s fruitful idea of information framesand their logics is much wider this particular approach. For instance, the context’slogical consequences within information frames could be of interest for algebraiclogicians; information frames formalism could find favorable applications in AIand CS. However, even regarding only pure modal logics framework, it seems thatby now only a door in a very attractive area is opened. For instance, we showed anyKripke complete normal modal logic is complete w.r.t. information frames. Doesthe converse hold? If not, how to specify which modal logics are complete w.r.t.information frames? To study this question we, very likely, would need to considercertain modal algebras associated with information frames rather than associatedframes as we did. Regarding the weak modal logics, for any Kripke complete normalmodal logic L, we cannot say immediately that L is representable as a weak modallogic of an appropriate class of information frames (as it was for usual modal logicsof information frames). This is a case because we verify validity for weak logicsnot by states of situations but by arbitrary states from St (�) and Lemma 3.10 doesnot work any more. So, another open questions are: which modal logics among,say, Kripke complete logics, logics complete w.r.t neighborhood semantics, all nor-mal modal logics, can be represented as weak modal logics of information frames.Or, say, less global question mentioned above: to find an axiomatization for theweak modal logic of all sound information frames. In the final part of this paper weprovided mathematical instruments which have a certain resemblance with generalcorrespondence theory in modal logics. We restricted ourselves only with toolswhich are necessary for our research concerning axiomatizations. But we feel thatthe theory could be developed much deeper towards general correspondence theoryas for the case of usual modal logic.

Acknowledgements. I am thankful to Professor Gerhard Jaeger for inspiring atmospherein Bern University to initiate my research presented in this paper, for his kind attention,discussions and advice.

Barwise’s information frames and modal Logics 277

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