Showing papers on "Normal modal logic published in 1994"

The proof theory and semantics of intuitionistic modal logic

Abstract: Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary meta-theory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic meta-theory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are self-justifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic first-order logic. It is also established that, in many cases, the natural deduction systems induce well-known intuitionistic modal logics, previously given by Hilbert-style axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their susceptibility to proof-theoretic techniques. Strong normalization (and confluence) results are proved for all of the systems. Normalization is then used to establish the completeness of cut-free sequent calculi for all of the systems, and decidability for some of the systems. Lastly, techniques developed throughout the thesis are used to establish that those intuitionistic modal logics proved decidable also satisfy the finite model property. For the logics considered, decidability and the finite model property presented open problems.

First Steps in Modal Logic

Abstract: Introduction Acknowledgements Part I. Preliminaries: 1. Survey of propositional logic 2. The modal language Part II. Transition Structures and Semantics: 3. Labelled transition structures 4. Valuation and satisfaction 5. Correspondence theory 6. The general confluence result Part III. Proof Theory and Completeness: 7. Some consequence relations 8. Standard formal systems 9. The general completeness result 10. Kripke-completeness Part IV. Model Constructions: 11. Bismulations 12. Filtrations 13. The finite model property Part V. More Advanced Material: 14. SLL logic 15. Lob logic 16. Canonicity without the fmp 17. Transition structures aren't enough Part VI. Two Appendices: Bibliography.

Sequent Calculi for Normal Modal Propositional Logics

Abstract: In this paper a systematic sequent-style proof theory for the most important systems of normal modal prepositional logic based on classical prepositional logic CPL is presented After discussing philosophical, methodological, and computational aspects of the problem of Gentzenizing modal logic, a variant of Belnap's display logic is introduced. It is shown that within this proof theory the modal axiom schemes D, T, 4, 5, and B (and some others) can be captured by characteristic structural inference rules For all sequent systems under consideration (i) cut is admissible, (n) the subformula property holds, and (in) all connectives are uniquely characterized Also modal systems based on substructural subsystems of CPL are briefly dealt with KeywordsSequent calculi, modal logic, display logic, substructural logics. Gentzen 's proof-theoretical methods have not yet been properly applied to modal logic. Serebriannikov [27, p. 79]

An Overview of Temporal and Modal Logic Programming

Abstract: This paper presents an overview of the development of the field of temporal and modal logic programming. We review temporal and modal logic programming languages under three headings: (1) languages based on interval logic, (2) languages based on temporal logic, and (3) languages based on (multi)modal logics. The overview includes most of the major results developed, and points out some of the similarities, and the differences, between languages and systems based on diverse temporal and modal logics. The paper concludes with a brief summary and discussion.

Strongly Analytic Tableaux for Normal Modal Logics

Abstract: A strong analytic tableau calculus is presentend for the most common normal modal logics. The method combines the advantages of both sequent-like tableaux and prefixed tableaux. Proper rules are used, instead of complex closure operations for the accessibility relation, while non determinism and cut rules, used by sequent-like tableaux, are totally eliminated. A strong completeness theorem without cut is also given for symmetric and euclidean logics. The system gains the same modularity of Hilbert-style formulations, where the addition or deletion of rules is the way to change logic. Since each rule has to consider only adjacent possible worlds, the calculus also gains efficiency. Moreover, the rules satisfy the strong Church Rosser property and can thus be fully parallelized. Termination properties and a general algorithm are devised. The propositional modal logics thus treated are K, D, T, KB, K4, K5, K45, KDB, D4, KD5, KD45, B, S4, S5, OM, OB, OK4, OS4, OM+, OB+, OK4+, OS4+. Other logics can be constructed with different combinations of the proposed rules, but are not presented here.

Refutation Systems in Modal Logic

Abstract: Complete deductive systems are constructed for the non-valid (refutable) formulae and sequents of some propositional modal logics. Thus, complete syntactic characterizations in the sense of Lukasiewicz are established for these logics and, in particular, purely syntactic decision procedures for them are obtained. The paper also contains some historical remarks and a general discussion on refutation systems.

On modal logics for qualitative possibility in a fuzzy setting

Abstract: Within the possibilistic approach to uncertainty modeling, the paper presents a modal logical system to reason about qualitative (comparative) statements of the possibility (and necessity) of fuzzy propositions. We relate this qualitative modal logic to the many-valued analogues MVS5 and MVKD45 of the well known modal logics of knowledge and belief S5 and KD45 respectively. Completeness results are obtained for such logics and therefore, they extend previous existing results for qualitative possibilistic logics in the classical non-fuzzy setting.

Cut-free sequent and tableau systems for propositional Diodorean modal logics

Abstract: We present sound, (weakly) complete and cut-free tableau systems for the propositional normal modal logicsS4.3, S4.3.1 andS4.14. When the modality □ is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points. Although cut-free, the last two systems do not possess the subformula property. But for any given finite set of formulaeX the “superformulae” involved are always bounded by a finite set of formulaeX* L depending only onX and the logicL. Thus each system gives a nondeterministic decision procedure for the logic in question. The completeness proofs yield deterministic decision procedures for each logic because each proof is constructive. Each tableau system has a cut-free sequent analogue proving that Gentzen's cut-elimination theorem holds for these latter systems. The techniques are due to Hintikka and Rautenberg.

An Intuitionistic Modal Logic with Applications to the Formal Verification of Hardware

Abstract: We investigate a novel intuitionistic modal logic, called Propositional Lax Logic, with promising applications to the formal verification of computer hardware. The logic has emerged from an attempt to express correctness ‘up to’ behavioural constraints — a central notion in hardware verification — as a logical modality. The resulting logic is unorthodox in several respects. As a modal logic it is special since it features a single modal operator O that has a flavour both of possibility and of necessity. As for hardware verification it is special since it is an intuitionistic rather than classical logic which so far has been the basis of the great majority of approaches. Finally, its models are unusual since they feature worlds with inconsistent information and furthermore the only frame condition is that the O-frame be a subrelation of the ⊃-frame. We provide the motivation for Propositional Lax Logic and present several technical results. We investigate some of its proof-theoretic properties, and present a cut-elimination theorem for a standard Gentzen-style sequent presentation of the logic. We further show soundness and completeness for several classes of fallible two-frame Kripke models. In this framework we present a concrete and rather natural class of models from hardware verification such that the modality O models correctness up to timing constraints.

Coming to terms with modal logic : on the interpretation of modalities in typed lambda-calculus

Abstract: ion G 1A: Bis G 1(>.x : C.d) : (ITx : C.D), an immediate consequence of (1) G, x : C 1d : D and (2) G 1(IIx: C.D) : s. By lH (on (1)) FV(d),FV(D) ç FV(G,x: C) and (on (2)) FV(ITx: C.D),FV(s) ç FV(G,x: C). Hence FV( C) Ç FV( G) and FV(Àx : C.d) Ç FV( G) since by definition FV(Àx : C.d) = FV(C) u (FV(d){x}) (FV(ITx : C.D) = FV(C) u (FV(D){x})). Notice that by Free Variabie Lemma (i) and (1), x (/: FV(G). Therefore FV(>.x : C.d),FV(ITx: C.D) Ç FV(G). The transfer, import and export-cases again use the Free Variabie definition FV( G IQ] t:) = FV( G). Proof of (iii). The cases forstart and weakening use the part (ii) of the Free Variabie Lemma, we do the case for start: Start G 1A: Bis G', x : C 1x : Ca direct consequence of G' 1C : s. By lH for all A; in G', FV(A;) Ç {x1, ... , x;_t}. We have to prove that FV( C) Ç { x1, ... , xi} = FV( G'). But si nee G' 1C : s, we have FV ( C) Ç F V ( G') by the previous clause (i i) of the Free Variabie Lemma, hence FV(Ai) Ç {x1, ... , x;_ I} for all A; in G. For the transfer, import and export-cases we need the simple observation that A; in G are the A; in G IQl E 3.2.3. LEMMA. Start Lemma Let G be a legal generalized context. Then {i) IJ s1 : s2 is a typing axiom {E A Ty1>e ), G 1s1 : s2 3.2. PRELIMINARIES 97 {ii} IJ c : A is a logica/ axiom, (E ALogi<: ), G f-c : A : Prop {iii} Ij (x : A) E r in G' tQJ r {where G ::: G' IQ! r }, then G' IQ! r f-x : A PROOF. Proof of (i),(ii) and (iii) . By assumption of G f-A: B forsome A: B. The result follows by induction on the derivation of G 1A : B. Proof of (i). The cases for transfer and import all use the transfeTJ.-rule: Transfer2 G f-A: B is G' tQJ t: fC : D where G = G'IQJ t:, an immediate consequence of G' fC : D : Type. By IH G' f-SI : s2, but then by transfer1 G' IQJ E fSJ : s2. Hence G fSJ : s2, for SJ : s2 E A Type . For the export-cases we have look further back in the derivation in applying the IH, we show the case for K -export: K -export G f-A : B is G f-kC : DD an immediate consequence of G tQJ E fC : D : Prop. Applying the IH to the last step in the derivation does not work bere, however since all derivations start from the context t:; e we can go up in the derivation tree to find the place where the IQ) was introduced going from G to G IQJ e for the first time. This means sarnething must have been derivable on G before, and since this derivation is shorter, IH gives that G f-SJ : s2, for SJ : s2 E A Type. The proof of (ii) is completely analogous. Proof of (iii). The proof is straightforward for non-modal cases, and trivia! for the transfer and import-rules: Transfer1 G fA : B is G'IQJ t: f-C : s where G = G'IQJ e, animmedia te consequence of G' f-C : s. Note that this cannot occur when G is 'non-blocked' ( G = r). Therefore we treat the case of the 'complex' context G' IQJ e. Since r = e, it contains no variables, and so trivially G f-x : C if (x : C) Er. For the export-cases we use an argument similar to the one given above in the proof of (i). Given the following definitions of substitution on, and concatenation of generalized contexts, a Substition Lemma can be proved: 3.2.4. DEFINITION. Subtitution, concatenation On a generalized context 1::. = !::.1 ~ ... IQI l::.n, the substitution of a term D fora variabie x yields !::.[x :=DJ = t:.t[x := D]IQJ . .. IQ! l::.n[x :=Dj. Given two generalized contexts G ::: f 1 IQJ . .. IQJ r m and t:. = 1::.1 IQJ ••• IQJ l::.n, their concatenation G, t:. = f1 IQJ ••• IQJ r m, !::.1 IQJ ••• IQJ l::.n . 3.2.5. LEMMA. Substitution Lemma Assume (1) G, x : C, 1::. 1A : B and (2) G 1D : C, where G and 1::. are generalized pseudo-contexts. Then G, !::.[x:= Dj f-A[x := D]: B[x :=DJ . 98 CHAPTER 3. META THEORY OF MPTSs PROOF. Proof. By induction on (the lengthof the) derivation of (1), where M* is used as an abbreviation for M[x := DJ. The non-modal cases are analogous to those in the proof of the Substitution Lemma for PTS's. The modal cases require some calculations with the definitions for substitution, e.g. : K-export G,x: C,/1 1A : Bis G,x: C, 1kE: OF an immediate consequence of G,x: C,/1 li:lle 1E: F: Prop. By lH G,(/1 li:lle)* 1E*: F*(: Prop). By definition (!1 fQJ e)* = !1* 101 e* = !1* 101 E, since E* = E. Hence G, (!1 [QJ E)> = G, f1• [QJ E, and so G, !1* fQJ E 1E*: F*(: Prop). Therefore by K-export G, !1* 1kE*: D(F*), and since (kE*) := (kE)* (FV(kE) = FV(E)) and D(F*) := (DF)*, G, !1* 1(kE)*: (DF)*. The proof of a Thinning Lemma is not completely straightforward. The transfer and import-rules are formulated in such a way that they yield a new generalized context of the iorm G fQJ e. However, to prove Thinning we have to show that there are derived versionsof these rules that yield generalized contexts G fQJ r, for an arbitrary 'non-blocked' context r. To prove this, the following lemma is needed: 3.2.6. LEMMA. Legality Lemma Ij G fQJ f 1 , x : C is leg al then G lQl f 1 r C : s. PROOF. By induction on the length of the derivation of G fQJ T1, x : C 1A : B . Except for the axiom cases which cannot occur (since G fQJ f 1, x : C t;. e) and start and weakening which are immediate, the non-madal cases are regular. The transfer and import-cases cannot occur: Transfer1 G 19 f 1, x : C 1A : B is G IQJ E 1D : s , an immediate consequence of G 1D : s. This case cannot occur: G IQJ f 1, x : C '1. G fQJ e. and the export-cases require some additional reasoning: K -export G fQJ f 1, x : C 1A : B is G 1kD : DE an immediate consequence of G 101 f 1 , x : c 101 E: 1D : E : Prop. Since all derivations start from E and are finite, we can go up in the tree to find the place where the 101 was introduced, going from G IQI f 1 , x : C, to G IJ f 1, x : C 101 e for the first time. This means that sarnething must have been derivable on G ~ f 1 , x : C before, and since this denvation is shorter IH gives us that G lQl r-' 1C : s. 3.2.7. LEMMA. Derived Rules Lemma The following are derived rules in an MPTS: 1 GI-A :s Transfer1 G r A IQI 1: s Tra , 1 G 1A : B : Set ns,er3 G 101 r 1A : B K . 1 G 1A : DB: Prop tmport _ · G IQI r 1kA: B . I G r A: -.OB: Prop 5 tmport • G lQl r 1SA : -.DB T 11 GI-A:B:Type rans,er2 G !QJ r 1A : B Th :j 1 G 1c : A : Prop ans er ax G fQJ r 1c : A 1 G 1A : DB : Prop 4 import • G 101 r 14A : DB B . 1 G 1A : B : Prop tmport G !QJ r 1bA : -.o-.B where r is a {non-blocked) pseudocontext such that G IQ! r is legal. 3.2. PRELIMINARIES 99 Given the original rules of Transfer and Import, proving the following is sufficient: 1 IfGIQJe 1A : s then G !dl r 1A : s . 2 IfGIQJe 1A : B(: Type) then G IQJ r 1A: B. 3 IfGIQJe 1A: B(: Set) then G l!:ll r 1A: B. 4 IfGIQJe 1c: A(: Prop) then G IQJ r 1c: A. 5 IfGIQJe 1kA : B then G IQJ r 1kA : B 6 IfGIQJe 14A : DB then G !bil r 14A : DB 7 lfGIQJe 1SA : -.OB then G IQJ r 1SA : -.OB 8 IfGIQJe 1bA : -,0-,B then G IQJ r 1bA : -,0-,B PROOF. By induction on the lengthof f . The basic case for r =: e is immediate by the above. The induction case where r =: f', x : C is the same for all cases, we show 1: 1 By lH G IQI r' 1A : s, and by the Legality Lemma ( G IQJ r' is legal) G IQJ r' 1C : s, hence weakening yields G IQI f', x : C 1A : s and G IQJ r fA: s. Now we can prove a Thinning Lemma for the modal systems, using the 'subset relation' for generalized contexts defined earlier. 3.2.8. LEMMA. Thinning Lemma Let G and /::;. be leg al generalized pseudocontexts such that G Ç /::;.. Th en if G 1A : B, ó. 1A: B. PROOF. By induction on the lengthof the derivation of G 1A : B. The cases for transfer and import require the derived forms of the these rules from the derived rules lemma. We show the case for K -import: K-import G 1A : B is G' !bil e 1kc : D where G = G' IQI e, an immediate consequence of G' 1C : DD : Prop. Since G = G' IQJ e and G Ç ó., it must be the case that ó. = ö.' IQJ r forsome r (since for all r, e Ç r) and G' Ç ó.'. Hence by lH ö.' 1C: DD: Prop , so by the derived rule K-import ' I::;.' IQ! r 1kC: D and therefore ~::;. 1kc: n. In the Export-cases we have to show that /::;. is legal before the IH can be applied: K -export G 1A : B is G 1kC : DD an immedia te consequence of G lbll e 1C : D : Prop. Since G Ç /::;. and e Ç e, by definition G IQJ e Ç /::;.IQJ e. Furthermore ó.lbll eis !ega!: /::;. is legal, hence by the Start Lemma (i) /::;. fs1 : s2 for s1 : s2 E A Type (note that A Type f= 0), and by trans j eT}, /::;. IQJ e 1s1 : s2. Therefore by lH /::;. IQJ e 1C : D : Prop, andsoó. 1kC:DD . 100 CHAPTER 3. META THEORY OF MPTSs 3.2.9. COROLLARY. Strong Thinning For terms {A) that are not proofs {nat A : B : Prop), we can prove a stronger result (jor cases where G is an intialpart of t:.) by combining Thinning with the transfer rule. Let G and t:. be /ega/ generalized pseudocontexts such that G ~ t:.. Th en (i) ij G 1A : sI t:. 1A : s. {ii) ij G 1A : B : Type, t:. 1A : B : Type. {iii) ij G 1A : B : Set, t:. 1A : B :Set. PROOF. Proof of (i), (ii), and (iii). By construction of t:. from G while preserving the derivability of A : s . We do the proof of (ii): Suppose that G 1A : B : Type for some terms A and B: (1) f1 IQ! ••• ~ r m 1A : B : Type Since 'v'i(1 ~i ~ m)(f; Ç r~), we can conclude by Thinning (2) r~ !':ll ••. !':ll r~ 1A : B : Type Now the transfer2 rule can be applied to obtain r~ IQJ • • • l!:ll r~ IQI f 1A : B and by transfer1 we have r~~CJ ... Ii:!lr~~f 1B:Type

A uniform tableau method for intuitionistic modal logics I

Abstract: We present tableau systems and sequent calculi for the intuitionistic analoguesIK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IKD5, IK45, IKD45 andIS5 of the normal classical modal logics. We provide soundness and completeness theorems with respect to the models of intuitionistic logic enriched by a modal accessibility relation, as proposed by G. Fischer Servi. We then show the disjunction property forIK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IK45 andIS5. We also investigate the relationship of these logics with some other intuitionistic modal logics proposed in the literature.

A modal logic for subjective default reasoning

Abstract: Introduces a logic endowed with a two-place modal connective that has the intended meaning of "if /spl alpha/, then normally /spl beta/". On top of providing a well defined tool for analyzing common default reasoning, such a logic allows nesting of the default operator. We present a semantic framework in which many of the known default proof systems can be naturally characterized, and prove soundness and completeness theorems for several such proof systems. Our semantics is a "neighborhood modal semantics", and it allows for subjective defaults, i.e. defaults may vary within different worlds that belong to the same model. The semantics has an appealing intuitive interpretation and may be viewed as a set theoretic generalization of the probabilistic interpretations of default reasoning. We show that our semantics is general in the sense that any modal semantics that is sound for some basic axioms for default reasoning is a special case of our semantics. Such a generality result may serve to provide a semantical analysis of the relative strengths of different proof systems and to show the nonexistence of semantics with certain properties. >

The Complexity of Propositional Modal Theories and the Complexity of Consistency of Propositional Modal Theories

Abstract: This paper is concerned with the computational complexity of the following problems for various modal logics L: (1). The L-deducibility problem: given a finite set of formulas S and a formula A, determine if A is in the modal theory T HL(S) formed with all theorems of the modal logic L as logical axioms and with all members of S as proper axioms. (2). The L-consistency problem: given a finite set of formulas S, determine if the theory THL(S) is consistent.

Criteria for admissibility of inference rules. Modal and intermediate logics with the branching property

Abstract: The main result of this paper is the following theorem: each modal logic extendingK4 having the branching property belowm and the effective m-drop point property is decidable with respect to admissibility. A similar result is obtained for intermediate intuitionistic logics with the branching property belowm and the strong effective m-drop point property. Thus, general algorithmic criteria which allow to recognize the admissibility of inference rules for modal and intermediate logics of the above kind are found. These criteria are applicable to most modal logics for which decidability with respect to admissibility is known and to many others, for instance, to the modal logicsK4,K4.1,K4.2,K4.3,S4.1,S4.2,GL.2; to all smallest and greatest counterparts of intermediate Gabbay-De-Jong logicsDn; to all intermediate Gabbay-De-Jong logicsDn; to all finitely axiomatizable modal and intermediate logics of finite depth etc. Semantic criteria for recognizing admissibility for these logics are offered as well.

KED: a deontic theorem prover

Abstract: Deontic logic (DL) is increasingly recognized as an indispensable tool in such application areas as formal representation of legal knowledge and reasoning, formal specification of computer systems and formal analysis of database integrity constraints. Despite this acknowledgement, there have been few attempts to provide computationally tractable inference mechanisms for DL. In this paper we shall be concerned with providing a computationally oriented proof method for standard DL (SDL), i.e., normal systems of modal logic with the usual possible-worlds semantics. Because of the natural and easily implementable style of proof construction it uses, this method seems particularly well-suited for applications in the AI and Law field, and though in the present version it works for SDL only, it forms an appropriate basis for developing efficient proof methods for more expressive and sophisticated extensions of SDL.

A modal view of linear logic

Abstract: We present a sequent calculus for the modal logic S4, and building on some relevant features of this system (the absence of contraction rules and the confinement of weakenings into axioms and modal rules) we show how S4 can easily be translated into full prepositional linear logic, extending the Grishin-Ono translation of classical logic into linear logic. The translation introduces linear modalities (exponentials) only in correspondence with S4 modalities. We discuss the complexity of the decision problem for several classes of linear formulas naturally arising from the proposed translations.

A Modal Logic for Quantification and Substitution

Abstract: The aim of this paper is to study the n-variable fragment of first order logic from a modal perspective We define a modal formalism called cylindric mirror modal logic, and show how it is a modal version of first order logic with substitution In this approach, we can define a semantics for the language which is closely related to algebraic logic, as we find Polyadic Equality Algebras as the modal or complex algebras of our system The main contribution of the paper is a characterization of the intended ‘mirror cubic’ frames of the formalisms and, a consequence of the special form of this characterization, a completeness theorem for these intended frames As a consequence, we find complete finite yet unorthodox derivation systems for the equational theory of finite-dimensional representable polyadic equality algebras

Propositional State Event Logic

Abstract: In this paper we propose the representation of concurrent events and causality between events in modal logic. This approach differs from previous approaches in the following directions: first, events enjoy the same attention as states. In the same way as states can be viewed as models of the formulae describing the facts that hold in them we think of events as models of the formulae describing the subevents. Second, instead of postulating just one set of states as primitive objects we use two sets, a set of states and a set of events. In terms of modal logic, the universe then becomes a set of pairs in which one component is a state and the other is one of the events following the state. The connection between two subsequent pairs is expressed by an accessibility relation.

Common knowledge: Relating anti-founded situation semantics to modal logic neighbourhood semantics

Abstract: Two approaches for defining common knowledge coexist in the literature: the infinite iteration definition and the circular or fixed point one. In particular, an original modelization of the fixed point definition was proposed by Barwise (1989) in the context of a non-well-founded set theory and the infinite iteration approach has been technically analyzed within multi-modal epistemic logic using neighbourhood semantics by Lismont (1993). This paper exhibits a relation between these two ways of modelling common knowledge which seem at first quite different.

A Modal Logic For Hypothesis Theory

Abstract: Hypothesis theory for nonmonotonic reasoning expresses notions of hypotheses and of known information. In this paper, we define these notions in the framework of a new modal system with two modalities, one for exprressing known information and the other for expressing possible hypotheses. A complete characterization of the new logic is given in terms of Kripke semantics. Moreover, our logic allows to characterize completely default logic, including a necessary and sufficient criterium for the existence and the non-existence of extensions. We also present a notion of nonmonotonic inference which is cumulative.

Modalities in Linear Logic Weaker than the Exponential "of Course": Algebraic and Relational Semantics

Abstract: We present a semantic study of a family of modal intuitionistic linear systems, providing various logics with both an algebraic semantics and a relational semantics, to obtain completeness results. We call modality a unary operator □ on formulas which satisfies only one rale (regularity), and we consider any subsetW of a list of axioms which defines the exponential “of course” of linear logic. We define an algebraic semantics by interpreting the modality □ as a unary operationμ on an IL-algebra. Then we introduce a relational semantics based on pretopologies with an additional binary relationr between information states. The interpretation of □ is defined in a suitable way, which differs from the traditional one in classical modal logic. We prove that such models provide a complete semantics for our minimal modal system, as well as, by requiring the suitable conditions onr (in the spirit of correspondence theory), for any of its extensions axiomatized by any subsetW as above. We also prove an embedding theorem for modal IL-algebras into complete ones and, after introducing the notion of general frame, we apply it to obtain a duality between general frames and modal IL-algebras.