In [12] the study of Positive Modal Logic is initiated using standard Kripke semantics and the positive modal algebras are introduced. The minimum system of Positive Modal Logic is the -fragment of the local consequence relation defined by the class of all Kripke models. It can be axiomatized by a sequent calculus and extensions of it can be obtained by adding sequents as new axioms. In [6] a new semantics for PML is proposed to overcome some frame incompleteness problems discussed (...) in [12]. The frames of this semantics consists of a set of indexes, a quasi-order on them and an accessibility relation. The models are obtained by using increasing valuations relatively to the quasi-order of the frame. This semantics is coherent with the dual structures obtained by developing the Priestley duality for positive modal algebras, one of the topics or the present paper, and can be seen also as arising from the Kripke semantics for a suitable intuitionistic modal logic. The present paper is devoted to the study of the mentioned duality as well as to proving some d-persistency results as well as a Sahlqist Theorem for sequents and the semantics proposed in [6]. Also a Goldblatt-Thomason theorem that characterizes the elementary classes of frames of that semantics that are definable by sets of sequents is proved. (shrink)

This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.

What is the minimal algebraic structure to reason about information flow? Do we really need the full power of Boolean algebras with co-closure and de Morgan dual operators? How much can we weaken and still be able to reason about multi-agent scenarios in a tidy compositional way? This paper provides some answers.

A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel-Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.

We give a set of postulates for the minimal normal modal logicK + without negation or any kind of implication. The connectives are simply , , , . The postulates (and theorems) are all deducibility statements . The only postulates that might not be obvious are.

Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponential-free linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also (...) have a 'cyclic' counterpart. These logics have been studied extensively and are quite well understood.Generalising further, one can start with intuitionistic Bi-Lambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bi-linear, Bi-relevant, Bi-BCK and Bi-intuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and some even have a 'cyclic' counterpart. These extensions of Bi-Lambek logic are not so well understood. Cut-elimination for Classical Bi-Lambek logic, for example, is not completely clear since some cut rules have side conditions requiring that certain constituents be empty or non-empty.The display logic of Nuel Belnap is a general Gentzen-style proof theoretical framework designed to capture many different logics in one uniform setting. The beauty of display logic is a general cut-elimination theorem, due to Belnap, which applies whenever the rules of the display calculus obey certain, easily checked, conditions. The original display logic, and its various incarnations, are not suitable for capturing bi-intuitionistic and bi-classical logics in a uniform way. We remedy this situation by giving a single Display calculus for the Bi-Lambek Calculus, from which all the well-known extensions are obtained by the incremental addition of structural rules to a constant core of logical introduction rules.We highlight the inherent duality and symmetry within this framework obtaining 'four proofs for the price of one'. We give algebraic semantics for the Bi-Lambek logics and prove that our calculi are sound and complete with respect to these semantics. We show how to define an alternative display calculus for bi-classical substructural logics using negations, instead of implications, as primitives.Borrowing from other display calculi, we show how to extend our display calculus to handle bi-intuitionistic or bi-classical substructural logics containing the forward and backward modalities familiar from tense logic, the exponentials of linear logic, the converse operator familiar from relation algebra, four negations, and two unusual modalities corresponding to the non-classical analogues of Sheffer's 'dagger' and 'stroke', all in a modular way.Using the Gaggle Theory of Dunn we outline relational semantics for the binary and unary intensional connectives, but make no attempt to do so for the extensional connectives, or the exponentials.Finally, we flesh out a suggestion of Lambek to embed intuitionistic logic using two unusual 'exponentials', and show that these 'exponentials' are essentially tense logical modalities, quite at odds with the usual exponentials.Using a refinement of the display property, you can pick and choose from these possibilities to construct a display calculus for your needs. (shrink)

This is a revised and expanded edition of a seminal work in the logic and philosophy of time, originally published in 1968. Arthur N. Prior (1914-1969) was the founding father of temporal logic, and his book offers an excellent introduction to the fundamental questions in the field. Several important papers have been added to the original selection, as well as a comprehensive bibliography of Prior's work and an illuminating interview with his widow, Mary Prior. In addition, the Polish logic which (...) made Prior's writings difficult for many readers has been replaced by standard logical notation. This new edition will secure the classic status of the book. (shrink)

In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist (...) axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions. (shrink)

In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives /, *, \, together with a package of structural postulates characterizing the resource management properties of the * connective. Different choices for Associativity and Commutativity yield the familiar logics NL, L, NLP, LP. Semantically, a simple Lambek system is a unimodal logic: the connectives get a Kripke (...) style interpretation in terms of a single ternary accessibility relation modeling the notion of linguistic composition for each individual system.The simple systems each have their virtues in linguistic analysis. But none of them in isolation provides a basis for a full theory of grammar. In the second part of the paper, we consider two types of mixed Lambek systems.The first type is obtained by combining a number of unimodal systems into one multimodal logic. The combined multimodal logic is set up in such a way that the individual resource management properties of the constituting logics are preserved. But the inferential capacity of the mixed logic is greater than the sum of its component parts through the addition of interaction postulates, together with the corresponding interpretive constraints on frames, regulating the communication between the component logics.The second type of mixed system is obtained by generalizing the residuation scheme for binary connectives to families of n-ary connectives, and by putting together families, of different arities in one logic. We focus on residuation for unary connectives, hence on mixed frames, as these already represent the complexities in full. We prove a number of elementary logical results for unary families of residuated connectives and their combination with binary families. The existing proposals for unary ‘structural modalities’ are situated within this general framework, and a number of new linguistic applications is given. (shrink)

We construct logical languages which allow one to represent a variety of possible types of changes affecting the information states of agents in a multi-agent setting. We formalize these changes by defining a notion of epistemic program. The languages are two-sorted sets that contain not only sentences but also actions or programs. This is as in dynamic logic, and indeed our languages are not significantly more complicated than dynamic logics. But the semantics is more complicated. In general, the semantics of (...) an epistemic program is what we call aprogram model. This is a Kripke model of ‘actions’,representing the agents' uncertainty about the current action in a similar way that Kripke models of ‘states’ are commonly used in epistemic logic to represent the agents' uncertainty about the current state of the system. Program models induce changes affecting agents' information, which we represent as changes of the state model, called epistemic updates. Formally, an update consists of two operations: the first is called the update map, and it takes every state model to another state model, called the updated model; the second gives, for each input state model, a transition relation between the states of that model and the states of the updated model. Each variety of epistemic actions, such as public announcements or completely private announcements to groups, gives what we call an action signature, and then each family of action signatures gives a logical language. The construction of these languages is the main topic of this paper. We also mention the systems that capture the valid sentences of our logics. But we defer to a separate paper the completeness proof. (shrink)