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Abstract. Let V be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if V is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety V is generated by an elementary class of relational structures. Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure. 1.

This chapter introduces modal logic as a tool for talking about graphs, or to use more traditional terminology, as a tool for talking about Kripke models and frames. We want the reader to gain an intuitive appreciation of this perspective, and a firm grasp of the key technical ideas (such as bisimulations) which underly it. We introduce the syntax and semantics of basic modal logic, discuss its expressivity at the level of models, examine its computational properties, and then consider what it can say at the level of frames. We then move beyond the basic modal language, examine the kinds of expressivity offered by a number of richer modal logics, and try to pin down what it is that makes them all ‘modal’. We conclude by discussing an example which brings many of the ideas we discuss into play: games.

We show how to reason, in the proof assistant Coq, about realistic programming languages using a combination of separation logic and heterogeneous multimodal logic. A heterogeneous multimodal logic is a logic with several modal operators that are not required to satisfy the same frame conditions. The result is a powerful and elegant system for reasoning about programming languages and their semantics. The techniques are quite general and can be adopted to a wide variety of settings.

This paper presents a survey of topological spatial logics, taking as its point of departure the interpretation of the modal logic S4 due to McKinsey and Tarski. We consider the effect of extending this logic with the means to represent topological connectedness, focusing principally on the issue of computational complexity. In particular, we draw attention to the special problems which arise when the logics are interpreted not over arbitrary topological spaces, but over (low-dimensional) Euclidean spaces.

We (1) define a logic , called CPL (for Cryptographic Protocol Logic), where truth is established on the grounds of evidence-based knowledge (as opposed to awarenessbased belief), spanning the dimensions of first-order, temporal, epistemic, deontic, and linear logic; (2) state a few of its key properties; and (3) illustrate how it can be used to compositionally specify and verify cryptographic protocols designed to establish trust in the security of communication (as opposed to security of storage) between protocolcompliant participants in a hostile environment. Our claim hereby is to give (1) the first formalisation of cryptographic discourse within the framework of multi-dimensional logic, (2) the most comprehensive, logically connected formal model of cryptographic protocols proposed so far, and (3) a rigourous clarification of the concepts constituting the common knowledge of the community of protocol designers.

1.2 Kripke frames and structures................................... 4 1.3 The standard translations into first- and second-order logic.................. 5 1.4 Theories, equivalence and definability.............................. 6 1.5 Polyadic modalities........................................ 8 2 Bisimulation and basic model constructions.............................. 8

Lax modalities occur in intuitionistic logics concerned with hardware verification, the computational lambda calculus, and access control in secure systems. They also encapsulate the logic of Lawvere-Tierney-Grothendieck topologies on topoi. This paper provides a complete semantics for quantified lax logic by combining the Beth-Kripke-Joyal cover semantics for first-order intuitionistic logic with the classical relational semantics for a “diamond ” modality. The main technique used is the lifting of a multiplicative closure operator (nucleus) from a Heyting algebra to its MacNeille completion, and the representation of an arbitrary locale as the lattice of “propositions ” of a suitable cover system. In addition, the theory is worked out for certain constructive versions of the classical logics K and S4. An alternative completeness proof is given for (non-modal) first-order intuitionistic logic itself with respect to the cover semantics, using a simple and explicit Henkin-style construction of a characteristic model whose points are principal theories rather than prime saturated ones. The paper provides further evidence that there is more to intuitionistic modal logic than the generalisation of properties of boxes and diamonds from Boolean modal logic.

1.2 Short overview of the different logics....................................... 9

Introduction Alethic modalities are the necessity, contingency, possibility or impossibility of something being true. Alethic means `concerned with truth'. [28, p. 132] The above dictionary characterization of alethic modalities states the central notions of alethic modal logic: necessity, and other notions that are usually thought of as being definable in terms of necessity and Boolean negation: impossibility, contingency, and possibility. The syntax of modal propositional logic is inductively defined over a denumerable set of sentence letters p 0 , p 1 , p 2 , . . . as follows: A ::= p | A | (A # B) | #A The other Boolean operations (#, #, #, # and #) are defined as usual. A formula<F10.9