Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1

The emergent Semantic Web, despite being in its infancy, has already received a lot of attention from academia and industry. This resulted in an abundance of prototype systems and discussion most of which are centred around the underlying infrastructure. However, when we critically review the work done to date we realise that there is little discussion with respect to the vision of the Semantic Web. In particular, there is an observed dearth of discussion on how to deliver knowledge sharing in an environment such as the Semantic Web in effective and efficient manners. There are a lot of overlooked issues, associated with agents and trust to hidden assumptions made with respect to knowledge representation and robust reasoning in a distributed environment. These issues could potentially hinder further development if not considered at the early stages of designing Semantic Web systems. In this perspectives' paper, we aim to help engineers and practitioners of the Semantic Web by raising awareness of these issues.

We provide algebraic semantics together with a sound and complete sequent calculus for information update due to epistemic actions. This semantics is flexible enough to accommodate incomplete as well as wrong information e.g. deceit. We give a purely algebraic treatment of the muddy children puzzle, which moreover extends to situations where the children are allowed to lie and cheat. Epistemic actions, that is, information exchanges between agents A, B,... ∈ A, are modeled as elements of a quantale, hence conceiving them as resources. Indeed, quantales are to locales what monoidal closed categories are to Cartesian closed categories, respectively providing semantics for Intuitionistic Logic, and for non-commutative Intuitionistic Linear Logic, including Lambek calculus. The quantale (Q, � , •) acts on an underlying Q-right module (M, � ) of epistemic propositions and facts. The epistemic content is encoded by appearance maps, one pair f M A: M → M and f Q A: Q → Q of (lax) morphisms for each agent A ∈ A. By adjunction, they give rise to epistemic modalities [12], capturing the agents ’ knowledge on propositions and actions. The module action is epistemic update and gives rise to dynamic modalities [20] — cf. weakest preconditions. This model subsumes the crucial fragment of Baltag, Moss and Solecki’s [6] dynamic epistemic logic, abstracting it in a constructive fashion while introducing resource-sensitive structure on the epistemic actions. Keywords: Multi-agent communication, knowledge update, resource-sensitivity, quantale, Galois adjoints, dynamic epistemic logic, sequent calculus, Lambek calculus, Linear Logic.

Abstract. Semantic interoperability has become a key issue for realizing the Semantic Web in its full potential. However, there is a lot of controversy regarding the meaning and scope of the term and scarce formal approaches to the problem of semantic heterogeneity. In this paper, we discuss these approaches and propose a formalisation of semantic interoperability based on the Barwise-Seligman theory of information flow. We argue for a theoretical framework that favours the analysis and implementation of semantic interoperability scenarios. We present an example case of such a scenario where our framework has been applied as well as variations of it in the domain of ontology mapping. 1

Modal logic becomes action logic by adding programs as in propositional dynamic logic or the µ-- calculus. Modal languages can be seen as decidable fragments of first-order logic that admit a natural bisimulation, and hence enjoy a good model theory. Recently, much stronger 'guarded fragments' of firstorder logic have been identified that enjoy the same pleasant features. The latter can serve as richer action languages as well. We will develop the logic of guarded fragments as a form of process theory. ln particular, moving from sequential to parallel process operations correlates with moving to first-order fragments that are close to, or perhaps just over the decidable--undecidable fence. 1 The modal dynamics of actions We will start by reviewing the basics. Standard polymodal logic is a decidable fragment of the first-order logic of process graphs (labeled transition systems, Kripke models). It can be characterized semantically as consisting, up to logical equivalence, of those firs...

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality. In part II, we consider lattice-ordered algebras (lattices with additional operators) , extending the J'onsson and Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the J'onsson-Tarski additive operators. Representation of `-algebras is extended to full duality. In part III we discuss applications in logic...

We advance a theoretical framework...

Default inheritance reasoning is a propositional approach to nonmonotonic reasoning designed to model reasoning with natural language generics. Inheritance reasoners model sets of natural language generics as directed acyclic graphs, and inference corresponds to the specification of paths through those networks. A proliferation of inheritance proof theories exist in the literature along with extensive debate about the most reasonable way to construct inferences, based on intuitions about interpretations of particular inheritance networks. There has not been an accepted semantics for inheritance which unifies the set of possible proof theories, which would help identify truly ill-motivated proof theories. This thesis attempts to clarify the inheritance literature in the three ways indicated in the title: psychological plausibility, proof theory and semantics. The thesis intends to displace debate about the best inferences to draw about a network from logicians ' introspections to empiri...

The B-Method provides a collection of structuring mechanisms which support information hiding, modularisation and compositionality of module operations, although, in order to achieve compositionality and independent (parallel) refinement, sharing is restricted in B. In this paper we elaborate some non-interference and compositionality assumptions that underlie structuring mechanisms such as uses, sees and imports and show how they may be violated by inducing emerging properties which alter the context of the used, seen or imported machine. We discuss how such situations can be avoided by considering necessary and sufficient conditions for logical conservativeness and modularisation. As proof obligations, these conditions ensure that the properties of the context of the seen, used or imported component are conserved, i.e. that they are preserved but not enriched. From a logical viewpoint, these proof obligations require that the uniform interpolant of the contextual extension axioms is implied by the base context.

We present a process-algebraic approach to situated multi-agent systems which incorporates syntax for agent systems as well as for the environments (workspaces) they live in. Agent states are characterized by (1) a belief state, (2) a goal (desire, need) state and (3) a capabilities state. Capabilities are determined by a subspace of the workspace the agent lives in, and they are represented in the agent state as a collection of types of objects (tools, in the agent's workspace) the agent has the expertise (or permission) to use. Action capabilities are then just permissible transformations of workspaces, restricted to the particular capabilities space of the agent. Workspaces (certain kinds of collections of objects, modelled here as simply typed records of labelled attributes) and workspace transformations (record update operations) are structured into a transition system. We use a simple model of objects as records and, therefore, of workspaces and thus we only deal in this report with deterministic environments. Transitions are determined by agent actions (modifications of properties of objects in their capabilities space, modelled as record update operations), thus allowing for a formal account of agent systems living in an ever changing environment. Beliefs and desires are formulae of a many-sorted, first-order multi-modal language of properties of workspaces, where sorts are object types and modalities are indexed by workspace transformations (record updates). The logical language is kept simple, allowing only for first-order beliefs (beliefs about properties of workspaces). We present a language of situated, cooperative, self-interested agent systems, proposing a basic collection of agent behaviors (including ground observation actions, commitments, communication via assertions and requests, recursive behaviors, choice, concurrent behaviors etc), we provide an operational semantics for this language and discuss some examples of useful definable agent behaviors, such as perceiving (performing observations triggered by a statement). We then present and discuss notions of (behavioral) agent preorder and equivalence relations. Using the operational semantics and our notion of behavioral preorder and equivalence we propose an inequational theory for reasoning about agents. We conclude with presenting and discussing a case study for a simple agent system.