Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Evidence and plausibility in neighborhood structures

The intuitive notion of evidence has both semantic and syntactic features. In this paper, we develop an {\em evidence logic} for epistemic agents faced with possibly contradictory evidence from different sources. The logic is based on a neighborhood semantics, where a neighborhood $N$ indicates that the agent has reason to believe that the true state of the world lies in $N$. Further notions of relative plausibility between worlds and beliefs based on the latter ordering are then defined in terms of this evidence structure, yielding our intended models for evidence-based beliefs. In addition, we also consider a second more general flavor, where belief and plausibility are modeled using additional primitive relations, and we prove a representation theorem showing that each such general model is a $p$-morphic image of an intended one. This semantics invites a number of natural special cases, depending on how uniform we make the evidence sets, and how coherent their total structure. We give a structural study of the resulting `uniform' and `flat' models. Our main result are sound and complete axiomatizations for the logics of all four major model classes with respect to the modal language of evidence, belief and safe belief. We conclude with an outlook toward logics for the dynamics of changing evidence, and the resulting language extensions and connections with logics of plausibility change

Logical dynamics meets logical pluralism?

Where is logic heading today? There is a general feeling that the discipline is broadening its scope and agenda beyond classical foundational issues, and maybe even a concern that, like Stephen Leacock’s famous horseman, it is ‘riding off madly in all directions’. So, what is the resultant vector? There seem to be two broad answers in circulation today. One is logical pluralism, locating the new scope of logic in charting a wide variety of reasoning styles, often marked by non-classical structural rules of inference. This is the new program that I subscribed to in my work on sub-structural logics around 1990, and it is a powerful movement today. But gradually, I have changed my mind about the crux of what logic should become. I would now say that the main issue is not variety of reasoning styles and notions of consequence, but the variety of informational tasks performed by intelligent interacting agents, of which inference is only one among many, involving observation, memory, questions and answers, dialogue, or general communication. And logical systems should deal with a wide variety of these, making information-carrying events first-class citizens in their set-up. The purpose of this brief paper is to contrast and compare the two approaches, drawing freely on some insights from earlier published papers. In particular, I will argue that logical dynamics sets itself the more ambitious diagnostic goal of explaining why substructural phenomena occur, by ‘deconstructing’ them into classical logic plus an explicit account of the relevant informational events

Morphological Computing as Logic Underlying Cognition in Human, Animal, and Intelligent Machine

This work examines the interconnections between logic, epistemology, and sciences within the Naturalist tradition. It presents a scheme that connects logic, mathematics, physics, chemistry, biology, and cognition, emphasizing scale-invariant, self-organizing dynamics across organizational tiers of nature. The inherent logic of agency exists in natural processes at various levels, under information exchanges. It applies to humans, animals, and artifactual agents. The common human-centric, natural language-based logic is an example of complex logic evolved by living organisms that already appears in the simplest form at the level of basal cognition of unicellular organisms. Thus, cognitive logic stems from the evolution of physical, chemical, and biological logic. In a computing nature framework with a self-organizing agency, innovative computational frameworks grounded in morphological/physical/natural computation can be used to explain the genesis of human-centered logic through the steps of naturalized logical processes at lower levels of organization. The Extended Evolutionary Synthesis of living agents is essential for understanding the emergence of human-level logic and the relationship between logic and information processing/computational epistemology. We conclude that more research is needed to elucidate the details of the mechanisms linking natural phenomena with the logic of agency in nature.Comment: 20 pages, no figure

Abduction for (non-ominiscient) agents

Among the non-monotonic reasoning processes, abduction is one of the most important. Usually described as the process of looking florexplantions, it has been recognized as one of the most commonly used in our daily activities. Still, the traditional definitions of an abductive problem and an abductive solution mention only theories and formulas, leaving agency out of the picture. Our work proposes a study of abductive reasoning from an epistemic and dynamic perspective, making special emphasis on non-ideal agents. We begin by exploring what an abductive problema is in terms of an agent’s information, and what an abductive solution is in terms of the actions that modify it. Then we explore the different kinds of abductive problems and abductive solutions that arise when we consider agents whose information is not closed under logical consequence, and agents whose reasoning abilities are not complete

Dynamic Logic of Common Knowledge in a Proof Assistant

Common Knowledge Logic is meant to describe situations of the real world where a group of agents is involved. These agents share knowledge and make strong statements on the knowledge of the other agents (the so called \emph{common knowledge}). But as we know, the real world changes and overall information on what is known about the world changes as well. The changes are described by dynamic logic. To describe knowledge changes, dynamic logic should be combined with logic of common knowledge. In this paper we describe experiments which we have made about the integration in a unique framework of common knowledge logic and dynamic logic in the proof assistant \Coq. This results in a set of fully checked proofs for readable statements. We describe the framework and how a proof can beComment: 15

Exploring the power of converse events

Dynamic epistemic logic as viewed by Baltag, Moss and Solecki (DEL) and propositional dynamic logic (PDL) offer different semantics of events. On the one hand, DEL adds dynamics to epistemic logic by introducing so-called event models as syntactic objects into the language. On the other hand, PDL has instead transition relations between possible worlds. This last approach allows to easily introduce converse events. In this paper we add epistemics to this, and call the resulting logic epistemic dynamic logic (EDL). We show that DEL can be translated into EDL thanks to this use of the converse operator: it enables us to translate the structure of the event model directly within a particular axiomatization of EDL, without having to refer to a particular epistemic event model in the language (as done in DEL). It follows that EDL is more expressive and general than DEL and we characterize semantically and syntactically in EDL this embedding of DEL

Dynamic consequence and public announcement

Junta de Andalucía P08-HUM- 04159European Research Council EPS 313360