Jeremy Seligman

Logic in the community

Communities consist of individuals bounds together by social relationships and roles. Within communities, individuals reason about each other’s beliefs, knowledge and preferences. Knowledge, belief, preferences and even the social relationships are constantly changing, and yet our ability to keep track of these changes is an important part of what it means to belong to a community.

Logical Dynamics of Belief Change in the Community

In this paper we explore the relationship between norms of belief revision that may be adopted by members of a community and the resulting dynamic properties of the distribution of beliefs across that community. We show that at a qualitative level many aspects of social belief change can be obtained from a very simple model, which we call ‘threshold influence’. In particular, we focus on the question of what makes the beliefs of a community stable under various dynamical situations. We also consider refinements and alternatives to the ‘threshold’ model, the most significant of which is to consider changes to plausibility judgements rather than mere beliefs. We show first that some such change is mandated by difficult problems with belief-based dynamics related to the need to decide on an order in which different beliefs are considered. Secondly, we show that the resulting plausibility-based account results in a deterministic dynamical system that is non-deterministic at the level of beliefs.

Internalization: The Case of Hybrid Logics

A sequent calculus for hybrid logics is developed from a calculus for classical predicat logic by a series of transformations. We formalize the semantic theory of hybrid logic using a sequent calculus for predicate logic plus axioms. This works, but it is ugly. The unnattractive features are removed one-by-one, until the final vestiges of the metalanguage can be set aside to reveal a fully internalized calculus. The techniques are quite general and can be applied to a wide range of hybrid and modal logics.

Channels: From Logic to Probability

Information arises in conditions of uncertainty. When we are unsure of what has happened, our uncertainty is reduced by gaining information. Central to any mathematical model of information is the representation of a state of uncertainty and the change of state brought about by the acquisition of a new piece of information.

The Scope of Turing's Analysis of Effective Procedures

Turings (1936) analysis of effective symbolic procedures is a model of conceptual clarity that plays an essential role in the philosophy of mathematics. Yet appeal is often made to the effectiveness of human procedures in other areas of philosophy. This paper addresses the question of whether Turings analysis can be applied to a broader class of effective human procedures. We use Siegs (1994) presentation of Turings Thesis to argue against Clelands (1995) objections to Turing machines and we evaluate her proposal to understand the effectiveness of procedures in terms of their reliability and precision. A number of conditions for effectiveness are identified and these are used to provide a general argument against the possibility of a Leibnizian decision procedure.

4 – Situation Theory

Publisher Summary This chapter provides an introduction to the main themes and technical contributions of situation theory. Situation theory has its fair share of neologisms, the most inspired of which is surely the term infon, coined by Keith Devlin. An infon is an item of information. The term is intended to be as neutral as possible about the form in which the information is represented. It is a commonplace that in foundational theories one must be very careful to distinguish between metalanguage and object language. Situation theory is no exception, and special difficulties arise with the word “relation.” The aim of the chapter is to present the existing literature on the subject and to provide a coherent statement of situation theory. It introduces to “simple information structures.” Structures in this class are intended to provide a naive model of the structure of information, as captured by the relational structures of first-order model theory. The class is axiomatized, and each of the axioms is discussed with a view to generalization. Many of the novelties of situation theory can be seen as generalizations of this sort.

Situations, Constraints and Channels (Update of Chapter 4)

A significant concept to emerge from early work in Situation Theory (see Seligman and Moss, this volume) is that of a constraint between situations, and its relation to the flow of information. This chapter surveys recent work on this topic. Situation theory has always held that information is carried relative to constraints, but the idea has been implemented in different ways in early channel theory, the theory of situated inference, and later channel theory. Early channel theory takes constraints to be connections between situations, which are also treated as situations. Situated inference takes constraints to be pieces of information contained in situations. And later channel theory constraints to work on two different levels: connections between specific events and regularities between event types. Consideration of the structural properties of the two-level account led to the development of Channel Theory, a general algebraic theory of information and information flow.

Towards a Logic of Tweeting

In this paper we study the logical principles of a common type of network communication events that haven’t been studied from a logical perspective before, namely network announcements, or tweeting, i.e., simultaneously sending a message to all your friends in a social network. In particular, we develop and study a minimal modal logic for reasoning about propositional network announcements. The logical formalisation helps elucidate core logical principles of network announcements, as well as a number of assumptions that must be made in such reasoning. The main results are sound and complete axiomatisations.

Boolean Network Games and Iterated Boolean Games

A Boolean Network Game is a game played on a network structure. Players choose actions depending on the actions of those in their neighbourhood and attempt to achieve some goal expressed in a modification of Linear Temporal Logic over an infinite run. Iterated Boolean Games are similar, but lack network structure. We define and give translations between these models, and give some complexity results.

Algebraic Semantics for Dynamic Dynamic Logic

Dynamic dynamic logic (DDL) is a generalisation of propositional dynamic logic PDL and dynamic epistemic logic. In this paper, we develop algebraic semantics for DDL without the constant program. We introduce inductive and continuous modal Kleene algebras for PDL and show the validity of reduction axioms in algebraic models and hence the algebraic completeness of DDL.