Logic of Belief Change

Perhaps the most frequently cited work is the creation of the so-called AGM account of the logic of belief change, with Carlos Alchourrón and Peter Gärdenfors. This was done in a variety of converging forms: postulational, in terms of partial meet operations, relations of epistemic entrenchment, and safe contraction (refs 20, 23, 24, 25, 26), with also a paper (54) reviewing the ways in which the logics of belief change and uncertain reasoning have led to new ways of doing logic.

Recent papers in the area examine the question of relevance in belief change in the light of the finest splitting theorem (refs 64, 65 with George Kourousias, 67). 

Logic of Uncertain Reasoning

In this area, my research has followed three main lines.

One was directed to clarifying the logical patterns to be found in qualitative uncertain reasoning, commonly (and rather misleadingly) known as nonmonotonic inference (refs 31, 42).

Another, carried out jointly with Peter Gärdenfors, established the basic relationships between nonmonotonic inference and belief revision (refs 35, 40).

More recently (refs 53, 58, 60, 66, B3), I have focussed on the relations between qualitative and probabilistic approaches to uncertain reasoning (ref 63 with Jim Hawthorne), conditional probability in the light of qualitative belief change (ref 68), the concept of a lossy inference rule (ref 70 and ref 74 again with Jim Hawthorne).  

Logic of Norms and Normative Systems

In the logic of norms (also known as deontic logic), earlier publications (refs 27 and 30) analyse the Hohfeld classification of rights relationships and its application to real-life rights claims (particularly collective rights)

More recent work reconstructs the logic of norms in accord with the philosophical position that norms lack truth values (ref 50), developing into a general theory of input/output logics as a framework for conditional directives and permissions (refs 51, 52, 54, 56).

Other Areas of Logic

Some of my research output does not fall squarely into any of the above categories.

One paper (ref 43) separates combinatorial from decision-theoretic components in Arrow's impossibility theorem and the closely related Blair/Bordes/Kelly/Suzumura theorem in the theory of collective preference, providing a particularly elegant proof of those results.

Another (ref 61) articulates the fascinating concept of logical friendliness, studying its implicit manifestations in the literature of the last hundred and fifty years, as well as its properties.

A third formulates the concept of parallel interpolation and continues Parikh's analysis of splitting in classical propositional logic (refs 64, 65, with George Kourousias, also 67).

More recently, (ref 69) undertakes a close formal examination of the generation of intuitionistic logic by means of introduction and elimination rules, developing into a general semantic study of intelim rules for classical connectives (ref 73)

Finally, (ref 72) examines the subtle relationships between classical and relevance logic showing how, even on the level of their consequence relations, the latter may be seen as a conservative extension of the former.

Early Work in Modal Logic

Early work focussed largely on modal logic. Perhaps the most cited contribution in this area was the adaptation, in my1965 D.Phil. thesis and a following publication (ref 6), of the maximal consistent set method, then well-known in classical propositional and predicate logic (Lindenbaum, Henkin), to serve as a tool for establishing completeness results in modal and other non-classical logics, where it is now a standard procedure.

Also often mentioned is the discovery (ref 8) of the first simple and natural propositional logic lacking the finite model property; and formulation (ref 10) of a generalised notion of relational model for modal logic, bringing the relational account into harmony with the algebraic one.

Another, quite ignored at the time of its publication in 1971 but often cited in recent years (ref 12), proves the first (and still the main) embedding theorems for modal logics.