Last update of this section: 2018-12-16

This annotated list also appears, more less complete up to June 2014, on ResearchGate ( and in print (complete up to March 2014) on pages 421-43 of Sven Ove Hansson ed, David Makinson on Classical Methods for Non-Classical Problems. Series: Outstanding Contributions to Logic (vol 3) Springer 2014.


(B4) Sets Logic and Maths for Computing. Springer. Second (revised) edition February 2012, ISBN 978-1-4471-2499-3.

An introductory text for those who need to be able to understand and use basic formal notions such as set, relation, function, recursion, counting, probability, trees, and logic. Directed in the first place to students of computer science, but also suitable for students of mathematics, philosophy, or linguistics. See preface and table of contents of the second edition. Further details here

(B3) Bridges from Classical to Nonmonotonic Logic (London: College Publications. Series: Texts in Computing, vol 5, 2005 ISBN 1-904987-00-1. 

A textbook for graduate students of philosophy, computer science, or mathematics. Based on the perspectives developed in the papers (53), (55), using material from the overview (60) plus pedagogical material suitable for a textbook.

Table of contents and preface

Reviewed in  Zentralblatt f. Math 2006:1084.03001, Bull. Symb. Logic 2006 vol 12.3 September 2006 499-502,Math. Reviews 2007b:03020001, Theoria December 2006 336-340, Logic and Logical Philosophy 2007, Studia Logica August 2008 437-439. 

Polish version with title Od Logiki Klasycznej do Niemonotonicznej, translated by Tomasz Jarmuzek, published 2008 by Wydawnictwo Naukowe Universytetu Mikolaja Kopernika, Torun.

(B2) Topics in Modern Logic (London: Methuen, 1973) ISBN 416-77440-7 (hardback) and 416-78100-4 (paperback). Also Italian translation published by Boringhieri (Turin, 1979), Japanese translation by Huritsu Bunka-Sha (Tokyo, 1980). Out of print.

Directed to the undergraduate student of philosophy with a limited background in mathematics but who wishes to go beyond the elementary level in logic, to obtain an overall picture and grasp some technical material of philosophical significance. 


(B1) Aspectos de la Logica Modal, series Notas de Logica Matematica, Universidad Nacional del Sur, Bahia Blanca, Argentina, 1970. Out of print.  Note in 2016: The text has recently been made available electronically at

Directed to students of mathematics, especially those with a background in abstract algebra, who wish to learn about modal logics and their associated algebraic and relational structures.


The following list begins with the most recent, including those in preparation or in press. Co-authors are indicated for joint publications. Internal technical reports are grouped with the full publications refining them. Each item is accompanied by a brief abstract/commentary. In two instances, items (73) and (75), the published paper is accompanied or replaced by a significantly revised and extended post-publication version.

If a blank page appears when you click the link, try clicking on "download" near the top. Please report any anomalies to david(dot)makinson(at)gmail(dot)com so that they can be corrected.

A few of the older papers have not yet been scanned and so are not available by clicking here. If you need one, it may be possible to send an offprint or photocopy: contact the author at the same address.

A file is available containing reviews of the papers that appeared in Mathematical Reviews up to 2008.

Papers 73, 75, 76 have been considerably revised and extended since publication. Links to both published and current updated versions are provided, and readers are advised to use the latter.


(77) "The scarcity of stable sets"

This text was written early in 2015, following an invitation to participate in a volume on the stability theory of beliefs. However the status of the volume is unclear and the paper remains unpublished.

The 'Humean thesis', formulated by Hannes Leitgeb, seeks to coordinate the plain beliefs of an ideal agent with its degrees of belief by requiring that they should jointly satisfy a certain equivalence condition expressing the idea of stability under conditionalization. We draw attention to a feature of the proposed condition, namely the scarcity of probability functions that can enter into it, which appears to impede it from playing the desired bridging role.


(76) "Relevance via decomposition". Australasian Journal of Logic 14:3 2017, Article n° 1, open access at

We report on progress and an unsolved problem in our attempt to obtain a clear rationale for relevance logic via semantic decomposition trees. Suitable decomposition rules, constrained by a natural parity condition, generate a set of directly acceptable formulae that contains all axioms of the well-known system R, is closed under substitution and conjunction, satisfies the letter-sharing condition, but is not closed under detachment. To extend it, a natural recusion is built into the procedure for constructing decomposition trees. The resulting set of acceptable formulae has many attractive features, but it remains an open question whether it continues to satisfy the crucial letter-sharing condition.

A considerably extended post-publication version, containing new material up to 2018-10-07, is available here Readers are advised to make use of this version. I continue to work on the subject, integrating material into the text.

A short version,giving only the essential ideas, is scheduled to appear, perhaps in 2018 or 2019, in a volume Logic in Question. Copies available from the author on request.

(75) "Gödel's Master Argument: what is it and what can it do?" IFCoLog Journal of Logic and its Applications 2015, 2: 1-16. 

A considerably extended post-publication text, last updated 2017-05-27, is available here. Readers are advised to make use of this version.

This text is expository. We explain Gödel’s ‘Master Argument’ for incompleteness as distinguished from the ‘official’ proof of his 1931 paper, highlight its attractions and shortcomings as well as ways of getting around some of the latter. 

(74) With J. Hawthorne “Lossy inference rules and their bounds: a brief review”. Pages 385-408 of The Road to Universal Logic, vol I, ed. A. Koslow & A. Buchsbaum. Springer 2014.

Reviews results that have been obtained about bounds on the loss of probability occasioned by applying classically sound, but probabilistically unsound, Horn rules for inference relations. It uses only elementary finite probability theory without appealing to linear algebra, and also provides some new results, in the same spirit, on non-Horn rules. 

(73) "Intelim rules for classical connectives". Pages 359-382 of Sven Ove Hansson ed., David Makinson on Classical Methods for Non-Classical Problems. Series: Outstanding Contributions to Logic. Springer, 2014.

A revised version extends some of the results, adds comments, corrects some minor errors, and is generally more reader-friendly. Last updated 2018-12-16, it is available here. Readers are advised to make use of this version.

Investigates introduction and elimination rules for truth-functional connectives, focusing on the general questions of the existence, for a given connective, of at least one such rule that it satisfies, and the uniqueness of a connective with respect to the set of all of them. The answers are straightforward in the context of rules using general set/set sequents of formulae, but rather complex and asymmetric in the restricted (but more often used) context of set/formula sequents.

Note: The text linked here extends the published one, and seeks also to be more readable.

(72) "Relevance logic as a conservative extension of classical logic". Pages 383-398 of Sven Ove Hansson ed., David Makinson on Classical Methods for Non-Classical Problems. Series: Outstanding Contributions to Logic. Springer, 2014.

An outline of the main results without proofs appeared as "Advice to the relevantist policeman" in The Logica Yearbook 2012 ed.Vit Puncochar & Petr Svarny, College Publications, London, 2013.

Relevance logic is ordinarily seen as a subsystem of classical logic under the translation that replaces arrows by horseshoes. If, however, we consider the arrow as an additional connective alongside the horseshoe and other classical connectives, another perspective emerges. Relevance logic, specifically the system R, may be seen as the output of a conservative extension of classical consequence into the language with arrow. 

(71) "On an inferential evaluation system for classical logic". Logic Journal of IGPL 22: 2014, 147-154.

We seek a better understanding of why an evaluation system devised by Tor Sandqvist yields full classical logic, despite its inferential character, by providing and analysing a direct proof of the fact using a suitable maximality construction.

(70) "Logical questions behind the lottery and preface paradoxes: lossy rules for uncertain inference". Synthese 186: 2012, 511-529.

We reflect on lessons that the lottery and preface paradoxes provide for the logic of uncertain inference. One of these lessons is the unreliability of the rule of conjunction of conclusions in such contexts, whether probabilistic or qualitative. This failure leads us to a formal examination of consequence relations without that rule, the study of other rules that may nevertheless be satisfied in unceratin inference, and a partial rehabilitation of conjunction as a ‘lossy’ rule. Another lesson is the possibility of rational inconsistent belief. This leads us to discuss criteria for deciding when an inconsistent set of beliefs may nevertheless reasonably be retained.

(69) With Lloyd Humberstone, "Intuitionistic logic and elementary rules", Mind 120: 2011, 1035-1051.

The interplay of introduction and elimination rules for propositional connectives is often seen as suggesting a distinguished role for intuitionistic logic. We prove three formal results about intuitionistic propositional logic that bear on that perspective, and discuss their significance.

(68) "Conditional probability in the light of qualitative belief change". Journal of Philosophical Logic 40: 2011, 121-153.

(A preliminary version, less complete than the final one, was published in Hykel Hosni and Franco Montagna (eds), Probability, Uncertainty and Rationality. CRM Series Vol 10, Edizioni della Scuola Normale Superiore, Pisa 2010). 

We explore ways in which purely qualitative belief change in the AGM tradition can throw light on options in the treatment of conditional probability. First, by helping see why we sometimes need to go beyond the ratio rule defining conditional from one-place probability. Second, by clarifying criteria for choosing between various non-equivalent accounts of the two-place functions. Third, by suggesting novel forms of conditional probability, notably screened and hyper-revisionary. Finally, we show how qualitative uncertain inference suggests another, very broad, class of 'proto-probability' functions.

(67) "Propositional relevance through letter-sharing" J. Applied Logic 7 December 2009 377-387.

The concept of relevance between classical propositional formulae, defined in terms of letter-sharing, began to take on a fresh life in the late 1990s when it was reconsidered in the context of the logic of belief change. two new ideas appeared in independent work of Odinaldo Rodrigues and Rohit Parikh: the relation of relevance was considered modulo the choice of a background belkief set, and the belief set was considered in a canonical form, called its most modular (or finest splitting) representation. This paper develops and extends these ideas in a systematic way.


(66) “Levels of belief and nonmonotonic reasoning”, pp 341-354 of Degrees of Belief ed. Franz Huber and Christoph Schmidt-Petri (Springer 2009), series Synthese Library vol 342. ISBN: 978-1-4020-9197-1.

Reviews the connections between different kinds of nonmonotonic logic and the general idea of varying degrees of belief. Summarizes the leading ideas of the book (B3).


(65) With George Kourousias, “Respecting relevance in belief change”. Análisis Filosófico 26.1 May 2006 (special issue commemorating the tenth anniversary of the death of Carlos Alchourrón) pp 53-61.

An explanation, without proofs, of the results of (64), focussing on the application to the logic of belief change and discussing the significance of the formal results.


(64) With George Kourousias, “Parallel interpolation, splitting, and relevance in belief change”. Journal of Symbolic Logic 72 September 2007 994-1002.

Devises a new ‘parallel interpolation’ theorem for classical propositional logic, strengthening standard interpolation, and uses it to extend Parikh's ‘finest splitting theorem’ from the finite to the infinite case. Shows that although AGM partial meet contraction and revision fail Parikh's criterion of relevance, the criterion is respected if we apply the operations to a normalized version of a belief set, obtained by applying the finest splitting theorem to it.


(63) With James Hawthorne, “The quantitative/qualitative watershed for rules of uncertain inference”. Studia Logica 86 September 2007 249-299.

Charts the ways in which closure properties of consequence relations for uncertain inference take on different forms according to whether the relations are generated in a quantitative or a qualitative manner. Leading themes are the different behaviour of distinct kinds of rule (finite-premise Horn, countable premise Horn, alternative conclusion) rules in this context, ‘watershed’ principles characterizing the difference, and the possibility of representation or completeness theorems for probabilistic consequence.


(62) “Completeness Theorems, Representation Theorems: What’s the Difference?” In Hommage à Wlodek: Philosophical Papers dedicated to Wlodek Rabinowicz, ed. Rønnow-Rasmussen et al. 2007, (electronic publication).

A discussion of the connections and differences between completeness and representation theorems in logic, with examples drawn from classical and modal logic, the logic of friendliness, and nonmonotonic reasoning.


(61) “Friendliness and sympathy in logic” pp 195-224 of Logica Universalis, 2nd edition, ed. J.-Y. Beziau (Basel: Birkhauser Verlag, 2007).

Defines and examines a notion of logical friendliness, a broadening of the familiar notion of classical consequence. Also reviews familiar notions and operations with which friendliness makes contact, providing a new light in which they may be seen. 

This is a considerably extended version of the paper appearing in the first edition of the collection (2005, pages 191-205) under the title “Logical friendliness and sympathy”. The extended version also appeared under the title “Friendliness for logicians”, in We Will Show Them! Essays in Honour of Dov Gabbay, vol 2, ed S. Artemov, H. Barringer, A. Garcez, L. Lamb, J. Woods (London: King’s College Publications, October 2005), pp 259-292.


(60) “How to go nonmonotonic”. In Handbook of Philosophical Logic, Second Edition, volume 12, ed. D. Gabbay and F. Guenthner (Amsterdam, Springer, 2005), pages 175-278.

An extended overview of the essentials of nonmonotonic logic. Based on the perspectives developed in the papers (53), (55).


(59) “Natural deduction and logically pure derivations”. PhiNews, April 2004 (

Systems of natural deduction transform logically pure second-level derivations into first-level ones, introducing annotations and procedural steps but losing logical purity. In this paper we investigate two questions: Are there general guidelines for minimizing the dangers of the naïve approach? Can we obtain logical purity without reconstructing everything in terms of sequent-calculi?

A preliminary version appeared under the title "Logically pure derivations" in the Electronic Festschrift for Professor Norman Foo, November 2003 (

(58) “Supraclassical inference without probability”. Chapter 6 (pp 95-111) in P. Bourgine & J-P. Nadal eds, Cognitive Economics: An Interdisciplinary Approach. Springer Verlag, 2003.

An overview of the basic ideas of nonmonotonic inference, written for a readership of economists with some background in probability but little if any in logic.

(57) “Conditional statements and directives”. Chapter 13 (pp 213-227) in P. Bourgine & J-P. Nadal eds, Cognitive Economics: An Interdisciplinary Approach. Springer Verlag, 2003.

An expository paper, describing some of the different kinds of conditionals that appear in ordinary discourse, and ways in which logicians have sought to model them. The first part looks at conditional propositions of various kinds, while the latter part turns to conditional directives, outlining the basic ideas of input/output logic as a means of handling them.

(56) With Leendert van der Torre, “Permission from an input/output perspective”. Journal of Philosophical Logic 32, 2003, 391-416.  

Shows how input/output operations provide a clear formal articulation of the well-known distinction between negative and positive permission. They also enable us, for the first time, to distinguish two distinct kinds of positive permission, dynamic and static, with quite different uses in practical life.

(55) “Ways of doing logic: what was different about AGM 1985?”. Journal of Logic and Computation 13 (2003) 3-13. Remark: This paper was published in the hardcopy issue of the journal, but due to a publisher's computer error was omitted from the electronic version as the author discovered some time later, too late to correct the omission. An electronic copy may be obtained via the highlighted link above.

Reflects on what, in 1985, was new or different about AGM belief revision as a way of doing logic, and what are the perspectives today.

(54) With Leendert van der Torre, What is input/output logic?”. Foundations of the Formal Sciences II: Applications of Mathematical Logic in Philosophy and Linguistics, pp163-174. Dordrecht: Kluwer, Trends in Logic Series, vol 17, 2003.

Explains the raison d’être and basic ideas of input/output logic, sketching the central elements with pointers to other publications for detailed developments. French translation appeared as an internal document of CREA (Centre de Rechereche en Epistemologie Appliquée, Paris, 2001).

(53)  Bridges between classical and nonmonotonic logic”. Logic Journal of the IGPL 11 (2003) 69-96. Web:

The purpose of this paper is to take some of the mystery out of what is known as nonmonotonic logic, by showing that it is not as unfamiliar as may appear on first sight. In effect, there are logics that act as natural bridges between the main nonmonotonic logics to be found in the literature and classical consequence. These bridge systems are perfectly monotonic, although like the nonmonotonic systems are supraclassical. They have an interest of their own as well as providing passage to the nonmonotonic case.

(52) With Leendert van der Torre, “Constraints for input/output logics”, Journal of Philosophical Logic 30 (2001) 155-185.

Studies ways of constraining input/output operations to avoid output that is inconsistent with a constraint, with particular attention to the case where the constraint is the input.

(51) With Leendert van der Torre, “Input/output logics”. Journal of Philosophical Logic 29 (2000) 383-408.

In a range of contexts, one comes across processes resembling inference, but where input propositions are not in general included among outputs, and the operation is not in any way reversible. Examples arise in contexts of conditional obligations, goals, ideals, preferences, actions, and beliefs. The purpose of the paper is to develop a general theory of propositional input/output operations. Particular attention is given to the special case where outputs may be recycled as inputs.

(50)  "On a fundamental problem of deontic logic", Norms, Logics and Information Systems. New Studies in Deontic Logic and Computer Science, edited by Paul McNamara and Henry Prakken (Amsterdam: IOS Press, Series: Frontiers in Artificial Intelligence and Applications, Volume: 49, 1999, ISBN 9051994273) pp 29-53.

The usual presentations of deontic logic, whether axiomatic or semantic, treat norms as if they could bear truth-values. A fundamental problem of deontic logic is to reconstruct it in accord with the philosophical position that norms direct rather than describe, and are neither true nor false. The approach takes seriously the warning: no logic of norms without attention to a system of which they form part. The formal development in this paper provides the basis for the more abstract mathematics of input/output logics in items (51), (52).

(49)  "Screened revision", Theoria 63 (1997) 14-23.

Develops a concept of revision, akin in spirit to AGM partial meet revision, but in which the postulate of 'success' may fail. The basic idea is to see such an operation as composite, with a pre-processor using a priori considerations to resolve the question of whether to revise, following which another operation revises in a manner that protects the a priori material.

(48)  With Ramón Pino Pérez and Hassan Bezzazi "Beyond rational monotony: on some strong non-Horn conditions for nonmonotonic inference operations" Journal of Logic and Computation 7 (1997) 605-632.

Explores the effect of adding to the rules of preferential inference a number of non-Horn rules stronger than or incomparable with rational monotony, but still weaker than plain monotony (or, in some cases, than conditional excluded middle), focussing on their representation in terms of special classes of preferential models. Also considers some curious Horn rules beyond those of preferential inference but weaker than monotony.

(47)  "On the force of some apparent counterexamples to recovery", in Normative Systems in Legal and Moral Theory: Festschrift for Carlos Alchourrón and Eugenio Bulygin, edited by E.Garzón Valdés et al (Berlin: Duncker & Humblot, 1997).

Examines the principal alleged counterexamples to the recovery postulate for operations of contraction on closed theories, and shows that the theories considered are implicitly ‘clothed’ with additional justificational structure. Recovery remains appropriate for ‘naked’ closed theories.

(46)   With Sven Ove Hansson "Applying normative rules with restraint", in Logic and Scientific Methods, M.L. Dalla Chiara et al eds (Dordrecht: Kluwer, 1997) 313-332.

Also appeared as Uppsala Preprint in Philosophy, 1995 n°7, and as part of an informal Festschrift to Sten Linström.

Investigates the logic of applying normative rules, and in particular those applications that are 'restrained', carried through as fully as is compatible with avoidance of contradictions or other consequences specified as undesirable.

 (45) "Carlos Eduardo Alchourrón - a memorial note", Rechtsphilosophie 27 (1996) 125-131.

Obituary notice with overview of the work of Carlos Alchourrón, based on (44) below, but focussing on his contributions to the logic of norms and of normative systems, corresponding to the readership of the journal.

(44)   "In memoriam Carlos Eduardo Alchourrón", Nordic Journal of Philosophical Logic 1, 1996, 3-10.

Obituary notice with overview of the work of Carlos Alchourrón, focussing on his contributions to the logic of theory change.

(43)  "Combinatorial versus decision-theoretic components of impossibility theorems", Theory and Decision 40, 1996, 181-190.

Separates the purely combinatorial component of Arrow's impossibility theorem in the theory of collective preference from its decision-theoretic part, and likewise for the closely related Blair/Bordes/Kelly/Suzumura theorem. Such a separation provides a particularly elegant proof of Arrow's result, via a new 'splitting theorem'.

(42) “General Patterns in Nonmonotonic Reasoning, in Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3, ed. Gabbay, Hogger and Robinson, Oxford University Press, 1994, pages 35-110.

An extended review of what is known about the formal behaviour of nonmonotonic inference operations, including those generated by the principal systems in the artificial intelligence literature. Directed towards computer scientists and others with some background in logic. Completed towards the end of 1991, this monograph was already widely cited in the literature before its publication early in 1994, and has become a standard reference.

 (41) With Karl Schlechta, "Local and global metrics for the semantics of counterfactual conditionals", Journal of Applied Non-Classical Logics 4, 1994, 129-140.

Also appeared in advance semi-published form as Technical Report RR37, September 1994, Lab. d'Informatique, URA CNRS 1787, Univ. Provence, Marseille, France.

Considers the question of how far the different 'closeness' relations, indexed by worlds, in a given model for counterfactual conditionals may be derived from a common source. Counterbalancing some well-known negative observations, we show that there is also a strong positive answer.

(40) With Peter Gärdenfors, "Nonmonotonic inference based on expectations", Artificial Intelligence 65, 1994, 197-245.

Shows how nonmonotonic inference relations may be generated in natural ways from sets of expected propositions, and also from relations of expectation between propositions. Adapts to the context of nonmonotonic reasoning ideas developed by the authors for the logic of theory change (references 23 and 29 below).

(39)   "Five faces of minimality", Studia Logica 52, 1993, 339-379.

An extended discussion of similarities and residual differences, within the general semantic framework of minimalisation, between defeasible inference, belief revision, counterfactual conditionals, updating, and conditional obligation. The purpose is not to establish new results, but to bring together existing material to form a clear overall picture.

(38) With Jürgen Dix, "The relationship between KLM and MAK models for nonmonotonic inference operations", Journal of Logic, Language, and Information 1 (1992), 131-140.

Also appeared as Interner Bericht 16/91 of the Institut für Logik, Komplexität und Deduktionssysteme of the University of Karlsruhe, Germany, 1991.

Sets out the exact relationship between two variants of the general notion of a preferential model for nonmonotonic inference: that of Kraus, Lehmann and Magidor, and that of Makinson (reference 31 below).

(37)  With Gerhard Brewka and Karl Schlechta, "Cumulative inference relations for JTMS and logic programming", in Dix, Jantke and Schmitt eds, Nonmonotonic and Inductive Logic, (Berlin: Springer Lecture Notes in Artificial Intelligence, n° 543, 1991, 1-12).

A preliminary version appeared as "JTMS and logic programming" in Nerode and others eds, Proceedings of the First International Workshop on Logic Programming and Nonmonotonic Reasoning (Cambridge, Mass: MIT Press, 1991).

Observes that JTMS and logic programming with negation (under the Gelfond-Lifschitz semantics) are non-cumulative; shows how to render them cumulative by an indexing device; notes residual drawback of floating conclusions.

 (36) With Karl Schlechta, "Floating conclusions and zombie paths: two deep difficulties in the directly skeptical approach to defeasible inheritance nets", Artificial Intelligence 48 (1991) 199-209.

Isolates and studies two recalcitrant phenomena in the theory of inference relations generated by defeasible inheritance nets.

A preliminary version appeared as an internal publication, under the title "On some difficulties in the theory of defeasible inheritance nets" in  Morreau ed., Proceedings of the Tübingen Workshop on Semantic Networks and Nonmonotonic Reasoning (SNS Bericht 89-48, 1989, 58-68). An almost final version appeared as an internal publication in Brewka & Freitag eds, Proceedings of the Workshop on Nonmonotonic Reasoning (Arbeitspapiere der GMD n°443, 1990, 201-216). Closely related material is also contained in the internal publication "On principles and problems of defeasible inheritance", Research report RR-92-59 of the DFKI, Saarbrücken, Germany.

Isolates and studies two recalcitrant phenomena in the theory of inference relations generated by defeasible inheritance nets.

(35) With Peter Gärdenfors, "Relations between the logic of theory change and nonmonotonic logic", in Fuhrmann & Morreau eds, The Logic of Theory Change (Berlin: Springer, 1991, 185-205).

Also appeared in two internal publications: Proceedings of the Workshop on Nonmonotonic Reasoning (Arbeitspapiere der GMD n°443, 1990) 7-28, and Report of RP2: First Workshop on Defeasible Reasoning and Uncertainty Management Systems (DRUMS), University of Toulouse, 1990 (France)

Examines the link between nonmonotonic inference relations and theory revision operations, focusing on the correspondence between abstract properties which each may satisfy.

 (34)  "The Gärdenfors impossibility theorem in nonmonotonic contexts", Studia Logica 49 (1990) 1-6.

Shows that the well-known impossibility theorem continues to be applicable in contexts where the background logic is nonmonotonic.

(33)   With Michael Freund and Daniel Lehmann, "Canonical extensions to the infinite case of finitary nonmonotonic inference relations", as an internal publication in Brewka & Freitag eds, Proceedings of the Workshop on Nonmonotonic Reasoning (Arbeitspapiere der GMD n° 443, 1990, 133-138).

Examines the problem of finding good ways of extending a nonmonotonic inference relation defined on finite sets of propositions, to one defined also on infinite sets of propositions.

 (32)  "Logique modale: quelques jalons essentiels", in L.Iturrioz & A.Dussauchoy eds, Modèles logiques et systèmes d'intelligence artificielle (Paris: Hermes, 1990, 75-97).

An expository review of central concepts and results of modal propositional logic, for a readership of computer scientists.

(31) "General theory of cumulative inference", in M.Reinfrank & others eds, Nonmonotonic Reasoning (Berlin: Springer-Verlag, Lecture Notes on Artificial Intelligence n°346, 1989, 1-17).

Studies the formal behaviour of inference operations which, whilst not monotonic, may nevertheless satisfy various other conditions, notably that of cumulativity. Very widely cited in the literature. Provides the germ from which the monograph General Patterns in Nonmonotonic Reasoning later developed.

 (30)  "Rights of peoples: point of view of a logician", in J.Crawford ed., The Rights of Peoples (Oxford University Press, 1988, 69-92).

A review of the main logical questions arising in connection with the notion of a right held by all peoples, focussing on problems of indeterminacy and of inconsistency.

A condensed version, omitting some of the documentation and focussing on the conclusions drawn, was published subsequently under the title "On attributing rights to all peoples: some logical questions", in Law and Philosophy (USA) 8 (1989) 53-63. A preliminary version was published in French translation under the title "Les droits des peuples: perspectives d'un logicien" in Cahiers de la Fondation Lelio Basso (Italy) 7 (1986) 53-60.

(29) With Peter Gärdenfors, "Revisions of knowledge systems and epistemic entrenchment", in M.Vardi ed., Proceedings of the Second Conference on Theoretical Aspects of Reasoning about Knowledge (Los Altos: Morgan Kaufmann, 1988, 83-95).

Relates the postulates for contraction and revision functions, introduced in (23) below, to a relation of "epistemic entrenchment" between propositions.

(28)   "On the status of the postulate of recovery in the logic of theory change", The Journal of Philosophical Logic 16 (1987) 383-394.

Describes and discusses the rather special behaviour of one of the postulates in the AGM account of theory change, as developed in (23) below.

(27)  "On the formal representation of rights relations: remarks on the work of Stig Kanger and Lars Lindahl", The Journal of Philosophical Logic 15 (1986) 403-425.

A discussion of work formalising Hohfeld's classic taxonomy of rights relations between two parties.

(26)  With Carlos Alchourrón, "Maps between some different kinds of contraction function: the finite case", Studia Logica 45 (1986) 187-198.

Studies the relationship, in the finite case, between the safe contractions of (25) and the partial meet contractions of (23).

 (25)  With Carlos Alchourrón, "On the logic of theory change: safe contraction", Studia Logica 44 (1985) 405-422.

Studies a special kind of contraction function, defined in terms of preservation of elements that are in a certain sense "safe".

(24)   "How to give it up: a survey of some recent work on formal aspects of the logic of theory change", Synthese 62 (1985) 347-363 and 68 (1986) 185-186.

An overview of the work of Alchourrón, Gärdenfors and the author in the area of the logic of theory change, covering in particular the central ideas of (20), (23), (25).

 (23)  With Carlos Alchourrón and Peter Gärdenfors, "On the logic of theory change: partial meet contraction and revision functions", The Journal of Symbolic Logic 50 (1985) 510-530. Russian translation to appear 2013 in E. Lisanyuk ed. Normative Systems and Other Works in Legal Philosophy and Logic of Norms (Russian title), Publishing House of Saint Petersburg State University.

An extended study of formal aspects of the logic of theory change, defining partial meet contraction functions and establishing a representation theorem in terms of suitable postulates. This paper is the locus classicus of what has since come to be known as the AGM approach to theory (or belief) change.

(22)  "Stenius' approach to disjunctive permission", Theoria 50 (1984) Festschrift issue for Erik Stenius, 136-145.

A critical review of Stenius' account of the logic of disjunctive permissions, leading to a proposal for a closely related approach in terms of "checklist conditionals".

 (21)  "Individual actions are very seldom obligatory", The Journal of Non-Classical Logic 2 (1983) 7-13.

Studies some striking differences between attributing deontic predicates to individual actions, on the one hand, and to generic actions on the other.

 (20)  With Carlos Alchourrón, "On the logic of theory change: contraction functions and their associated revision functions", Theoria 48 (1982) 14-37.

A study in the logic of theory change, examining the properties of maxichoice contraction and revision operations.

, Publishing House of Saint Petersburg State University..

(19)   With Carlos Alchourrón, "Hierarchies of regulations and their logic", in Hilpinen ed., New Studies in Deontic Logic (Dordrecht: Reidel, 1981, 125-148). Russian translation to appear 2013 in E. Lisanyuk ed. Normative Systems and Other Works in Legal Philosophy and Logic of Norms (Russian title), Publishing House of Saint Petersburg State University.

Investigates the resolution of contradictions and ambiguous derogations in a code, by means of the imposition of partial orderings. Although formulated as a study in the logic of norms, it provided the initial ideas for work on the logic of theory (or belief) change, developed by the authors in a series of papers beginning with (20).

(18)  "Quantificational reefs in deontic waters", in Hilpinen ed., New Studies in Deontic Logic (Dordrecht: Reidel, 1981, 87-91).

On the prevalence of implicit quantification in deontic assertions in ordinary language.

 (17) "Non-equivalent formulae in one variable in a strong omnitemporal modal logic", Zeitschrift für math. Logik und Grundl. der Math. 27 (1981) 111-112.

A minor technical observation in modal propositional logic. The last of the author's papers in modal logic.

(16) "A characterization of structural completeness of a structural consequence operation", Reports on Mathematical Logic 6 (1976) 99-102.

A lattice-theoretic characterisation of the notion of structural completeness of a consequence operation.

 (15)  With Krister Segerberg, "Post completeness and ultrafilters", Zeitschrift für math. Logik und Grundl. der Math. 20 (1974) 355-358.

A cardinality result in modal propositional logic.

(14) "Strengths and weaknesses of the Ajdukiewicz approach to grammar" (in Portuguese) Discurso (Brazil) 4 155-165.

Reviews the powers and limitations of a simple mathematical model of grammatical structure.

(13)   "A warning about the choice of primitive operators in modal logic", The Journal of Philosophical Logic 2 (1973) 193-196.

Draws attention to some unexpected consequences of using a zero-ary connective in modal propositional logic.

 (12)  "Some embedding theorems for modal logic", Notre Dame Journal of Formal Logic 12 (1971) 252-254.

Some results on the upper end of the lattice of all modal propositional logics.

(11)  With Luisa Iturrioz, "Sur les filtres premiers d'un treillis distributif et ses sous-treillis", Comptes Rendus de l'Academie des Sciences de Paris 270A (1970) 575-577.

Solution to a cardinality question in the theory of distributive lattices, generalising the result of (9).

 (10)  "A generalization of the concept of a relational model for modal logic", Theoria 36 (1970) 331-335.

Generalises the concept of a relational model for modal logic, due to Kripke, so as to obtain a closer correspondence between relational and algebraic models. The generalisation obtained is essentially equivalent to the notion of a "first-order" model that was defined independently by S.K.Thomason.

(9)   "On the number of ultrafilters of an infinite Boolean algebra", Zeitschrift für math. Logik und Grundl. der Math. 15 (1969) 121-122.

Solution of a cardinality question in the theory of Boolean algebras. Later generalised in (11) to distributive lattices.

(8)    "A normal modal calculus between T and S4 without the finite model property", The Journal of Symbolic Logic 34 (1969) 35-38.

The first example of an intuitively meaningful propositional logic without the finite model property, and still the simplest one in the literature. The question of its decidability appears still to be open.

(7)   "Remarks on the concept of distribution in traditional logic", Nous 3 (1969) 103-108.

Uses ideas of modern mathematical logic to clarify and to a certain extent defend a controverted and rather obscure notion of traditional syllogistic.

(6)   "On some completeness theorems in modal logic", Zeitschrift für math. Logik und Grundl. der Math. 12 (1966) 379-384.

Gives the first published adaptation of the Lindenbaum/Henkin method of maximal consistent sets for establishing the completeness of modal propositional logics with respect to the relational models of Kripke. This paper is perhaps the most widely known of the author's publications in modal logic, and its proof has become the standard textbook one, as for example in the text of Hughes and Cresswell. It is based on sections of the author's D.Phil. thesis, as are (5) and (4) below.

(5)    "How meaningful are modal operators?", Australasian Journal of Philosophy 44 (1966) 331-337.

A philosophical discussion of the intuitive meaning of the formalism of modal propositional logics.

(4)   "There are infinitely many Diodorean modal functions", The Journal of Symbolic Logic 31 (1966) 406-408.

A cardinality result in the theory of modal logics.

(3)   "An alternative characterization of first-degree entailment", Logique et analyse (1965) 305-311, with misprints corrected in 9 (1966) 394.

Gives an axiomatisation of the first-degree entailments of the Anderson-Belnap system, using the notion of a formula in which no variable occurs more than once.

(2)    "The paradox of the preface", Analysis 25 (1965) 205-207.

By means of an example, shows the possibility of beliefs that are separately rational whilst together inconsistent.

(1)    "Nidditch's definition of verifiability", Mind 74 (1965) 240-247.

Shows the failure of an attempt, which appears to have been the last in a series of increasingly complex efforts following Ayer, to define the notion of verifiability in purely deductive terms.


(2) "Qu'est-ce que la complétude structurale?", abstract of paper n° 16 above, Proc. Conf. Soc. Française de Logique, Méthodologie et Philosophie des Sciences (Lyon: Publ.Dept. Math. Lyon, 16 1979, 65-66).

(1) "Some properties of the lattice of all modal logics", in Abstracts from the Fourth International Congress for Logic, Methodology and Philosophy of Science (Bucharest, 1971).

Other texts

4. Extended review of N.J.Smith The Laws of Truth in Studia Logica 2015.

3.  "A tale of five cities". Pages 19-32 of Sven Ove Hansson ed., David Makinson on Classical Methods for Non-Classical Problems. Series: Outstanding Contributions to Logic. Springer, 2014.

An autobiographical sketch: The author looks back at his work in logic under the headings: early days in Sydney, graduate studies in Oxford, adventures in Beirut, midnight oil in Paris, and last lap in London.

2. "Reflections on the contributions". Pages 409-420 of Sven Ove Hansson ed., David Makinson on Classical Methods for Non-Classical Problems. Series: Outstanding Contributions to Logic. Springer, 2014.

Reflects on some general issues prompted by a reading of the contributions to the volume.

1. Foreword to the 2008 reprint by College Publications of Peter Gärdenfors Knowledge in Flux.

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