David Makinson

Last updated 08 July 2024


Currently Honorary Associate Professor in the School of Historical and Philosophical Inquiry, University of Queensland, Australia. Previously, Guest Professor in the Department of Philosophy, Logic and Scientific Method, London School of Economics (LSE). Before that, counting backwards, Senior Research Fellow in the Department of Computer Science, King's College London, Programme Specialist with Unesco, and Assistant to Full Professor in the American University of Beirut, Lebanon.

A short intellectual autobiography up to 2014 "A tale of five cities", was published in Sven Ove Hansson ed., David Makinson on Classical Methods for Non-Classical Problems (Series: Outstanding Contributions to Logic) Springer 2014, pp 19-32.  See also an interview with Hykel Hosni in The Reasoner 2014. The version of "A tale..." on this website includes a postscript taking the story from 2014 to 2024.

Photo 2018-04-05      Photo 2009-05-25

Contact info

Email: david(dot)makinson(at)gmail(dot)com. Postal address:  Les Etangs, B2, Domaine de la Ronce, 92410 Ville d'Avray, France.

Areas of research

In general, my research has been in logic and its relations with neighbouring disciplines. 

Most recent 

Currently, I am working mainly on relevance-sensitive truth-trees, and in some topics in the history of logic concerning Boole, Hamilton, Frege, and Orlov. In recent years I have also  returned to topics that had for a long time been on shelves at the back of my mind. These include Gödel's 'master argument' for his first incompleteness theorem, and a semantic study of intelim rules for classical connectives. 

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 known as nonmonotonic inference. Another established the basic relationships between nonmonotonic inference and belief revision. More recently, I have investigated the relations between qualitative and probabilistic approaches to uncertain reasoning, the concept of a lossy inference rule, and conditional probability in the light of qualitative belief change.  

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, with also a paper reviewing the ways in which the logics of belief change and uncertain reasoning have led to new ways of doing logic. More recent papers in this area, with George Kourousias, examine the question of relevance in belief change in the light of the finest splitting theorem. 

Logic of norms and normative systems

In the logic of norms (also known as deontic logic), earlier publications 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, developing into a general theory of input/output logics as a framework for conditional directives and permissions.

Other areas of logic

Some of my work does not fall squarely into any of the above categories. One paper 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 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.

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, 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 of the first simple and natural propositional logic lacking the finite model property; and formulation of a generalised notion of relational model for modal logic, bringing the relational account into harmony with the algebraic one. Another item, quite ignored at the time of its publication in 1971 but often cited in recent years, proves the first (and still the main) embedding theorem for modal logics.


If any links to publications below fail to function, do not hesitate to contact me at david.makinson@gmail.com so that I can dispatch pdfs (and repair the links). 

The annotated list below appears, more less complete up to 2014 only and without links, on pages 421-43 of David Makinson on Classical Methods for Non-Classical Problems, Series: Outstanding Contributions to Logic, Springer 2014 and on ResearchGate (https://www.researchgate.net/profile/David_Makinson3).

As from 01/01/2021, Zentralblatt für Math has placed its reviews online with open access at https://www.zbmath.org/. This contains reviews of many of the items below. We provide here links to reviews of books and monograph, but not to reviews of journal papers, book chapters etc; they may easily be located via Zentralblatt's database. 


Sets, Logic and Maths for Computing, 3rd edition, Springer 2020 . 

The text introduces the student to basic concepts of set theory, discrete mathematics and formal logic that are needed to work with abstract structures. Although the book appears in a series for students of informatics and computer science, it is written to be equally accessible and useful to students of philosophy and theoretical linguistics.

Readers are strongly advised to make use of the third edition, which considerably revises the first and second editions of 2008 and 2012, with an additional chapter on relevance-sensitive truth-trees, solutions to all exercises. 

 Preface to the 3rd edition.    Table of contents of 3rd edition.   Errata and additional remarks for third edition - last revised 2023-12-18. Most of the errata arise in solutions to exercises.  As some of them could give rise to misunderstandings, readers are ecouraged to download the list for consultation as needed. Almost all of the corrections have been contributed by Ireneusz Szpilewski, to whom the author extends his deepest thanks. Of course, some items may have been missed and contributions from readers are welcome! 

A Chinese translation of the first edition of 2008 was published by Tsinghua University Press in 2010.  Review of 3rd edition: Zentralblatt. Review of 2nd edition: Zentralblatt. Reviews of first edition: Zentralblatt,  ACM Computing Reviews, Amazon Books customer review.               

Bridges from Classical to Nonmonotonic Logic, College Publications 2005. A Polish translation Od Logiki Klasycznej do Niemonotonicznej was published in 2008 by Wydawnictwo Naukowe Universytetu Mikolaja Kopernika, Torun.              

This is a graduate textbook on nonmonotonic logic. Its purpose  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 easy passage to the nonmonotonic case. 

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. 

Topics in Modern Logic, Methuen 1973, republished by Routledge in 2020. An Italian translation was published by Boringhieri, Turin, 1979, a Japanese translation by Huritsu Bunka-Sha, Tokyo, 1980. 

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

Table of contents, preface, first chapters, index


Aspectos de la Logica Modal, monograph without isbn in the series Notas de Logica Matematica, Universidad Nacional del Sur, Bahia Blanca, Argentina, 1970. Accessible at http://inmabb.conicet.gob.ar/static/publicaciones/nlm/nlm-28.pdf.

Directed to students of mathematics with some background in abstract algebra, wishing to learn about modal logics and their associated algebraic and relational structures.

Review in Zentralblatt

Journal papers, book chapters

Almost all entries in the list below are linked to copies of the items concerned. Some papers have been significantly improved and extended since initial publication. In those instances, which are marked by # alongside the list number, a link is given to the post-publication version alongside (or instead of ) the published one.  

To Appear

(84)  "Attitudes to limiting cases in ancient logic".  To appear in Filosofiska Notiser. Final draft here. Last revised 30 June 2024. 

Draws attention to a tendency of ancient Peripatetic logic echoing into the middle ages and beyond, to disregard limiting cases. The text reviews the variety of ways in which the tendency manifested itself and its apparent absence or at least reduced presence in Stoic logic, notes some exceptions, and speculates on the reasons for its existence.


(83) With Werner Stelzner, "Orlov ninety-six years on: a guide to 'The calculus of compatibility of propositions'".  Australasian J. of Logic 2024, issue 1: open access at ojs.victoria.ac.nz/ajl/article/view/8098/8395.

Orlov's paper “The calculus of the compatibility of propositions”, published in Russian in 1928, is fascinating for anyone interested in the early history of relevance logic, also modal and intuitionistic logic. The guide is published conjointly with an English translation by Werner Stelzner; it is intended to help the reader get to grips with Orlov's paper and its relations with subsequent literature.

(82) "Display conventions for octagons of opposition"Axioms 2024 13, 287 (open access) https://doi.org/10.3390/axioms13050287. Also accessible here with minor post-publication additions (11 June 2024) and the author's preferred formatting (endnotes rather than asides in main text, reference calls by author and date rather than by number).

As usually presented, octagons of opposition are rather complex objects and can be difficult to assimilate at a glance. We show how, under suitable conditions, alternative display conventions can simplify the diagrams, making them easier to grasp without loss of information.  

(81)# "Hamilton's cumulative conception of quantifying particles: an exercise in third-order logic". J. Logic and Computation online 01 December 2023, DOI 10.1093/logcom/exad072 (open access).  A post-publication version with additional material, last revised 20 June 2024, is maintained here. 

Shows how to give formal shape, using the language of third-order logic, to Hamilton's cumular (aka collective) conception of quantitative particles in their application to terms in his 'doubly quantified' categorical propositions. It also reviews a number of variants, some equivalent and others not so, as well as their reductions to second-order logic, and situates historical sources, both before and after Hamilton, with respect to the web of formal constructions.

(80) with Guillermo Badia, "First-order friendliness".  Review of Symbolic Logic published online 07 June 2023 , doi:10.1017/S175502032300014X (open access). 

Studies a first-order counterpartof the notion of logical friendliness, introduced in the propositional context by Makinson in 2007. Although compactness and interpolation fail dramatically, other interesting properties are preserved from the propositional case.

(79)# "Boole's indefinite symbols re-examined".  Australasian Journal of Logic 2022, 19: 165-181 (open access). The link here is to a post-publication text with some additional material in section 4, last revised 24 April 2024. The published text itself may be downloaded with open access from the journal.

We show how one can make good sense of Boole’s notorious "auxiliary symbols" by treating them formally as variables that range over functions from classes to classes rather than just over classes while, at the same time, following Hailperin’s proposal of binding them existentially.

(78)# "The relevance logic program: failed or just stalled?". Logic in Question, ed. J-Y. Beziau, 2023  (Birkhauser) pp. 709-720. The link is to a post-publication version last updated 2022-03-27.

A brief presentation of the essential ideas of "Relevance-sensitive truth-trees" for an audience of philosophers not specializing in logic. It omits many of the details, as well as the proofs, historical background, and pointers to the literature. 

(77)# "Frege's ontological diagram completed".  Logica Universalis (first online, August 2022). The link is to a post-publication version with additions made following discussions with Göran Sundholm and André Fuhrmann.  

In a letter of 1891, Frege drew a diagram to illustrate his logical ontology. We observe that it omits features that play a role in his thought on the matter, propose an extension of the diagram to include them, and compare with a diagram of the ontology of current first-order logic.

(76)# "Crashing with parity". Last updated 2024-05-01. This is a considerably extended version of  "Relevance-sensitive truth-trees", published in Alasdair Urquhart on Non-Classical and Algebraic Logic and Complexity of Proofs, Series Outstanding Contributions to Logic, Springer, 2021. That in turn was a more developed version of "Relevance via decomposition: a project, some results, an open question". Australasian Journal of Logic 14:3 2017, https://ojs.victoria.ac.nz/ajl/article/view/4009/3617. The essential ideas are also  presented for a general audience in the paper "The relevance logic program: failed or just stalled?" (item 78# above), as well as for the classroom with many exercises and their solutions, in chapter 11 of the third edition of the author's Sets Logic and Maths for Computing, Springer, Springer, 2020 (see under "Books" above).

The goal is to articulate a clear rationale for relevance-sensitive propositional logic. The method: truth-trees. Familiar decomposition rules for truth-functional connectives, accompanied by novel ones for the for the arrow, together with a recursive rule, generate a set of ‘acceptable’ formulae that properly contains all theorems of the well-known system R and is closed under substitution, conjunction and detachment. We conjecture that it satisfies the crucial letter-sharing condition. 

(75)# "The phenomenology of second-level inference: perfumes from the inference garden". Bulletin of the Section of Logic 49.4 (December 2020) 327-342 . The link is to a post-publication text with some additional remarks.

Comments on certain features that second-level inference rules commonly used in mathematical proof sometimes have, sometimes lack: suppositions, indirectness, goal-simplification, goal-preservation and premise-preservation. The emphasis is on the roles of these features, which we call ‘perfumes’, in mathematical practice rather than on the space of all formal possibilities, deployment in proof-theory, or conventions for display in systems of natural deduction. The link is to the current version, last updated 2022-01-05 , which improves on the published one.

(74)# "Gödel's Master Argument: what is it and what can it do?" IFCoLog Journal of Logic and its Applications 2015, 2: 1-16. Readers are advised to make use of an extended post-publication text, last updated 2020-10-27. 

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. 

(73) 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. 

(72)# "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. Readers are advised to make use of a post-publication text, last updated 2024-02-12. It extends results, adds comments, corrects minor errors, and is much more reader-friendly. 

Investigates introduction and elimination rules focusing on the general questions of the existence, for a given truth-functional 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 quite complex  in the restricted  contexts of set/formula and set/formula-or-empty sequents, with some questions remaining open.

(71) "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-sensitive 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: it may be seen, at least in the case of the system R, as a conservative extension of classical consequence. 

(70) "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.

(69) "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.

(68) 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.The formal work underlying this paper was taken much further in the author's paper "Intelim rules for classical connectives" (see above).

(67)# "Conditional probability in the light of qualitative belief change". Journal of Philosophical Logic 40: 2011, 121-153. A preliminary version was published in Hykel Hosni and Franco Montagna (eds), Probability, Uncertainty and Rationality. CRM Series Vol 10, Edizioni della Scuola Normale Superiore, Pisa 2010. Readers are advised to make use of a post-publication text, last edited in September 2019, which contains some additions.

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.

(66) "Propositional relevance through letter-sharing" J. Applied Logic 7: 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 belief 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. 

(65) “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 connections between different kinds of nonmonotonic logic and the general idea of varying degrees of belief. Roughly speaking, it summarizes the leading ideas of the 2005 book Bridges from Classical to Nonmonotonic Logic (see above).

(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 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 ofthe authors' JSL publication of 2007, "Parallel interpolation, splitting, and relevance in belief change” (see above), focussing on the application to the logic of belief change and discussing the significance of the formal results.

(62) With James Hawthorne, “The quantitative/qualitative watershed for rules of uncertain inference”. Studia Logica 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.

(61) “Completeness Theorems, Representation Theorems: What’s the Difference?” In Hommage à Wlodek: Philosophical Papers dedicated to Wlodek Rabinowicz, ed. Rønnow-Rasmussen et al. 2007, www.fil.lu.se/hommageawlodek (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.

(60) “Friendliness and sympathy in logic” pp 195-224 of Logica Universalis, 2nd edition, ed. J.-Y. Beziau (Basel: Birkhauser Verlag, 2007). 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 (College Publications 2005), pp 259-292. This is a revised and extended version of the paper appearing in the first edition of the collection (2005, pages 191-205) under the title “Logical friendliness and sympathy”.  

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. 

(59) “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. Roughly speaking, this is a slightly abridged version of the book Bridges from Classical to Nonmonotonic Logic (see above) without the exercises.

(58) “Natural deduction and logically pure derivations”. PhiNews, April 2004. A preliminary version appeared under the title "Logically pure derivations" in the Electronic Festschrift for Professor Norman Foo, November 2003 (http://www.cse.unsw.edu.au/~ksg/Norman).

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? Later comment: The paper introduces the idea of a 'split-level' derivation rule, which is also explained in chapter 11 of the 2020 textbook Sets, Logic and Maths for Computing, 3rd edition.

(57) “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.

(56) “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.

(55) 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.

(54) “Ways of doing logic: what was different about AGM 1985?”. Journal of Logic and Computation 13 (2003) 3-13. This paper was published in the hardcopy issue of the journal but, due to a publisher's computer error, it 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.

(53) 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. A French translation appeared as an internal document of CREA (Centre de Rechereche en Epistemologie Appliquée, Paris, 2001).

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

Outlines the basic ideas of the 2005 book of the same name (see above).

(51) 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.

(50) 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.

(49)  "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) 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. Later comment: The formal development in this paper provides the basis for the more abstract mathematics of input/output logics in the subsequent paper with Leendert van der Torre, "Input-output logics" (see above).

(48)  "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.

(47)  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.

(46)  "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.

(45)   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.

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

Obituary notice with overview of the work of Carlos Alchourrón, focussing on his contributions to the logic of norms and of normative systems.

(43)   "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.

(42)  "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'. Later comment: This is the author's only paper in the area of decision theory.

(41) “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. Later comment: 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.

(40) 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.

(39) 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 and Carlos Alchourron for the logic of theory change.

(38)   "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.

(37) 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.

(36)  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.

(35) 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. 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.

(34) 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.

(33)  "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.

(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. Later comment: Very widely cited in the literature. Provides the germ from which the text General Patterns in Nonmonotonic Reasoning (see above) 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 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 (no copy of this preliminary version currently available to the author; if you have one, kindly send me a copy to put here).

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.

(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 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.

(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 safe contractions and partial meet contractions.

 (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.

(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. Republished as chapter 13 of H. Arló-Costa et al. eds, Readings in Formal Epistemology, 2015, in the series Springer Graduate Texts in Philosophy. A Russian translation appeared in 2013 in E. Lisanyuk ed. Normative Systems and Other Works in Legal Philosophy and Logic of Norms (translation of 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. Later comment: 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. 

Draws attention to 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.

The first of a series of studies, with Carlos Alchourrón and/or Peter Gärdenfors on the formal logic of belief (theory) change. This paper examines the properties of maxichoice contraction and revision operations.

(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 in E. Lisanyuk ed. Normative Systems and Other Works in Legal Philosophy and Logic of Norms (translation of Russian title), 2013, 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. Later comment: 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 and/or Peter Gärdenfors on the formal logic of belief (theory) change. 

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

Draws attention to 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. The last of the author's papers in modal logic, apart from the expository paper "Logique modale: quelques jalons essentiels" of 1990 (see above). 

A technical observation in modal propositional logic: there are infinitely many mutually non-equivalent formulae in a single variable in a certain strong omnitemporal 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 the structural completeness of a consequence operation.

 (15)  With Krister Segerberg, "Post completeness and ultrafilters", Zeitschrift für math. Logik und Grundl. der Math. 20 (1974) 385-388. 

A cardinality result in modal propositional logic.

(14) "Vantagens e limitacoes da abordagem adjukiewicziana da grammatica" (in Portuguese) Discurso (Brazil) 4 (1973) 155-165.  Accessible electronically at  http://www.revistas.usp.br/discurso/article/view/37766.

Reviews the powers and limitations of a simple mathematical model of grammatical structure. Later comment: This is the author's only paper in formal linguistics.

(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 primitive 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. Later comment: Largely ignored at the time of publication, it gradually became a frequently cited item in the following decades as investigations in modal logic took on a more abstract and generalizing perspective. The title is rather misleading, as the word "embedding" suggests a semantic content; the result is purely syntactic although the methods of proof are semantic. A better word would be "subsumption" or "extension" or "upper bound".

(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, generalizing an earlier result for boolean algebras (see below). The result: in any infinite distributive lattice there are at least as many prime filters as elements.

 (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. Later comment: The generalisation obtained is essentially equivalent to the notion of a "first-order" model that was defined independently by S.K. Thomason; however subsequent researchers have found Thomason's formulation more convenient to work with because it does not use the substitution functions that form part of this presentation.

(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, generalised by the author and Luisa Iturrioz in the following year to distributive lattices (see above).

(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 mathematical logic to clarify and to a certain extent defend a controverted and rather obscure notion of traditional syllogistic. See also #Postscript of 2022.

(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. Later comment: 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 a section of the author's 1965 D.Phil. thesis.

(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. Later comment: Based on a section of the author's 1965 D. Phil. thesis. 

(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: there are infinitely many non-equivalent formulae in a single variable in the so-called Diodorean modal logic. Later comment: Based on a section of the author's 1965 D. Phil. thesis. 

(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. Later comment: This was the first timid foray of the author into the area of relevance-sensitive logics; he returned to the subject only in 2014 with his major contribution in 2017, further extended in 2021. The 1965 paper passed almost entirely unnoticed in the literature, and its result is difficult to generalize beyond the first-degree fragment of relevance-sensitive logic.  

(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, therby forcing one into the dilemma of either accepting, in certain contexts, an inconsistent belief as rational or rejecting the closure of the set of rational beliefs under conjunction. Later comment: Little cited at the time of its publication it gradually became, after the turn of the century, a standard reference in literature on formal epistemology along with Kyburg's discussion of the closely related lottery paradox.

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

Shows the failure of Nidditch's attempt to repair A.J. Ayer's proposal to define the notion of verifiability in purely deductive terms, given the failure of Ayer's original definition and a series of earlier attempts to repair it. Later comment: To the author's knowledge, Nidditch's proposal was the last in the series, and Ayer's project appears to have been universally abandoned.

Other texts

Workshop proceedings

With Michael Freund and Daniel Lehmann, "Canonical extensions to the infinite case of finitary nonmonotonic inference relations", in Brewka & Freitag eds, Proceedings of the Workshop on Nonmonotonic Reasoning (Arbeitspapiere der GMD n° 443, 1990, 133-138). Sorry, no electronic or paper version of this paper currently available to the author; if you have access to one, kindly send me a copy to put here.

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.


(2) "Qu'est-ce que la complétude structurale?", abstract of the 1976 paper "A characterization of structural completeness of a structural consequence operation" (see 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). Its ideas were developed in the paper "A warning about the choice of primitive operators in modal logic" (see above). Sorry, no electronic or paper version of this paper currently available to the author; if you have access to one, kindly send me a copy to put here.

Selected book reviews

Fermé & Hansson  Belief Change: Introduction and Overview Zentralblatt für Math 2021

Parent & van der Torre  Introduction to Deontic Logic and Normative Systems Zentralblatt für Math 2020

Schlechta  Formal Methods for Nonmonotonic and Related Logics, vols I, II  Zentralblatt für Math 2019  

McNamara  Deontic Logic Math. Reviews 2017

 Smith  The Laws of Truth Studia Logica 2015

Gabbay & Schlechta  Conditionals and Modularity in General Logics Philosophy in Review 2012

Antonelli  Grounded Consequence, Math Rev or Zentralblatt für Math 2007

Christensen Putting Logic in its Place: Formal Constraints on Rational Belief Zentralblatt für Math, 2006

Leitgeb  Inference on the Low Level, Zentralblatt für Math  2004

Bochman  A Logical Theory of Nonmonotonic Inference and Belief Change, Zentralblatt für Math 2001 

Feldman  Doing the Best We Can: An Essay in Informal Deontic Logic J. Symbolic Logic 1987  

Rescher & Brandom  The Logic of Inconsistency J. Symbolic Logic 1982

Lewis Counterfactuals Studia Logica 1974, reprint also reviewed in Zentralblatt 2001

Reviews of journal papers

No attempt here at selection from the reviews of journal papers etc in Math. Reviews (more than four hundred, according to a list prepared by Math. Rev), Zentralblatt (around the same number), J. Symbolic Logic and other reviewing journals.  Those published by Zentralblatt now  freely accessible from its webpage.

Ms left unpublished

"The scarcity of stable sets"  Written early in 2015, following an invitation to participate in a volume on Leitgeb's stability theory of beliefs. However the volume did not appear and the paper remains unpublished. The points made in this paper were subsequently taken up, with acknowledgements, by Hans Rott in a more detailed critical examination of Leitgeb's proposals, and so the author has not felt the need to proceed beyond the ms stage.

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.  

Volume introductions

"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.

Discusses some general issues prompted by a reading of the contributions to the volume mentioned.

Foreword to the 2008 reprint by College Publications of Peter Gärdenfors's book Knowledge in Flux.

Recalls the genesis of AGM 1985 and its evolution up to 1988.


"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. The version on this website contains a postscript that takes the story from 2014 to 2024. See also an Interview with Hykel Hosni in The Reasoner 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. Well, that was not quite the last lap; more has happened since then ...

A collection of papers discussing some of my work

David Makinson on Classical Methods for Non-Classical Problems, ed. Sven Ove Hansson,  Series Outstanding Contributions to Logic, Springer 2014. 

This is a collection of papers by well-known authors discussing aspects of my work in various areas of logic. It also contains two of my papers (see above).