- (with Christopher Schabel) "Wulderstorff’s Quodlibet", in Arts Quodlibeta at the University of Vienna: The 1429 Quodlibet under Urban of Melk, edited by Christopher Schabel and Ota Pavlicek, in the Brepols series Intellectual Practice and Thought at Late Medieval and Early Modern Universities.
Abstract: In Vienna in 1429 Urban of Melk was in charge of giving questions to the other arts masters for a public performance over several days. In a volume in preparation, these Quodlibetal questions are each edited by a particular scholar together with the editors, Ota PavlĂÄek and Christopher Schabel. It is accompanied in each case by a commentary by that scholar on the master’s response to this challenge. The particular Quodlibet discussed in this paper focuses on the challenge given to Thomas de Wulderstorff: “Whether all the moods of the first figure hold thanks to their form” (Utrum omnes modi prime figure teneant gratia forme).
- "Marsilius of Inghen on Insolubles and the English Tradition"
Abstract: In the course of presenting his own solution to the insolubles (logical paradoxes such as the Liar), Marsilius of Inghen criticises four earlier theories, which appear to be those of Albert of Saxony, (the early) Buridan, Roger Swyneshed and a modification of William Heytesbury’s solution which we find in many textbooks and anonymous treatises known as presentations of the Logica Oxoniensis. Marsilius’s solution bears interesting resemblances to all four, but has its own distinctive features. The core idea of his solution is that all propositions have a two-fold signification, a material signification and a formal one. The material signification, also called the primary or direct signification, is what most would call the proposition’s usual signification; e.g., the material signification of ‘This proposition is false’ is that that proposition is false. Its formal, aka indirect or reflexive, signification is, in the case of affirmative propositions, that the subject and predicate supposit for the same thing, and in the case of negative propositions, that they do not. This reflexive signification derives from the meaning of the (affirmative resp. negative) copula. Thus the reflexive signification of ‘This proposition is false’ is that ‘this proposition’ and ‘false’ supposit for the same thing, that is, that it is false that that proposition is false. Presenting Marsilius’s formal signification in such cases as stating of that proposition’s being false, for example (which is the material signification of ‘This proposition is false’), that it is false (that is, falls under the supposition of ‘false’) suggested to Paul Spade that Marsilius’s solution was a development of Gregory of Rimini’s account. I will argue that any resemblance here is, in the absence of any external evidence, superficial and coincidental, and that Marsilius’s view is much closer to the Oxford solutions and Albert’sâAlbert and Marsilius being, after all, members of the English Nation at Paris. Marsilius’s arguments in favour of his theory, and his application of the solution to a range of insolubles, are well worth looking at in detail, which I do, though not at the length which Marsilius devotes to it.
- "Roger Swyneshed's Insolubles: an English translation" (also available at )
Abstract: An English translation of Roger Swyneshed’s Insolubilia, edited by Paul Vincent Spade in âRoger Swyneshedâs Insolubilia: edition and commentsâ, Archives dâhistoire doctrinale et littĂ©raire de moyen Ăąge 46 (1979), 177-220 (reprinted in idem, Lies, Language and Logic in the Late Middle Ages, London 1988), and by Lorenzo Pozzi in Il Mentitore e il Medioevo (Parma 1987), 180-99. For biographical details on Swyneshed, see, e.g., J. Weisheipl, 'Roger Swyneshed OSB, Logician, Natural Philosopher, and Theologian,' in Studies Presented to Daniel Callus (Oxford 1964), 231-252.
- ""
Abstract: Although the theory of the assertoric syllogism was Aristotle's great invention, one which dominated logical theory for the succeeding two millenia, accounts of the syllogism evolved and changed over that time. Indeed, in the twentieth century, doctrines were attributed to Aristotle which lost sight of what Aristotle intended. One of these mistaken doctrines was the very form of the syllogism: that a syllogism consists of three propositions containing three terms arranged in four figures. Yet another was that a syllogism is a conditional proposition deduced from a set of axioms. There is even unclarity about what the basis of syllogistic validity consists in. Returning to Aristotle's text, and reading it in the light of commentary from late antiquity and the middle ages, we find a coherent and precise theory which shows all these claims to be based on a misunderstanding and misreading.
At Press
- "Jennifer Ashworth: An Appreciation In Memoriam", Oxford Studies in Medieval Philosophy 12 (forthcoming).
Abstract: Earline Jennifer (âJennyâ) Ashworth, who died on 23 July 2024 at the age of 84, was a leading figure in the study of logic in the late Middle Ages and Renaissance at least from the publication of her first book, Language and Logic in the Post-Medieval Period in 1974.
Published Books and Papers
- , critical edition and English translation by E. Jennifer Ashworth with the assistance of Stephen Read (Auctores Britannici Medii Aevi 46), The British Academy, 28 April 2026.
Abstract: Ralph Strode came to be one of the most successful Oxford logicians of the mid-fourteenth century, especially in Italy, where manuscripts of his treatises on consequences and on obligations were widely circulated in the fifteenth century. Strode was the author of six works on logic. These works consist of two introductory tracts, Tractatus Primus and Tractatus Secundus: De Principiis Logicalibus, along with tracts on supposition, consequences, obligations and insolubles. Obligations (obligationes) constitute a unique genre of logical treatise, in which a Respondent is challenged by the Opponent to grant, deny or express uncertainty about a succession of statements regarding an agreed case or scenario, usually counterfactual. Over the past fifty years or so, three questions have dominated research on Obligationes disputations: first, what are they? Secondly, what was their function and purpose? Thirdly, when and why did they begin and develop? Although much studied over the past fifty years, Strode’s treatise on obligations is here edited for the first time from the numerous surviving manuscripts and early printed texts, together with an English translation and substantial introduction to the genre and Strode’s treatise in particular.
- "", Logica Universalis 19 (2025), 1â22 (Open Access). doi.org/10.1007/s11787-025-00369-3
Abstract: Forty years ago, Niels Green-Pedersen listed five different accounts of valid consequence, variously promoted by logicians in the early fourteenth century and discussed by Niels Drukken of Denmark in his commentary on Aristotle's Prior Analytics, written in Paris in the late 1330s. Two of these arguably fail to give defining conditions: truth preservation was shown by Buridan and others to be neither necessary nor sufficient; incompatibility of the opposite of the conclusion with the premises is merely circular if incompatibility is analysed in terms of consequence. Buridan was perhaps the first to define consequence in terms of preservation of what we might dub verification, that is, signifying as things are. John Mair pinpointed a sophism which threatens to undermine this proposal. Bradwardine turned it around: he suggested that a necessary condition on consequence was that the premises signify everything the conclusion signifies. Dumbleton gave counterexamples to Bradwardine's postulates in which the conclusion arguably signifies more than, or even completely differently from the premises. Yet a long-standing tradition held that some species of validity depend on the conclusion being in some way contained in the premises. We explore the connection between signification and consequence and its role in solving the insolubles.
- , critical edition and English translation by Barbara Bartocci and Stephen Read, , published (Open Access) on 17 October 2024.
Abstract: Paradoxes, such as the Liar (‘What I am saying is false’), fascinated medieval thinkers. What I said can’t be true, for if it were, it would be false. So it must be falseâbut then it would be true after all. Attempts at a solution to this contradiction led such thinkers to develop their theories of meaning, reference and truth. A popular response, until it was attacked at length by Thomas Bradwardine in the early 1320s, was to dismiss such self-reference as impossible: no term (here, ‘false’) could refer to (or in medieval terms, “supposit for”) a whole, e.g., a proposition, of which it is part. In light of Bradwardine’s criticisms, Walter Segrave, writing around 1330, defended so-called restrictivism (restrictio) by claiming that such paradoxes exhibited a fallacy of accident. The classic example of this fallacy, the first of Aristotle’s fallacies independent of language, is the Hidden Man puzzle: you know Coriscus, Coriscus is the one approaching, but you don’t know the one approaching since, e.g., he is wearing a mask. But Aristotle’s account is unclear and Segrave, building on ideas of Giles of Rome and Walter Burley, shows how the fallacy turns on an equivocation over the supposition of the middle term or one of the extremes in a syllogism. Thereby, Segrave is able to counter Bradwardine’s arguments one by one and defend the restrictivist solution. In this volume, Segrave’s text is edited from the three extant manuscripts, is translated into English, and is preceded by a substantial Introduction.
- "",
in Interpreting Buridan: Critical Essays, edited by Henrik Lagerlund and Spencer Johnston (Cambridge University Press, 2024), 37â50.
Abstract: Bertrand Russell famously remarked that “a logical theory may be tested by its capacity for dealing with puzzles, and it is wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible.” John Buridan, like many medieval logicians, adopted the same methodology: he developed and refined his theories of truth and other semantic notions against an array of paradoxes. Indeed, in an Appendix to his Summulae de Dialectica Buridan gave a succinct but comprehensive introduction to his logical philosophy in the form of responses to a collection of sophisms (his Sophismata). Among these sophisms are the logical paradoxes, the so-called “insolubles”, such as the Liar paradox (is one who says he is lying, lying or telling the truth?). But before he comes to these especially difficult examples, Buridan tackles puzzles which lead him to reject standard accounts of meaning, truth, logical consequence, tense, hyperintensionality and other semantic notions. In the final chapter of the Sophismata, he deals with various problems involving self-referential propositions, not all of them insolubles. When he turns at last to the insolubles, he elaborates a revision he had already suggested in his Questions on the Sophistical Refutations to the general solution he had developed for them over a long career in the Arts Faculty at the University of Paris dealing with philosophical puzzles.
- (edited, with Barbara Bartocci), (College Publications, 2023).
- "Theories of Paradox from Thomas Bradwardine to Paul of Venice",
in Theories of Paradox in the Middle Ages (College Publications, 2023), 11-42.
Abstract: We can divide medieval discussions of the insolubles—logical paradoxes such as the Liar—into two main periods, before and after Bradwardine, who wrote his treatise on Insolubles in Oxford in the early 1320s. Bradwardine’s aim was to develop a solution to the insolubles which, unlike the then dominant theories, restrictio and cassatio, placed no restriction on self-reference or the theory of truth. He claimed to be able to prove that insolubles signify not only that they are false but also that they are true, and so are false. Few subsequent writers on insolubles followed him completely. Nonetheless, Heytesbury’s solution agrees with Bradwardine’s that there is an additional signification, though he was agnostic what that additional signification was; and a popular solution commonly found in the teaching manuals at Oxford modified Heytesbury’s solution to incorporate aspects of Bradwardine’s. There were remarkably similar developments at Paris, where Buridan’s solution also claimed that propositions have an additional signification or implication of their own truth. Gregory of Rimini claimed that (spoken and written) insolubles correspond to a conjunction of two mental propositions, one of which says that the other is false. Gregory’s solution was taken up and adapted by Peter of Ailly, arguing that the phenomena are better explained by realising that insolubles are equivocal, both true and false, corresponding to two different mental propositions. In contrast, Roger Swyneshed’s aim was to provide a solution without the postulation of hidden meanings, but taking the expressions at face value. At the end of the century, Paul of Venice subscribes to the modified Heytesbury solution in his Logica Parva, but in his Logica Magna he defends a version of Swyneshed’s solution.
- "Walter Segrave's ‘Insolubles’: A Restrictivist Response to Bradwardine",
in Theories of Paradox in the Middle Ages (College Publications, 2023), 43-66.
Abstract: Walter de Segrave was at Merton College, Oxford from 1321 until at least 1338.
Segrave’s ‘Insolubles’ is his only known work, which appears to have been composed at Oxford in the late 1320s or early 1330s, consistent with the fact that it is clearly a response to Bradwardine’s own ‘Insolubles’, composed when Bradwardine was regent master at Balliol College, that is, from 1321-23, before he moved to Merton in 1323. The dominant theory at the time Bradwardine was writing was restrictivism, the claim that a part cannot supposit for the whole of which it is part (and consequently, for its contradictory or anything convertible with it), at least in the presence of a privative term, in particular, privative alethic and epistemic terms such as ‘false’ and ‘unknown’. Accordingly, Bradwardine spends two and a half chapters attacking restrictivist theories, in particular, that of Walter Burley. Segrave’s treatise is an extensive and detailed response to Bradwardine,. He defends restrictivism by presenting a well-thought out reason for the restriction of supposition required to avoid contradiction. Where Burley and Bradwardine both attributed the fallacy in insolubles to what Aristotle described as the fallacy of the conditional and the unconditional (secundum quid et simpliciter), Segrave attributed it to the fallacy of accident, turning on a variation in the supposition of the middle term and the extremes in what might otherwise appear to be a sound syllogism.
- Paul of Venice, Logica Magna: Treatise on Insolubles, critical edition and English translation, with Systematic Introduction and Commentary, by Barbara Bartocci and Stephen Read, , published on 25 October 2022.
Abstract: The Logica Magna was the subject of a British Academy project, begun over forty years ago and involving a number of scholars, to edit and translate into English the whole work in twenty volumes. Eventually, seven volumes appeared, but the final treatise, on Insolubles, was not among them, and the project was abandoned in the 1990s. Whether that was the right outcome for the other projected volumes is moot, but the treatise on insolubles is an important record for possible approaches to the paradoxes (still a live debate today, as noted above), and it is important that it receive a modern edition. Although Paul’s compendium was printed in 1499, most modern readers need an apparatus explaining the discussion, and many need a translation. Moreover, although the 1499 text is fairly reliable, it needs some correction against the sole surviving manuscript.
- (with Barbara Bartocci) "", in Logic and modalities in the tradition of the Oxford Calculators, special issue of Noctua 9(3): 48-88 (2022).
Abstract: The manuscript 397 Scaff. XVIII of the Biblioteca Antoniana in Padua ends with an anonymous Epitome (Compendium) of John Dumbleton’s solution to the semantic paradoxes (insolubilia). Bottin edited this Epitome of Dumbleton’s Insolubles more than forty years ago, when knowledge of the insolubilia-literature was still quite limited. Since then, much research has been done and many primary sources have been studied and edited, including Dumbleton’s Summa Logice, which is the source for the Epitome. In preparing the edition of the Insolubles from Dumbleton’s Summa Logice recently, we realized that a new edition of the short Epitome was needed, in which on the one hand, some corruptions in Bottin’s version could be corrected, and which, on the other hand, could serve to elucidate the Epitome’s content and provide some context by outlining Dumbleton’s own solution. Accordingly, in the first part of this paper, we briefly present Dumbleton’s cassationist solution to the semantic paradoxes. In the second part, we analyse the solution to various types of insolubles proposed by the anonymous author of the Epitome. In the third part, we provide a critical edition of the Latin text and an English translation.
- "The Calculators on the Insolubles: Bradwardine, Kilvington, Heytesbury, Swyneshed and Dumbleton", for Quantifying Aristotle: the Impact, Spread and Decline of the 'Calculatores' Tradition, ed. Edith D. Sylla & Daniel A. Di Liscia (Brill 2022), 126-52.
Abstract: The most exciting and innovative period in the discussion of the insolubles (i.e., logical paradoxes) before the twentieth century occurred in the second quarter of the fourteenth in Oxford, and at its heart were many of the Calculators. It was prompted by Thomas Bradwardine's iconoclastic ideas about the insolubles in the early 1320s. Framed largely within the context of the theory of (logical) obligations, it was continued by Richard Kilvington, Roger Swyneshed, William Heytesbury and John Dumbleton, each responding in different ways to Bradwardine's analysis, particularly his idea that propositions had additional hidden and implicit meanings. Kilvington identified an equivocation in what was said; Swyneshed preferred to modify the account of truth rather than signification; Heytesbury exploited the respondent's role in obligational dialogues to avoid Bradwardine's tendentious closure postulate on signification; and Dumbleton relied on other constraints on signification to give new life to two long-standing accounts of insolubles that Bradwardine had summarily dismissed. The present paper focusses on the central thesis of each thinker's response to the insolubles and their interaction.
- , History and Philosophy of Logic 43 (2022), 332-46 (Open Access).
Abstract: In his Quadratura, Paul of Venice considers a sophism involving time and tense which appears to show that there is a valid inference which is also invalid. Consider this inference concerning some proposition A: A will signify only that everything true will be false, so A will be false. Call this inference B. Then B is valid because the opposite of its conclusion is incompatible with its premise. But he proceeds to argue that it is possible that B's premise could be true and its conclusion false, so B is not only valid but also invalid. Thus A and B are the basis of an insoluble—that is, a Liar-like paradox. Like the sequence of statements in Yablo’s paradox, B looks ahead to a moment when A will be false, yet that moment may never come. In his Logica Parva, Paul follows the solution to insolubles found in the collection of elementary treatises known as the Logica Oxoniensis, which posits an implicit assertion of its own truth in insolubles like B. However, in the treatise on insolubles in his Logica Magna, Paul develops and endorses a different solution that takes insolubles at face value, meaning no more than is explicit in what they say. On this account, insolubles imply their own falsity, and that is why, in so falsifying themselves, they are false. We consider how both types of solution apply to A and B: on both, B is valid. But on one, B has true premises and false conclusion, and contradictories can be false together; on the other (following the Logica Oxoniensis), the counterexample is rejected.
- "Modality in Medieval Philosophy", in The Routledge Handbook of Modality, ed. Otavio Bueno and Scott Shalkowski, 344-54.
- "John Buridan's Theory of Consequence and his Octagons of Opposition", Around and Beyond the Square of Opposition, edited by
J.-Y.Beziau and D.Jacquette,
BirkhÀuser, Basel, 2012, pp. 93-110.(DOI: 10.1007/978-3-0348-0379-3)
Abstract:
One of the manuscripts of Buridan’s Summulae contains three figures, each in the form of an octagon. At each node of each octagon there are nine propositions. Buridan uses the figures to illustrate his doctrine of the syllogism, revising Aristotle's theory of the modal syllogism and adding theories of syllogisms with propositions containing oblique terms (such as ‘man’s donkey’) and with ‘propositions of non-normal construction’ (where the predicate precedes the copula). O-propositions of non-normal construction (i.e., ‘Some S (some) P is not’) allow Buridan to extend and systematize the theory of the assertoric (i.e., non-modal) syllogism. Buridan points to a revealing analogy between the three octagons. To understand their importance we need to rehearse the medieval theories of signification, supposition, truth and consequence.
- "", (Colombia) (2011) 12(18), 229-35.
Abstract: In his article "Verdades antiguas y modernas" (in the same issue, pp. 207-27), David Miller criticised Thomas Bradwardine’s theory of truth and signification and my defence of Bradwardine’s application of it to the semantic paradoxes. Much of Miller’s criticism is sympathetic and helpful in gaining a better understanding of the relationship between Bradwardine’s proposed solution to the paradoxes and Alfred Tarski’s. But some of Miller’s criticisms betray a misunderstanding of crucial aspects of Bradwardine’s account of truth and signification.
- "", Synthese 187 (2012), 899-912 (in a special issue on "Logical Consequence and Inference", edited by Sten Lindström).
Abstract: The recovery of Aristotle’s logic during the twelfth century was a great stimulus to medieval thinkers.
Among their own theories developed to explain Aristotle’s theories of valid and invalid reasoning was a theory of consequence, of what arguments were valid, and why.
By the fourteenth century, two main lines of thought had developed, one at Oxford, the other at Paris. Both schools distinguished formal from material consequence,
but in very different ways. In Buridan and his followers in Paris, formal consequence was that preserved under uniform substitution.
In Oxford, in contrast, formal consequence included analytic consequences such as ‘If it’s a man,
then it’s an animal’. Aristotle’s notion of syllogistic consequence was subsumed under the treatment of formal consequence.
Buridan developed a general theory embracing the assertoric syllogism, the modal syllogism and syllogisms with oblique terms.
The result was a thoroughly systematic and extensive
treatment of logical theory and logical consequence which repays investigation.
- (edited, with Amirouche Moktefi) , 15, April 2011.
- (with Wilfrid Hodges) "Western Logic", Journal of the Indian Council for Philosophical Research 27(1), special issue on "", ed. A. Gupta and J. van Benthem, January-March 2010, 13-45.
Abstract: The editors invited us to write a short paper that draws together the main themes of logic in the Western tradition from the Classical Greeks to the modern period. To make it short we had to make it personal. We set out the themes that seemed to us either the deepest, or the most likely to be helpful for an Indian reader.
-
. (Dallas Medieval Texts and Translations 10.) Leuven: Peeters 2010, 236 pp.[Reviewed in 42(4), Winter 2011, pp. 1229-31]
The fourteenth-century thinker Thomas Bradwardine is well known in both the history of science and the history of theology. The first of the Merton Calculators (mathematical physicists) and passionate defender of the Augustinian doctrine of salvation through grace alone, he was briefly archbishop of Canterbury before succumbing to the Black Death in 1349. This new edition of his Insolubilia, made from all thirteen known manuscripts, shows that he was also a logician of the first rank. The edition is accompanied by a full English translation. In the treatise, Bradwardine considers and rejects the theories of his contemporaries about the logical puzzles known as "insolubles," and sets out his own solution at length and in detail. In a substantial introduction, Stephen Read describes Bradwardine's analysis, compares it with other more recent theories, and places it in its historical context. The text is accompanied by three appendices, the first of which is an extra chapter found in two manuscripts (and partly in a third) that appears to contain further thoughts by Bradwardine himself. The second contains an extract from Ralph Strode's Insolubilia, composed in the 1360s, repeating and enlarging on Bradwardine's text; and the third consists of an anonymous text that applies Bradwardine's solution to a succession of different insolubles. |
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Here is a list of corrections and improvements.
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- "The validity paradox", , ed. Michal Peliš, College Publications 2010.
- "", Journal of Philosophical Logic 39, 2010, 557-76. (DOI: 10.1007/s10992-010-9133-7)
Abstract: Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In particular, logical expressions are autonomous if given meaning by their introduction-rules, rules specifying the grounds for assertion of propositions containing them. If the elimination-rules do no more, and no less, than is justified by the introduction-rules, the rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between rules leads to a general form of elimination-rule, and when the rules have this form, they may be said to exhibit “general-elimination” harmony. Ge-harmony ensures that the meaning of a logical expression is clearly visible in its I-rule, and that the I- and E-rules are coherent, in encapsulating the same meaning. However, it does not ensure that the resulting logical system is normalizable, nor that it satisfies the conservative extension property, nor that it is consistent. Thus harmony should not be identified with any of these notions.
- "", Kriterion 51, 2010, 47-67. DOI: 10.1590/S0100-512X2010000100003. [Reprinted in The Realism-Antirealism Debate in the Age of Alternative Logics, ed. Shahid Rahman, Giuseppe Primiero and Mathieu Marion, Springer 2012, pp. 251-67. DOI: 10.1007/978-94-007-1923-1_13.]
Abstract: What makes necessary truths true? I argue that all truth supervenes on how things are, and that necessary truths are no exception. What makes them true are proofs. But if so, the notion of proof needs to be generalized to include verification-transcendent proofs, proofs whose correctness exceeds our ability to verify it. It is incumbent on me, therefore, to show that arguments, such as Dummett's, that verification-truth is not compatible with the theory of meaning, are mistaken. The answer is that what we can conceive and construct far outstrips our actual abilities. I conclude by proposing a proof-theoretic account of modality, rejecting a claim of Armstrong's that modality can reside in non-modal truthmakers.
- "", History and Philosophy of Logic 31(2), 2010, 161-76. Winner of the for History and Philosophy of Logic in 2010.
Abstract: Hartry Field’s revised logic for the theory of truth in his new book, Saving Truth from Paradox, seeking to preserve Tarski’s T-scheme, does not admit a full theory of negation. In response, CrispinWright proposed that the negation of a proposition is the proposition saying that some proposition inconsistent with the first is true. For this to work, we have to show that this proposition is entailed by any proposition incompatible with the first, that is, that it is the weakest proposition incompatible with the proposition whose negation it should be. To show that his proposal gave a full intuitionist theory of negation, Wright appealed to two principles, about incompatibility and entailment, and using them Field formulated a paradox of validity (or more precisely, of inconsistency).
The medieval mathematician, theologian and logician, Thomas Bradwardine, writing in the fourteenth century, proposed a solution to the paradoxes of truth which does not require any revision of logic. The key principle behind Bradwardine’s solution is a pluralist doctrine of meaning, or signification, that propositions can mean more than they explicitly say. In particular, he proposed that signification is closed under entailment. In light of this, Bradwardine revised the truth-rules, in particular, refining the T-scheme, so that a proposition is true only if everything that it signifies obtains. Thereby, he was able to show that any proposition which signifies that it itself is false, also signifies that it is true, and consequently is false and not true. I show that Bradwardine’s solution is also able to deal with Field’s paradox and others of a similar nature. Hence Field’s logical revisions are unnecessary to save truth from paradox.
- "Inferences", in The Cambridge History of Medieval Philosophy, ed. R. Pasnau, Cambridge UP 2010, 173-85.
- (with Paul Vincent Spade), "",
The Stanford Encyclopedia of Philosophy (Winter 2009 Edition), Edward N. Zalta (ed.),
URL = <>.
Abstract: The medieval name for paradoxes like the famous Liar Paradox (âThis proposition is falseâ) was âinsolublesâ or insolubilia.[1] From the late-twelfth century to the end of the Middle Ages and beyond, such paradoxes were discussed at length by an enormous number of authors. Yet, unlike twentieth century interest in the paradoxes, medieval interest seems not to have been prompted by any sense of theoretical âcrisis.â The history of the medieval discussions can be divided into three main periods: (a) an early stage, from the late-twelfth century to the 1320s; (b) a period of especially intense and original work, during roughly the second quarter of the fourteenth century; (c) a late period, from about 1350 on.
- "", Philosophical Studies 145, 2009, 363-75.
Abstract: In recent years, speech-act theory has mooted the possibility that one utterance can signify a number of different things. This pluralist conception of signification lies at the heart of Thomas Bradwardine's solution to the insolubles, logical puzzles such as the semantic paradoxes, presented in Oxford in the early 1320s. His leading assumption was that signification is closed under consequence, that is, that a proposition signifies everything which follows from what it signifies. Then any proposition signifying its own falsity, he showed, also signifies its own truth and so, since it signifies things which cannot both obtain, it is simply false. Bradwardine himself, and his contemporaries, did not elaborate this pluralist theory, or say much in its defence. It can be shown to accord closely, however, with the prevailing conception of logical consequence in England in the fourteenth century. Recent pluralist theories of signification, such as Grice's, also endorse Bradwardine's closure postulate as a plausible constraint on signification, and so his analysis of the semantic paradoxes is seen to be both well-grounded and plausible.
- (with Andrew Aberdein) "The Philosophy of Alternative Logics", in The Development of Modern Logic, ed. L. Haaparanta, Oxford UP 2009, pp. 613-723.
- "Harmony and Modality", in Dialogues, Logics and Other Strange Things: Essays in Honour of Shahid Rahman, ed. C. Dégremont, L. Kieff and H. Rückert, College Publications 2008, pp. 285-303.
Abstract. It is argued that the meaning of the modal connectives must be given inferentially, by the rules for the assertion of formulae containing them, and not semantically by reference to possible worlds. Further, harmony confers transparency on the inferentialist account of meaning, when the introduction-rule specifies both necessary and sufficient conditions for assertion, and the elimination-rule does no more than exhibit the consequences of the meaning so conferred. Hence, harmony is not to be identified with normalization, since the standard modal natural deduction rules, though normalizable, are not in this sense harmonious. Harmonious rules for modality have lately been formulated, using labelled deductive systems.
- "Further Thoughts on the Truth-scheme and the Liar", in Unity, Truth and the Liar: The Modern Relevance of Medieval Solutions to the Liar Paradox, ed. S. Rahman, T. Tulenheimo and E. Genot, Springer Verlag 2008, pp. 205-225. (Logic, Epistemology, and the Unity of Science , Vol. 8.) (The article is my response to the other contributions to the volume.)
- "The Truth-Schema and the Liar", in Unity, Truth and the Liar: The Modern Relevance of Medieval Solutions to the Liar Paradox, ed. S. Rahman, T. Tulenheimo and E. Genot, Springer Verlag 2008, pp. 3-17. (Logic, Epistemology, and the Unity of Science , Vol. 8.) (The article is the subject of discussion by the other contributions to the volume.) [Reprinted in (Colombia) 11, 2010, pp. 119-37.]
- (with Catarina Dutilh Novaes) ‘’, Vivarium 46(2), 2008, 175-91.
- "Bradwardine's Revenge", in Revenge of the Liar, ed. J.C. Beall, Oxford UP 2007, 250-61.
- "Farewell to Opacity", in Current Topics in Logic and Analytic Philosophy, ed. C. Martinez, J.L. Falguera and J.M. Sagüillo, Universidade de Santiago de Compostela 2007, 233-46.
- "William Ockham's The Sum of Logic" (review article), 26(2), 2007, 271-7.
- "Symmetry and paradox", 27, 2006, 307-18.
- "Monism: the one true logic", in A Logical Approach to Philosophy: Essays in Honour of Graham Solomon, ed. D. DeVidi and T. Kenyon, Springer 2006, 193-209.
- (with Crispin Wright) ,
Aarhus Universitetsforlag 2006, 93 pp.
- "The Unity of the Fact”, Philosophy 80 (2005), pp.
317-42.
- "In Defence of the Dog: Reply to Restall", in Logic,
Epistemology and the Unity of Science, ed. S. Rahman and J. Symons,
Kluwer 2004, pp. 175-80.
- (with Graham Priest) "Intentionality—Meinongianism and the Medievals", , September 2004, pp. 416-35.
- "Identity and Harmony", 64 (2004),
pp. 113-119. Revised version:
"Identity and Harmony revisited"
Abstract: The standard rules for identity do not seem to lie in harmony, where the elimination-rule is justified by the meaning given to a logical operator by its introduction-rule. However, harmonious rules can be given for identity, showing that identity is a logical notion even on the inferentialist conception. These new rules are shown to be sound and complete. The revised paper omits §3 of the original published version, which responded to an objection by the referee, shown to be mistaken by Michael Kremer in 'Read on identity and harmony: a friendly correction and simplification', Analysis, 67, April 2007, pp. 157-59.
- "Logical consequence as truth-preservation", Logique et Analyse 183-4 (2003), pp. 479-93.
- "Freeing Assumptions from the Liar Paradox", 63 (2003),
pp. 162-6.
- "The Liar Paradox from John Buridan back to Thomas Bradwardine",
40 (2002), pp. 189-218.
- (jointly with M. Astroh and I. Grattan-Guinness) "A survey of the life
of Hugh MacColl (1837-1909)",
22 (2001), 81-98.
- "Medieval Theories of Properties of Terms", in the .
- "Truthmakers, Entailment and Necessity", in Essays on Non-classical
Logic , ed. H. Wansing, World Scientific 2001, pp. 91-110.
- (with E.P. Bos) , Peeters (Louvain), 2001, XII + 148 pp.
- "Self-Reference and Validity Revisited", an expanded version of
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Yrjonsuuri, Kluwer 2001, pp. 183-96.
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Logic, 29 (April 2000), pp. 123-54.
- "Truthmakers and the Disjunction Thesis", 109 (January 2000),
pp. 67-79.
- "How is material supposition possible?",
8 (1999), pp. 1-20.
- "Sheffer's stroke: a study in proof-theoretic harmony", Danish
Yearbook of Philosophy, 34 (1999), pp. 7-23.
Abstract:
In order to explicate Gentzen's famous remark that the
introduction-rules for logical constants give their meaning, the
elimination-rules being simply consequences of the meaning so given,
we develop natural deduction rules for Sheffer's stroke, alternative
denial. The first system turns out to lack Double Negation.
Strengthening the introduction-rules by allowing the introduction of
Sheffer's stroke into a disjunctive context produces a complete system
of classical logic, one which preserves the harmony between the rules
which Gentzen wanted: all indirect proof reduces to direct proof.
- "Hugh MacColl and the algebra of strict implication", 3 (December 1998), 59-83.
- "Quotation and Reach's Puzzle", 19 (1997), pp. 9-20.
- "Completeness and and Categoricity", 18 (1997), pp. 79-93.
- "The Slingshot Argument", Logique et Analyse, 143-4 (1993), pp. 195-218.
- "Conditionals and the Ramsey test", Proceeding of the Aristotelian
Society, Supplementary Volume 69 (1995), pp. 47-65.
- , Oxford
U.P.: Oxford Paperbacks University Series (OPUS), 1995, viii
+ 262 pp.
- translated as Rowohlts Enzyklopadie, 1997,
312 pp.
- also translated into ,
1998
- and into Swedish 1999 (Att Tänka
På Logik, Thales, ISBN: 91-87172-94-1)
- and into Hungarian, 2001 (Bevezetés
a logika filozofiájába, Kossuth Kiado, ISBN:
963-09-4272-0)
- "Formal and Material Consequence", Journal of Philosophical
Logic, 23 (1994), pp. 247-65.
- (edited
with an introduction) Kluwer Academic Publishers: Nijhoff International
Philosophy Series vol. 48, 1993, xvii + 421 pp.
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in Argumentationstheorie: Scholastische Forschungen zu den logischen und
semantischen Regeln korrekten Folgerns, ed. K. Jacobi (Leiden 1993), pp.233-59.
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Peirce", Analysis, 52 (1992), pp. 5-12.
- "Thomas of Cleves and Collective Supposition", Vivarium,
29 (1991), pp. 50-84.
- "Descensus copulatim: Albert of Saxony vs. Thomas Maulfelt", in
Itineraires d'Albert de Saxe: Paris-Vienne au XIVe, ed. J. Biard (Paris,
1991), pp. 71-85.
- "Homophone Semantik fuer die relevante Aussagenlogik", Conceptus,
23 (1989), pp. 77-89.
- Relevant Logic,
Blackwells 1988, viii + 199 pp. (now out of print, but available to download by clicking on the title).
