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Workshop on Theories of Paradox in the Middle Ages

21st October 2020 - 23rd October 2020

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We can now make available the recordings from this workshop. Full details of the event are beneath.

N.B. The Workshop will now take place wholly online.

Giorgione, Three Philosophers

Paradoxes seized the attention of logicians in the middle ages, and were used both as tests for the viability of theories of logic, language, epistemology, and possibly every philosophical issue, and also in the specific genre of insolubles as needing a theoretical solution, usually involving issues about signification, truth, knowledge and modality. Numerous theories were developed, not only in the Latin West, but also in the Islamic world and in the Byzantine tradition. Some of these theories are well known, others barely investigated, if at all. This workshop is an opportunity to discuss and contrast a range of these theories and consider their advantages and drawbacks, and their relation to more recent theories of paradox and antinomy. It will also be an occasion to hear and discuss what has been achieved locally in the Leverhulme-funded project ‘Theories of Paradox in Fourteenth-Century Logic: Edition and Translation of Key Texts‘.


The workshop will be held using Zoom software. It will run from 13.45 – 18.45 BST (British Summer Time = GMT+1) on 21 and 22 October 2020, and from 13.45 – 16.15 BST on 23 October.


Registration is free but required for attendance. To register, please email Dr Barbara Bartocci (bb66@st-andrews.ac.uk) with your Full Name, email address, affiliation and whether you are a student.



Wednesday 21 October Thursday 22 October Friday 23 October
13.45 (Virtual) Doors Open (Virtual) Doors Open (Virtual) Doors Open
13.50 Opening Remarks
14.00-15.00 Stephen Read (Arché Research Centre): ‘Theories of insolubles from Thomas Bradwardine to Paul of Venice’ Alessandro Conti (Università L’Aquila): ‘Wyclif and Paul of Venice on the Liar’s Paradox: a Comparison’ Barbara Bartocci (Arché Research Centre): ‘John Dumbleton on signification and semantic paradoxes’
15.15-16.15 Harald Berger (Karl-Franzens-Universität Graz): ‘An unknown version of Albert of Saxony’s De insolubilibus (= Logica, tr.VI, pt.1)’ Miroslav Hanke (Academy of Sciences of the Czech Republic): Tractatus insolubilium and fifteenth-century Cologne scholasticism’ Stephen Read: ‘Walter Segrave on insolubles: a restrictivist response to Bradwardine’
16.30-17.30 Manuel Dahlquist (Universidad Nacional Del Litoral): ‘Nulla propositio est negativa (and the paradox of validity)’ Graziana Ciola (Radboud University Nijmegen): ‘Marsilius of Inghen on insolubles’
17.45-18.45 David Sanson (Illinois State University) and Ahmed Alwishah (Pitzer College): ‘Al-Dawānī on truth, grounding, and the Liar’ Mohammad Saleh Zarepour (University of Birmingham): ‘Abharī’s Solution to the Liar paradox: a logical analysis’


    • Stephen Read: ‘Theories of insolubles from Thomas Bradwardine to Paul of Venice’

We can divide medieval discussions of the insolubles—logical paradoxes such as the Liar—into two main periods, before and after Bradwardine. Thomas Bradwardine wrote his treatise on Insolubles in Oxford in the early 1320s and it marks a sea change in the solutions which were mainly favoured. Up until then two types of solution were the focus of attention, restrictio and cassatio. Bradwardine attacked several versions of restrictivism mercilessly in chs.3-4 of his Insolubles. While this was the pre-eminent proposal for solving the insolubles before his attack, it had few proponents thereafter. It was Bradwardine’s own proposal which initiated a whole new era in discussions of the insolubles. His aim was to develop a solution to the insolubles which placed no restriction on self-reference or the theory of truth.
However, although most if not all subsequent writers on insolubles owe a debt to Bradwardine, few followed him completely. Two alternative proposals presented in Oxford in the 1330s, both responding to Bradwardine’s idea but in different ways, dominated subsequent discussion of the insolubles. They were due to William Heytesbury and Roger Swyneshed. Heytesbury’s solution agrees with Bradwardine’s that there is a hidden signification, but feels under no obligation to specify what it is. A later author, Robert Eland, presented Bradwardine’s and Heytesbury’s solutions and invited the reader to choose between them. What seems to have happened, however, is that the popular solution was to combine them. It is the solution commonly found in the teaching manuals at Oxford now known as the Logica Oxoniensis. Swyneshed’s solution was at root very different. His aim was to provide a solution without the postulation of hidden meanings, but taking the expressions at face value. He proposed that truth should require that a proposition not entail its own falsehood. 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.
There were remarkably similar developments at Paris, remarkable not least for the fact that their differences suggest that it is hard to detect any direct influence. Buridan’s solution to the insolubles is nowadays perhaps the most famous of all medieval solutions, again claiming that propositions have an additional signification or implication. Gregory of Rimini claimed that (spoken and written) insolubles correspond to a conjunction of two mental propositions, the first of which captures the primary or customary signification of the insoluble, and the second of which says that the first conjunct is false. Gregory’s solution was taken up and adapted by Peter of Ailly. Where Gregory claimed that insolubles were false, corresponding to false mental conjunctions, Peter argued that the phenomena are better explained by realising that insolubles are equivocal, both true and false, corresponding to two different mental propositions.

    • Harald Berger: ‘An unknown version of Albert of Saxony’s De insolubilibus (= Logica, tr.VI, pt.1)’

Albert of Saxony’s well-known usual version of the partial treatise on Insolubles starts with four descriptions, six suppositions, and seven conclusions. On this basis, he subsequently discusses and solves a lot of exempla, the first of which is, as he himself puts it, “illud insolubile commune ‘Ego dico falsum’”.
There is, however, a single witness of Albert’s Logica which hands down a completely different version of the Insolubilia (and the Obligationes as well), namely Milano, Veneranda Biblioteca Ambrosiana, Cod. B 36 sup., fol. 118va-131rb. This version first presents five previous opinions regarding insolubles, an introduction which is quite common (as, e.g., with Bradwardine, Heytesbury, Brinkley, Wyclif, Marsilius of Inghen, Paul of Venice, Strode), but is completely lacking in Albert’s usual version. Subsequently, different sorts of insolubles are treated in four chapters. Finally, there is an additional chapter on apparent insolubles (“quae apparent insolubilia et non sunt”, which wording has a parallel in the definition of paralogisms as argumentations “quae apparent esse consequentiae et non sunt”, tr. V, ch. 1). This last chapter discusses four examples, the previous chapters include 29 examples.
In my talk, I shall discuss the pros and cons of the authenticity of this unique treatise and give an overview of its contents. Up to now, I am more inclined to regard this treatise as a genuine work of Albert’s than to the contrary.

    • Manuel Dahlquist: ‘Nulla propositio est negativa (and the paradox of validity)’

Jean Buridan gives us two examples of inferences that do not meet the requirements of the intuitive notion of consequence, even though they should be considered intuitively valid. The first of the two examples appears in the Tractatus de Consequentiis; (1) No proposition is negative; therefore, no ass is running (TC 2015, 67) and the second in the Sophismata: (2) Every proposition is affirmative; therefore, no proposition is negative (SD 2001, 952). Both (1) and (2) have problems to be accepted as good inferences for Buridan. Both of them have a common sentence: “No proposition is negative”. Nulla Propositio est negativa (hereinafter NPN) was studied by Jean Buridan always linked to the notion of logical consequence and has some logical characteristics that make it special: (a) NPN is self-referential; (b). NPN has a syntactic character; (c). NPN does not behave like a classical paradoxical sentence regarding negation; (d). NPN is a case of a proposition that is always false.
Both inferences have been studied in the literature on medieval logic; (1) where NPN appears as a premise, in works such as Dutilh Novaes (2005), Klima (2004), Read (2012); (2) where NPN appears as a conclusion, in works such as (Yrjönsuuri, 2008) (Hughes, 1982) (Scott, 1966) and Prior (1969), who develops Buridan’s solution to insolubles. My work is about the latter.
When Buridan analyzes the insoluble Omnis propositio est affirmativa; ergo, nulla propositio est negativa, like all medieval logicians, he argues for and against it, but makes an unusual move: he proposes his own solution within the critiques of the validity of the inference (Hughes 1982): after doing so, he definitively departs from these reasons, to found the validity of the insoluble on a modal distinction. The difference between my approach and Buridan´s (and Prior´s) is that I am going to give all the weight they deserve to the first three arguments in favor of the validity of the insoluble that Buridan presents to us at the beginning of the chapter.
As we said before, Sophismata’s first insoluble aims to problematize the notion of logical consequence. I am going to argue that there are three notions of logical consequence under consideration in the First Insoluble: Two are well described by Hughes (1982) as Theory A and Theory B (Theory B is the one used by Prior in his paper); the third – which I will call Theory C – is proposed by Buridan himself in his first three arguments in favor of the validity of the insoluble and reflect the validity conditions proposed by the medievals logicians of the School of Paris for the Formal consequence. I am going to present these three notions of consequence; then, I will show: a) that in Sophismata Buridan accepts only two of the three notions of consequences that are being considered; b) that accepting these two notions together leads to paradox.
Buridan, J. 2001. Summulae de Dialectica, New Haven & London: Yale University Press.
Buridan, J. 2015. Treatise on consequences, S. Read (tr.), New York: Fordham University Press.
Dutilh Novaes, C. 2005. ‘Buridan’s Consequentia: Consequence and Inference Within a Token-Based Semantics’, History and Philosophy of Logic, 26, 4: 277-297.
Hughes, G. E.. 1982, Chapter Eight of Buridan´s Sophismata Translated, with an Introduction and a philosophical Commentary, Cambridge University Press.
Klima, G. 2004. ‘Consequences of a closed, token-based semantics: the case of John Buridan’, History and Philosophy of Logic, 25(2), 95-110.
Prior, A. 1969 “The Possibly-True and the Possible” in Mind, New Series, Vol. 78, No. 312 (Oct., 1969), pp. 481-492.
Read, S. 2012. “The medieval theory of consequence”, Synthese, Vol. 187: 899-912.
Scott, T. 1966. Sophisms on Meaning and Truth, (tr. Sophismata, by Jean Buridan), New York: Meredith Publishing Company.
Yrjönsuuri, M. 2008. “Treatments of the Paradoxes of Self-reference” in Handbook of the History of Logic Volume 2 Mediaeval and Renaissance Logic, Ed. D.Gabbay and J. Woods, ELSIEVER.

    • David Sanson and Ahmed Alwishah: ‘Al-Dawānī on truth, grounding, and the Liar’

In his treatise, The Final Word on Solving the [Liar Paradox], Jalāl al-Dīn al-Dawānī (d. 908/1502) offers a novel solution to the Liar. At the core of his solution is that idea that, for a sentence to be truth apt, it must be an imitation (ḥikāya) of a fact. When I say, “This sentence is composed”, my sentence imitates itself, and also corresponds to itself, and so is non-paradoxically true. But when I say, “This sentence is false”, my sentence attempts to imitate its own imitating, which it cannot do, so it fails to imitate anything, and so fails to be truth apt. He uses this to explain why only some self-referential sentences are pathological, and he uses it to solve range of Liar and Liar-like Paradoxes, including the Truthteller (“This sentence is true”), the Simple Liar (“This sentence is false”), Contingent Liars (“Everything I say this hour is false”), and Liar Cycles (“Everything I say on Tuesday is true”, “Everything I say on Wednesday is true”, “Everything I said on Monday is false”).
We reconstruct al-Dawānī’s position and compare it to contemporary grounding solutions to the Liar, as found in Herzberger (1970) and Kripke (1975). This comparison helps clarify al-Dawānī’s reasons for thinking that attempted self-imitation fails, and helps clarify what is distinctive about his use of imitation as a semantic relation. We then look at al-Dawānī’s argument that truth aptness requires imitation, and the role he thinks imitation plays in explaining the distinction between declarative (khabar) and constructative (≈performative) (inshāʾ) sentences. Tracing a bit of the history of that debate suggests that al-Dawānī is synthesizing two debates: a debate about the Liar, self-reference, and the correspondence theory of truth that has its roots in al-Ṭūsī, and a debate about declaratives and constructatives that has its roots in the grammatical tradition, going back (at least) to al-Astarābādī (d. 686/8, 1287/9).

    • Alessandro Conti: ‘Wyclif and Paul of Venice on the Liar’s Paradox: a Comparison’

In this paper I will consider two different solutions to the paradox of the liar, the one proposed by Wyclif in chapter eight of the third treatise of his Continuatio logicae (on conditional propositions and insoubles) and the one developed by Paul of Venice in the fifteenth treatise of the second part of his Logica Magna (which deals with insolubles). The two different solutions seem to me more interesting than others, since they are emblematic of two different ways of approaching the problem of paradoxes: semantic the former (via the philosophy of languge) and formal the latter (via the logic). Wyclif places his solution within a larger, semantic context, namely his theory of meaning and truth of the propositions, while Paul, on the basis of Roger Swyneshed’s own theory, tries to come up with a technical solution – a sort of dialetheism ante litteram, since he seems to admit that there are cases in which from true it can follow the false and vice versa and that, for this reason, the classical liar sentences are both true and false. Paradoxically, this was the same general thesis that Wyclif himself had mantained (p. 205): “Dico quod omnia vocata communiter insolubilia sunt tam vera quam falsa. Claudit enim contradiccionem quod aliqua sit proposicio insolubilis.” Yet, the explanation and justification that Wyclif works out for supporting this thesis are totally different from those of Swyneshed and Paul, as well as the general context of the solution. As a consequence, Wyclif’s thesis and Paul of Venice’s thesis on the liar’s paradox are opposite. In fact, Wyclif distinguishes three different levels of truth and consequently three corresponding levels of falsehood, and no (so called) insoluble is both true and false according to truth and falsehood of the same level.

    • Miroslav Hanke: Tractatus insolubilium and fifteenth-century Cologne scholasticism’

Scholastic analysis of self-referential paradoxes seems to have peaked in the fourteenth century, as far as establishing basic paradigms is concerned, and the subsequent tradition, including the so-called “Second Scholasticism” either focused on further technical development, or limited the debate on insolubles, sometimes to a bare minimum. The period between 1400s and 1550s predominately instantiates the first tendency, making the texts from this period interesting both historically and systematically. The paper will analyse texts produced at Thomist and Albertinist “bursae” in fifteenth century Cologne and printed together with commentaries on Peter of Spain’s Summulae. These consist of an anonymous treatise on insolubles (the core of this textual tradition) supplemented by commentaries. The paper aims to reconstruct the strategy of solving insolubles endorsed in these texts, to describe their connections to the earlier scholastic tradition and discuss the degree of innovation.

    • Graziana Ciola: ‘Marsilius of Inghen on Insolubilia

This presentation is centred on Marsilius of Inghen’s still unedited treatise on insolubilia and on the theory presented there.
First, I introduce Marsilius’ Tractatus de insolubilibus, outlining its structure and its manuscript tradition. In doing so, I suggest a tentative date of composition for the text and situate it in the context of the production of Marsilius’ other treatises on parva logicalia. Secondly, I offer an overview of the most relevant features of Marsilius’ approach to insolubilia. And, finally, I situate it within the framework of the 14th century on the subject as well as in the context of Marsilius’ own most peculiar philosophical positions advanced throughout his other treatises on parva logicalia.

    • Mohammad Saleh Zarepour: ‘Abharī’s Solution to the Liar paradox: a logical analysis’

The medieval Islamic solutions to the liar paradox can be categorized into three different families. According to the solutions of the first family, the liar sentences are not well-formed truth-apt sentences. The solutions of the second family are based on a violation of the classical principles of logic (e.g. the principle of non-contradiction). Finally, the solutions of the third family render the liar sentences as simply false without any contradiction. In the Islamic tradition, almost all the well-known solutions of the third family are inspired by the solution proposed by Aṯīr al-Dīn al-Abharī (d. 1265). Providing a logical analysis of his discussion of the liar paradox, I show that his solution is based on a conception of truth according to which every sentence signifies, usually among other things, its own truth. This makes Abharī’s solution of the same spirit as certain solutions that were later developed in the Latin tradition, in particular by John Buridan (d. 1358) and Albert of Saxony (d. 1390).

    • Barbara Bartocci: ‘John Dumbleton on signification and semantic paradoxes’

John Dumbleton was active in Oxford from the end of the 1330s, where he was a member of the group of the Merton Calculators and perhaps composed the nine volumes of his Summa Logicae et Philosophiae Naturalis, which was left unfinished probably due to Dumbleton’s decease in the Black Death. In the Summa Logicae, which is the first volume of his extensive work, Dumbleton presents an original agent-centred theory of signification, according to which terms signify only when they actualize a thing’s intention (intentio) through an act of comprehension and “propositions and all proper signification subsist only in the soul”. The semantic theory sustains and underpins Dumbleton’s solution to the logical paradoxes which combines the nullifiers’ (cassantes) and restrictivists’ (restringentes) approaches.

In the first part of my talk, I will introduce Dumbleton’s semantics of terms, particularly demonstratives, and propositions. And in the second part, I will present Dumbleton’s view on the nature of paradoxical sentences and his solution to paradoxes of the Liar family (‘This proposition is false’; ‘Every truth is one of these’, supposing it exists together with two true propositions to which it refers) and to Curry’s Paradox (‘This inference is valid, therefore a man is an ass’).

    • Stephen Read: ‘Walter Segrave on insolubles: a restrictivist response to Bradwardine’

Walter de Segrave (or de Sexgrave) was at Merton College, Oxford from 1321 until at least 1338, and had obtained his Magister Artium by 1336. From 1340-42 Segrave was Chancellor to Richard de Bury, Bishop of Durham, who famously gathered around him some of the very best minds in the kingdom, including Burley, Bradwardine and Kilvington. Segrave subsequently became Dean of Chichester, but was dead by 1349. 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 the 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, defending 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 (simpliciter et secundum quid), 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.


Through the generosity of the British Society for the History of Philosophy, and in order to guarantee a demographically diverse participation we are able to offer financial assistance with childcare during the Workshop. Please email Dr Barbara Bartocci (bb66@st-andrews.ac.uk) with details of the childcare needed.

We are grateful to the University of St Andrews, via the Arché Research Centre, to the British Society for the History of Philosophy, and to the Scots Philosophical Association for financial support.

Scots Philosophical Association


21st October 2020
23rd October 2020


A virtual workshop by Zoom
The University
St Andrews, KY16 9L United Kingdom


Stephen Read
Barbara Bartocci