University of St Andrews

School of Philosophical and Anthropological Studies

Arché Research Centre

Theories of Paradox in Fourteenth-Century Logic: Edition and Translation of Key Texts

St Andrews Institute of Mediaeval Studies

Leverhulme Trust

Medieval Logic Reading Group

Principal Investigator: Stephen Read

Research Fellow: Barbara Bartocci

Summary

The project, funded by a Leverhulme Research Project grant to Professor Read, began on 1 August 2017 and continues until 31 July 2020. It consists of preparing an edition of the Latin text, together with an English translation and commentary, of the late 14th-century treatise on Insolubles (logical paradoxes) by Paul of Venice from his Logica Magna; together with editions and English translations of two treatises on insolubles from earlier in the century, by Walter Segrave and John Dumbleton.

The main and most direct aim is scholarly and historical, to provide scholars and students with access to important and interesting texts from the 14th century on the logical paradoxes. The logical paradoxes have played a significant role in the development of philosophical ideas, not just in logic but also in philosophy of language, epistemology, metaphysics and even ethics and political philosophy, throughout the 20th and 21st centuries. They played a no less significant role in later medieval philosophy and were the subject of much debate and the spur to original ideas, arguably reaching their zenith in the 14th century. Much has been learned about the medieval debate in the past fifty years, in the writings of John Buridan, Thomas Bradwardine and others. But other interesting treatises remain unedited, many only surviving in contemporary manuscripts. Among these is the treatise on insolubles (logical paradoxes) by Paul of Venice, summarizing and developing theories and solutions from his predecessors in the 14th century, constituting the final treatise of his Logica Magna. Seven of the treatises from this huge work were edited and translated into English between 1978 and 1991. But the treatise on insolubles was not among them. The current project aims to edit and translate this treatise, which describes fifteen other theories which it rejects, then develops its own at length, together with a commentary; and to edit and translate two further treatises, those of Walter Segrave and John Dumbleton, writing in Oxford in the second quarter of the century, which Paul mentions and which remain unedited, containing rich ideas about alternative solutions, restrictio and cassatio respectively. Publication of these texts will allow a better overview of the development of solutions to the paradoxes through the 14th century, as well as giving further insight into the nature of the paradoxes and their possible solution.

In sum, the following publications are planned:

The Logical Paradoxes

Logical paradoxes dominated 14th-century logic just as they did 20th-century logic. The semantic paradoxes, such as the Liar Paradox (see below), have been known for millennia. But it was the discovery of the set-theoretic paradoxes, at the heart of the foundational project in the philosophy of mathematics which had resulted from the discovery of the problems concerning infinity in mathematics itself during the 19th century, which sparked the advances in logic and philosophy of language in the early 20th century. Although general opinion favours distinct solutions, by avoidance of absolute infinity in set theory, and by rejection of bivalence in the theory of truth, few think these solutions are fully satisfactory and debate continues in search for a coherent and complete account in each case. (See, e.g., [9] and [10]).

The Liar Paradox itself arguably goes back at least to Eubulides (fl. 4th century BCE) and perhaps to Epimenides (fl. 7th or 6th century BCE). Eubulides is reported to have puzzled over the self-reflexive claim ‘I am lying’ (the simplest version of the Liar Paradox): if he was lying when he said he was lying, he was telling the truth and not lying; while if he was telling the truth when he said he was lying, then he really was lying and not telling the truth. Was he both lying and telling the truth?—surely impossible; or was he neither lying nor telling the truth?—is that plausible? Epimenides the Cretan is reported by St Paul (in the epistle to Titus) and others as commenting sourly on his compatriots: ‘All Cretans are liars’. If he was lying about all Cretans, including himself, then he was not lying, so not all Cretans were liars and so he was not lying; but if he too was lying, then not all Cretans were liars and so some other Cretan must have been telling the truth. Simply by claiming that all Cretans were liars, Epimenides seems to have shown, as if by magic, that some other Cretan must have told the truth.

The rediscovery of such paradoxes in the 12th century CE seems to have started a lively investigation into logic and the philosophy of language, initially concentrating on theories of fallacy [8], extending Aristotle’s analysis of fallacy in his Sophistical Refutations, newly available in translation in the Latin West from the 1120s [3]. This investigation gathered pace over the next hundred years or so, with new ideas on the theory of logical consequence, the development of technical notions in the philosophy of language to describe the semantic properties of expressions, and the application of these ideas to the analysis of so-called “insolubles”, semantic paradoxes such as the Liar and epistemic paradoxes such as the Knower (‘You do not know this proposition’).

Dominant theories of insolubles in the 13th century were mostly variants on so-called “restrictivism” (restrictio), the claim that no expression can refer to a whole of which it is itself a part ([35] §2.4). So, e.g., when Eubulides said he was lying, he could not (as at first appears, and as leads to paradox) be referring to his own claim to be lying. Similarly, when Epimenides said all Cretans were liars, he could not include his own remark in that claim, for then his statement would be referring to a whole (i.e., all Cretan claims) of which it was a part.

That leading view seems to have been largely overturned by a sustained attack on it by Thomas Bradwardine in his treatise on Insolubles composed at Oxford in the early 1320s [5]. His new approach, involving two revisions to the traditional account of signification (meaning) and truth, dominated the succeeding century, though few adopted Bradwardine’s theory in its entirety. These revisions were, first, that an utterance, such as the Liar, may mean or signify more than at first appears; secondly, that truth requires that everything an utterance signifies must obtain. Applied to the Liar Paradox, ‘I am lying’, he claimed that it also signifies of itself that it is true (as well as signifying more obviously that it is false); and so it is (simply) false, since not everything it signifies can obtain—nothing can be both true and false.

Bradwardine’s proof that every utterance which signifies its own falsehood also signifies its own truth depended on a further postulate, that signification is closed under consequence: that is, that an utterance signifies everything that follows from what it signifies. Most subsequent theories about the insolubles balked at adopting this postulate. The two most important of these, those of Roger Swyneshed [33] and William Heytesbury [11], both writing in Oxford in the 1330s, agreed with Bradwardine that expressions could refer to a whole of which they were part, and rejected restrictivism. They also revised the account of truth, but not as Bradwardine had done. Swyneshed’s proposal was the simplest: an utterance is true, he said, if things are as it signifies and it does not falsify itself; otherwise, it is false. So the Liar sentence is false because it clearly falsifies itself, since one can infer from it that it is false—indeed, that is essentially what it says. Heytesbury’s account was more subtle, and framed within a peculiarly medieval logic practice, namely, the art of (logical) obligations [12]. In an obligational disputation, there are two parties, an Opponent and a Respondent. Briefly, the Opponent describes a background scenario, puts forward a (usually false, but possible) proposition, and if they are accepted by the Respondent (as they should be, if they describe a real possibility), the Respondent must then defend the given proposition by granting, denying or expressing doubt about, in accordance with set rules, a succession of further propositions the Opponent fires at him, while remaining consistent with that initial proposition. Heytesbury’s proposal for the insolubles was that they must signify more than at first appears (as Bradwardine had claimed), indeed, something that fails to obtain, but that (within the terms of the obligational disputation), the Respondent is under no obligation to specify what that extra component of its signification is.

Similar ideas were developed at Paris by John Buridan [31], though to what extent they were influenced by the debate at Oxford, if at all, is unclear. Oxford and Paris were the two most important centres of new work in logic throughout the 14th century, and a succession of writers in both places developed Bradwardine’s, Swyneshed’s and Heytesbury’s ideas on the one hand, and Buridan’s on the other. Two leading thinkers in Paris were Albert of Saxony [1,2] and Peter of Ailly [27]. Their theories and others were brought together and discussed in the final treatise of a logical compendium attributed to Paul of Venice and composed after his time in Oxford in the 1390s, his Logica Magna (see [6]).

Although Bradwardine’s own treatise on insolubles has received a modern edition and English translation (by the leader of the current project), an edition of Swyneshed’s Latin text has been published, Heytesbury’s has been published in both Latin and English ([11], [29], [30]; but see [34]), Albert’s has appeared in a modern edition in Latin, English and German ([1],[2]) and Ailly's at least in English [27], many of the original texts of those theories Paul discusses have not appeared in a modern edition in either Latin or English, or indeed been printed at all.

Paul of Venice's Logica Magna

In this treatise, running to over 20,000 words (in a work that totals more than 500,000 words), Paul presents sixteen different theories, rejecting most of them, and arguing in favour of a version of Swyneshed’s view. 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 ([18]-[24]), 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.

Paul of Venice wrote at least four treatises, of varying lengths, on the insolubles: a short treatise (about 6500 words) in his Logica Parva ([16],[17]), a longer treatise (a little over 20,000 words, the main object of the current project) in his Logica Magna [13], a brief discussion in sophism 50 of his Sophismata Aurea [15], and further discussion in his Quadratura [14]. It is by no means certain that these four works have a single author. In particular, Alan Perreiah (who edited and translated the Logica Parva) has repeatedly argued that the Logica Magna is not by Paul, on account of inconsistencies between this treatise and the others ([16] pp.327-43, [17] p.xviii, [25] §1, [26] pp.73-127). However, his arguments have not been generally persuasive, and for the present the assumption is that Paul was the author of the Logica Magna [4]. In any event, the work is an interesting one as providing a compendium of views and arguments covering much of 14th-century logic, regardless of who in fact wrote it.

Here is an analysis of the whole of the Logica Magna, showing the place of the treatise on Insolubles in it (where E is the 1499 incunable, M the manuscript, and 'Fasc' refers to the divisions proposed in the incomplete British Academy edition). Note also the different position of the treatise on Syllogisms in M and in E: in M it is placed straight after the discussion of subject-predicate (or categorical) propositions; in E it comes much later, after the discussion of molecular propositions and the account of truth, falsehood, signification and modality, even though Paul concentrates only on assertoric, that is, non-modal, categorical syllogisms:

Folios in E

Folios in M

Sections

Chapters

Fasc.

Treatises and chapters in E

Part I

2ra

3ra

1: On Terms

i: Categorematic and Syncategorematic Terms

I:1

1: de terminis

3vb

9vb

ii: Naturally Significant and Arbitrarily Significant Terms

c.ii: secunda divisio terminorum

8ra

10vb

iii: Common and Discrete Terms

c.iii: tertia divisio terminorum

12va

16rb

iv: Immediate and Mediate Terms

c.iv: quarta divisio terminorum

16ra

20rb

2: On Supposition

i: Material Supposition

I:2

2: de suppositionibus terminorum

18ra

23ra

ii: Simple Supposition

20ra

25vb

iii: Personal Supposition

22ra

28va

iv: Supposition in respect of ampliative words

26vb

34va

v: Supposition of Relatives

31va

40va

3: On Terms that Render Supposition Confused

i: On the Term ‘differt’

I:3

3: de terminis confundentibus

34ra

43vb

ii: Exclusive Expressions

4: de dictionibus exclusivis: “in hoc capitulo”

35vb

46ra

5: de regulis exclusivarum

38ra

48va

iii: Exceptive Expressions

6: de dictionibus exceptivis

39ra

50ra

7: de regulis exceptivarum

41ra

52ra

iv: Reduplicatives

I:4

8: de reduplicativis

42ra

53rb

v: On the ‘just as’ construction

9: de dictione sicut

43vb

55vb

vi: On Comparatives

10: de comparativis

47va

60va

vii: On Superlatives

11: de superlativis

49rb

62vb

viii: On ‘Maximum’ and ‘Minimum’

12: de maximo et minimo

52rb

68ra

13: de obiectionibus et solutionibus argumentorum

56ra

71rb

4: On Terms that can be taken Categorematically or Syncategorematically

i: On ‘Whole’ taken Categorematically

I:5

14: de toto cathegorematice tento

57vb

73ra

ii: On ‘Always’ and ‘Eternal’

15: de semper et aeternum: “Secundus terminus qui”

58vb

74va

iii: On ‘Infinite’

16: de isto termino infinitum

61ra

77va

iv: On ‘Immediate’ and ‘Mediate’

17: de isto termino immediate

63rb

80rb

v: On ‘Begins’ and ‘Ceases’

18: de incipit et desinit

70vb

89vb

5: On Intensionality

i: On Hyper-intensional Terms

I:6

de terminis officiabilibus

71ra

90va

ii: On Modal Propositions

19: de propositione exponibili

73ra

93ra

iii: On Hyper-intensional Propositions

20: de propositione officiabili

76rb

97va

iv: On Compounded and Divided Senses

21: de sensu composito et diviso

82ra

104vb

6: On Knowledge, Doubt, Necessity and Contingency

i: On Knowledge and Doubt

I:7

22: de scire et dubitare

91ra

115rb

ii: On Future Contingents

I:8

23: de necessitate et contingentia futurorum

Part II

101ra

128rb

1: On Subject-Predicate Propositions

i: On the Definition of ‘Proposition’

II:1

1: de propositione

101vb

129va

ii: On Subject-Predicate Propositions

2: de propositione cathegorica

109ra

139rb

iii: On the Quantity of Propositions

3: de propositione in genere

111ra

141vb

4: de quantitate propositionum

113rb

144rb

iv: On the Square of Opposition

II:2

5: de figuris propositionum (capitulum de figura)

114rb

145rb

v: On the Equivalence of Propositions

6: capitulum de equipollentiis

115rb

146ra

vi: More on the Square of Opposition

7: capitulum de natura situatorum in figura (de lege positorum in figura)

122vb

155va

vii: On the Conversion of Propositions

8: de conversione propositionum

172ra

158rb

2: On Syllogisms

i: On the First Figure

II:7

13: de syllogismis capitulum

174rb

161ra

ii: On the Second Figure

174vb

161vb

iii: On the Third Figure

176rb

163rb

iv: On Irregular Syllogisms

124va

164rb

3: On Molecular Propositions

i: On ‘When’-Propositions

II:3

9: de hypotheticis propositionibus

125vb

165vb

ii: On ‘Where’-Propositions

126ra

166ra

iii: On ‘Why’-Propositions

127ra

167ra

iv: On ‘Although’-Propositions

127vb

168ra

v: On Properly Molecular Propositions

129va

170rb

vi: On Conjunctive Propositions

131rb

172va

vii: On Disjunctive Propositions

134rb

175va

viii: On Conditional Propositions

II:4

139rb

180va

ix: On Entailment Propositions

147rb

189rb

x: Rules of Inference

II:5

158rb

201va

4. On Truth, Signification and Modality

i: On Truth and Falsehood

II:6

10: de veritate et falsitate propositionum

162ra

205va

ii: On the Significate of the Proposition

11: de significato propositionis capitulum

167vb

211vb

iii: On the Modality of Propositions

II:7

12: de necessitate contingentia possibilitate et impossibilitate propositionum

177ra

216va

5. On Obligations

i: On Obligations in General

II:8

14: de obligationibus

181ra

221va

ii: On Positio

191va

234vb

iii: On Depositio

192rb

236ra

6. On Insolubles

i: On Previous Opinions

II:9

15: de insolubilibus

194rb

238va

ii: The Author’s Opinion

195vb

240rb

iii: Objections and Replies

197rb

241rb

iv: On the Famous Insoluble ‘Socrates says a falsehood’

197vb

243ra

v: On Covert Insolubles

198rb

243va

vi: On Quantified Insolubles

198vb

244rb

vii: On Non-Quantified Insolubles

199ra

244va

viii: On Merely Apparent Insolubles

In the treatise on insolubles in the Logica Magna, after presenting and discussing fifteen alternative solutions, Paul presents his own, a development of Roger Swyneshed’s account. Paul adds to Swyneshed's view that a proposition is true if things are as it signifies, unless it falsifies itself, his theory of the exact (or core) signification of a proposition (its significatum adaequatum): a proposition is true if its exact significate is true, unless that is incompatible with its truth.

Segrave and Dumbleton

Walter Segrave was writing in Oxford in the 1320s or ‘30s, defending a restrictivist theory explicitly in response to Bradwardine’s criticisms. His treatise is preserved in three mss, one incomplete ([32] pp.113-15). He claims to apply the fallacy of accident to diagnose the error in insolubles; e.g., the term ‘false’ in the Liar sentence cannot stand for itself, since it signifies itself to be true and not false (echoing Bradwardine’s core idea). This is a version of restrictivism which Bradwardine does not include in his discussion.

John Dumbleton was, like Bradwardine, one of the famous Oxford Calculators, whose main interest was in mathematical physics. His discussion of insolubles occurs as the second chapter of Part I (Summa Logicae) of his magnum opus, Summa Logicae et Philosophiae Naturalis, a huge work running to some 400,000 words ([32] pp.63-65). The whole work was transcribed by James Weisheipl from a single ms (Vat.lat. 6750) in the early 1950s when preparing his Oxford D.Phil. thesis on Dumbleton’s natural philosophy, but that transcription was never published, and exists, it seems, in a single copy in the Library of the Pontifical Institute of Medieval Studies in Toronto. Useful as it is, it is in a very preliminary state, with many insecure, and arguably mistaken, readings, and needs comparison with the texts of the other extant mss of Dumbleton’s Summa which also contain this early section on insolubles. (Two mss are incomplete in lacking Part I; all are incomplete in lacking Part X, which Dumbleton refers to but arguably never completed before he succumbed to the Black Death in 1348 or 1349.) The chapter on insolubles is preceded by an extended discussion of signification, which is important for understanding Dumbleton’s solution to the insolubles and so needs to be included in the edition. It is followed by two chapters on knowledge and doubt, the four chapters making up Part I, on logic. Thus it makes sense to edit Part I as a whole. In his theory of insolubles, Dumbleton revives a solution much criticised by Bradwardine and others, cassationism, otherwise advocated only in a single treatise from the early 13th century [7], which claims that insolubles are not in fact propositions at all.

In addition, five of the mss contain a separate treatise on Insolubles, arguably by Dumbleton himself, at various places in the text of Part I of the Summa Logicae.

Paul Spade argues ([11] p.73) that Cajetan’s identification (in his 15th-century commentary on Heytesbury’s Insolubles) of the second view criticised by Heytesbury, and consequently the eighth discussed by Paul of Venice, as Dumbleton’s cannot be right, since Dumbleton’s treatise itself argues against Heytesbury’s view. But this is a weak argument, for Heytesbury, Swyneshed, Dumbleton and others were all working together in Oxford in the 1330s and would have been aware of each others’ ideas and so could easily end up criticising each other.

The Source Material

The known manuscripts and early printed texts to be used are as follows:

References

  1.    Albert of Saxony. ‘Insolubles’. In The Cambridge Translations of Medieval Philosophical Texts, vol. I: Logic and the Philosophy of Language, trans. N. Kretzmann and E. Stump. Cambridge: Cambridge University Press, 1988, 338-68.
  2.    Albert of Saxony. Logik: Lateinisch-Deutsch (Perutilis Logica), ed. and tr. H. Berger. Hamburg: Meiner 2010.
  3.    Aristotle. De Sophisticis Elenchis, Translatio Boethii, Fragmenta Translationis Iacobi et Recensio Guillelmi de Moerbeke (Aristoteles Latinus VI 1-3), ed. B. Dod. Leiden: Brill 1975. 
  4.    Ashworth, E. Jennifer. ‘Paul of Venice on Obligations:  The Sources for both the Logica Magna and the Logica Parva Versions’, in Knowledge and the Sciences in Medieval Philosophy, Vol. 2, ed. Simo Knuuttila et al. Publications of Luther-Agricola Society 1990, 407-415. 
  5.    Bradwardine, Thomas. Insolubilia. Edition, English translation and Introduction by Stephen Read. (Dallas Medieval Texts and Translation 10.) Leuven: Peeters 2010.
  6.    Conti, Alessandro, ‘Paul of Venice’, The Stanford Encyclopedia of Philosophy. Ed. E. N. Zalta (Summer 2017 Edition)
  7.    De Rijk, Lambertus M. ‘Some Notes on the Mediaeval Tract De insolubilibus, with the Edition of a Tract Dating from the End of the Twelfth Century.’ Vivarium 4 (1966), 83-115
  8.    De Rijk, Lambertus M. Logica Modernorum: A Contribution to the History of Early Terminist Logic. Vol. 1: On the Twelfth Century Theories of Fallacy. Assen: Van Gorcum 1962.
  9.    Field, Hartry. Saving Truth from Paradox. Oxford: Oxford UP 2008.
  10.    Hauser, Kai. ‘Gödel’s program revisited: Part I: The turn to phenomenology’, Bulletin of Symbolic Logic 12 (2006), 529-90.
  11.    Heytesbury, William. On “Insoluble” Sentences: Chapter One of His Rules for Solving Sophisms. Tr. Paul Vincent Spade. “Mediaeval Sources in Translation,” vol. 21. Toronto: Pontifical Institute of Mediaeval Studies, 1979.
  12.    Martin, Christopher J. ‘Obligations and Liars.’ In Sophisms in Medieval Logic and Grammar. Ed. Stephen Read. Dordrecht: Kluwer 1993, 357-81; reprinted in Medieval Formal Logic. Ed. M. Yrjönsuuri, Kluwer 2001, 63-94.
  13.    Paul of Venice. Logica Magna. Venice 1499.
  14.    Paul of Venice. Quadratura. Venice 1493. 
  15.    Paul of Venice. Sophismata Aurea. Venice 1493. 
  16.    Paul of Venice. Logica Parva. Tr. A.R. Perreiah. Munich/Vienna: Philosophia Verlag 1984.
  17.    Paul of Venice. Logica Parva. Ed. A.R. Perreiah. Leiden: Brill 2002.
  18.    Paul of Venice. Logica Magna, Secunda Pars: Tractatus de Veritate et Falsitate Propositionis et Tractatus de Significato Propositionis. Ed. F. del Punta and tr. M.M. Adams. Oxford UP for the British Academy, 1978.
  19.    Paul of Venice. Logica Magna, Prima Pars: Tractatus de Terminis. Ed. N. Kretzmann. Oxford UP for the British Academy, 1979.
  20.    Paul of Venice. Logica Magna, Prima Pars: Tractatus de Scire et Dubitare. Ed. P. Clarke. Oxford UP for the British Academy, 1981.
  21.    Paul of Venice. Logica Magna, Secunda Pars: Tractatus de Obligationibus. Ed. E.J. Ashworth. Oxford UP for the British Academy, 1988.
  22.    Paul of Venice. Logica Magna, Secunda Pars: Capitula de Conditionali et de Rationali. Ed. G. Hughes. Oxford UP for the British Academy, 1990.
  23.    Paul of Venice. Logica Magna, Secunda Pars: Tractatus de Hypotheticis. Ed. A. Broadie. Oxford UP for the British Academy, 1990.
  24.    Paul of Venice. Logica Magna, Prima Pars: Tractatus de Necessitate et Contingentia Futurorum. Ed. C.J.F. Williams. Oxford UP for the British Academy, 1991. 
  25.    Perreiah, A.R. ‘Insolubilia in the Logica Parva of Paul of Venice.’ Medioevo 4 (1978), 145-71.
  26.    Perreiah, A.R. Paul of Venice: a Bibliographical Guide. Philosophy Documentation Center 1986.
  27.    Peter of Ailly. Concepts and Insolubles: An Annotated Translation. Tr. Paul Vincent Spade. “Synthese Historical Library” vol. 19. Dordrecht: Reidel 1980.
  28.    Peter of Mantua. Logica. Padua 1477.
  29.    Pironet, Fabienne. ‘William Heytesbury and the treatment of Insolubilia in 14th-century England.’ In Unity, Truth and the Liar: The Modern Relevance of Medieval Solutions to the Liar Paradox. Ed. Shahid Rahman et al. Berlin: Springer-Verlag 2008, 251-327.
  30.    Pozzi, Lorenzo. Il Mentitore e il Medioevio. Edizioni Zara 1987. 
  31.    Read, Stephen. ‘The Liar paradox from John Buridan back to Thomas Bradwardine.’ Vivarium 40 (2002), 189-218.
  32.    Spade, Paul Vincent. The Medieval Liar. Toronto: Pontifical Institute of Medieval Studies 1975.
  33.    Spade, Paul Vincent. ‘Roger Swyneshed's Insolubilia: Edition and Comments.’ Archives d'histoire doctrinale et littéraire du moyen âge, 46 (1979), 177-220.
  34.    Spade, Paul Vincent. ‘The manuscripts of William Heytesbury’s Regulae solvendi sophismata: conclusions, notes and descriptions’, Medioevo 15 (1989), 271-314.
  35.    Spade, Paul Vincent and Read, Stephen. ‘Insolubles’, The Stanford Encyclopedia of Philosophy. Ed. E. N. Zalta. (Fall 2018 Edition).
  36.    Strobino, Riccardo. ‘Truth and Paradox in Late XIVth Century Logic: Peter of Mantua's Treatise on Insoluble Propositions.’ Documenti e studi sulla tradizione filosofica medievale, 23 (2012), 475-519. 

Steering Committee

[Updated 12 January 2019]