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"--P. [4] of cover. (shrink)

Il volume intende presentare in una prospettiva organica la letteratura medievale sugli Insolubilia. Gli insolubili sono stati definiti come proposizioni autoreferenziali che hanno origine direttamente dalla tradizione del paradosso del mentitore. L’impostazione del presente lavoro vuole essere sia filologica che teorica in modo che da un lato si possano inserire le analisi speculative sugli Insolubilia in un contesto critico–storico più ampio che restituisca la peculiarità della trattazione medievale, e dall’altro, possa essere effettuata una analisi precisa dei tentativi di (...) soluzione proposti anche attraverso gli strumenti della logica moderna (quando possibile e non fuorviante). (shrink)

Thomas Bradwardine makes much of the fact that his solution to the insolubles is in accordance with Aristotle's diagnosis of the fallacy in the Liar paradox as that of secundum quid et simpliciter. Paul Spade, however, claims that this invocation of Aristotle by Bradwardine is purely "honorary" in order to confer specious respectability on his analysis and give it a spurious weight of authority. Our answer to Spade follows Bradwardine's response to the problem of revenge: any proposition saying of itself (...) that it is false says more than does Bradwardine's proposition saying of it that it is false, and so follows from that other proposition only in respect of part of what it says, and not simpliciter. (shrink)

A further reformulation of Naive Set Comprehension related to that proposed in “Resolving Insolubilia: Internal Inconsistency and the Reform of Naive Set Comprehension” is possible in which contradiction is averted not by excluding sets such as the Russell Set but rather by treating sentences resulting from instantiation of such sets as the Russell Set in their own descriptions as invalid. So the set of all sets that are not members of themselves in this further revision is a valid set (...) but the claim that that set is or is not a member of itself is not validly expressible. Such an approach to set comprehension results in a set ontology co-extensive with that permitted by the Naive Set Comprehension Principle itself. This approach has as strong a claim to consistency as that formulated in “Resolving Insolubilia: Internal Inconsistency and the Reform of Naive Set Comprehension.”. (shrink)

Abstract:Russell’s substitutional theory conferred philosophical advantages over the simple type theory it was to emulate. However, it faced propositional paradoxes, and in a 1906 paper “On ‘Insolubilia’ and Their Solution by Symbolic Logic”, he modified the theory to block these paradoxes while preserving Cantor’s results. My aim is to draw out several quandaries for the interpretation of the role of substitution in Russell’s logic. If he was aware of the substitutional (p0a0) paradox in 1906, why did he advertise “ (...) class='Hi'>Insolubilia” as a solution to the Epimenides? If he was dissatisfied with the solution, as his correspondence suggests, why did he go on to publish it? Why did substitution reappear with orders in “Mathematical Logic as Based on the Theory of Types” if he had rejected a hierarchy of orders as intolerable? I offer the following as possible explanations: he construed the “logical Epimenides” as a version of the p0a0 paradox; his dissatisfaction with the “Insolubilia” solution was philosophical, not technical; and substitution re-emerged because he hoped for a new philosophical gloss on orders. Whether or not my explanations are correct, these issues must be addressed in accounting for Russell’s reasons for ramification. (shrink)

This article examines by textual criticism a manuscript of the early fifteenth century preserved at the Stadtbibliothek Lübeck. Being a textual witness of Peter of Ailly’s Insolubilia, the Lübeck manuscript, then unknown, is of special interest for my critical edition of Peter’s treatise (Turnhout: Brepols 2022). However, this witness turns out to be of minor importance for the constitution of the critical text. As an addendum to my edition, this article also provides the reader with a full documentation of (...) all variant readings in the Lübeck manuscript. (shrink)

The manuscript Venice, Biblioteca Nazionale Marciana, Class XI n. 12, Zanetti Latini 301 (= 1576), contains on fols. 1r–24v a seemingly unique copy of a series of fifteen logical questions, ten on obligationes and the remaining five on insolubilia.1 The series on obligationes is untitled and unattributed in the manuscript, but the questions on insolubilia begin (fol. 18r11) “Incipiunt quaestiones super insolubilibus,” and are attributed at the end to a certain John of Wesel (fol. 24v41): “Ergo expletae sunt (...) quaestiones insolubilium a Johanne de Vesalia Parixius2 disputatae. Deo gratias. Amen.”. (shrink)

Paul of Venice joined the Austin Friars at an early age and was sent by them from Padua to study at Oxford in 1390. When he returned, full of ideas and laden with books, he began his prodigious writing career with several books on logic, including the Logica Magna, which runs to some half a million words. The current volume contains the final treatise, on insolubles - that is, logical paradoxes. After surveying fifteen previous solutions, Paul develops his own, based (...) on the idea that such propositions falsify themselves. Besides a critical edition of the Latin text, the volume also contains an English translation, a detailed commentary, excerpts from two other logical works of Paul, and a substantial introduction. The introduction describes the fourteenth-century background to Paul's treatise; it also gives a detailed rebuttal of a recent claim that the Logica Magna is not by Paul because its content clashes with genuine works of his. All in all, the volume greatly enhances our understanding of the development of logic, in particular of the semantics of propositions, during a crucial century in its history. (shrink)

Jan Salamucha was born on the 10th of June 1903 in Warsaw and murdered on the 11th of August 1944 in Warsaw during the Warsaw Uprising very early on in his scholarly career. He is the most original representative of the branch of the Lvov-Warsaw School known as the Cracow Circle. The Circle was a grouping of scholars who were interested in reconstructing scholasticism and Christian philosophy in general by means of mathematical logic. As Jan Lukasiewicz’s successor in the area (...) of logic and Konstanty Michalski’s student in the area of the history of medieval thought, Salamucha had an excellent preparation for this task. His main achievements include a masterful logical analysis of the proof _ex motu _for the existence of God, a modern interpretation of analogical notions and a comprehensive approach to the problem of essence. He also contributed several historical studies: he examined Aristotle’s theory of deduction, he reconstructed William Ockham’s propositional logic and established the authenticity of his treatise on _insolubilia,_ and he identified the historical sources of the antinomies in Antiquity and the Middle Ages. He did not shy away from popularizing philosophy, and in that work he was able to elucidate rather than oversimplify the complexities of philosophy. (shrink)

This paper revisits Buridan’s Bridge paradox (Sophismata, chapter 8, Sophism 17), itself close kin to the Liar paradox, a version of which also appears in Bradwardine’s Insolubilia. Prompted by the occurrence of the paradox in Cervantes’s Don Quixote, I discuss and compare four distinct solutions to the problem, namely Bradwardine’s “just false” conception, Buridan’s “contingently true/false” theory, Cervantes’s “both true and false” view, and then the “neither true simpliciter nor false simpliciter” account proposed more recently by Jacquette. All have (...) in common to accept that the Bridge expresses a truth-apt proposition, but only the latter three endorse the transparency of truth. Against some previous commentaries I first show that Buridan’s solution is fully compliant with an account of the paradox within classical logic. I then argue that Cervantes’s insights, as well as Jacquette’s treatment, are both supportive of a dialetheist account, and Jacquette’s in particular of the strict-tolerant account of truth. I defend dialetheist intuitions (whether in LP or ST guise) against two objections: one concerning the future, the other concerning the alleged simplicity of the Bridge compared to the Liar. (shrink)

This article presents for the first time complete descriptions of four codices of the Austrian National Library at Vienna, viz. 1617, 5237, 5248 and 5377. Cod. 1617 is a fragment of Henry Totting of Oyta’s 13 Quaestiones Sententiarum, comprising part of q.7 and qq.8-13 in 198 ff.. The other three manuscripts contain mainly logical texts, e.g., Albert of Saxony’s Sophismata in Cods. 5237 and 5377, his Insolubilia in Cod. 5248, and his Quaestiones Posteriorum in Cod. 5377; 11 of the (...) 12 Tractatus by Peter of Spain in Cod. 5248; an apparently unnoticed copy of Porphyry’s Isagoge translated by Boethius and treatises by Thomas Manlevelt, Richard Billingham and John of Holland along with several anonymous texts in Cod. 5248; a fragment of John Buridan’s Quaestiones breves veteris artis in Cod. 5377. The latter codex has also two interesting sets of anonymous Quaestiones quodlibetales and an erratic leaf with questions on Peri heremeias. Charters or charter-like texts are to be found in Cods. 1617 and 5248 , which may be of value for the prosopography of the fourteenth century. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:January 22, 2009 (8:41 pm) G:\WPData\TYPE2802\russell 28,2 048red.wpd russell: the Journal of Bertrand Russell Studies n.s. 28 (winter 2008–09): 127–42 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631; online 1913-8032 YABLO’S PARADOX AND RUSSELLIAN PROPOSITIONS Gregory Landini Philosophy / U. of Iowa Iowa City, ia 52242–1408, usa [email protected] Is self-reference necessary for the production of Liar paradoxes? Yablo has given an argument that self-reference is not necessary. He (...) hopes to show that the indexical apparatus of self-reference of the traditional Liar paradox can be avoided by appealing to a list, a consecutive sequence, of sentences correlated one-one withnaturalnumbers.Yabloopens his “Paradox without Self-Reference” (Analysis, 1993) with the assumption that there is a sequence such that: Snz: “(;k)(k > n.!. True Sk)” Each sentence on Yablo’s list is supposed to be correlated one-one with number n. Each sentence is supposed to say that for every natural number kz greater than n, the k-th sentence on the list is not true. By comparing Yablo’s construction to an analogous construction with early Russellian propositions, we show that Yablo has failed to generate a paradox. 1. vicious circularity in yablo’s paradox S elf-reference has been taken to be the source of paradoxes. Poincaré and Russell are commonly associated with this thesis. In an amusing passage, the mathematician Jourdain recalls something of the dialogue between them: Nearly all mathematicians agreed that the way to solve these paradoxes was simply not to mention them; but there was some divergence of opinion as to how they were to be unmentioned. It was clearly unsatisfactory merely not to mention them. Thus Poincaré was apparently of the opinion that the best way of avoiding such awkward subjects was to mention that they were not to be mentioned. But [as Russell put it] “one might as well, in talking to a man with January 22, 2009 (8:41 pm) G:\WPData\TYPE2802\russell 28,2 048red.wpd 128 gregory landini 1 Philip Jourdain, The Philosophy of Mr. B*rtr*nd R*ss*llz (London: Allen & Unwin, 1918), p. 77. Russell is quoted from “Mathematical Logic as Based on the Theory of Types” (1908), LK, p. 63. 2 “On ‘Insolubilia’ and Their Solution By Symbolic Logic”, in B. Russell, Essays in Analysis, ed. Douglas Lackey (London: Allen & Unwin, 1973), p. 196. 3 I believe that this is Russell’s view in 1906. See Gregory Landini, “Russell’s Separation of the Logical and Semantic Paradoxes”, Revue internationale de philosophie 58 (2004): 257–94 (issue titled “Russell en héritage / Le centenaire des Principes”, ed. Philippe de Rouilhan). a long nose, say: ‘When I speak of noses, I except such as are inordinately long’, which would not be a very successful eTort to avoid a painful topic.”1 Poincaré maintained that one should exclude the oTending cases and therebyavoid“viciouscircles”ofself-referencewhichgenerateparadoxes. Russell objected: We may illustrate this by what M. Poincaré says concerning Richard’s paradox. Having Wrst put E = “all numbers deWnable in a Wnite number of words” we arrive at a paradox, due, says M. Poincaré, to our having included a number only deWnable in a Wnite number of words by means of E. This vicious circle he proposes to avoid by deWning E as “all numbers deWnable in a Wnite number of words without mentioning Ey”. To the uninitiated, this deWnition looks more circular than ever.2 Russell held that some paradoxes such as those of classes require, in order to exclude oTending cases without mentioning them, a “reconstruction of logical Wrst principles”. Others, such as Richard’s, are to be dismissed because they involve confused and viciously circular notions of “deWnability ”.3 Is self-reference necessary for paradoxes? The question is not well crafted. There are quite diTerent notions ofz “self-reference” involved in paradoxes. The self-reference involved in the Liar “This sentence is false” is provided by an apparatus of indexicals. This apparatus is quite distinct from the self-reference involved in Russell’s early ontology of propositions (as mind- and language-independent states of aTairs). This is an ontological self... (shrink)

This paper has a three-Fold aim: (1) to show that wittgenstein gave a logical solution to the logico-Mathematical paradoxes, Which has not been taken into consideration only because of its brevity; (2) to prove that this solution does not reject the theory of types but interprets it in the simplest way; (3) to show that wittgenstein's solution is the solution given to the paradoxes called 'insolubilia', By the famous scholastic logician, Petreisde alligaco.

Zbigniew Tworak The early Leśniewski and the Liar AntinomyIn his early, prelogistic article „Critique of the Logical Principle of Excluded Middle” (1913) Stanislaw Leśniewski presents a certain solution to the Liar Antinomy. He argues that the Logical Principle of Excluded Middle is false but he defends the so-called Principle of Contradictory Sentences (the weaker version of the Logical Principle of Excluded Middle) and the Logical Principle of Contradiction. The paper discusses this solution. Leśniewski’s solution to the Liar antinomy differs from (...) Tarski’s in several important details. On the one hand, there is a connection between the proposition and a medieval solution to insolubilia, the so-called restrictio theory. A restrictio is a doctrine which has it that self-reference is illegitimate (either in all cases or in some cases only). On the other hand, there is a connection between the proposition and the contextual approach (what is communicated by the sentence varies with the context of use of the sentence). Keywords: Stanislaw Leśniewski, Liar Antinomy, restrictio theory. (shrink)

Thomas Bradwardine's solution to the semantic paradoxes, presented in his Insolubilia written in Oxford in the early 1320s, turns on two main principles: that a proposition is true only if things are wholly as it signifies; and that signification is closed under consequence. After exploring the background in Walter Burley's account of the signification of propositions, the question is considered of the extent to which Bradwardine's theory is compatible with the distribution of truth over conjunction, disjunction, negation and the (...) conditional. (shrink)

In the first half of the fifteenth century, the Italian logician, natural philosopher, and doctor of medicine Cajetan of Thiene wrote a commentary on William Heytesbury’s Regulae solvendi sophismata, which later became a part of the printed edition of Heytesbury’s treatises. Several late fifteenth century reprints sustained its circulation and further influence. Following Heytesbury, Cajetan listed four alternative treatments of paradoxes, where the first three were formulated in general logico-semantic terms and the last one in terms of obligationes. The present (...) analysis reconstructs the first three positions in terms of the theories of logical operators endorsed as part of the solution to paradoxes. This reconstruction uncovers different underlying views of operators, namely context-sensitive, value-functional, and supervaluationist.Priore dimidia parte saeculi 15 Caietanus de Thiena, logicus, physicus et medicus, commentarium super G. Hentisberi Regulis solvendi sophismata conscripsit, quod posterius una cum Hentisberi tractatibus typis impressum est. Cuius commentarii notitiam auctoritatemque continuam iteratae nonnullae eius editiones in fi ne 15 saeculi factae sustinebant. Caietanus quattuor vias tractandi insolubilia distinxit, quarum tres primae conceptibus generalibus logico-semanticis, quarta doctrina de obligationibus innixae sunt. In analysi hic proposita auctor primas tres vias reconstruit, doctrinas varias de logicis coniunctionibus vel notis reserans, super quibus illae viae solvendi paradoxa fundantur. Quarum prima vim notarum a contextu sermonis dependentem facit. Altera notas pure “compositionaliter” tractat. Tertia iuxta modum doctrinae de “supervaluatione” omnes formales tautologias servat, aliis considerationibus neglectis. (shrink)

The property common to three kinds of paradoxes (logical, semantic, and cultural) is the underlying presence of an exclusive disjunction: even when it is put to a check by the paradox, it is still invoked at the level of implicit discourse. Hence the argumentative strength of paradoxical propositions is derived. Logical paradoxes (insolubilia) always involve two contradictory, mutually exclusive, truths. One truth is always perceived to the detriment of the other, in accordance with a succession which is endlessly repetitive. (...) A check is put on the principle of the excluded middle by the logical paradoxes, because self-reference leads to an endlessly repeating circle, out of which no resolution is conceivable. Logical paradoxes are to be compared with the `objective ambiguity' prevalent in oracles (Gallet, 1990). Semantic paradoxes are contextually-determined occurrences, whose resolution at the metalinguistic level is made possible by the discovery of a middle term. They express a wilful ambiguity, in which the interlocutor is invited to take an active part in the construction of sense, since what must be found is the unexpected sense thanks to which A and not-A can be asserted simultaneously. Cultural paradoxes play about doxa (`common sense') and openly challenge common opinion because of their character as inopinata (`unexpected'). My aim is to show that even cultural paradoxes hide sometimes a flaw of argumentation similar to logical or semantic paradox; they too imply an exclusive disjunction leading to the disappearance of the middle terms. Finally, basing myself on the theory of topoi (Anscombre and Ducrot, 1983), a tentative resolution of the cultural paradoxes will be suggested. (shrink)

Choice without preference : the problem of "Buridan's ass" -- Nicholas of Cusa on the Koran : a fifteenth-century encounter with Islam -- On learned ignorance and the limits of knowledge -- Unanswerable questions and insolubilia -- Omniscience and our understanding of God's knowledge -- Issues of infinite regress -- Being qua being -- Nonexistents then and now -- Thomism : past, present, and future -- Respect for tradition and the Catholic philosopher today.

The papers in this volume - written by well-known experts in the field - examine the rules for valid argument discovered and formulated in the works of medieval scholasticism and show their significance to modern discussions in logic and the philosophy of language. The editor's introductions make the papers interesting and comprehensible even to non-specialists.

La teoria delle obbligazioni rappresenta un significativo esempio di come un insieme di tecniche logiche possa essere applicato, in maniera feconda, a contesti disputazionali in cui la dimensione dialogica e l’interazione tra soggetti coinvolti in un confronto dialettico hanno un ruolo di primo piano.L’attenzione che numerosi studiosi hanno manifestato nei confronti di questa parte della logica modernorum negli ultimi quarat’anni testimonia la profondità e la difficoltà dei problemi teorici che stanno alla base della teoria.Molte interpretazioni sono state avanzate circa le (...) motivazioni che spingono i logici medievali a scrivere trattati in cui il portato delle loro dottrine logiche – come, ad esempio, la teoria delle consequentiae - viene declinato ed elaboratamente applicato al gioco dialettico tra due disputanti. La gamma di tali letture comprende differenti ipotesi: la teoria delle obbligazioni potrebbe fornire un nucleo di regole per l’impostazione di un metodo assiomatico rudimentale così come rappresentare un tentativo di elaborare esercizi volti a valutare le capacità logiche degli studenti o, ancora, un gioco teorico, svincolato dall’effettiva applicazione a contesti concreti, la cui finalità sarebbe il mantenimento della coerenza logica.E’ stata sostenuta da più interpreti la funzione pedagogica svolta dalla teoria, al di là del fatto che si tratti o meno di un metodo per la conduzione effettiva di dispute reali. Un’altra linea interpretativa collega la teoria delle obbligazioni al contesto della letteratura dei sophismata e degli insolubilia, dal cui repertorio, in effetti, sono spesso tratti molti degli esempi che si incontrano nella lettura dei trattati de obligationibus. In questo senso, la teoria offrirebbe regole e schemi di risposta che un interlocutore deve utilizzare per risolvere o rispondere a enunciati problematici o paradossali. Non si può escludere, inoltre, che nella teoria si realizzi un intreccio delle differenti motivazioni sopra esposte: potrebbe trattarsi di uno strumento che viene sviluppato tanto per la risoluzione di sophismata e insolubilia quanto per indagare il rapporto tra linguaggio e metalinguaggio o altre questioni teoriche. Altri studiosi hanno infine sostenuto che lo scopo della teoria delle obbligazioni consisterebbe nell’elaborazione di un metodo adatto alla costruzione e alla discussione di esperimenti mentali e che questo costituirebbe una sorta di metametodologia della scienza medievale sulla linea del ragionamento secundum imaginationem. (shrink)

A further reformulation of naïve set comprehension related to that proposed in _‘_Resolving Insolubilia: Internal Inconsistency and the Reform of Naive Set Comprehension’_ _ is possible in which contradiction is averted not by excluding sets such as the Russell Set but rather by treating sentences resulting from instantiation of such sets as the Russell Set in their own descriptions as invalid. So the set of all sets that are not members of themselves in this further revision is a valid (...) set but the claim that that set is or is not a member of itself is not validly expressible. Such an approach to set comprehension results in a set ontology co-extensive with that permitted by the Naïve Set Comprehension Principle itself. This approach has as strong a claim to consistency as that formulated in _‘_Resolving Insolubilia: Internal Inconsistency and the Reform of Naive Set Comprehension’. (shrink)

We present Prior's discussion of a puzzle about valditity found in the writings of the fourteenth-century French logician Jean Buridan and show how Prior's study of this puzzle may have provided the conceptual inspiration for his development of hybrid logic.

This paper offers an analysis of a hitherto neglected text on insoluble propositions dating from the late XiVth century and puts it into perspective within the context of the contemporary debate concerning semantic paradoxes. The author of the text is the italian logician Peter of Mantua (d. 1399/1400). The treatise is relevant both from a theoretical and from a historical standpoint. By appealing to a distinction between two senses in which propositions are said to be true, it offers an unusual (...) solution to the paradox, but in a traditional spirit that contrasts a number of trends prevailing in the XiVth century. It also counts as a remarkable piece of evidence for the reconstruction of the reception of English logic in italy, as it is inspired by the views of John Wyclif. Three approaches addressing the Liar paradox (Albert of Saxony, William Heytesbury and a version of strong restrictionism) are first criticised by Peter of Mantua, before he presents his own alternative solution. The latter seems to have a prima facie intuitive justification, but is in fact acceptable only on a very restricted understanding, since its generalisation is subject to the so-called revenge problem. (shrink)