Medieval Logic

Roger Swyneshed, in his treatise on insolubles (logical paradoxes), dating from the early 1330s, drew three notorious corollaries of his solution. The third states that there is a contradictory pair of propositions both of which are false. This appears to contradict the Rule of Contradictory Pairs, which requires that in every such pair, one must be true and the other false. Looking back at Aristotle's treatise De Interpretatione, we find that Aristotle himself, immediately after defining the notion of a contradictory (...) pair, gave counterexamples to the rule. Thus Swyneshed's solution to the logical paradoxes is not contrary to Aristotle's teaching, as many of Swyneshed's contemporaries claimed. Dialetheism, the contemporary claim that some propositions are both true and false, is wedded to the Rule, and in consequence divorces denial from the assertion of the contradictory negation. (shrink)

1st GPMR Workshop on Logic and Semantics: Medieval Logic and Modern Applied Logic, Reinische Friedrich Wilhelms Universität Bonn, Germany, 2007.

N. Germann and S. Harvey, editors. The Origin and Nature of Language and Logic: Perspectives in Medieval Islamic, Jewish, and Christian Thought. Turnhout: Brepols, 2020. xiii +422 pp. €71,46, ISBN...

In “Lectio 55” of his Notule libri Priorum, Robert Kilwardby discussed various objections that had been raised against Aristotle’s Theses. The first thesis, AT1, says that no proposition q is implied both by a proposition p and by its negation, ∼p. AT2 says that no proposition p is implied by its own negation. In Prior Analytics, Aristotle had shown that AT2 entails AT1, and he argued that the assumption of a proposition p such that (∼p → p) would be “absurd”. (...) The unrestricted validity of AT1, AT2, however, is at odds with other principles which were widely accepted by medieval logicians, namely the law Ex Impossibili Quodlibet, EIQ, and the rules of disjunction introduction. Since, according to EIQ, the impossible proposition (p ∧ ∼p) implies every proposition, it also implies ∼(p∧∼p), in contradiction to AT2. Furthermore, by way of disjunction introduction, the proposition (p∨∼p) is implied both by p and by ∼p, in contradiction to AT1. Kilwardby tried to defend AT1, AT2 against these objections by claiming that EIQ holds only for accidental but not for natural implications. The second argument, however, cannot be refuted in this way because Kilwardby had to admit that every disjunction (p ∨ q) is naturally implied by its disjuncts. He therefore introduced the further requirement that, in order to constitute a genuine counterexample to AT1, (p → q) and (∼p → q) have to hold “by virtue of the same thing”. In a recently published paper, Spencer Johnston accepted this futile defence of AT1 and developed a formal semantics that would fit Kilwardby’s presumably connexive implication. This procedure, however, is misguided because the remaining considerations of Lesson 55  which were entirely ignored by Johnston  show that Kilwardby eventually recognized that AT2 is bound to fail. After all he concluded: “So it should be granted that from the impossible its opposite follows, and that the necessary follows from its opposite”. (shrink)

If an argument is valid, it is impossible for its premises to be true, and its conclusion false. But how should we understand these notions of truth and impossibility? Here, I present the answers given by John Buridan (ca. 1300-60), showing (i) how he understands truth in his anti-realist metaphysics, and (ii) how he understands modality in connection with causal powers. In short: if an argument exists and is valid, there does not exist a power capable of making the premises (...) true and, at the same time, making the conclusion false. (shrink)

To help frame the Festschrift for Gyula Klima (Metaphysics Through Semantics: The Philosophical Recovery of the Medieval Mind), this appreciation offers perspective on the scholar's person and project. Drawing on biographical details and reflecting on signal contributions, it seeks to honor a distinguished philosopher who deserves to be celebrated by friends and introduced to a new generation of readers. Download with frontmatter from: link[dot]springer[dot]com/content/pdf/bfm:978-3-031-15026-5/1?pdf=chapter%20toc.

Gyula Klima’s distinctive work recovering medieval philosophy has inspired a generation of scholars. Klima’s attention to the distinctive terms, problems, and assumptions that constitute alternative historical conceptual frameworks has informed work in philosophy of language and logic, cognition and philosophical psychology, and metaphysics and theology. This volume celebrates Klima’s project by collecting new essays by colleagues, collaborators, and former students. Covering a wide range of thinkers (Plotinus, Anselm, Aquinas, Buridan, Ockham, and others) and various specifc questions (e.g., about language, cognition, (...) the soul, and God), it is unifed by a common interest in applying historically sensitive, hermeneutically sophisticated, and logically rigorous philosophical interpretation to recover, and appreciate, lost perspectives. (shrink)

This paper explores an idea of Stoic descent that is largely neglected nowadays, the idea that an argument is valid when the conditional formed by the conjunction of its premises as antecedent and its conclusion as consequent is true. As it will be argued, once some basic features of our naıve understanding of validity are properly spelled out, and a suitable account of conditionals is adopted, the equivalence between valid arguments and true conditionals makes perfect sense. The account of validity (...) outlined here, which displays one coherent way to articulate the Stoic intuition, accords with standard formal treatments of deductive validity and encompasses an independently grounded characterization of inductive validity. (shrink)

This paper has been presented at the Charles S. Peirce Society’s 10-Minute Thesis Initiative: “His Glassy Essence in Relation” on February 18, 2023, where papers were also presented by Professor Doctor António Manuel Martins and Professor Doctor Mohammad Shafiei, respectively affiliated to the Coimbra Institute for Philosophical Studies and Shahid Beheshti University. -/- The edition “His Glassy Essence in Relation” of the Charles S. Peirce Society’s 10-Minute Thesis Initiative has been jointly organized by Aaron Wilson, António Manuel Martins, Mohammad Shafiei, (...) Richard Kenneth Atkins, and Robert Junqueira. The second, the third, and the last all presented papers, while the first one presided over the proceedings. The second to last was responsible for creating the bridge that enabled it all to take place. -/- The 10-Minute Thesis Initiative is an ongoing cycle of the Charles S. Peirce Society. This edition in particular benefited from the collaboration of the Coimbra Institute for Philosophical Studies and the Shahid Beheshti University. This text has been published in PhicaRe (Philosophy & Care Repository) of the Institute for Philosophical Studies of the Faculty of Arts and Humanities of the University of Coimbra. -/- How to cite (Chicago Manual of Style 17th Edition): Junqueira, Robert. “Peirce and the Coimbra Jesuit Course: A Bond Far More Pervasive Than Commonly Believed.” In PhicaRe (Philosophy & Care Repository), 1–12. Coimbra: Institute for Philosophical Studies, 2023. DOI: 10.5281/zenodo.7607786. (shrink)

In the first of the Insolubles in Chapter 8 of his Sophismata, Buridan contends that the inference Omnis propositio est affirmativa; ergo, nulla propositio est negativa (PS) is valid, even though it appeals to the self-reference in the conclusion to show that what we (following Read 2001) call the classical conception of validity (CCV) fails. This requires that we accept that there are good inferences in which a false conclusion follows from true premises. Partially following Hughes’ proposal (1982), we argue (...) that the First Sophism (PS) involves three different notions of validity. Two of them correspond to the ones described by Hughes (1982, 80–86), who calls them Theory A and Theory B. The third one—that will we call Theory C—is not mentioned by Hughes; instead, it is suggested by Buridan himself in the first three arguments in favor of the validity of PS. We show that: a) from what Buridan says in his Theory C it follows that PS is a formal and material consequence, and hence, a valid one. Then we show that: b) the rejection of CCV and the acceptance of Nulla propositio est negativa (NPN) as a (formal) consequence of Omnis propositio est affirmativa (OPA) leads to a paradox that bears similarities with the one put forward by Pseudo Scotus—which has been studied by Read (2001) and is related to Curry’s paradox. However, there are enough differences to merit considering this paradox separately, especially in relation to the so-called validity paradoxes. Interestingly, our work suggests that Buridan was aware of these problems, which explains why he introduced a new criterion for validity, one that is not based on truth-preservation but on what Spade (1988) calls firmness, and Klima (2016) correspondence. (shrink)

Aristotle divided arguments that persuade into the rhetorical (which happen to persuade), the dialectical (which are strong so ought to persuade to some degree) and the demonstrative (which must persuade if rightly understood). Dialectical arguments were long neglected, partly because Aristotle did not write a book about them. But in the sixteenth and seventeenth century late scholastic authors such as Medina, Cano and Soto developed a sound theory of probable arguments, those that have logical and not merely psychological force but (...) fall short of demonstration. Informed by late medieval treatments of the law of evidence and problems in moral theology and aleatory contracts, they considered the reasons that could render legal, moral, theological, commercial and historical arguments strong though not demonstrative. At the same time, demonstrative arguments became better understood as Galileo and other figures of the Scientific Revolution used mathematical proof in arguments in physics. Galileo moved both dialectical and demonstrative arguments into mathematical territory. (shrink)

The phrase _secundum quid et simpliciter_ is the Latin expression translating and labelling the sophism described by Aristotle as connected with the use of some particular expression “absolutely or in a certain respect and not in its proper sense.” This paper presents an overview of the analysis of this fallacy in the history of dialectics, reconstructing the different explanations provided in the Aristotelian texts, the Latin and medieval dialectical tradition, and the modern logical approaches. The _secundum quid_ emerges as a (...) strategy that is based on the pragmatic dimension of arguments, and in particular the complex passage from an utterance (what is said) to its logical form (a proposition in an argument). The medieval and modern logical theories attempted to explain from different philosophical perspectives how the pragmatically enriched semantic representation can be achieved, justified, and most importantly manipulated. The different analyses of this fallacy bring to light various dimensions of the pragmatics of arguments, and the complex interdependence between context, meaning, and inferences. (shrink)

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. 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 the Quadratura, 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. 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. (shrink)

Multiple generality has long been known to cause confusion. For example, “Everyone has a donkey that is running” has two readings: either (i) there is a donkey, owned by everyone, and it is running; or (ii) everyone owns some donkey or other, and all such donkeys run. Medieval logicians were acutely aware of such ambiguities, and the logical problems they pose, and sought to sort them out. One of the most ambitious undertakings in this regard is a pair of massive (...) diagrams (magnae figurae) which map out the logical interrelations of two sets of doubly-general forms. These appear in a fourteenth-century MS of John Buridan’s Summulae de Propositionibus. In this paper, I present these diagrams, and determine the truth conditions of their different forms. To that end, I have developed a bespoke system of diagrams to display their truth conditions. As we will see, such forms present significant difficulties for an all-encompassing account of the role form plays in logic. Accordingly, they can tell us important things about the role logical form plays in Buridan’s account of logical foundations. (shrink)

In Formale Logik, published in 1956, J. M. Bocheński presented his first proposal for the solution to the liar paradox, which he related to Paul of Venice's argumentation from Logica Magna. A formalized version of this solution was then presented in Formalisierung einer scholastischen Lösung der Paradoxie des ‘Lügners’ in 1959. The historical references of the resulting formalism turn out to be closer to Albert de Saxon's argument and the later solution by John Buridan. Bocheński did not pose the question (...) of the consistency of his theory. The case was taken up by B. Sobociński in his private letter to Bocheński from 12 August 1954. Sobociński used a smart translation of the language of Bocheński's theory into the classical propositional language with the notions of truth and falsehood inverted. The translation preserves the structure of the formalized solution. We explore Sobociński's idea and reconstruct his original proof. (shrink)

Jean Buridan developed his theory of consequence within a semantical framework compatible with what we now call token-based semantics. In his Treatise on Consequences and Sophismata, Buridan showed...

A number of medieval Arabic logicians discussed inferences that combine the principles of propositional and term logic, for example: Whenever H is Z then Every J is DNo D is AWhenever H is Z then S...

Johannes Steuchius’ disputatio uses Arabic logic to present an historical account of the development of philosophical thought in Arabia before and after the emergence of Islam. Steuchius first proposes that philosophy drew its origins from the East. His evidence for this claim is that many of the Greek philosophers, considered the forefathers of European philosophy, began cultivating their philosophical thinking as a result of exposure to ancient Eastern philosophy. After the introduction of Greek philosophy, it is agreed that dialectic was (...) among the first of the arts the Arabs practiced. (shrink)

In his disputatio, Johann Peter von Ludewig provides a history of rational philosophy among the Arabs and sets out to contextualize the Turks’ attitude to it. Like many Lutheran scholars of the time, Ludewig believed that Islam, as a religion, impeded the development of rational philosophy in the Arab world. However, unlike those philosophers, he examines external influences that may have fed the interest of Arab Muslims in rational philosophy, especially dialectic. Unlike Orthodox Lutherans, such as Pfeiffer and Kromayer, in (...) his conclusion, Ludewig prays that Muslim philosophers cultivate reason to overcome the "deceit of Muhammadanists" toward rational worship. The use of the contentious and seemingly oxymoronic phrase “rational worship” (logikē latreia: λογική λατρεία, from Rom. 12:1), combining “rationality” and “worship” together refects, to a certain extent, the Enlightenment belief that reason could be used to understand the nature of God against any type of irrational religiosity. The idea of worshipping God rationally and equating God with reason shows the intellectual engagement of Protestant scholars with Enlightenment rationalism. (shrink)

The article deals with the Lutheran–Catholic religious colloquy at Regensburg of 1601. It points out that the event was of importance not only for political and religious reasons but also in terms...

This paper attempts to bring the traditional theodicy on the question of evil and the Divine Providence, to its logical conclusion, in such a way that a believer is challenged to totally accept the implication of his or her faith in God. To have faith is to completely surrender to Divine Providence. It is to completely surrender ones free will to the rational conclusions or consequences of faith in the Divine Providence. Hence, this paper is for those who are perplexed (...) due to the Pandemic. Especially for Christians whose faith in God is perplexed because of their existential experience of the COVID 19 pandemic as evil. The existential reality of dread and misery caused by this pandemic is globally and more subjectively experienced due to the digitalization of this age. This shows how disastrously powerful the digitalization of our world can be in the proliferation of dread, misery and hopelessness. (shrink)

What makes a valid argument valid? Generally speaking, in a valid argument, if the premisses are true, then the conclusion must necessarily also be true. But on its own, this doesn’t tell us all that much. What is truth? And what is necessity? In what follows, I consider answers to these questions proposed by the fourteenth century logician John Buridan († ca. 1358). My central claim is that Buridan’s logic is downstream from his metaphysics. Accordingly, I treat his metaphysical discussions (...) as the key to his logic. As has been often noted, Buridan’s metaphysics are radically anti-realist about universals, though I think the depth and scope of his anti-realism has at times been papered over. Buridan constructs his logic on an amazingly spartan ontology, and this accomplishment is overdue for reconsideration. To the foregoing questions: truth is a feature of propositions, and of propositions only—not of proposition-like states of affairs, or anything like that. It is a function of the reference or supposition (suppositio) of their terms, which depends on predication in a propositional context. And necessary truth is grounded in causation: a predication is necessary if it cannot be falsified by any power, natural or supernatural, without ii annihilating what its terms stand for. “Socrates is a human”, for instance, can only be falsified by annihilating Socrates, and so it is a necessary truth. -/- These considerations provide an ample theoretical basis for a thorough examination of Buridan’s modal logic. This is what the thesis culminates with. In the final chapter, I set forth some novel and surprising findings, chief of which is this: Buridan’s modal syntax and semantics are nothing like Kripke’s—and indeed are incompatible with them. This has significant implications not only for modal logic and metaphysics, but for how we think about medieval logic and philosophy more generally. -/- Throughout the thesis, I advocate a methodology of emphasising the differences, rather than the similarities, between past and present thought. Buridan is wildly unlike what we’re used to, and we should let him speak for himself. (shrink)

Brentano’s account of intentionality has often been traced back to its scholastic sources. This is justified by his claim that objects of thought have a specific mode of being—namely, “intentional inexistence” —and that mental acts have an “intentional relation” to these objects. These technical terms in Brentano do indeed recall the medieval notions of esse intentionale, which is a mode of being, and of intentio, which is a “tending towards” of mental acts. However, within the lexical family of intentio there (...) is another distinction that plays an important role in medieval philosophy—namely, the distinction between first and second intentions, which are, roughly speaking, concepts of things and concepts of concepts respectively. What is less well-known is that Brentano explicitly borrowed this distinction as well, and used it in his account of intentionality. This paper explores this little-known chapter in the scholastic-Austrian history of intentionality by evaluating both the historical accuracy and the philosophical significance of Brentano’s borrowing of the scholastic distinction between first and second intentions. (shrink)

L'elaborato intende analizzare lo statuto ontologico dell'ente a partire dal binomio esse e id quod est, sviluppato da Tommaso nel commento al De Hebdomadibus di Boezio, specificando i concetti di partecipazione come causalità forte, e di essere come massimo estensivo e più generale.

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 At_īr 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). (shrink)

The present endeavour aims at the clarification of the concept of the logical consequence. Initially we investigate the question: How was the concept of logical consequence discovered by the medieval philosophers? Which ancient philosophical foundations were necessary for the discovery of the logical relation of consequence and which explicit medieval contributions, such as the notion of the formality (formal validity), led to its discovery. Secondly we discuss which developments of modern philosophy effected the turn from the medieval concept of logical (...) consequence to its most recent conceptions, such as the semantic, syntactic, axiomatic and natural deductive approaches? Thirdly we examine which are the similarities and the differences between the logical concepts of consequence, inference, implication and entailment? Furthermore, we ask what kind of relation signifies the concept of the logical consequence? That is to say, which is the analytic definition of the consequence relation R between the premises p1, p2...pn and the conclusion c of a formally valid argument? Finally, we focus on the respective answers given through the developments in proof theory by David Hilbert and Gerhard Gentzen. (shrink)

This special issue contains three papers on evolutions in logic during the so-called ‘post-medieval’ period. The papers discuss the following topics: traditions of...

This special issue contains three papers on evolutions in logic during the so-called ‘post-medieval’ period (roughly the years 1450–1700). The papers discuss the following topics: (1) traditions of logic in the British Isles during the sixteenth and seventeenth centuries; (2) approaches to validity in medieval and post-medieval logic; and (3) scholastic and humanist interpretations of the prologue to Galen’s Art of Medicine. All papers provide an original contribution to research on post-medieval logic, which to date is still in the early (...) stages. Together they testify to the great diversity of the discourse on logic during the period under consideration, something that has not been sufficiently appreciated in the scholarship. (shrink)

Scholastic logic provided us with a variety of accounts of validity. With some degree of simplification, validity translated into truth-preservation evaluated against a set of worlds and wa...

This paper seeks to articulate the relationship between medieval logic and theology. Reviewing modern scholarship, we find that the purpose of medieval logic, when it is even inquired about, has proven difficult to articulate without reference to theology. This prompts reflection on the metaphors of logic as a “tool” and a “game”: a tool is not merely instrumental, insofar as it can have its own intrinsic goods and can shape and be shaped by that which it serves; likewise a game, (...) with its own intrinsic goods, may yet contribute to extrinsic goods as well. After reviewing some distinctive ways in which theology shaped developments of medieval logic, this paper summarizes key examples from the work of Thomas Aquinas where medieval logic shaped the articulation of, and is therefore crucial to a proper understanding of, theological arguments and claims. The conclusion suggests implications for future philosophical and theological work. (shrink)

Robert Kilwardby occupies an important place in the history of logic, and the history of western thought more generally. Perhaps best known to scholars for his Oxford condemnations of 1277...

North-American philosophy was bolstered with the doctrines of the Jesuits. The penetration of the Coimbra Jesuits in the United States of America can be examined through the paradigmatic case of Charles Sanders Peirce. The extent to which Peirce was affected by the Coimbra Jesuits has not yet been researched. However, it is known that Peirce was acquainted with the Coimbra Jesuit Aristotelian Course.

Roger Swyneshed, in his treatise on insolubles, dating from the early 1330s, drew three notorious corollaries from his solution. The third states that there is a contradictory pair of propositions both of which are false. This appears to contradict what Whitaker, in his iconoclastic reading of Aristotle’s De Interpretatione, dubbed “The Rule of Contradictory Pairs”, which requires that in every such pair, one must be true and the other false. Whitaker argued that, immediately after defining the notion of a contradictory (...) pair, in which one statement affirms what the other denies of the same thing, Aristotle himself gave counterexamples to the rule. This gives some credence to Swyneshed’s claim that his solution to the logical paradoxes is not contrary to Aristotle’s teaching, as many of Swyneshed’s contemporaries claimed. Insolubles are false, he said, because they falsify themselves; and their contradictories are false because they falsely deny that the insoluble itself is false. Swyneshed’s solution depends crucially on the revision he makes to the acount of truth and falsehood, brought out in his first thesis: that a false proposition can signify as it is, or as Paul of Venice, who took up and developed Swyneshed’s solution some sixty years later, puts it, a false proposition can have a true significate. Swyneshed gave a further counterexample to when he claimed that some insolubles, like future contingents, are neither true nor false. Dialetheism, the contemporary claim that some propositions are both true and false, is wedded to the Rule, and in consequence divorces denial from the assertion of the contradictory negation. Consequently, Swyneshed’s logical heresy is very different from that found in dialetheism. (shrink)

John Wyclif’s logical works have lain under a kind of fog since they were first published in the 1890s. My first aim is to clear up some long-standing confusions by dispelling this fog once and for all. A partial identification of Wyclif’s source material then allows me to make a more dramatic claim about persistent misunderstandings of what is thought to be his earliest work.

This paper investigates the pre-history of Nicolò Leoniceno's De tribus doctrinis ordinatis secundum Galeni sententiam. It has been often maintained that Leoniceno's treatise broke with scho...

In this paper, we would show how the logical object “square of opposition”, viewed as semiotic object, has been historically transformed since its appearance in Aristotle’s texts until the works of Vasiliev. These transformations were accompanied each time with a new understanding and interpretation of Aristotle’s original text and, in the last case, with a transformation of its geometric configuration. The initial textual codification of the theory of opposition in Aristotle’s works is transformed into a diagrammatic one, based on a (...) new “reading” of the Aristotelian text by the medieval scholars that altered the semantics of the O form. Further, based on the medieval “Neo-Aristotelian” reading, the logicians of the nineteenth century suggest new diagrammatic representations, based on new interpretations of quantification of judgements within the algebraic and the functional logical traditions. In all these interpretations, the original square configuration remains invariant. However, Nikolai A. Vasiliev marks a turning point in history. He explicitly attacks the established logical tradition and suggests a new alternation of semantics of the O form, based on Aristotelian concepts that were neglected in the Aristotelian tradition of logic, notably the concept of indefinite judgement. This leads to a configurational transformation of the “square” of opposition into a “triangle”, where the points standing for the O and I forms are contracted into one point, the M form that now stands for particular judgement with altered semantics. The new transformation goes beyond the Aristotelian logic paradigm to a new “Non-Aristotelian” logic, i.e. to paraconsistent logic, although the argumentation used in support of it is phrased in Aristotelian style and the context of discovery is foundational. It establishes a bifurcation point in the development of logic. No unique logic is recognized, but different logics concerning different domains. One branch of logic remains to be the “Neo-Aristotelian” one, while the new logic is “Non-Aristotelian”. (shrink)