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.

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...

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)

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 forma...

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” (intentionale Inexistenz) – and that mental acts have an “intentional relation” (intentionale Beziehung) 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” (tendere in) 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 (intentio prima and intentio secunda), 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)

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...

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...

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.

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...

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...

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)

A counterpossible conditional, or counterpossible for short, is a conditional proposition whose antecedent is impossible. The filioque doctrine is a dogma of western Christian Trinitarian theology according to which the Holy Spirit proceeds from the Father and the Son. The filioque doctrine was the principal theological reason for the Great Schism, the split between Eastern Orthodoxy and western Christianity, which continues today. In the paper, I review one of the earliest medieval defenses of the doctrine in Anselm of Canterbury, and (...) I show that Anselm’s treatment of counterpossible conditionals concerning the procession of the spirit from the son in Trinitarian theology represent an early foray into default logic. Thus, the mutual estrangement of eastern and western positions on the matter may not lie fundamentally in a change in dogma, but rather in a change in logic. (shrink)

Alberic of Paris put forward an argument, ‘the most embarrassing of all twelfth-century arguments’ according to Christopher Martin, which shows that the connexive principles contradict some other logical principles that have become deeply entrenched in our most widely accepted logical theories. Building upon some of Everett Nelson’s ideas, we will show that the steps in Alberic of Paris’ argument that should be rejected are precisely the ones that presuppose the validity of schemas that are nowadays taken as some of the (...) most trivial logical truths: (A∧B) → A and (A∧B) → B, i.e. Simplification. (shrink)

While classical sources including Aristotle, Cicero and Boëthius addressed different notions of probability, medieval contributions to probability seem rather scarce. The situation changes during the Second Scholasticism with the post-Tridentine debates on “probable opinion” in moral theology and the introduction of “moral necessity” and “moral implication” in the debates on compatibilism and theological optimism. The eighteenth-century transformation of scholastic philosophy was marked, among other characteristics, by a gravitation towards the early modern scientific revolution. In his Philosophia Pollingana ad normam Burgundicae, (...) the renowned moral theologian Eusebius Amort addressed the basic issues of probabilistic logic from the philosophical, logical, and mathematical points of view in an attempt to synthesise earlier scholastic conceptual analyses of probability and probabilistic epistemic logic with the cutting-edge mathematical calculus introduced by Jacob Bernoulli. (shrink)

This paper makes two main arguments. First, that to understand analogy in St. Thomas Aquinas, one must distinguish two logically distinct concepts he inherited from Aristotle: one a kind of likeness between things, the other a kind of relation between linguistic functions. Second, that analogy (in both of these senses) plays a relatively small role in Aquinas's treatment of divine naming, compared to the realist semantic framework in which questions about divine naming are formulated and resolved, and on which the (...) coherence of the doctrine of divine simplicity—which is what gives rise to the questions of divine naming in the first place—depends. ***E-mail author for off-print.***. (shrink)

We reconstruct as much as we can the part of al-Fārābī's treatment of modal logic that is missing from the surviving pages of his Long Commentary on the Prior Analytics. We use as a basis the quotations from this work in Ibn Sīnā, Ibn Rushd and Maimonides, together with relevant material from al-Fārābī's other writings. We present a case that al-Fārābī's treatment of the dictum de omni had a decisive effect on the development and presentation of Ibn Sīnā's modal logic. (...) We give further evidence that the Harmonisation of the Opinions of Plato and Aristotle was not written by al-Fārābī. (shrink)

Nicholas of Cusa was first of all a theologian but he was interested also in mathematic and natural sciences. In fact philosophico-theological and mathematical ideas were intertwined by him, theological and philosophical ideas influenced his mathematical considerations, in particular when he considered philosophical problems connected with mathematics and vice versa, mathematical ideas and examples were used by him to explain some ideas from theology. In this paper we attempt to indicate this mutual influence. We shall concentrate on the following problems: (...) the role and place of mathematics and mathematical knowledge in knowledge in general and in particular in theological knowledge, ontology of mathematical objects and their origin, in particular their relations to God and their meaning for the description of the world and physical reality, infinity in mathematics versus infinity in theology and their mutual relations and connections. It will be shown that—according to Nicholas—mathematics and mathematical thinking are tools of rationalization of theology and liberating it in a certain sense from the trap of apophatic theology. (shrink)

This light revision of Bornholdt's doctoral thesis (Würzburg, 2015) is effectively a medievally-oriented follow-up to Richard Gaskin’s 'The Sea Battle and the Master Argument' (1995). The book is stimulating from a philosophical point of view, but the exegesis is disappointingly unreliable.

The article aims to show how the acceptance of non-monotonic logic enables arguments to be held between science and religion in a way that does not exclude either of these two spheres. The starting point of the analyses is the idea of the 13th century Danish philosopher, Boethius of Dacia, who states that it is both acceptable that: a natural scientist negates that the world had a beginning, and a Christian theologian asserts that the world had a beginning, because each (...) of them is basing their statements on the principles of their respective discipline: the first on the principles of nature, and the latter on knowledge supplemented by divine revelation. What is more, analogically: a metaphysician, when limited to his principles, cannot settle the issue, as he takes into account supranatural beings and their powers, but cannot know what God or another powerful supranatural would have decided in a such a case. The paper shows that Boethius’s approach: violates the rule of monotonicity, cannot be finally interpreted in terms of classical logic and assumes certain non-monotonic logic as its inference framework. Other presented examples of arguments between religious beliefs and the statements of natural science are resolved in the same way. Thus, it is shown how non-monotonic thinking allows us to seriously treat both scientific and religious inference as compatible. (shrink)

This paper was presented at the Second World Congress on Logic and Religion on the sub-topic of logics vis-à-vis illogicalities in religion. It deals with fourteenth-century-logician Thomas Manlevelt’s Ockhamist approach to logic and its ontological outcome: rejection of the existence of substance. Although God, the Trinity, the Blessed Virgin, the Antichrist and the consecrated host keep popping up within the domain of logic, it is argued that Manlevelt kept clear water between logic and theology. Things divine, and especially the transubstantiation (...) of the consecrated host, do however play a crucial role in the development of his idea of the non-existence of substance. It is suggested that this might imply the miraculous nature of the whole universe. (shrink)

_ Source: _Volume 56, Issue 3-4, pp 292 - 319 With William of Ockham and John Buridan, Walter Burley is often listed as one of the most significant logicians of the medieval period. Nevertheless, Burley’s contributions to medieval logic have received notably less attention than those of either Ockham or Buridan. To help rectify this situation, the author here provides a comprehensive examination of Burley’s account of consequences, first recounting Burley’s enumeration, organization, and division of consequences, with particular attention to (...) the shift from natural and accidental to formal and material consequence, and then locating Burley’s contribution to the theory of consequences in the context of fourteenth-century work on the subject, detailing its relation to the earliest treatises on consequences, then to Ockham and Buridan. (shrink)

_ Source: _Volume 56, Issue 3-4, pp 201 - 221 This paper summarizes medieval definitions and divisions of consequences and explains the import of the medieval development of the theory of consequence for logic today. It then introduces the various contributions to this special issue of _Vivarium_ on consequences in medieval logic.

This paper studies John Buridan's octagons of opposition for the de re modal propositions and the propositions of unusual construction. Both Buridan himself and the secondary literature have emphasized the strong similarities between these two octagons (as well as a third one, for propositions with oblique terms). In this paper, I argue that the interconnection between both octagons is more subtle than has previously been thought: if we move beyond the Aristotelian relations, and also take Boolean considerations into account, then (...) the strong analogy between Buridan's octagons starts to break down. These differences in Boolean structure can already be discerned within the octagons themselves; on a more abstract level, they lead to these two octagons having different degrees of Boolean complexity (i.e. Boolean closures of different sizes). These results are obtained by means of bitstring analysis, which is one of the key tools from contemporary logical geometry. Finally, I argue that this historical investigation is directly relevant for the theoretical framework of logical geometry, and discuss how it helps us to address certain open questions in this framework. (shrink)

Relatively recent literature on supposition theory seems to use different modern logical tools of interpretation that can be generally described as formalizations. Since the act of formalizing may be understood as a process of changing its object in the sense of making it more formal, an assessment of this kind of approaches is necessary. Accordingly, our main goal in this paper is to analyze the formality of Peter of Spain’s theory of supposition and to evaluate its interpretation as a quantification (...) theory. Our main thesis is that although Peter’s theory presents certain weak notions of formality, the formality presupposed by the quantificational interpretation is nowhere to be found in his considerations. (shrink)

This thesis addresses three challenges posed by intentionality - the ability of our mental states and language to be about something - to a logician: an apparent reference to non-existent objects, intentional indeterminacy and the failure of substitutivity of coextensive terms in an intentional context. Since intentionality plays an important role in our everyday reasoning, a proper formal account of it is highly desirable, yet it requires a departure from classical logic. One can modify classical logic and adapt the formal (...) apparatus to account for the aforementioned problems (Graham Priest’s logic for intentionality serves as an example of such an approach in this work) or one can make an even more radical shift and seek for inspiration in a different logical tradition like the terminist logic developed in the Late Middle Ages by figures like William of Ockham or John Buridan. The second path is explored here as a modest attempt to show that once we abandon the bias against the history of logic as irrelevant, we can actually get access to firm logical solutions inaccessible from a classical perspective, while still practising logic proper. (shrink)

In this paper, the author offers an introduction to Marsilius of Inghen’s treatment of expositiones of sentences de incipit and de desinit in his treatise on Consequentiae, with an analysis of the various modi exponendi presented by Marsilius and an edition of the text. The author argues that, in the split between physical and logical approaches to the issues arising in analyses of incipit and desinit, Marsilius’ theory presents some hybrid features, but tends towards the logical end of the spectrum.

In order to elucidate his logical analysis of modal quantified propositions (e.g. ‘all men are necessarily mortal’), the 14th century philosopher John Buridan constructed a modal octagon of oppositions. In the present paper we study this modal octagon from the perspective of contemporary logical geometry. We argue that the modal octagon contains precisely six squares of opposition as subdiagrams, and classify these squares based on their logical properties. On a more abstract level, we show that Buridan’s modal octagon precisely captures (...) the interaction between two classical squares of opposition, viz. one for the quantifiers and one for the modalities. Finally, we argue that several aspects of our contemporary formal analyses were already hinted at by Buridan himself. (shrink)

_ Source: _Volume 55, Issue 4, pp 307 - 339 This article puts forward an analysis of the theory of signs contained in the _Prodidagmata ad logicam Aristotelis_, a compendium on logic written by the Flemish philosopher and Louvain professor Laurentius Ghiffene. Focusing on Ghiffene’s definition and division of a sign and his account of the problem of self-reference, the author argues that Ghiffene positioned himself in the tradition of the Conimbricenses and relied extensively on their influential commentary on Aristotle’s (...) _Organon_, published in 1606. The aim of the present contribution is to shed new light on the teaching of logic at Louvain during the years immediately preceding the rise of modern philosophy, an episode in the university’s history which remains almost entirely unstudied. (shrink)

Thomas Aquinas maintained that God foreknows future contingent events and that his foreknowledge does not entail that they are necessarily the case. More specifically, he stated that if God knows a future contingent event, this future contingent event will be necessarily the case de sensu composito, but not de sensu diviso. After emphasizing the unified nature of Aquinas’ notion of necessity, we propose an interpretation of his theses by restating them within the framework of non-normal modal logics. In this framework, (...) the K-axiom does not hold, i.e. the necessity operator does not distribute over the material implication. Moreover, assuming that Aquinas rejected the K-axiom is not only consistent, but also leads to a logical framework that allows us to understand other theses maintained by the Doctor Angelicus. In particular, we argue that Aquinas’ remarks on the principle of non-contradiction rest on an impossible worlds semantics for non-normal modal logics. (shrink)

_ Source: _Volume 55, Issue 4, pp 273 - 306 Jan Dullaert was a direct student of John Mair and a teacher of Gaspar Lax, Juan de Celaya, and Juan Luis Vives. His commentary on Aristotle’s _Peri Hermeneias_ addresses the foundations of propositional logic, including a detailed analysis of conditionals and the semantics of logical connectives. Dullaert’s propositional logic is limited to the immediate implications of the semantics of these connectives, i.e., their introduction and elimination rules. In the same context, (...) he discusses several alternative treatments of semantic paradoxes, paying most attention to the approaches derived from Martin Le Maistre and John Mair. (shrink)

In this paper we will argue that the ontology implicit in John Buridan’s modal octagon commits him to a form of contingentism. In particular, we will argue that Buridan is committed to denying the validity of the Barcan and converse Barcan formulae.

I argue that the essence that is actualized by existence is the essence that is a determinate nature in an individual and not the essence absolutely considered. This essence in individuals has a potential being that is actualized by existence. This thesis has important consequences for the essence/existence distinction in Thomas Aquinas.

Thomas Aquinas a quodam nostri temporis viro docto de potentiis inquirenti “dispositionalis modalitatis propugnator fortissimus” nominatus est. Huius tractationis scopus est, hanc assertionem criticae subicere analysi. Imprimis autem nonnulla Aquinatis de potentiis doctrinae elementa exponuntur, ea disceptationibus, quae nostro tempore aguntur, conferendo. Deinde duae de potentiarum modalitatis natura sententiae contrariae explicantur: scil. “modalitas dispositionalis” et “necessitas conditionata”. Quo exacto Aquinatis textus examinantur inquirendo, utram illarum sententiarum ille docuerit. Testimonia demum postremae faventia inveniuntur. Loco conclusionis auctor suadet, quomodo Aquinas exempla a (...) “modalitate dispositionali” sectatoribus proposita explicare posset. Ad quaestionem in titulo positam igitur auctor respondit opinionem de Aquinate ut fortissimo modalitatis dispositionalis propugnatore esse falsam, idque tropaeum cuidam alteri esse tribuendum.Aquinas has been labelled “the greatest advocate of dispositional modality”, by one contemporary power theorist. This paper’s goal is to critically analyse this claim. Before doing so, however, it first explicates some components of Aquinas’s ontology of powers, putting him in dialogue with contemporary discussions. Next it explicates the two competing views of the modality of powers, dispositional modality and conditional necessity, and proceeds to examine the textual basis as to which of the two Aquinas held. Ultimately the paper finds evidence in favour of the latter. The paper then concludes with a suggestion as to how Aquinas would explain examples given by those who advocate the dispositional modality position. In answer to the title, therefore, the paper argues that thinking of Aquinas as the greatest advocate of dispositional modality is a fiction, and that this award belongs to someone else. (shrink)

One of the founding myths of analytic philosophy is that the predicate logic that was developed in the late 19th century was far more powerful than its predecessors. This ambitious book argues on the contrary that medieval philosophers developed "a system of logic that is similar to the predicate calculus in richness and power" – or that, as Parsons put it in his presidential address to the APA, "the core of medieval logic is as accurate and as expressive as the (...) core of contemporary logic.". (shrink)