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)

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)

n this paper, a non-classical axiomatic system was introduced to classify all moods of Aristotelian syllogisms, in addition to the axiom "Every a is an a" and the bilateral rules of obversion of E and O propositions. This system consists of only 2 definitions, 2 axioms, 1 rule of a premise, and moods of Barbara and Datisi. By adding first-degree propositional negation to this system, we prove that the square of opposition holds without using many of the other rules of (...) classical logic (including double negation elimination). We then show that the Propositional Substructural Logic SLe is the best logic to study Aristotelian Syllogisms. Also, based on the IFLe square of opposition, the rules of conversation and the rules of negation are completely proved in Muzaffar's logic. For this purpose, we used the monadic first-order logic with the same standard deductive apparatus of quantifiers in classical logic, plus the axioms of "some a is an a" and "some not-a is a not-a". Finally, to show that there is no existential commitment to general terms in categorical logic, the Strong Four-Valued Relevant-classical Logic KR4 was used. With the same existential interpretation of the quantifiers and the standard translation of the quarter quantified. (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. -/- . . . Johannes Steuchius was a prominent Swedish Lutheran theologian and academic, a descendant of celebrated Lutheran bishops and academics. Born in 1676 in Härnösand in northern Sweden, Steuchius moved with his family to Lund in 1694 when his father, Matthias Steuchius, the renowned academic and theologian, was appointed Bishop of Lund. After completing his studies in logic and metaphysics at Uppsala University, Steuchius was given the opportunity to attend some of Europe’s foremost Protestant academic institutions, a luxury afforded to few. He continued his theological and philosophical studies as a visiting student at the universities of Rostock, Hamburg, Wolfenbüttel, Helmstedt, Wittenberg, Altdorf, London, Oxford, Amsterdam, Haarlem, and Leiden. During this period, he studied under many prominent Lutheran academics, such as Professor Johann Fecht, one of Germany’s leading representatives of Lutheran orthodoxy. He also met Gottfried Wilhelm Leibniz and visited the famous library in Wolfenbüttel during this time. (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...

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)

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)

In this essay, I consider some logical treatises and commentaries from the first decades of the 12th century (many of which are still unedited) which contain a discussion on modalities and modal logic. After presenting a short catalogue of these sources and a description of their common features, I shall focus on some definitions of the modal term “necessarium” which are provided in them. As we will see, Abelard and logicians of his time advanced three different characterizations of this term: (...) necessity was either defined in terms of unavoidability (ineuitabilitas), or in terms of immutability and omnitemporality (impermutabilitas, sempiternitas), or again in terms of absolute necessity as opposed to conditioned one (necessitas absoluta vs. determinata). I argue that the temporal understanding of necessity in terms of omnitemporality, inherited from ancient sources and extensively used by Abelard and others in the first years of the twelfth-century, started to disappear in texts datable from around the 1120, perhaps due to several difficulties that were related to this definition when applied in logical contexts. I also discuss how the notion of necessitas determinata was used by Abelard’s contemporaries to qualify the modal status of present and past events, which were generally believed to be necessary only in a “weak” and harmless sense that did not prevent them from being contingent. -/- . (shrink)

A sound historiographical account of the role of logic in the development of medieval philosophical and theological reflection requires a thorough examination of its historical roots and its theoretical implications. An apparent historiographical bias, due to the idea that only the development of contemporary formal logic enables a proper reconstruction of the whole history of logic, can be exposed by taking into account the case of the medieval discussions on the topics, starting from their late-antiquity legacy. An attentive inspection of (...) the successive stages of the gradual formalisation process of medieval topical logic brings to the fore significant theoretical aspects in the philosophy of logic, all too often left aside in common contemporary discussions. In particular, the distinction between topical maxims and topical differences, or principles according to and principles from which an inference is drawn, highlights the specific nature of a major part of medieval logic, that deals with meta-logical issues by means of higher-order logical resources. It is precisely this special character of medieval logic that proves to be of the highest significance in the discussion of the most subtle theological—and just as well philosophical—questions that engaged medieval authors, thus testifying to the essential role of logic in the overall development of medieval thought. (shrink)

In this paper I examine Avicenna’s conception of the consequence relation. I will consider in particular his categorical and hypothetical logics. I will first analyse his definition of the implication and will show that this relation is not a consequence relation in his frame. Unlike the medieval logicians, he does not distinguish explicitly between material and formal consequences. The arguments discussed in al-Qiyās, where the conclusion is true only in some matters, and would seem close to a material consequence for (...) that reason, are rejected explicitly as not syllogistic. He also rejects the ‘enthymemes’ unless they are complemented by their missing premise and the superfluous premises which, according to him, should promptly be ruled out. It seems then that the consequence relation in his theory is formal. It can be characterized as being ‘productivity in all matters’ or ‘necessary truth preserving’. It is illustrated by some single premise arguments, and above all by all kinds of syllogisms which, in his theory, are more numerous and various than in Aristotle’s one. The syllogism may contain two or more premises, including disjunctive ones. When it is hypothetical, it may lead to several conclusions. The premises may be in conflict, but then, the conclusion is false. He thus rejects the principle according to which ‘anything follows from a contradiction’. But, unlike what some scholars say, he does not admit any connexive principle. In the compound syllogisms, the conclusion follows by steps, each step taking two premises at once. (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)