Medieval Logic

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)

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)

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)

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)

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)

A short reconstruction of the analytic-synthetic method, a key idea in Medieval and Early Modern Logic.

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)

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)

Machine generated contents note: -- Series Editors' PrefaceAcknowledgementsNotes on ContributorsHow Things Are Elsewhere; W. Schwarz Information Change and First-Order Dynamic Logic; B.Kooi Interpreting and Applying Proof Theories for Modal Logic; F.Poggiolesi & G.Restall The Logic(s) of Modal Knowledge; D.Cohnitz On Probabilistically Closed Languages; H.Leitgeb Dogmatism, Probability and Logical Uncertainty; B.Weatherson & D.Jehle Skepticism about Reasoning; S.Roush, K.Allen & I.HerbertLessons in Philosophy of Logic from Medieval Obligations; C.D.Novaes How to Rule Out Things with Words: Strong Paraconsistency and the Algebra of Exclusion; (...) F.Berto Lessons from the Logic of Demonstratives; G.RussellThe Multitude View on Logic; M.Eklund Index. (shrink)

The influence of Cajetan’s De Nominum Analogia is due largely to its first three chapters, which introduce Cajetan’s three modes of analogy: analogy of inequality, analogy of attribution, and analogy of proportionality. Interpreters typically ignore the final eight chapters, which describe further features of analogy of proportionality. This article explains this neglect as a symptom of a failure to appreciate Cajetan’s particular semantic concerns, taken independently from the question of systematizing the thought of Aquinas. After an exegesis of the neglected (...) chapters, which describe the semantics of analogy through the three levels of cognition, the article concludes with observations about the relationship between Cajetan and Aquinas and the philosophical and historical signifi cance of Cajetan’s approach to the semantics of analogy. (shrink)

The common view that Aquinas changed his mind about analogy (before and after De Veritate 2.11) is unwarranted. Dialectical context, and clarifications about the logic of analogy and the implications of proportionality, reveal consistency in Aquinas's teaching on the analogy of divine names.

Review of the English translation of Bernard Montagnes' influential 1963 monograph on analogy in Aquinas. (Pre-publication copy -- please cite final version.).

What role does Scotus‘s understanding of univocity play in Cajetan‘s development of a theory of analogy? In this paper I examine three relevant texts from Cajetan (question 3 of his commentary on Aquinas‘s De Ente et Essentia, his treatise De Nominum Analogia, and his commentary on question 13, article 5 of Aquinas‘s Summa Theologiae) in which Cajetan articulates his understanding of analogy at least in part through dialectical engagement with Scotus‘s arguments about univocity.

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)

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)

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)

In reply to the question whether articles of faith can be demonstrated, William of Ockham articulated a peculiar thesis: Even though it is impossible to prove articles of faith in this life, the blessed in heaven could demonstrate our creditive propositions. In contrast to traditional views, William held that both conclusions drawn in heaven and conclusions drawn in this life are subject to the same criteria. This assumption led to a controversy between William's contemporaries, namely, Walter Chatton and Adam Wodeham, (...) who discussed whether theories of propositions are applicable to articles of faith. This essay tries to explain why the problem of demonstrating creditive propositions arose and in what way William's answer gave a new meaning to the dispute. Document Type: Research article. (shrink)

Insolubilia novissima is a short anonymous fifteenth-century treatise on semantic paradoxes printed in the Cambridge compendium Libellus sophistarum. Along with a working edition of this treatise, basic information about its content and historical and systematic context is offered. Insolubilia novissima endorses the Swyneshedian contextualist solution to paradoxes based on distinguishing between compositional and contextual meaning of sentences.

John Mair was an influential post-medieval scholar. This paper focuses on his Tractatus insolubilium, in which he proposed semantic analysis of self-referential phenomena, in particular on his solution to alethic and correspondence paradoxes and his treatment of their general semantic aspects as well as particular applications. His solution to paradoxes is based on the so-called “network evaluation”, i.e. on a semantics which defines the concepts of truth and correspondence with reality in contextual terms. Consequently, the relation between semantic valuation, synonymy (...) and contradiction must be redefined. (shrink)

John Mair was an influential post-medieval scholar. This paper focuses on his Tractatus insolubilium, in which he proposed semantic analysis of self-referential phenomena, in particular on his solution to alethic and correspondence paradoxes and his treatment of their general semantic aspects as well as particular applications. His solution to paradoxes is based on the so-called “network evaluation”, i.e. on a semantics which defines the concepts of truth and correspondence with reality in contextual terms. Consequently, the relation between semantic valuation, synonymy (...) and contradiction must be redefined. (shrink)

In this essay, an attempt is made to formalize the idea of analogy in a way which is as faithful as possible to Thomas Aquinas’ theory of analogy. To accomplish this, we must first present Aquinas’ theory of analogy as it appears in his main works; we then express the contents of Aquinas’ theory of analogy using Leśniewski’s Ontology, a symbolic language which is both rigorous and true to the spirit of Aquinas’ philosophy. In doing this we present definitions and (...) theorems lying outside the scope of Leśniewski’s Elementary Ontology and we demostrate that the notion of “to be the definition of” is not an extensional functor. (shrink)

This paper addresses the question of whether there is a proper analogy of being according to both meaning and being. I disagree with Ralph McInerny’s understanding of how things are named through concepts and argue that McInerny’s account does not allow for the thing represented by the name to be known in itself. In his understanding of analogy, only ideas of things may be known. This results in a wholesale inability to name things at all and thereby forces McInerny to (...) relegate naming to a purely logical concern. As a consequence, for McInerny, since naming becomes only a logical concern, being itself cannot be known as analogous according to being and meaning since naming only involves the naming of ideas, not of things. (shrink)

This paper is consecrated to the problems of the semantics in the Anselm’s philosophy of language – one of the most important parts of his philosophical inquiry. The main care is focused to the analysis of terms veritas and rectitudo, mainly because of significatio and the semantics – e.g. significatio with respect to names. Special passage refers to denominative names, because in their case Anselm of Canterbury makes differences between significatio and appellatio in his semantic analysis.

The article deals with various modal versions of the ontological argument from N. Malcolm’s to P. Tichý’s interpretation of Anselm’s second proof. Three key presuppositions of the modal proof are pin-pointed and examined. The principal problem with the proof seems to be the notion of necessary existence attributed to God. More precisely, the question is whether this is not too strong an attribute, for then there would not be a situation, i.e. a possible world, consistently thinkable which precludes the existence (...) of God. However, this seems to be wrong. (shrink)

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