1st GPMR Workshop on Logic & Semantics

Medieval Logic and Modern Applied Logic

Rheinische Friedrich-Wilhelms-Universität Bonn, Germany

June 28-30, 2007


Franciscus de Prato on Reduplication

Fabrizio Amerini (Parma, Italy) and Massimo Mugnai (Pisa, Italy)

Our talk is divided into two parts. (1) In the first part, we mean to give a general characterisation of Franciscus de Prato's tract on reduplication (status of edition, manuscripts, historical sources), a short presentation of his content, and a brief description of our project of publication of some fourteenth-century anonymous treatises on reduplications. More particularly, we shall illustrate the relationship Franciscus's exposition of reduplication bears to Burley's and his strategy of reconnecting some scattered intuitions of Hervaeus regarding this issue to the treatment offered by Burley in the De puritate artis logicae. (2) In the second part, we shall attempt to provide a philosophical assessment of Franciscus's theory by comparing more closely it to Burley's and Ockham's logical analyses of reduplicative propositions. Specifically, our focus will be on the logical relationship holding between a reduplicative proposition and its exponentes.


  • Mugnai [1982]: M. Mugnai, La 'Expositio reduplicativarum' chez Walter Burleigh and Paulus Venetus, in A. Maierù (a cura di), English Logic in Italy in the 14th and 15th Centuries, Bibliopolis, Napoli 1982, 305-320.
  • Bäck [1996]: A. Bäck, On Reduplication: Logical Theories of Qualification, Brill, Leiden- New York 1996.

The Fate of the Fallacy of Accident

Alan Bäck (Kutztown, PA, USA)

Current accounts of the fallacy of accident do not attempt a very formal analysis. They find the fallacy to come about because a claim holding of a particular case on account of its special features is taken to apply to appeal to all the cases of the same type. We avoid the fallacy, we are told, if we take care to learn the material facts about the extent to which that claim holds. This approach has as its ancestor that traditional understanding of the fallacy of accident as being occasioned by taking an accidental attribute essentially.

Elsewhere I have argued that Aristotle, who originally named this fallacy, thought otherwise. Here however I wish to consider a medieval treatment of the fallacy of accident that analyzes it formally. On this view, the fallacy arises on account of the fallacy having a syllogistic form with a non-universal major premise. Hence its inference is not valid. I end by considering whether this syntactic approach offers a better method for solving the fallacy of accident.

Aquinas on the Modes of Predication: Deriving the Aristotelian Categories

Brad Berman (Philadelphia, PA, USA)

By late antiquity, Aristotle's Categories had long been a prominent member of the philosophical canon. Indeed, through to the early modern period, it regularly served to introduce budding young students to philosophy and, in general, would remain an object of serious reflection for them (Frede 1983/1987, p. 11). Commentaries, naturally enough, were just shy of ubiquitous. But in spite of all that attention, and perhaps often because of it, the Categories continued to pose a veritable labyrinth of interpretative challenges.

One such challenge, the heated issue of the sufficientia praedicamntorum, stems from Aristotle's well-known list of ten categories at Cat. 4 (1b25-2a4). The trouble is that, while Aristotle provides lists of categories on over fifty occasions in the surviving corpus, he only once elsewhere, at Topics I.9, offers the full set of ten, and he apparently intends many of those less robust lists as complete. Interpreters were thus quite concerned to determine whether and why the categories were ten in number.

Their investigations usually took the form of a derivation--i.e., an attempt to justify some subset of the ten by linking them to a purportedly independent investigation of a related subject.1 By the thirteenth century, a number of competing routes into such a derivation had already been advanced. Of them, it is Simplicius' method of examining the diverse modes of predication, as employed by St. Thomas Aquinas, that I here single out for scrutiny.

Aquinas, although he wrote no commentary devoted the Categories, took up the problem of the sufficientia praedicamntorum on two separate occasions, offering two distinct, if related, derivations (In Met. V, nn. 891-892 and In Phys. III, n. 322). His efforts have increasingly becoming the object of scholarly attention (e.g., Scheu 1944; Wippel 1987 and 2000). Yet, significant features of Aquinas' treatments have routinely gone either unnoticed or unappreciated.

In particular, I argue that the Platonic method of collection and division, widely employed by Aristotle, grounds much of Aquinas' theory of predication and analysis of 'being.' This, while somewhat complicating Aquinas' derivations, ensures that his classification of predicates via the system of Aristotelian categories will be exhaustive and neat. It further, however, is indicative of Aquinas' view that logic is a systematic enterprise, or scientific body of knowledge. And it is only in this light, I argue, that Aquinas' theory of categories and of predication can be properly understood.

Note 1: Medieval philosophers were generally more optimistic in their appraisal of Aristotle's methods than was Kant, who famously charged Aristotle of conducting his search for the categories in the absence of any sort of guiding principle (Critique of Pure Reason, A81/B107). Of late, interpreters' sympathies have begun to shift back into accord with the medievals (see, e.g., Ackrill 1963 and Kahn 1978).

Richard Billingham and Henry of Coesfeld on the Principles of the Expository Syllogism

Bert Bos (Leiden, The Netherlands)

In the fourteenth century logicians concentrate on the analysis of propositions. This approach can especially be found in England. The propositions are analysed with the help of an analysis of the terms that constitue them and that are distinguished in 'mediate' and 'immediate'. In this analysis, supposition theory was used. It was meant to be a tool in determining truth and falsity of propositions. Apparently it was thought to be useful in physics especially. The theoriy of probatio focussed on propositions about changens of qualities, about 'begins' and 'ceases' etcetera.

Richard Billingham's Speculum puerorum is a handbook in which such a probatio propositionis, or: propositional analysis, is presented, and which was influential in the whole of Europe. In my contribution I would like to discuss aspects of Billingham's handbook, and on a fairly large (ca. 60.000 words) commentary on it by a Henry of Coesfeld. The latter commentary is preserved in two manuscripts, giving his comments in two versions. In the introduction to his commentary,. Henry explais the use of Richard's book, its formal subject; he defines 'probare' and 'improbare', enumerates the ways in which a proposition can be analysed. He lays special emphasis on the expository syllogism, the 'empirical' syllogism, on account of which a proposition can be analysed.

Naming and Signifying in Anton Marty (1847-1914), as Exemplified by his Understanding of Medieval Supposition Theory

Laurent Cesalli (Freiburg, Germany)

At two points in his major work, Untersuchungen zur Grundlegung der allgemeinen Grammatik und Sprachphilosophie (1908, sec. 38 and 124), Anton Marty is expressly concerned with medieval supposition theory. Both cases serve to illustrate his own thoughts about the semantic functions of names, which Marty refers to as their naming and signifying (nennen und bedeuten). The present investigation wishes to pursue a twofold question: (1) What is Marty's semantics of names? (2) In what relationship does it stand to medieval supposition theory and in particular to the distinction between significatio, suppositio and appellatio?

Regarding (1): Names are called 'presentation-suggestives' (Vorstellungssuggestive) by Marty. In other words, their meaning consists on the one hand in the manifestation of a certain presentation in the mind of the speaker, and on the other hand in leading to the formation a presentation of the same object in the mind of the listener. In addition, names name objects: the name 'Man' signifies the presentation [Man] and names men (Marty is a nominalist with respect to universals).

Regarding (2): a) 'Signification' (Bedeutung), to Marty, agrees partially with 'significatio' according to the medieval terminists, since significatio is understood as intellectum constituere or intellectui repraesentare. They differ however in the fact that most terminists understand significatio (as opposed to suppositio) as a pre-contextual semantic function, going back to the impositio nominum, while in Marty such a distinction is absent. Signification, to Marty, seems thus to correspond both to significatio and suppositio. b) Also in the case of the pair appellatio/Naming (Nennung), there exists an important difference between Marty and the terminists, which becomes evident only upon clarification of their respective notions of existence: Marty and the terminists are united on the fact that appellatio/Naming is present when a term stands for something .presently-existing., however Marty works with a remarkable notion of existence which, for example, allows that the past object Socrates be named by the name 'Socrates' (and not only signified). Contrary to medieval authors, Marty differentiates between existence and reality, such that for him not all that exists is real: the contents of judgments, as well as the objects of certain presentations are not real, albeit existing, objects.

Nicholas of Cusa's Logical Way of Arguing Interpreted and Re-Constructed According to Modern Logic

Antonino Drago (Pisa, Italy)

20th Century's studies on mathematical logic concluded that the validity or not of the law of double negation constitutes the very borderline between classical logic and most kinds of non-classical logic. (1) Hence, this distinction is more basic than whatsoever foundational law of a single kind of logic, classical one too.

As a fact, several scientific theories essentially include double negated sentences, whose corresponding positive sentences lack of scientific evidence (DNSs). For instance, chemists inquiring about the unknown reality of atoms, had as basic tenet 'Matter is not divisible at a non finite extent'; thermodynamicists, owing to their ignorance of heat's nature, followed the principle 'It is not true that heat is not work', although heat cannot be entirely converted in work. (2)

As a further fact, even ancient scholars questioned classical logic, although in a partial way. (3) In particular, Cusanus rejected classical logic, but without qualifying Cusano's new logic. (4)

By considering DNSs as introducing an author to argue in a different logical world from the classical one, I examine, interpret and re-construct (5) Cusanus' first book of De Docta Ignorantia through the great number of DNSs occurring inside his exposition.

Cusanus' basic notions result to be DNSs, by at most restoring in each one the suppressed negative word; for instance, the oxymore 'docta ignorantia' is restored as 'ignorance informed by the infinite beings' - a sentence giving the exact meaning intended by Cusano.

Moreover, the studies on the more advanced theories of modern mathematics evidentiated, beyond potential infinity, different degrees of infinity; the least degree can be considered Cavalieri's infinity; it is generated by the geometrical intuition of either the existence of a point at infinity in a straight line, or the totality of real objects (omnes) (6); i.e., this degree of infinity is generated by applying one time only on real objects each of the two quantifiers (there exists, all). By equating this geometrical intuition to Cusanus' intellectualis visus, I analyse and re-construct as much as possible his exposition of the 'principle of transfert' (transferre).

  1. D. Prawitz and P.-E. Melmnaas: 'A survey of some connections between classical intuitionistic and minimal logic', in H. A. Schmidt, K. Schuette and H.-J. Thiele (eds.): Contributions to Mathematical Logic, North-Holland, Amsterdam, 1968, pp. 215-229. M. Dummett: Principles of Intuitionism, Oxford U.P., Oxford, 1977.
  2. A. Drago: 'Incommensurable scientific theories: The rejection of the double negation logical law', D. Costantini e M. G. Galavotti (eds.): Nuovi problemi della logica e della filosofia della scienza, CLUEB, Bologna, 1991, I, 195-202.
  3. Notice that Leibniz '.Letter to Arnauld', 14-7-1686, Gerh. II, Q., Opusc. 402, 513) suggested that two basic logico-philosophical principles underlay our theoretical activity, i.e. the non-contradiction principle and the principle of sufficient reason, itself stated by means of a DNS: 'Nothing is without reason, or everything has its reason, although we are not always capable of discovering this reason..' See also my papers: 'The birth of an alternative mechanics: Leibniz' principle of sufficient reason', in H. Poser et al. (eds.): Leibniz-Kongress. Nihil Sine Ratione, 2001, Berlin, vol. I, 322-330; 'La monade di Leibniz alla luce dello sviluppo della scienza moderna', in B.M. D'Ippolito, A. Montano and F. Piro (eds.): Monadi e monadologie. Il mondo degli individui tra Bruno, Leibniz e Husserl, Rubettino, Soveria Mannelli, 2005, 291-313.
  4. E.g., see E. Cassirer: Individuum und Kosmos in der Philosphie der Renaissance, 1927 (tr. It. Nuova Italia, 1950, 25-28, 31). F.A. Wyller: Identitaet und Kontradiction, MFCG 15 (1981) 104-120 tried to interpret Cusanus' thinking through non-classical logic.
  5. It is well-known that 'Nicholas did not express himself clearly. Many of his works were written in haste. [he] generates many different ideas [without working] out their implications.' Hence there is '... a gap between what he means and what he says...one has to interpret his unclear passages.' J. Hopkins: 'Nicholas of Cusa: First Modern Philosopher?', in P.A. French and H.K. Wettstein (eds.): Midwest Studies in Philosophy, XXVI (2002) p. 13.
  6. K. Andersen: 'Cavalieri's Method of Indivisibles', Arch. Hist. Ex. Sciences, 31 (1985) 291-367.
  7. My paper 'The introduction of actual infinity in modern science', Ganita Bharati, 25 (2003) 79-93 showed that these properties are shared by the elementary mathematics invented in 1918 by H. Weyl; see S. Feferman: "Weyl vindicatus; Das Kontinuum sixty years after", in C. Cellucci and G. Sambin (eds.): Temi e prospettive della Logica e dellla Filosofia della Scienza contemporanee, CLUEB, Bologna, 1988, 59-93 (also in Phil. Topics, 17, 1990).

Ockham's Supposition Theory as a Forerunner of Computational Semantics

Catarina Dutilh Novaes (Amsterdam, The Netherlands)

Supposition theories in general, and Ockham's in particular, are traditionally seen as the medieval counterpart of modern theories of reference. Elsewhere I have argued -- on the basis of historical, textual and conceptual arguments -- against this identification and in favor of an alternative interpretation of these theories; Ockham's supposition theory in particular can be described as a formal theory for the interpretation of sentences belonging to a certain class (known as categorical sentences). In other words, it is best seen as a forerunner of current computational semantics. In this talk, I present a formalization of this theory based on this core idea.

Divine Predetermination in Thomism: Infallibility and Necessity

Petr Dvořák (Prague, Czech Republic)

The Jesuit doctrine of middle knowledge and the so-called conditionals of freedom have attracted a great amount of attention in recent literature. Yet the other party in the old scholastic debate on free will, the Thomist theory of divine predetermination through physical premotion, has been almost entirely neglected. Nevertheless, this theory, hinging on the distinctions between necessity and infallibility and in sensu diviso, in sensu composito, is interesting in its own right and is capable of a fresh re-formulation in the possible worlds language of current modal logic. The paper will reconstruct (from a systematic point of view) the Thomist theory in its mature form in some leading 16th-18th century followers of D. Banez from different religious orders (Dominican, Cistercian, Benedictine): D. Alvarez, John of St. Thomas, J. Caramuel and L. Babenstuber in order to illustrate the breath and vibrancy of the late scholastic debates on the issue as well as to demonstrate the usefulness of the contemporary semantic and metaphysical conceptual apparatus once again.

Modes of Signifying - Interaction between Grammar and Logic

Sten Ebbesen (Copenhagen, Denmark)

Philology or Philosophy?: The first chapters of Moses Maimonides' 'Guide for the Perplexed' and their Latin translation

Görge Hasselhoff (Bonn, Germany)

Roughly one third of the first part of Maimonides' 'Guide for the Perplexed' (ca. 1190) is concerned with the explanation of several (Biblical) terms (e.g. 'place', 'throne', 'stand'). Most of these terms are normally applied on God. Maimonides' intention is to show that these terms are inadequate for a characterisation of God. The Latin translation of the originally Arabic-written work (probably Paris, 1242) leaves out several of Maimonides' explanations and definitions. In my paper I intend to show with three or four examples how Maimonides argued, what the Latin translator made of Maimonides' explanations, and what this meant for mediaeval philosophy.

The Enterprise of Mediaeval Logic

Peter King (Toronto, Canada)

Contemporary logicians study properties (such as completeness or compactness) of logical systems, taken as abstract structures usually defined with regard to a consequence-relation. That was not -- could not -- have been the concern of mediaeval logicians. So what was the enterprise of mediaeval logic? I look at the topics that were counted as "logical" in the Middle Ages, as well as explicit theoretical claims, to offer an account of mediaeval logic. This will allow us to have a much clearer idea of the extent to which modern techniques (such as formalization) are successful in capturing the mediaeval enterprise.

Via Antiqua vs. Via Moderna Semantics: Two Ways of Constructing Semantic Theory

Gyula Klima (New York, NY, USA)

This paper will take on the inherently anachronistic task of discussing the sort of theories we could get if we were to construct formal semantics based on intuitions coming from medieval realist, 'via antiqua', vs. medieval nominalist, 'via moderna', thinkers (rather than based on our own post-Fregean/Tarskian/etc. intuitions), in hopes of arriving at some philosophically intriguing conclusions about the theoretical advantages and disadvantages of constructing semantic theory in one way or the other.

Three Medieval Theories of Modal Syllogistic

Simo Knuuttila (Helsinki, Finland)

I shall deal with the theories of modal syllogistic by Robert Kilwardby, Richard Campsall and Jean Buridan. These were based on various approaches to modal semantics which gave rise to considerable variations in understanding the nature of modal logic. While Kilwardby's influential commentary on the Prior Analytics codified the central tenets of Aristotelian essentialism, Buridan was a representative of fourteenth-century semantics of modality as alternativeness. Campsall's work was influenced by late thirteenth-century developments. Contrary to some resent suggestions, I shall argue that Campsall was not a forerunner of specifically fourteenth-century modal semantics.

The Art and Obligation of Doubting

Jaap Maat (Amsterdam, The Netherlands), Katherina Rybalko (Amsterdam, The Netherlands), and Sara L. Uckelman (Amsterdam, The Netherlands)

Oxford MS Canon misc 281 contains a tract on obligatio which can be tentatively dated and placed in early 13th century France (the text was edited by de Rijk in Vivarium). The tract is divided into three sections, positio, dubitatio, and depositio. We are currently translating this text into English and will present work in progress concerning the middle section, on the obligation of doubting. After introductory remarks concerning the history and nature of obligatio material, plus a discussion of the different types of obligatio discussed in this tract, we discuss briefly how dubitatio differs from positio. The main part of our talk is devoted to a discussion of the rules for dubitatio. These rules are not straightforward and can be interpreted in various ways. We present a formalization of an interpretation of the rules which is coherent and consistent and provide examples demonstrating and applying these rules.

The Expository Syllogism

Calvin Normore (Los Angeles, CA, USA)

Richard Billingham claims (in his Speculum Puerorum) that the expository syllogism in the third figure is 'the foundation of all affirmative syllogisms' and that all negative syllogisms are founded on the negative expository syllogism. I examine these bold claims and Billingham's discussion of them, relate them to work by Parsons, Smith, Thom, Wolff and others on the nature of Aristotelian ecthesis, explore some of the relations between expository syllogism and the procedures of universal instantiation and existential generalization in Quantification Theory, and draw some conclusions for medieval conceptions of proof. And all in twenty minutes, promise!

Entia Rationis in Suárez: From the Decline of Logic to the Rise of Ontology

Daniel Novotny (Buffalo, NY, USA)

Famously, I. M. Bocheński has observed that the period 1400-1800 was a period of the decline of formal logic. Three tendencies can be observed at that time (a) humanism . an attempt to throw away logic; (b) the rise of the 'classical' logic; (c) a few attempts to extend the latter. Francisco Suárez (1548-1617), the father of modern metaphysics, does not fit well into any of these tendencies. He is a humanist in language but preserves medieval subtleties and rigorosity of argumentation; he respects logic (dialectics) yet he thinks that it takes up various questions of metaphysics that need to be re-claimed. One such cluster of questions that metaphysicians - and not logicians - need to address concerns the nature, the causes and the division of beings of reason (one species of which, namely the second intentions, were considered by the Thomists to be the object of logic). In my presentation I would like to present Suárez's complex theory of beings of reason and criticize its coherency. I would also like to show that post-Suárezian scholastic philosophers did not leave problems of Suárez's views unnoticed. Thus, it transpires in passing that Suárez should not be called 'the last original scholastic philosopher'.

Richard of Lavenham's Analysis of the Future Contingency Problem Represented in Terms of Modern Tempo-modal Logic

Peter Øhrstrøm (Aalborg, Denmark)

The literature on the problem of the contingent future is very comprehensive, and any attempt to cover it all seems hopeless. However, one may still gain a vantage point by trying to focus on some salient features and questions. This was exactly the approach taken by the medieval logician and philosopher Richard Lavenham (c.1380), who in his treatise De eventu futurorum gave a clear overview over the basic approaches to the problem within scholasticism. Lavenham herein addressed the problem already mentioned, the apparent contradiction between God's foreknowledge and the free will of men and women. His central idea is quite clear: If two dogmas are seemingly contradictory, then one must first of all accept or reject the reality of the contradiction. If the contradiction is accepted as real, then solving the problem implies that one must deny at least one of the dogmas. If the contradiction is rejected it must be demonstrated that the contradiction is only apparent and not real.

Denial of the dogma of human freedom leads to fatalism (1st possibility). Denial of the dogma of God's perfect foreknowledge can either be based on the claim that God does not know the truth about the contingent future (2nd possibility), or the assumption that no truth about the contingent future has yet been decided (3rd possibility). Rejection of the reality of the contradiction between the two dogmas must be based on the formulation of a system according to which the two dogmas, rightly understood, can be united in a consistent way (4th possibility).

Lavenham considered a central argument leading from God's foreknowledge to the necessity of the future and the lack of human freedom. The main structure of this argument is very close to what is believed to have been the Master Argument of Diodorus Kronos. It is clear from Lavenham's text that he had some knowledge of this old Stoic or Megaric argument through his reading of Cicero's De Fato. In a modern context it greatly facilitates discussion to represent the argument using a bit of formalism. In the following a version of the classical argument regarding the contingent future will be presented - here formulated in a non-theological manner.

In this paper, I intend to show how Lavenham's argument can be reformulated using the kind of modern tense logical formalism invented by A.N. Prior (1914-69), and how the four positions mentioned in Lavenham's text can be presented and discussed in terms of Priorean tempo-modal logic (including branching time semantics).

Ontology of Truth and Norms: A Paraconsistent and Paracomplete Lecture of Scotism

Luca Parisoli (Calabria, Italy)

I want to propose arguments for these assertions : 1) usually, the principle of explosion (ex falso sequitur quodlibet) is attributed to Pseudo-Scotus - I believe that Pseudo-Scotus - "Scotist" -, doesn't believe in the principle of explosion, at the contrary the same text is a critical argumentation against it - Scotist is stressing for a relevant concept of implication : 2) Scotism, by a strong ontological conception of norms as the issue of God's or human's will, refuses the universal validity of the principle of bivalence, in that Scotism choose a synchronical conception of contingence "when A is happening at t0, not-A is possible at t0" - so, it is also possible to stress that there are genuine moral dilemmas, i.e. true contradictions. There are metaphysical and practical Scotist reasons for a paraconsistent and paracomplete philosophy.

In In librum primum analyticorum, q. 10 is stated the principle of Explosion, but it is stressed that it is valuable only for formal contradiction (as ( is an object, and ((((), and the "exploding" conclusions are also formal. I'm stressing that this Scotist passage is at the contrary asserting the negation of the principle. The misunderstanding is in the classical paradigm of formal logic as a very dominant one, producing a reader of this Scotist passage looking only for lexical confirmation of a very obvious (for him) idea. So, it is very easy to ignore the adjective "formal": at the contrary, I want to stress that the adjective "formal" is a capital one.

The metaphysical structure of formalitates, possible worlds and reality (it is real every thing independent from intellect, and there are things real but not-existing in actual world) gives to Scotus the opportunity to create a link between God's omnipotence and human lawgiver omnipotence, by building a theory of normativity grounded on will, and nothing other than will. The principle of contradiction is a normative one in a legal system: you are obliged to consistency only if there is a positive norm establishing so. God is tied by nothing, so no principle, no idea, no virtue can be a limitation for Him. He can realise in our actual world everything, but everything He realises must be compossible to other things realised (Chimera is not such a thing in that it is absolutely simple and contradictory): Universal Law doesn't change, but God can will for me something against an universal norm. In the global ontology of all possible worlds, God can will to punish murderer and God can will that I murder my son and to praise me. This is a true contradiction, very different from Chimera, that is a super-Contradiction, always false.

  • E. H. Cousins, Bonaventure and the Coincidence of Opposites, Chicago 1978
  • Luca Parisoli, La contraddizione vera, Roma 2005
  • G. Priest, R. Routley, 'Lessons from Pseudo Scotus', in Philosophical Studies 42 (1982) 189-199

Positio impossibilis and concept formation: Duns Scotus on the concept of infinite being

Roberto Hofmeister Pich (Bonn, Germany)

The purpose is to investigate Duns Scotus's uses of the technique of positio impossibilis for building up his concept of ens infinitum or (ontological) infinitas intensiva in Quodl. q. 5 n. 2-4 (his mature version of that concept). Even though there are several studies about his understanding of infinity as proper notion of God and the fundamental intrinsic mode of being, there is no definite account of his conscious application of that kind of logical tool for the objective of concept formation - in the case of "ens infinitum" not primarily, or not only, as an exercise of logical consistency and valid consequences. In this sense, I would like to show that Scotus goes far beyond the "classical view" of the nature of concepts (shareable constituents of thought as abstracted formalities through which real aspects of the world are representatively known by the intellect), and even far beyond the classical empirical accounts of concept acquisition at all. In order to acquire a complex concept like "infinite being", Scotus is concerned first of all with compatibility of contents and conceivability. The way he starts with mathematical potential infinity according to Aristotle and arrives via positio impossibilis at ontological actual infinity through analysis of the notions of "whole" (totum) and "perfection" (perfectum) might be compared to specific contemporary theories of concepts like (i) conceptual analysis (beginnig with G. E. Moore and C. I. Lewis), since Scotus uses positio impossibilis to account for the constitutiveness of infinity for ontological purposes, and (ii) the semantics of counterfactual conditionals, therefore also the semantics of possible worlds. In this last case I mean particularly that part of intensional logics which takes "intension" as a kind of function which locates a possible world to the extension of the concept under definition in that world (for example, the formal semantics of D. Lewis and R. Stalnaker). Finally, it is well known that Scotus's concept of infinitas intensiva has been loosely compared to Georg Cantor's notion of the transfinite (Grundlagen (1883) and Math. Ann. 5 (1871), establishing a difference between conceptus rei proprius ex propriis and conceptus proprius ex communibis). Is this comparison consistent? The answer depends surely on how much Scotus's techniques of concept formation secundum imaginationem can be related in any possible sense to Cantor's accounts of irrational numerical quantities in mathematics.

Thomas Bradwardine and Epistemic Paradox

Stephen Read (St. Andrews, Scotland)

The most famous epistemic paradox is Fitch's paradox. In it, Frederic Fitch offered a counterexample to the Principle of Knowability (PK), namely, that any true proposition can be known. His example is any proposition which is true but not known. This proposition is not paradoxical or contradictory in itself, but contradicts the principle, PK, which many have found appealing.

Thomas Bradwardine, writing in the early 1320s, developed a solution to the semantic paradoxes (insolubilia) based on a closure principle for signification: every proposition signifies whatever is implied by what it signifies. In ch. 9 of his treatise, he extends his account to deal with various epistemic paradoxes. Comparison of Fitch's paradox with one of these paradoxes, the Knower paradox ('You do not know this proposition') explains the puzzlement caused by Fitch's paradox. Bradwardine's argument shows that the Knower paradox signifies its own truth, and is false.

Modal Models for Bradwardine's Theory of Truth

Greg Restall (Melbourne, Australia)

In this paper, I present modal models for Stephen Read's reconstruction of Thomas Bradwardine's theory of truth [1,2]. This model theory gives us tools to perform a number of tasks.

(1) We can show that this theory is consistent with relatively strong principles governing the truth of grounded sentences. The demonstration of consistency uses a novel application of a familiar fixed-point construction.

(2) We can show that theories not involving semantic notions may be conservatively extended with a Bradwardine-style truth predicate. This is an application of the previous result, showing how models for modal or classical theories may be extended with the apparatus required to interpret the truth predicate.

(3) We can compare Read's reconstruction of Bradwardine's theory with other theories of truth.

(4) We can vindicate Read's account that the insolubilia are all simply untrue by providing models in which all ungrounded sentences are untrue.

(5) We can examine Read's reconstruction of Bradwardine's argument to the effect that if x says of itself that it is untrue, then it says of itself that it is true as well. We provide models Read's theory in which this conclusion fails: we may models in which there is some x that says of itself that it is untrue, which demonstrably does not say of itself that it is true. This hinges on substantial issues on the interaction between signification, truth and modality.

In this talk I hope to introduce the model theory and to sketch a number of these applications -- given time, (1), (2) and (5).

  • [1] Stephen Read. 'The Liar Paradox from John Buridan back to Thomas Bradwardine'. Vivarium, 40(2):189.218, 2002.
  • [2] Stephen Read. 'Symmetry and Paradox'. History and Philosophy of Logic, 27:307.318, 2006.

Ibn Khaldun and Epistemic Logic

Hans van Ditmarsch (Otago, New Zealand)

My modest investigation in medieval logic started when I was reading a history of Cairo, while camping in Australia's Snowy Mountains, surrounded by hundreds of quietly grazing kangaroos. This rekindled the flame once lighted, in the 1970s, by De Rijk's most fascinating and well-remembered lectures on medieval logic.

Ibn Khaldun was a 14th century historiographer and author of the well-known Prolegomena. From a family originating in Seville, prior to its conquest ('reconquest') by the king of Castille, he lived an itinerant life serving as a magistrate for Spanish and Moroccan Islamic courts, including negotiating treaties with the Christian Spanish crown. He ended up in Cairo at the renowned Al-Azhar University. His career appears to mirror that of Leo Africanus, who lived a century later, in Granada, prior to that town's reconquest, and who ended his itineries in Rome at the court of the Pope.

I consulted Ibn Khaldun's Prolegomena from A to Z searching for references to logic or knowledge (in the authorative French translation from the 1930s - the first complete edition of the Prolegomena in a western language). There also appears to be a lost treatise on logic by Ibn Khaldun. The naive hypothesis I was trying to confirm or reject was whether Ibn Khaldun considered the three properties of knowledge as formalised in the logic S5: truthfulness, positive introspection, and negative introspection. My recent publication - not accidentally - entitled 'Prolegomena' refers to the existence of such text fragments, and suggests that the answer to that tripartite question is: yes, yes, no. I am now less certain of the two 'yes's.

The two relevant parts in the Prolegomena are the chapters 'on reflection', and 'on the nature of human and angelic knowledge' in volume 2 (426-430 and 433-435), and a chapter 'logic' in volume 3 (149-160). They can be summarized as follows. Reflection is the faculty that distinguishes humans from animals, who only possess the faculty of perception. Reflection provides proof of the existence of the human soul, because it allows us to know things that are not directly observed. Reflection also allows us to interact with the sphere of angels. The power of reflection can be measured as the maximum length of a cause-effect chain: "some people can still follow a series of five or six," and as the ability to avoid actions that result in unpleasant consequences.

It is tempting to see such reflection on acquired knowledge as a form of introspection in the modern epistemic logical sense. It is then comforting for a modal logician that awareness of knowledge provides proof of the existence of the soul. That knowledge of something corresponds to its being true, seems also easily read into various phrases. I did not find a reference to negative introspection.

  • M. de Slane (translator and commentator), Les prolegomenes d'Ibn Khaldoun (three volumes) Librairie Orientaliste Paul Geuthner, Paris 1934/1936/1938
  • H. van Ditmarsch, "Prolegomena to Dynamic Logic for Belief Revision", Synthese 147:229-275 2005

Logic and Ontological Commitment: St. Vincent Ferrer's Theory of Natural Supposition

Thomas Ward (Los Angeles, CA, USA)

Relative to other leading fourteenth century logicians, Vincent Ferrer (1350-1419) has received scant scholarly attention. Yet his De suppositionibus deserves more attention. Among anglophone philosophers, Ferrer has in recent decades been well-served by John Trentman, who published a critical edition of De suppositionibus, as well as several articles on Ferrer's logic. Ivo Thomas wrote a short paper on his theory of supposition, and Ferrer gets a brief mention by William and Martha Kneale in The Development of Logic. Yet Ferrer has yet to be given an amount of philosophical scrutiny proportionate to what the brighter lights of medieval logic-Ockham, Buridan, Burley et al-have received.

My paper tries to rectify this by focusing on Ferrer's theory of natural supposition. Walking what he calls a "middle way" between nominalist and realist logicians, Ferrer insists that logical theorizing can only proceed within a framework of specific ontological commitments. Following St. Thomas Aquinas (De ente et essentia, c.II), Ferrer distinguishing three modes of being of an essence: in individuals, in the soul, and "absolutely considered." Ferrer divides the modes of supposition along these lines: personal, simple, and natural, respectively. Ferrer argues that a proposition whose subject has natural supposition can be true even if the subject term is empty. But, also like Aquinas, Ferrer denies that the essence "absolutely considered" exists. Thus, Ferrer's theory of natural supposition seems to commit him to reference to nonexistent objects.

Ferrer's assertion that an universal affirmative categorical proposition whose subject supposits naturally can be true even when the subject term is empty, comports with the truth conditions of the material condition in modern formal logic: (A)x(Fx->Gx). But modern logicians consider it ontologically innocuous that the antecedent of a material conditional be false, and the conditional itself be true. But what makes the conditional true in such circumstances? Logicians such as Carnap would argue that such statements are true in virtue of meaning. Ferrer shows awareness of propositions that are true simply in virtue of meaning ( e.g., mortuus expiravit), but he denies that a proposition such as pluvia est aqua guttatim cadens, which can be true even if there is no rain, is true just in virtue of meaning. For Ferrer, a universal, e.g., rain, is a potential being which has rational structure even when not actualized in thought or in rain. This moderate realism is taken to provide the ontological backing to Aristotelian accounts of real definition. Were there no rain, certain predications of rain would still be true by virtue of this rationally structured potential being.

I close the paper by reflecting on aspects of Ferrer's semantics that should strike modern logicians as odd. For example, when a subject term has natural supposition, Ferrer argues that it always supposits for everything that falls under the term, even if the subject has an existential quantifier, such as quidam. I conclude that concerns about science and ontology, moreso than syntax, motivate Ferrer's conclusions.

Bradwardine's Theorem in a Relevant Framework

Elia Zardini (St. Andrews, Scotland)

The essence of Thomas Bradwardine's theory of truth consists in the claim that an utterance is true (if and) only if everything the utterance says is the case. The essence of his solution to the Liar paradox flows from this, and consists in the claim that a Liar utterance is false as, in addition to its saying of itself that it is false, it also says of itself that it is true, and so, by contravalence (no utterance is both true and false), not everything it says is the case. What we may call 'Bradwardine's theorem' in turn establishes on the basis of Bradwardine's definition of truth that the Liar utterance does say of itself that it is true.

We owe the revival of interest in Bradwardine's theory of truth to Steve Read's careful and penetrating interpretation and elaboration of Bradwardine's texts. I propose to focus on Read's reconstruction of the proof of Bradwardine's theorem. The crux of the proof consists in observing that if a Liar utterance λ is false, then something λ says is not the case. Letting Q be the weakest proposition entailing everything λ says that is not entailed by the proposition that λ is false, we have (ut nunc) that it is not the case that [l is false and Q]. Thus, if Q, then it is not the case that λ is false, and so, by bivalence (every utterance is either true or false), λ is true. By closure of saying-that under (ut nunc) logical conseQuence, λ's saying that λ is false and λ's saying that Q jointly entail λ's saying that λ is true, as desired.

Strictly speaking, in its implicit use of modus ponendo tollens, the above reasoning is relevantly invalid. This specific flaw might be thought to be remedied by reading .and' as intensional conjunction (fusion) rather than extensional conjunction. In the main part of the paper, I will study the semantic conditions under which the conclusion of the theorem can be expected to be had on this reading of .and'. The technical framework will be that of the (simplified) possible-world semantics for relevant logics. I will assume the constraints on the accessibility relation characterizing system R. The standard semantic structures will have to be modified with an extra accessibility relation for the saying-that operator and with definitions of propositions and operations on them (both extensional and intensional). In this framework, I will offer two semantic proofs of the conclusion of the theorem, one modelling closely Bradwardine's reasoning, the other one streamlining to the essential point behind it. Both proofs make use of a weak principle of closure of saying-that under (ut nunc) logical conseQuence and of a principle of constancy of saying-that. I will in particular defend the principle of constancy from possible objections. I will close by casting doubt on the legitimacy of another assumption implicit in the proofs of the theorem which the model-theoretical framework helps to bring out.