History of Logic

website logic logoeBook version

My other websites:

Theory and History of Ontology

website ontology logoeBook version 

 

Bibliographia

website bibliographia logoeBook version  

History of Logic from Aristotle to Gödel by Raul Corazzon | e-mail: rc@ontology.co

 

Writings of E. J. Ashworth on the History of Logic. Second Part: Articles from 1973 to 1979

Articles 1973-1979

  1. Ashworth, Earline Jennifer. 1973. "Are There Really Two Logics?" Dialogue. Canadian Philosophical Review no. 12:100-109.

    "As a historian of logic, I am frequently puzzled by the things which people have to say about the relationship between mathematical logic and some other kind of logic which is variously described as ‘intentional’ and ‘traditional.’ Part of my puzzlement arises from my failure to understand precisely what kind of system is being offered under the guise of intentional logic. I have always taken it that logic is concerned with valid inferences, with showing us how we may legitimately derive a conclusion from a set of premisses; yet the validation of inferences seems to be the least of the concerns of the intentional logician. He says that it can be done, but he does not bother to show us how. My purpose in this paper is to list some of the sources of my puzzlement in the hope that an exponent of intentional logic will show me how they can be resolved, and how their resolution will contribute to the building of a system (however informal) in which different types of argument can be validated."

  2. ———. 1973. "Existential Assumptions in Late Medieval Logic." American Philosophical Quarterly no. 10:141-147.

    "There are three types of existential assumption that are commonly made by logicians: (1) that subject terms refer to non-empty classes; (2) that proper names have referents; and (3) that formulas are to be interpreted only within non-empty domains. In the standard first-order quantificational calculus with constants, the second and third of these assumptions are retained, but the first, which is attributed to traditional syllogistic, has been abandoned.

    Subject terms may refer to empty classes, and a distinction can be drawn within the system between those inferences which are valid only for non-empty classes and those which are valid for both empty and non-empty alike. For instance, given the assumption that universally quantified propositions whose subject terms refer to empty classes are true, but that existentially quantified propositions whose subject terms refer to empty classes are false, it turns out that the inference from "All As are B" to "Some As are B" only holds with the addition of the premiss, "There is at least one A." More recently, systems have been constructed in which the other two assumptions have also been discarded. Their valid formulas are valid in both empty and non-empty domains, and non-denoting constants are admitted. Any inference whose validity depends on the assumption that the domain of interpretation is non-empty, or that a constant denotes, is distinguished from the others by the presence of an extra premiss.(1) Thus, what was an assumption implicitly applied to all cases, is now made explicit and is shown to apply only to a subset of formulas within the system.

    It is frequently assumed that medieval logic operated with a group of implicit existential assumptions similar to those I have mentioned, but this view is erroneous. Late medieval logicians were just as concerned as contemporary logicians to deal with non-denoting terms within their systems, and to draw explicit distinctions between those inferences whose validity involves existential assumptions and those whose validity does not involve existential assumptions. It is inappropriate to ask whether they took their formulas to be valid within the empty domain or not, both because they worked with ordinary language rather than with formal systems, and because they did not use the notion of interpretation within a domain. When they interpreted a sentence such as "All men are animals," they did not speak of a domain of individuals some of whom were men and some of whom were animals, but only of those individuals who were either men or animals.

    However, they explicitly concerned themselves with the other two existential assumptions, and they admitted both non-denoting constants and terms referring to empty classes to their system. In this paper I intend to examine how some logicians of the late 15th and early 16th centuries interpreted sentences containing non-denoting terms, how they assigned truth values to them, and how they dealt with those inferences which needed an existential premiss to ensure validity." (p. 141)

    (...)

    "My discussion has been necessarily somewhat sketchy, and I have not examined all the contexts in which constantia was used,(28) but it should have become clear by now not only that late medieval logicians had clear views about the existential import of various types of sentences, but that they used their initial decisions about the truth and falsity of sentences containing non-denoting terms to build a consistent system. It is to be regretted that the vast majority of logicians after the third decade of the 16th century ceased to discuss these matters, with the result that modern readers tend to think of traditional logic as lacking a sophistication which it did indeed possess." (p. 147)

    (1) See, for example, W. V. O. Quine, "Quantification and the Empty Domain" in Selected Logic Papers (New York, 1966), pp. 220-223; Hugues Leblanc and Theodore Hailperin, "Non-Designating Singular Terms," The Philosophical Review, vol. 68 (1959), pp. 239-243; B. C. van Fraassen, "Singular Terms, Truth Value Gaps and Free Logic," The Journal of Philosophy, vol. 63 (1966), p. 481-495. There is a large and growing body of literature on the topic of logics which are free from existential suppositions.

    (29)

  3. ———. 1973. "Andreas Kesler and the Later Theory of Consequence." Notre Dame Journal of Formal Logic no. 14:205-214.

    "In another paper I examined the theory of consequence presented by a number of later fifteenth and early sixteenth century writers, ending with Javellus, an Italian who died in 1538. (1) For this earlier period, there was an abundance of material, containing much sophisticated discussion of semantical issues; but the next hundred years do not offer more than a few sources, and these are of limited value. The only really outstanding figure, so far as I can see, is that of Andreas Kesler. He was a Protestant theologian who was born at Coburg in 1595, educated at Jena and Wittenberg, and died in 1643 after a long career in education. In 1623 he published a book entitled De Consequentia Tractatus Logicae which is unique, both for its own time, and as compared to the products of this earlier period, in that it explicitly subsumes the whole of formal logic under the theory of consequence. The laws of opposition and conversion, the categorical and hypothetical syllogism, were all seen as different types of consequence. Moreover, no extraneous material was included. Instead of starting with the categories, like the Aristotelians, or with the invention of arguments, like the Ramists, he devoted his first chapter to the definition of consequence. Topics, informal fallacies and other such subjects found no place, whereas some rarely discussed matters like exclusive and reduplicative propositions and the modal syllogism did appear. Thus he stands out for his contents as well as for his organization." (p. 205)

    (...)

    "After this brief survey one can only conclude that the theory of consequence suffered an abrupt decline after the first part of the sixteenth century. The one outstanding writer on the subject was Andreas Kesler, but he stands out for a single insight, rather than for any awareness of the ramifications of the theory. Unlike his sources, he saw that all of formal logic could be subsumed under the basic notion of consequence, and he was able to exclude extraneous material, but that was as far as he went. About the definition and division of consequence, and about consequential rules, he had nothing to say but what had been said before him by Fonseca and Regius. Nor did he betray any knowledge of earlier writers, although some at least must have been available to him in Wittenberg. For once those who deplore the loss of mediaeval insights during the sixteenth century seem to be justified." (p. 210)

    (1) See my paper "The Theory of Consequence in the Late Fifteenth and Early Sixteenth Centuries,", to appear in Notre Dame Journal of Formal Logic, vol. XIV (1973), No. 3, pp. 289-315.

  4. ———. 1973. "The Theory of Consequences in the Late Fifteenth and Early Sixteenth Centuries." Notre Dame Journal of Formal Logic no. 14:289-315.

    "In this paper I intend to examine the treatment accorded to consequences by a group of writers from the late fifteenth and early sixteenth centuries, although I shall make some reference to earlier periods. The subject of consequences (or valid inference) is of central importance to the historian of logic because those who discussed it covered such a wide range of logical issues, including criteria for validity, problems of self-reference, the status of the so-called paradoxes of strict implication, and the systematization of valid inference forms. Indeed, a large part of semantics and the whole of formal logic could be subsumed under this general heading. Whether the authors themselves fully appreciated that this was so is unfortunately not such an easy question to answer, for those I am concerned with frequently leave the reader in doubt as to their view of the relation of consequences to the rest of logic. So far as they discussed the matter, syllogistic was seen to be consequential in nature,(1) but they certainly did not make the subordinate position of the syllogism as clear as Burleigh had in the fourteenth century, or indeed as Andreas Kesler was to do in the seventeenth century.(2) A good guide to the way they viewed the problem is to see where consequences were discussed.

    A very few authors, including J. Major, A. Coronel and J. Almain, devoted a whole treatise to them, but generally speaking they came in on the coat-tails of other topics so far as separate treatises were concerned.

    They appear at the beginning of Dolz's treatise on the syllogism, at the end of Celaya's treatise on supposition and under 'hypothetical propositions' in the treatises on opposition written by R. Caubraith and F. Enzinas. The best places to look for a discussion of consequence turn out to be commentaries on Peter of Spain, where they appear either as an appendage to the Parva Logicalia or under the heading of 'hypothetical propositions', and, of course, general textbooks of logic. In these, a separate tract was sometimes devoted to consequences, as it was by C. Javellus, but more usually they were associated with the syllogism, whether as an introduction to it or, sometimes, as an appendix to it. Savonarola, for instance, said all he had to say of consequences in a section on the powers of the syllogism.

    The bibliography at the end of this paper should give a fairly clear picture of the situation; though it must be noted that the majority of commentaries and textbooks belonging to the sixteenth century did not mention consequences at all." (pp. 289-290)

    (1) Enzinas, Tractatus Syllogίsmorum, fo.I vo, said "syllogismus est consequentia bona et formalis . . . omnis consequentia formalis que non tenet gratia alicuius regule logicalis tenebit syllogistice." Cf. Heirich Greve, Parva Logicalia nuper disputata, Leipzig (149-).fo. lxxi.

    (2) Andreas Kesler, De Consequentia Tractatus Logicus (Wittenberg, 1623). See my paper, '*Andreas Kesler and the later theory of consequence," Notre Dame Journal of Formal Logic, vol. XIV (1973), pp. 205-214.

  5. ———. 1973. "The Doctrine of Exponibilia in the Fifteenth and Sixteenth Centuries." Vivarium no. 11:137-167.

    Reprinted as essay IX in: Studies in Post-Medieval Semantics.

    "One of the most neglected parts of late medieval logical theory is that devoted to exponibilia, or those propositions which need further analysis in order to lay bare their underlying logical form and to make clear under what conditions they can be said to be true or false. My main intention in this paper is to examine the rich array of printed sources which are available to us from the later fifteenth and early sixteenth centuries, but I will consider some texts written before the invention of printing, and I will also give some account of what happened to the theory in the late sixteenth and seventeenth centuries. The sources fall into three main groups. There are separate treatises on exponibles, especially those written by Peter of Ailly(*) and later Parisian logicians; there are commentaries on the treatise on exponibles attributed to Peter of Spain; and there are those parts of longer works which were devoted to ‘Proofs of Terms’, as in Paul of Venice and his followers. These groupings are not, of course, exhaustive. For instance, Marsilius of Inghen and George of Brussels discussed exponibles in the second part of a treatise on consequences, and Albert of Saxony included exponibles in the part of Perutilis Logica devoted to different kinds of proposition. As might be expected, the authors of the separate treatises on exponibles were considerably more detailed and careful in their analysis than were those authors who treated exponibles as a subsidiary matter. In my view, the two most outstanding treatises are those written by Peter of Ailly (d. 1420) and by Domingo de Soto (d. 1560). The latter is not original, but it is a very acute and thorough survey of the doctrines which were current in late fifteenth and early sixteenth century Paris, where de Soto had studied under and with such logicians as Major, Celaya and Lax, whose names will frequently occur in my text. Outside treatises devoted to exponibles, good brief treatments are to be found in the anonymous commentator on Marsilius of Inghen, and in George of Brussels, (both of the later fifteenth century) and in Hieronymus of St. Mark (of the early sixteenth century). The earlier writers are often disappointing.

    For instance, although Paul of Venice’s Logica Magna is sometimes described as an encyclopedia of medieval logic, the section on exponibles lacks the precise analysis of types and sub-types of exponible propositions found in other authors, and the examples are frequently confusing.

    Similarly, the treatise wrongly attributed to Peter of Spain lacks detail, and derives most of its value from the remarks of commentators." (pp. 137-138)

    (...)

    "To conclude, one can say that the history of exponible propositions mirrors the history of medieval logical doctrines in general. At the end of the fifteenth and beginning of the sixteenth centuries there was a sudden surge of activity, during which such topics as exponibilia, insolubilia and suppositiones were analyzed, clarified and elaborated in works which are highly respectable from the logician’s point of view, even if they contain little that is original. This period of activity was followed by a period of decline, in which medieval doctrines continued to receive some attention, especially in Spain, but they are clearly subordinated to the main business of expounding Aristotelian logic. By the end of the seventeenth century they cease ever to be mentioned." (p. 165)

    [* Peter of Ailly, Tractatus Exponibilium, Paris c. 1495?]

    [** Domingo de Soto, Opusculus Exponibilium, in Introductiones dialectice, Burgis 1529]

  6. ———. 1973. "Priority of Analysis and Merely Confused Supposition." Franciscan Studies no. 33:38-41.

    "In a recent article John J. Swiniarski discusses William of Ockham's use of merely confused supposition.(1) He claims that, in the case of universal affirmative propositions, Ockham's method of attributing merely confused supposition to the predicate accomplishes much the same result as Peter Thomas Geach's method of attributing determinate supposition to the predicate and using a priority of analysis rule, whereby the subject is always analysed first. However, he notes, Ockham's analytical procedures when applied to particular negative propositions can lead to erroneous results,which are only avoided by the adoption of a priority of analysis rule. Since such a rule renders merely confused supposition unnecessary, he concludes that Geach was right and that Ockham ought to have employed only distributive and determinate supposition to get her with a priority of analysis rule in his treatment of standard categorical propositions. I do not wish to criticize what Swiniarski has to say about the interpretation of Ockham. Instead, I wish to make a few remarks about the use of merely confused supposition by sixteenth century logicians in order to show that it is not in general so easily dispensed with. (2)" (p. 38)

    (...)

    "In the light of these two examples, I conclude that there was good reason for sixteenth century logicians to retain merely confused supposition, and to use Domingo de Soto's priority of analysis rule rather than Geach's." (p. 41)

    (1) Swiniarski, John J., "A New Presentation of Ockham's Theory of Supposition with an Evaluation of Some Contemporary Criticisms," Franciscan Studies, 30 (1970), 209-217. Those readers who are not familiar with supposition theory should be reminded that merely confused supposition involves an analysis into a disjunctive subject or predicate, whereas distributive supposition involves an analysis into a conjunction of propositions and determinate supposition involves an analysis into a disjunction of propositions.

    (2) For further details about supposition theory in the sixteenth century, see my paper: " The Doctrine of Supposition in the Sixteenth and Seventeenth Centuries, " Archiv für Geschichte der Philosophie, 51 (1969), 260-285.

  7. ———. 1974. "Some Additions to Risse's Bibliographia Logica." Journal of the History of Philosophy no. 12:361-365.

    "One of the greatest contributions to the history of logic in recent years was the publication in 1965 of Wilhelm Risse's Bibliographia Logica, Vol. I, which covers the years from 1472 to 1800. However, despite the fact that Risse's monumental work lists an estimated 8,000 logical works, it is still far from comprehensive, as Mr. Hickman pointed out in an earlier article in this journal. Why this should be the ease immediately becomes apparent when one starts to work in a library such as the Bodleian at Oxford with its handwritten catalogue of books printed before 1920 and its lack of any specialized bibliographies such as the British Museum has provided for early printed books. Even in well catalogued libraries such as the University Library at Cambridge it can be difficult to locate texts, and one often stumbles across a new logical work through the accident of its being bound in the same volume as better known works. As a result of my researches over the last few years, I have put together a list of works which do not appear in Risse in the hope that other historians of logic may benefit from my discoveries. I cannot, however, claim that I have exhausted the resources of the libraries which I have visited. Doubtless there are still not only new editions but new authors left to be discovered." (p. 361)

    "This paper concerns logic texts published between 1472 and 1800. I list 20 items whose authors do not appear in Risse, 12 items whose authors appear in Risse in connection with another title or other titles, and 58 items which appear in Risse in another edition or in other editions. I indicate the libraries in which all these items are to be found, and I also list some useful bibliographical works."

  8. ———. 1974. "Classification Schemes and the History of Logic." In Conceptual Basis of the Classification of Knowledge / Les fondements de la classification des savoirs, edited by Wojciechowski, Jerzy A., 275-283. New York - München - Paris: K. G. Saur.

    Proceedings of the Ottawa Conference on the Conceptual Basis of the Classification of Knowledge, October 1st to 5th, 1971.

    "Logic is one of the most important means of classification we have, for it enables us to appraise our reasoning by drawing the distinction between valid and invalid inferences. Its aim is a simple one, and easily stated, but when we get down to the task of specifying under precisely what conditions a true premiss set will entail a true conclusion, it seems that a whole range of different types of classification is necessary.

    Logicians commonly start by drawing the distinction between informal or natural languages and formal or artificial languages. Even at this point, divergent interpretations are possible. One can argue with the early Wittgenstein that natural language has a hidden ideal structure, which it is the task of the logician to uncover; or one can argue with the later Wittgenstein that natural language involves a series of games with different structures, any one of which the logician can choose to present as a formal language. Given both the complexities of natural languages and the variety of formal languages which have been developed, the latter interpretation is by far the most pausible. Once the notion of a formal language has been isolated, one can go on to draw the distinction between syntax, or the study of the relations of signs among themselves, and semantics, or the study of signs as interpreted, as having meaning and as being true or false. In turn we can obtain the notion of different types of logical calculi. For instance, a propositional calculus has one set of signs with certain limited transformations of these signs, and it is interpreted by the assignment of truth values to its constituent parts; whereas a quantificational calculus has a more elaborate set of signs with transformations to match, and it is normally interpreted by means of the assignment of members or sets of members of domains to its constituent parts.

    The teacher of logic is often tempted to claim that these types of classification are integral to the study of logic. This is true when logic is seen as the foundation of mathematics, but to say that only, through these distinctions can one sensibly talk about valid and invalid inferences is a much larger claim, and a more dubious one. I intend to look at selected aspects of the history of logic in order to throw some light on the problem of just what kinds of classification are necessary to the isolation of valid inferences, which I take to be the true task of logic. In particular, I shall look at the definitions of valid inference offered by the Scholastic logicians of the late fifteenth and early sixteenth centuries, since this is the historical period with which I am best acquainted. (1)" (p. 275)

    (...)

    "Are we now to conclude that elaborate classification schemes are irrelevant to the pursuit of logic, so long as we have an adequate definition of a valid inference? The answer to this question will depend in part on how much one wants out of logic.

    If one wishes to study the metalogical properties of formal systems, to obtain a complete set of rules, or to relate logic to mathematics, scholastic logic is necessarily inadequate. However, if one wishes to classify those inferences which are used in ordinary language, then one can argue that an elaborate classificatory apparatus combined with the development of formal systems will be a hindrance rather than a help. Even the simplest sentence contains subtleties which will be lost in symbolization. Moreover, there is the grave problem of which system to choose when one is symbolizing and assessing an inference. This problem has two facets. First, one may pick a system which is inadequate to one’s purposes. If one attempts to show that a relational inference is valid in terms of the standard monadic predicate calculus, one will fail. Yet one has not proved that the inference in question is not valid. Second, one may pick a system whose standard interpretation is alien to one’s purposes. A logician who wishes to show that ’—P’, therefore ’P’ holds would be ill advised to choose the intuitionist propositional calculus. Similarly a logician who wishes to show that ” ’Fa’ therefore ’(Ex)Fx’ ” should not choose a version of the quantificational calculus which admits non-denoting constants.

    The more systematic one’s approach to formal logic, the more arbitrary the choise of system seems to be, and hence the less relevant to the normal day to day task of assessing arguments. Scholastic logic, on the other hand, seems perfectly adapted to normal requirements. It is both unpretentious and powerful; it does not violate normal intuitions; and it is non-arbitrary. Or so one might think.

    However, let us look a little more closely. What are we to make of the following claims? ”An impossible proposition implies any other proposition.” ”A necessary proposition follows from any other proposition.” ”If you come to me I will turn you into an ass” is true provided that you do not come to me.” ’’All chimeras are chimeras” is false because there are no chimeras, but ”No chimeras are chimeras” is true for the same reason.” The first two examples, the paradoxes of strict implication, follow straight from the definition of a valid inference. The third example is a consequence of the truth-functional interpretation given to promissory conditionals. The last examples are a consequence of the arbitrary decision to save the square of opposition by counting all affirmative propositions with non-referring subjects as false. Yet none of the examples corresponded to the normal intuitions of the sixteenth century. They all gave rise to acrimonious debate, and were accepted only because of the exigencies of the desired system of rules and the desired interpretation of that system. Thus even the scholastics, operating within the framework of ordinary language, were forced to make some of the arbitrary decisions which people tend to blame modern logic for. One may still prefer scholastic logic to modern logic for various reasons, but that it enshrines a true and completely non-arbitrary system of picking out valid inferences cannot be one of them.

    In the last resort, the presence or absence of modern classification schemes logic does not make so much difference as one would like to think." (pp. 282-283)

    (1) I intend to use the term ’scholastic logician’ more narrowly than is proper, to refer to those men whom 1 am concerned with.

  9. ———. 1974. "For Riding is Required a Horse: A Problem of Meaning and Reference in Late Fifteenth and Early Sixteenth Century Logic." Vivarium no. 12:146-172.

    Reprinted as essay I in: Studies in Post-Medieval Semantics.

    "One of the most interesting features of the works of the logicians associated with the University of Paris in the late fifteenth century and the first part of the sixteenth century is their application of medieval logical doctrines to the discussion of actual examples. In this paper I intend to present a detailed study of one specific example, "For riding is required a horse" [Ad equitandum requiritur equus]. I shall first discuss each of the arguments that was used, showing its place in the general body of logical doctrine; then I shall present three typical texts, together with an analysis of the pattern of argument found in each. One text will deal with the problem in the context of contradiction, one in the context of conversion, and one in the context of supposition theory.In this way I hope to deepen our understanding both of the theories and of the techniques of medieval and post-medieval logic." (p. 146)

    (...)

    "The claim that the gerund 'riding' implies a reference to particular acts of riding,which can in turn be identified with individual horses, solved the problem of "For riding is required a horse" at the expense of raising further philosophical problems about both language and the world. However, the claim that the sentence should be regarded as equivalent either to a simple conditional or to some kind of modal proposition solved all the problems very neatly without, apparently, raising new ones. In the light of such an analysis one could maintain the truth of "For riding is required a horse" without at the same time having to argue that the sentence had the same truth-value as its contradictory or a different truth-value from its simple conversion, since these related sentences would have undergone a similar analysis, thus turning out to have the desired truth-values. Whichever solution one prefers, it seems to have been amply demonstrated that the simple minded approach in terms of personal supposition alone was inadequate. To speak of horses being required for riding is to do more than to make reference, successful or otherwise, to individual horses or any other identifiable objects in the world." (pp. 157-158)

  10. ———. 1975. "Descartes' Theory of Objective Reality." New Scholasticism no. 49:331-340.

    "In the Third Meditation Descartes, who is at the beginning sure only of his own existence, presents a complex proof for the existence of God which is based on the fact that he finds within himself an idea of God. I intend to ignore the supplementary proof which deals with the conservation of his existence, and to focus on his discussion of the properties of ideas, for it is here that Descartes is most difficult to comprehend yet most vulnerable to criticism. With the exception of Gassendi's remarks in the fifth objection, I shall concentrate upon what Descartes himself had to say, for a thorough survey of all the secondary sources often serves only to obscure the main issue." (p. 331)

    (...)

    "Descartes reinforced his arguments with various claims about the nature of predicates and the way in which we come to understand them. He thought, mistakenly, that one could not only distinguish between negative and positive predicates, but that one could demonstrate the logical priority of such positive predicates as 'infinite' or 'perfect' by showing that one can only understand the finite or imperfect in the light of a prior acquaintance with the infinite or perfect. (29) However, although he seems now to be talking about epistemology rather than ontology, it turns out that his claims rest upon the same assumptions about the content and causation of ideas as are involved in the main proof, so they do not need to be discussed further.

    However liberal one is in granting Descartes his desired premises, I think it is fair to conclude that his arguments do not prove what they purport to prove. This seems to be a strong indication that one will lose nothing by being illiberal from the very beginning." (p. 340)

    (29) E. Haldane and G. Ross, The Philosophical Works 0f Descartes (Cambridge, 1968), I, 166.

  11. ———. 1976. "I Promise You a Horse. A Second Problem of Meaning and Reference in Late Fifteenth and Early Sixteenth Century Logic (First Part)." Vivarium no. 14:62-79.

    Reprinted as essay II (first part) in: Studies in Post-Medieval Semantics.

    "The logicians associated with the University of Paris in the late fifteenth and the first part of sixteenth century were at one with their medieval predecessors in their attempt to formulate a unified theory of the reference of such general terms as 'horse'. To be successful, any such theory has to give a plausible account of what happens to general terms in modal and intentional sentences, and the logicians I am concerned with clearly tried to deal with this problem. However, because of the rather standard way in which logic texts tended to be organized, the relevant material has to be sought in various places. In an earlier paper, I made a detailed study of the reference the word 'horse' was said to have in the modal sentence, "For riding is required a horse"; and in order to carry out that study, I had to draw material from the discussion of contradiction, of conversion, and of supposition. (1) In this paper, I intend to make a detailed study of the reference the word 'horse' was said to have in the intentional sentence "I promise you a horse", and my material will be drawn from the discussion of contradiction, of conversion, of supposition and of appellation. (2) I shall first examine each of the arguments that was used, showing its place in the general body of logical doctrine; then I shall present four typical texts, together with an analysis of the pattern of argument found in each.

    One text will deal with the problem in the context of contradiction, one in the context of conversion, one in the context of supposition, and one in the context of appellation. In this way I hope to show what problems intentional sentences were seen to raise for the standard theory of reference, and how these problems were dealt with." (pp. 62-63)

    (...)

    "On the whole, it seems fair to say that the logicians I have examined failed to produce a theory of the reference of general terms which applied with equal success to all contexts. Some, like Sbarroya, found themselves forced to emphasize the difference between intentional and non-intentional contexts by postulating completely different types of referent. Some, like Heytesbury, overlooked the difference altogether in their appeal to personal supposition. Some, like Buridan, recognized that terms in an intentional context have a function which goes beyond that of referring to individual objects; but they were unable to say with precision just how this broader function was to be reconciled with the referential function. However, one thing is common to those who struggled with the logical problems caused by "I promise you a horse". That is, they managed to save the validity of those inferences they were concerned with, either by so interpreting sets of sentences that they were not to be counted as instances of the inferences in question, or by so interpreting sets of sentences that they came out to have the desired truth-values, and could no longer be cited as counter-examples to a general rule. Thus, they were successful as logicians, if not as philosophers of language." (p. 78)

    (1) " "For Riding is Required a Horse": A Problem of Meaning and Reference in Late Fifteenth and Early Sixteenth Century Logic". Vivarium, XII (1974) pp. 146-72.

    (2) Enzinas, Pardo and de Soto discussed the matter in the context of their discussion of contradiction; Celaya, Coronel (Prima Pars), Sbarroya and de Soto discussed the matter in the context of their discussion of conversion; Hieronymus of St. Mark and Martinez Siliceo discussed the matter in the context of their discussion of supposition; Tartaretus discussed the matter in the context of his discussion of descent; and Coronel (Secunda Pars), Dorp, Hieronymus of St. Mark, Major, Manderston, Mercarius and Pardo discussed the matter in the context of their discussion of appellation. It will be noted that some authors discussed the matter in more than one place. For details of the texts, see the bibliography at the end of the paper. Of the authors cited, Hieronymus of St. Mark and Sbarroya are not, so far as I know, specifically associated with Paris, though they are clearly influenced by Parisian logicians.

    For medieval discussions of the problem, see P. T. Geach, "A Medieval Discussion of Intentionality" in Logic Matters (Oxford, 1972) 129-138, and J. Trentman, "Vincent Ferrer and His Fourteenth Century Predecessors on a Problem of Intentionality" in Arts Libéraux et Philosophie au Moyen Âge (Montréal/Paris,1969) 951-956.

  12. ———. 1976. "I Promise You a Horse. A Second Problem of Meaning and Reference in Late Fifteenth and Early Sixteenth Century Logic (Second Part)." Vivarium no. 14:139-155.

    Reprinted as essay II (second-part) in: Studies in Post-Medieval Semantics.

    Parto Two: Texts and Analyses.

  13. ———. 1976. "Agostino Nifo's Reinterpretation of Medieval Logic." Rivista Critica di Storia della Filosofia no. 31:353-374.

    "A year ago, if I had been asked to give a brief account of medieval logic and its relationship to Renaissance logic, I would probably have said something like this. In the medieval period, logicians had made great advances in the areas both of semantics and of formal logic. In the area of semantics, we find lengthy and sophisticated discussions of terms, of propositions, of supposition theory, which dealt with the reference of terms in various contexts, and of insolubilia, or semantic pardoxes, with their farreaching implications for our ordinary assumptions about the truth and reference of propositions. In the area of formal logic, we find equally lengthy and sophisticated discussions of consequentiae,

    or valid inference forms for both unanalyzed and analyzed propositions, and of exponibilia, those propositions whose logical form needs to be uncovered by means of analysis. A natural result of these advances was a relative down-grading of Aristotle. Aristotelian syllogistic was put in a subordinate place, as just one variety of valid inference, and in general

    the logical works of Aristotle did not receive as much attention as one might have expected. Medieval logicians were as likely to comment on Peter of Spain or to write independent treatises on particular topics as they were to comment on Aristotle; and unless they were directly discussing Aristotle, they were unlikely to pay much attention to the matters

    treated of in the Analytica Posteriora, Topica and De Sophisticis Elenchis.

    All this, however, was to change with the coming of the Renaissance

    Ignoring those at the University of Paris and at various Spanish universities who consciously continued the medieval tradition (2), we find two completely new developments. On the one hand there is Humanism, with its bitter attacks on medieval sophistry, its dropping of virtually all formal logic, and its emphasis on the topics. On the other hand, there is Aristotelianism, with its emphasis on the pure text of Aristotle, freed from medieval accretions, and to be interpreted either directly or with the aid of Greek and Arab commentators. These two schools certainly differed in important respects, but they were united in their rejection of what I have described as the great advances of the medieval

    period. Supposition theory, insolubilia, consequentiae and exponibilia were to be discussed no more; and terms and propositions were to appear only as described by Aristotle or by the grammarians and rhetoricians.

    My view of the medieval advances remains unchanged, but I am not now so sure about the abruptness of the change from medieval to Renaissance logic in the works of the Aristotelians of the period. In this paper, I intend to present a case study of the transition as it appears in the works of one Aristotelian, namely Agostino Nifo (or Augustinus Niphus). I intend to show that medieval doctrines were still relatively well-known to him, and were discussed by him at length; but that he presented them in a way which diminished their value and hence made them easier to abandon. Someone who knew of the theory of terms or of supposition theory, to mention just two examples, only through Nifo could well wonder what use these doctrines were, despite the apparent care with which they had been expounded, and could therefore decide to abandon them completely in his own work. Whether this is indeed what happened in the sixteenth century can, of course, only be established after a good deal of further investigation; and I present the possibility here only as a tentative hypothesis.

    (1) I would like to thank Dr. C. B. Schmitt of the Warburg Institute, University of London, for inviting me to read an earlier version of this paper as part of a series devoted to Renaissance Aristotelianism. I would also like to thank the Canada Council for the generous financial support which made the research for this paper possible.

    (2) For further discussion and bibliography, see E. J. Ashworth, Language and Logic in the Post-Medieval Period, Dordrecht (Holland) - Boston (U.S.A.) 1974.

  14. ———. 1976. "Will Socrates Cross the Bridge? A Problem in Medieval Logic." Franciscan Studies no. 14:75-84.

    Reprinted as essay XII in: Studies in Post-Medieval Semantics.

    "In their treatises on insolubilia, or semantic paradoxes, medieval logicians frequently mentioned other cases in which the assumption that a proposition was true led to the conclusion that it was false, and the assumption that it was false led to the conclusion that it was true. Some of these cases were easily solved. If one considers the proposition "Socrates will enter a religious order" in relation to Socrates' vow, "I will enter a religious order if and only if Plato does," and to Plato's vow, "I will enter a religious order if and only if Socrates does not," one sees at once that the problem stems from contradictory premises.(1). But not all cases were of this sort. Consider the favourite example, "Socrates will not cross the bridge," when said by Socrates, in relation to the two premises, "All those who say what is true will cross the bridge" and "All those who say what is false will not cross the bridge."(2) It is easily demonstrated that "Socrates will not cross the bridge" is true if and only if it is false, but what is not so easily demonstrated is the source of the paradox. Certainly it is not a paradox just like "What I am now saying is false," since the key proposition does not speak of its own semantic properties, but the premises do indeed speak of truth and falsity in a way which has implications for the truth-value of "Socrates will not cross the bridge." The question thus arises whether "Socrates will not cross the bridge" is to be counted as a semantic paradox, to be dissolved in the same way as the Standard Liar is dissolved, or whether it is to be seen as needing another kind of solution, perhaps less radical in its implications for our common-sense notions about such matters as the legitimacy of self-reference or the definition of truth." (pp. 75-76)

    (...)

    "In conclusion, I would like to say that Paul of Venice's reputation as the last of the great medieval logicians seems to me to be vastly overrated. Several logicians of the late fifteenth and early sixteenth centuries, including Bricot, Eckius, Major and de Soto, offer more acute discussions of logical problems and more satisfactory solutions, as I hope I have demonstrated by this examination of the bridge paradox." (p. 83)

    (1) Thomas Bricot, Tractatus Insolubilium (Parisius, 1492) sign. b. viii and sign, c i; Johannes Eckius, Bursa Pavonis (Argentine, 1507) sign, k v; John Major, Insolubilia (Parrhisiis, 1516) sign, c ii ff. Cf. Albert of Saxony, Perutilis Logica (Venetiis, 1522) fo. 46 vo; Robertus de Cenali, Insolubilia in Liber Prioris Posterioris (Parisius, 1510) sign, o iiii.

    One should note here that vows, promises and the like were treated as propositions with truth-values rather than as performative utterances with no truth-values. This view was combined with a realization that there are certain conditions which have to be met before a vow is binding. For instance, the vower must genuinely intend to do what he vows to do, and what he vows to do must be both moral and within his power. These extra conditions were not thought relevant to the question whether "Socrates will enter a religious order" was true or false.

    To the slightly different question of whether Socrates would be bound by his vow, Major, for instance, held that he would not, on the grounds that his vow was conditional and that the condition, given Plato's vow, could not be fulfilled.

    For references to Major's text and to other discussions of vows and promises, see below, note 15.

    (2) Paul of Venice, Logica Magna (Venetiis, 1499) fol. 198 and Paul of Venice, Tractatus Summularum Logice Pauli Veneti (Venetiis, 1498) sign. e i vo. The latter work which appeared in many editions, is known as the Logica Parva. See also John Buridan, Sophisms on Meaning and Truth, translated and with an introduction by T. K. Scott (New York, 1966), pp. 219-220; Cenali, loc. cit.; David Cranston, Tractatus Insolubilium et Obligationum [Paris, c. 1512] sign. e iiii; Eckius, op. cit., sign, k iiii vo; Robert Holkot, Super Quattuor Libros Sententiarum Questiones (Lugduni, 1497) sign. E ii; Major, op. cit., sign. c ii vo; Peter of Ailly, Conceptus et Insolubilia (Parisius, 1498), sign. b. viii; Peter of Mantua, Logica (Venetiis, 1492), sign, o vivo; Domingo de Soto, Opusculum Insolubilium in Introductiones Dialectice (Burgis, 1529), fol. cxlvi f. Bricot, op. cit., sign. b. vii vo f. speaks of giving a penny to the truth-teller rather than of allowing him to cross a bridge, but the principle is the same. Some authors (e.g. Eckius, op. cit., sign. k v) gave both versions of the paradox, as did Paul of Venice himself (Logica Magna fol. 197 vo f., Lógica Parva, sign. e i f.) It should be noted that there are many variations in the names of the characters and in the phrasing of the propositions. Some authors substituted "You will throw me into the water" for "Socrates will not cross the bridge."

    (15) It is because of this association with promising that we find the bridge paradox and others similar to it discussed in theological works as well as logical, e.g. Holkot, op. cit., sign. C iiii ff., sign. D viii vo ff.; John Major, In Quartum Sententiarum Questiones ([Paris], 1519), sign. ccxcii vo ff. For general discussions of promising see, e.g., Gratian, Decretum, Chapter XXII (various editions) and Richard Mediavilla (or Middletown) Scriptum Super Quarto Sententiarum ([Venice], 1489) Book IV, distinction 38.

  15. Ashworth, Earline Jennifer. 1977. "Thomas Bricot (D. 1516) and the Liar Paradox." Journal of the History of Philosophy no. 15:267-280.

    Reprinted as essay XI in: Studies in Post-Medieval Semantics.

    "Preliminary Remarks.

    No one interested in the history of the Liar Paradox will gain a just appreciation of the variety and sophistication of the solutions that were offered unless he pays attention to the logicians working at the University of Paris at the end of the fifteenth century and the beginning of the sixteenth.(1) The study of semantic paradoxes, known as insolubilia, formed a significant part of the first-year logic curriculum, which was of course the curriculum for all arts students; and those who taught the subject by no means confined themselves to a repetition of earlier views. Although Ockham, Buridan, Paul of Venice and Peter of Ailly certainly were read and discussed, original work was also produced. One of the earliest and most influential of these original treatises was written by Thomas Bricot. First published in 1491, it received its eighth edition in 1511, (3) and was still being read as late as 1529 when Domingo de Soto discussed it in his own work on semantic paradoxes. (4)

    (...)

    The only original self-contained works of Bricot that I know are his Tractatus Insolubilium and his Tractatus Obligationum, which were always printed together. In this paper I intend to discuss only the Tractatus Insolubilium.

    The book’s organization is worthy of some preliminary comment. Its main division is into three Questiones, each phrased in a similar manner. In the first Questio Bricot inquires whether there is a way of saving the possibilities, impossibilities, contingencies, necessities, truths and falsities of self-referential propositions; the second and third Questiones ask simply whether there is another Way of saving, that is, justifying, the attribution of these modalities. The first Questio contains what is apparently Bricot’s own solution to the problem. In the second Questio he discusses the solution that stems from Ockham, and the solution derived from Peter of Ailly is covered in the third. Each Questio has exactly the same internal organization. At the beginning, Bricot poses five main questions concerning the proposed solution. He then divides the subsequent discussion into three sections: the first, headed notabilia, setting out the main principles of the proposed solution; the second, conclusiones, giving a brief list of conclusions; and the third, dubia, taking up a series of problems arising from the proposed solution. Each doubt is aimed at one of the notabilia and gives rise to a series of arguments against the proposed solution. Once these have been stated, they are refuted one by one.

    After all the doubts have been dealt with, Bricot offers replies to the five main questions posed at the beginning of the section. At no point are the separate solutions compared to one another, though arguments drawn from one view may be used in the critical discussion of another view; and at no point does he actually claim that the first solution is his own and hence to be preferred. Bricot is more forthright in a note on insolubilia he added to George of Brussels’s commentary on De Sophisticis Elenchis, where he says that the solution in question is “omnium probabilissimus.” (6) The first solution is explicitly attributed to Bricot by de Soto,(7) who himself studied at Paris, and there is also indirect evidence to support de Soto’s claim. The first solution does not figure among the fifteen solutions described by Paul of Venice in his Logica Magna (8) yet it does appear in the works of Parisian logicians contempory with or junior to Bricot. For example, Tartaretus discussed it in a treatise on insolubilia first published in 1494, (9) and it was also discussed by John Major and David Cranston.(10) Unfortunately these authors followed the normal practice of mentioning names only in a few outstanding cases. The German Trutvetter did both recommend Bricot by name and describe the view I attribute to him, but without specifically linking the two.(11)"

    (2) For further details and a bibliography, see E. J. Ashworth, Language and Logic in the Post-Medieval Period (Dordrecht Holland and Boston: Reidel, 1974). For the medieval background, see P. V. Spade, The Mediaeval Liar: A Catalogue of the Insolubilia-Literature (Toronto: Pontifical Institute of Mediaeval Studies, 1975).

    (3) Tractatus Insoiubilium (Paris, 1491; reprinted, Paris, 1492; Paris, 1494; Lyons, 1495; Lyons, 1496; Paris, 1498; Paris, 1504; Paris, 1511). I have examined copies of each printing and have prepared an edition of the text, on which I base my discussion.

    (4) Opusculum Insolubilium, in Introductiones Dialectice (Burgis, 1529), fol. cxliii-cxlix vo.

    (5) I draw my material from H. Élie, “Quelques maîtres de l’université de Paris vers l’an 1500,” Archives d’histoire doctrinale et littéraire du moyen âge 18 (1950-1951): 197-200; and A. Renaudet, Préréforme et humanisme à Paris pendant les premières guerres d’Italie 1494-1517 (Paris, 1916), pp. 96 ff.

    (6) George of Brussels, Expositio in logicam Aristotelis: una cum Magistri Thome bricoti textu de novo inserto nec non cum eiusdem questionibus in cuiusvis fine libri additis (Lugduni, 1504), fol. cclxxv.

    (7) Opusculum Insolubilium, fols, cxl vo and cxlvii.

    (8) (Venetiis, 1499), fols. 192ff.

    (9) Petrus Tartaretus, Sumularum Petri de hispania explanationes (Friburgensi, 1494), sign, k v-l i vo.

    (10) John Major, Insolubilia (Parrhisiis, 1516); David Cranston, Tractatus insolubilium et obligationum [Paris, c. 1512].

    (11) Jodocus Trutvetter, Summule totius logice (Erphurdie, 1501), sign. UUUvi vo-XXX i vo. Trutvetter’s reference is to Bricot’s note in George of Brussels, cclxxiii“vo ff.

  16. ———. 1977. "Chimeras and Imaginary Objects: A Study in the Post-Medieval Theory of Signification." Vivarium no. 15:57-79.

    Reprinted as essay III in: Studies in Post-Medieval Semantics.

    "I. Prefatory Note.

    In the following paper I shall be discussing a particular problem of meaning and reference as it was formulated by a group of logicians who studied and/or taught at the University of Paris in the early sixteenth century.(1) In alphabetical order they are: Johannes Celaya (d. 1558) who was in Paris from 1500 or 1505 until 1524; Ferdinandus de Enzinas (d. 1528) who was in Paris from about 15x8 until 1522; John Major (1469-1550) who was in Paris from 1492 or 1493 until 1517 and again from 1525 to 1531; William Manderston who taught at Sainte-Barbe from about 1514 and returned to Scotland in or shortly before 1530; Juan Martinez Siliceo (1486-1556) who left Paris in about 1516; Hieronymus Pardo (d. 1502 or 1505); Antonius f Silvester who taught at Montaigu ; and Domingo de Soto (1494-1560) who left Paris in 1519. I shall also discuss the work of the Spaniard Augustinus Sbarroya and the Germans Jodocus Trutvetter (d. 1519) and Johannes Eckius (1486-1543). Both Sbarroya and Eckius were well acquainted with the works of the Paris-trained logicians. Further material is drawn from the fifteenth-century Johannes Dorp and the anonymous author of Commentum emendatum et correctum in primum et quartum tractatus Petri Hyspani. The work of the medieval authors Robert Holkot, John Buridan and Marsilius of Inghen will appear as it was described by early sixteenth-century authors.

    II. Introduction.

    One of the main features of late medieval semantics was the attempt to formulate a unified theory of the reference of general terms. It is true that this attempt was not explicitly discussed, but many of the problems which arose in the context of such topics as signification, supposition, ampliation, appellation, and the logical relations between sentences clearly owed their existence to the assumption that general terms always referred to spatio-temporal individuals; and in the solutions offered to these problems, much ingenuity was employed to ensure that this assumption was modified as little as possible, if at all. I have already shown in two earlier papers how some logicians dealt with reference in the modal context “For riding is required a horse” and in the intentional context “I promise you a horse.” (2) At the end of this paper, I shall discuss another intentional sentence, "A man is imaginarily an ass”, which was thought to present a difficulty. However, it would be a mistake to think that context was the only complicating factor, for there were general terms which placed an obstacle in the path of those seeking a unified theory, not only by virtue of the contexts in which they appeared, but by virtue of their meaning. The favourite example of such terms was “chimera”, but “irrational man”, “braying man”, and “golden mountain” also served as illustrations. The problem was not merely that they failed to refer, but rather that they were thought to be incapable of referring because the objects which they apparently denoted were impossible just as, for the modern reader, a round square is impossible. The main purpose of the present paper is to explore the way in which the problem was presented, and some of the solutions which were offered." (pp. 57-58)

    (...)

    "VI. Conclusion.

    This survey of the way some early sixteenth century logicians treated the problem of chimeras reveals very clearly the alternatives faced by any philosopher who wants to give a unified theory of the reference of general terms. If one adopts a purely extensionalist interpretation of propositions, and allows only ordinary spatio-temporal entities into one’s universe of discourse, then one is faced with the choice between rejecting as false many sentences, such as “I imagine a chimera”, which one would wish to accept as true, and accepting as true many sentences, such as “ “Chimera” signifies an ass”, which one would wish to reject as false. If one extends one’s universe of discourse to include imaginary objects which are not just ordinary objects regarded in a certain way, one faces grave ontological problems. On the other hand, to appeal to appellation theory is to acknowledge that no purely extensionalist interpretation of all propositions can be given and that no unified theory of reference is possible; and to adopt Holkot’s solution is to admit that sentences which seem to be structurally similar are not in fact similar and that some sentences which appear to be about objects in the world are in fact about the contents of our own minds. On the whole my sympathies lie with those who abandoned the belief that both general terms and subject-object sentences can be given a uniform treatment, but I have great respect for the subtlety and sophistication with which arguments for a uniform treatment were presented. Post-medieval logicians were by no means mindless followers of their medieval predecessors." (p. 79)

    (2) E. J. Ashworth, 'For Riding is Required a Horse’: A Problem of Meaning and Reference in Late Fifteenth and Early Sixteenth Century Logic, in : Vivarium 12 (1974), 94-123; E. J. Ashworth, Ί Promise You a Horse’: A Second Problem of Meaning and Reference in Late Fifteenth and Early Sixteenth Century Logic, in: Vivarium 14 (1976), 62-79, 139-155." (pp. 57-58)

  17. ———. 1977. "An Early Fifteenth Century Discussion of Infinite Sets." Notre Dame Journal of Formal Logic no. 18:232-234.

    "In the opening years of the fifteenth century, or perhaps a little earlier, John Dorp (1) wrote a commentary on Buridan's Compendium Totius Logicae (2) and it is here that one finds a discussion of infinite sets which is not only quite unexpected (3) but which suggests that other thinkers of that period were interested in the same topic.

    The question of infinite sets arose in the context of the theory of reference. Medieval logicians assumed that affirmative sentences were true only if the subject and object terms had reference, but this assumption conflicted with their intuitions about such sentences as "I imagine a chimera" and "The word 'chimera' refers to a chimera". These sentences seem to be true, but "chimera" cannot refer to actual or possible chimeras, since a chimera is an impossible object, just as a round square is an impossible object. The question then arose of how such sentences were to be treated, and one obvious answer was to postulate a class of imaginary objects which included impossible objects and to which reference could be made in intentional contexts. (4) In his discussion of this answer, Dorp presented several arguments against the claim that one could refer to impossible objects." (p. 232)

    (...)

    "Historians of logic must always be wary of taking isolated passages out of context and reading modern developments into them. However, in the case of Dorp there do seem to be good grounds for claiming that he was aware of something describable as a non-denumerably infinite set. It is a great pity that he does not give us more detail about the reasoning that lay behind his assertions, but it is to be hoped that further research into late fourteenth and early fifteenth century mathematics will reveal it to us." (p. 233)

    (1) Dorp received his M.A. from the University of Paris in 1393 and he was last heard of at the University of Cologne in 1418. The dates of his birth and death are not known.

    (2) Johannes Buridanus, Compendium Totius Logicae, Venedig (1499). Facsimile edition: Frankfurt/Main, Minerva G.m.b.H. (1965). This edition contains Dorp's commentary.

    (3) For another medieval reference to infinite sets, see I. Thomas, "A 12th century paradox of the infinite," The Journal of Symbolic Logic, vol. 23 (1958), pp. 133-134.

    (4) For further discussion and references, see E. J. Ashworth, *'Chimeras and Imaginary Objects: A Study in the Post-Medieval Theory of Signification" [1977].

  18. ———. 1978. "A Note on Paul of Venice and the Oxford Logica of 1483." Medioevo no. 4:93-99.

  19. ———. 1978. "Multiple Quantification and the Use of Special Quantifiers in Early Sixteenth Century Logic." Notre Dame Journal of Formal Logic no. 19:599-613.

    Reprinted as essay X in: Studies in Post-Medieval Semantics.

    "I have three reasons for writing this paper. In the first place, I want to explain the early sixteenth century practice of using the letters 'a', 'b', 'c', and 'd' as special signs governing the interpretation of terms within sentences. In the second place, I want to investigate the analysis which logicians in the medieval tradition gave of such sentences as "There is somebody all of whose donkeys are running", "Everybody has at least one donkey which is running", and "At least one of the donkeys which everybody owns is running".(2) In the third place, I want to show that, despite what Geach has suggested, (3) logicians in the medieval tradition were capable of offering good reasons for rejecting such inferences as "Every boy loves some girl, therefore there is some girl that every boy loves". My discussion will be based mainly on the work of a group of logicians who were at the University of Paris in the first two decades of the sixteenth century, in particular Fernando de Enzinas, Antonio Coronel, and Domingo de Soto." (p. 599)

    (...)

    "Although the logicians whose work I have examined display considerably more flexibility and subtlety than scholastic logicians have usually been credited with, their discussion reveals two important weaknesses. In the first place, they can only cope with the relations expressed in certain kinds of sentences, particularly those containing genitives; and in the second place, they do not give adequate instructions for distinguishing the case in which one is speaking of all members of a class such as donkeys from the case in which one is speaking only of the members of a subclass, such as the donkeys belonging to a particular man. On the other hand, they are clearly sensitive to the different facets of such relationships as donkey-ownership, and they are also sensitive to the kinds of inference which have to be debarred. A complete account of these strengths and weaknesses will have to await further research." (pp. 610-611)

    (2) Cf. P. T. Geach, Reference and Generality, Ithaca, New York (1962), p. 15 ff.

    (3) P. T. Geach, "History of a fallacy" in Logic Matters, Oxford (1972), pp. 1-13.

  20. ———. 1978. "Theories of the Proposition: Some Early Sixteenth Century Discussions." Franciscan Studies no. 38:81-121.

    Reprinted as essay IV in: Studies in Post-Medieval Semantics.

    "I. Prefatory Notes

    In his excellent book, Theories of the Proposition. Ancient and Medieval Conceptions of the Bearers of Truth and Falsity, (1) Gabriel Nuchelmans carries the story up to Paul of Venice, who died in 1429.

    In this paper I intend to consider the discussions of propositional sense and reference found in the works of a group of authors connected with the University of Paris in the last decade of the fifteenth century and the first three decades of the sixteenth century. I confine myself to this group not only because it is a group, but because I know of few other sustained discussions of the problem by logicians after Paul of Venice. Two fifteenth century authors, Stephanus de Monte and Andreas Limos raised the matter in the context of insolubilia; (2) the Italian Agostino Nifo (1470-1538) discussed it in two places; (3) and various other authors, such as the German Jodocus Trutvetter, mentioned the topic only in passing. (4) Nor is the matter pursued in any of the early printed Sentence Commentaries I have examined, including those written by such authors as Celaya and Major. (5)

    The authors I shall discuss are first the two Frenchmen, Thomas Bricot (d. 1516) who did his main logical work in the last decade of the fifteenth century, and Jean Raulin (1443-1514) who entered the Benedictine Order at Cluny in 1497. Second, there is one German, Gervase Waim (c. 1491-1554) who began his studies at Paris in 1507 and was Rector of the University in 1519. Third, there is one Belgian, Pierre Crockaert (Peter of Brussels) (d. 1514) who became a Thomist. Fourth, there are two Scotsmen, John Major (1469-1550) who was taught by Bricot and Pardo before teaching at Paris himself from 1505 to 1517 and again from 1525 to 1531, and George Lokert (d. 1547) who was a pupil of Major. Fifth, there are five Spaniards, Hieronymus Pardo (d. 1505), Juan Celaya (d. 1558), Antonio Coronel, Juan Dolz and Fernando de Enzinas. Finally, there is Hieronymus de Sancto Marco about whom I know little except that he was at one time connected with Oxford, and that he studied theology at Paris. (6)

    The contexts in which theories of the proposition were discussed varied. Bricot and Major discussed the matter in their works on insolubilia, (7) though what Major had to say was reprinted, somewhat amplified, as a separate section in complete editions of his works.(8) This context was a natural one, since in order to solve the problem of semantic paradoxes it was necessary to ask what it was that was true or false, and how these properties were to be defined. Dolz inquired about the total significate of the proposition in his treatise on Terms, (9) while Coronel and Pardo asked the same question in more general logic treatises. (10) In a commentary on Peter of Spain, Enzinas asked whether a proposition was true or false by indicating and if so, what it indicated. (11) Similar questions were posed by Raulin and Hieronymus de Sancto Marco. (12) Enzinas also asked about the total significate of a proposition in his work on mental propositions, and Waim asked the same question in his Tractatus Noticiarum. (13) Lokert raised the matter in his Tractatus Noticiarum by asking about the object of judgment. (14) Bricot, Celaya and Coronel asked about the objects of science, judgment and assent in their commentaries on the Analytica Posteriora. (15) Finally, Pierre Crockaert asked about the truth of a proposition in the second quodlibet attached to his commentary on Peter of Spain. (16) Presumably because of the nature of a quodlibet his discussion is a good deal more elliptical and allusive than that found in the other sources." (pp. 81-83)

    (...)

    "Conclusion.

    One of the main features of late medieval logic was the heavy emphasis placed on the notion of reference. Another feature was the ontological parsimony which led logicians to reject impossible and imaginary objects, including complexe and incomplexe significabilia of the sort proposed by Gregory of Rimini. When we consider the theory of terms, we can see these two influences joining to produce a series of attempts to explain the reference of terms in intentional and modal contexts without abandoning either the view that entities must not be multiplied beyond necessity or the view that a unified theory of the reference of general terms is possible. Similar attempts were directed toward the explanation of the reference of such terms as “chimera,” which were thought of as having no possible extension. In other places I have argued that one of the main virtues of Parisian logicians of the early sixteenth century was their recognition that the views mentioned above were irreconcilable, and that a purely ex-tensionalist approach to the signification of terms would have to be abandoned.165 Their achievements with respect to the theory of the proposition itself are very similar. A number of them saw that both the attempt to postulate special objects of reference for propositions and the attempt to argue that propositions referred to things in the world had failed; and they also saw that the way of escape lay in the acknowledgement that the referential role of propositions is not after all primary. Their function is not to name or to refer, but to make an assertion which can only be further described by a that-clause or a paraphrase. In the last resort, one can only see what the meaning of a proposition is by understanding what claim has been made; pointing to an object or group of objects will never serve as an answer." (pp. 120-121)

    (1) Gabriel Nuchelmans, Theories of the Proposition. Ancient and Medieval Conceptions of the Bearers of Truth and Falsity (Amsterdam/London: North Holland Publishing Company 1973). This work should be consulted for discussion, references and bibliography pertaining to Buridan, Ockham, Gregory of Rimini, and other medieval authors. For further bibliography see E. J. Ashworth, The Tradition of Medieval Logic and Speculative Grammar from Anselm to the End of the Seventeenth Century: A Bibliography from 1836 Onwards (Toronto: Pontifical Institute of Mediaeval Studies, 1978). [Note added to the reprint: "Since I wrote this paper, Nuchelmans's book dealing with the period from 1450 to 1650 has appeared, see Gabriel Nuchelmans, Late-Scholastic and Humanist Theories of Proposition,. Amsterdam: North-Holland, 1980.]"

    (2) Andreas Limos, Dubia in Insolubilibus (Parisiis 1499) sig. a ii rb-b iv vb. Stephanus de Monte, Ars Sophistica [Paris, c. 1490?] sign, a vi r-b i r.

    (3) Agostino Nifo, Dialectica Ludicra (Venetiis, 1521) fo. 50V-53V, cited as DL. Agostino Nifo, Super Libros Priorum Aristotelis (Venetiis 1554), fols. 6ν-γτ. Nifo follows Pseudo-Scotus very closely, especially in the latter source: cf. Super Librum I Priorum Quaestio VIII in John Duns Scotus Opera Omnia (Parisius, L. Vivès, 1891) II, 98-101.

    (4) Jodocus Trutvetter, Summule Totius Logice (Erphurdie, 1301) sign. AA vi r.

    (5) In the earlier editions of his commentary on Sentences I, John Major gives a very brief discussion of some of the main views about objects of faith and knowledge, but he declines to discuss complexe significabilia on the grounds that they are ‘‘voluntarie ficta et sine auctoritate et sine ratione.” John Major, In Primum Senlentiarum (Parisiis 1519) fol. xvi ra. However in the Paris 1530 edition even this brief discussion has been excluded. He explains in the preface that he has revised the work so as to exclude many Arts topics such as the intension and remission of forms, and he refers to the struggle against Lutheranism as a reason for concentrating on theology.

    (6) I draw my information from the title and end pages of Hieronymus de Sancto Marco, Opusculum de Universali Mundi Machina ac de Metheoricis Impressionibus, s.l. [1505?], which also tells us that he was a Franciscan.

    (7) Thomas Bricot, Tractatus Insolubilium (Parisius, 1492), cited as TI. John Major, Insolubilia (Parrhisiis, 1516).

    (8) John Major, Inclytarum Artium ac Sacre Pagine Doctoris Acutissimi Joannis Maioris Scoti Libri Quos in Artibus in Collegio Montis Acuti Parisius Regentando in Lucem Emisit (Lugduni, 1516). All references will be to this edition.

    (9) Juan Dolz, Termini (Parisius [c. 1511)].

    (10) Antonio Coronel, Prima Pars Rosarii (Paris, s.a.), cited as PPR. Hieronymus Pardo, Medulla Dyalectices (Parisius, 1505).

    (11) Fernando de Enzinas, Primus Tr[actatus Summularum] (Compluti, 1523) cited as PT.

    (12) Jean Raulin, In Logicam Aristotelis (Parisiaca Urbe, 1500). Hieronymus de Sancto Marco, Compendium Preclarum quod Parva Logica seu Summule Dicitur (Impressum in alma Coloniensi universitate, 1507). All references are to this work.

    (13) Fernando de Enzinas, Tractatus de Compositione Propositionis Mentalis (Lugduni, 1528) cited as PM. Gervase Waim, Tractatus Noticiarum ([Paris] 1519).

    (14) George Lokert, Scriptum in Materiam Notitiarum (Parisius, 1524).

    (15) Thomas Bricot, Logicales Questiones Subtiles ac Ingeniose super Duobus Libris Posteriorum Aristotelis (Parisius, 1504) cited as AP. Juan Celaya, Expositio in Libros Priorum Aristotelis [Paris, c. 1516]. Antonio Coronel, Expositio super Libros Posteriorem Aristotelis (Parisius [1510]), cited as AP. Since completing this paper, I have discovered a similar discussion in the Aristotle commentary of another Scotsman, David Cranston: Questiones super Posteriorum ([Paris], 1506) sign, g iv ra-h iii ra. He mentions by name Andreas de Novo Castro, Buridan, Gregory of Rimini, Andreas Limos, Peter of Mantua and Hieronymus Pardo.

    (16) Pierre Crockaert, Summularum Artis Dialetice Utilis Admodum Interpretatio... una cum Fructuosis Quibusdam Quotlibetis ab Eodem Fratre Petro Compilatis in Conventu Parisiensi (Parisius, 1508).

    (165) See Ashworth, "Chimeras and Imaginary Objects: A Study in the Post-Medieval Theory of Signification", Vivarium, 15 (1977), 57-77. See also E. J. Ashworth, “ 'For Riding is Required a Horse’: A Problem of Meaning and Reference in Late Fifteenth and Early Sixteenth Century Logic,’’ Vivarium, 12 1x974), 94-123; and E. J. Ashworth, “ Ί Promise You a Horse’: A Second Problem of Meaning and Reference in Late Fifteenth and Early Sixteenth Century Logic,’’ Vivarium, 14 (1976), 62-79; continued: Ibid., 14 (1976), 139-155.

  21. ———. 1979. "The Libelli Sophistarum and the Use of Medieval Logic Texts at Oxford and Cambridge in the Early Sixteenth Century." Vivarium no. 17:134-158.

    "I. Introduction

    In this paper I intend to analyze two early printed logic texts, the Libellus Sophistarum ad Usum Cantabrigiensium (or Cantabrigiensem) published four times between 1497 and 1524, and the Libellus Sophistarum ad Usum Oxoniensium, published seven times between 1499 and 1530. I also intend to demonstrate the origin of these books in the manuscript tradition of the early fifteenth century.

    A complete description of the various editions of the printed texts, together with their reference numbers in the new Short Title Catalogue, (2) will be found in the appendix." (p. 134)

    (...)

    "Obviously I do not pretend to have done more than sample the available manuscript sources, but the interested reader will find a large number of further references in De Rijk’s invaluable studies of the Logica Cantabrigiensis and the Logica Oxoniensis (7).

    The importance of my task stems from the fact that the Libelli Sophistarum provide the main evidence we have for the nature of logic teaching at Oxford and Cambridge in the first three decades of the sixteenth century. The nature of this evidence can best be brought into focus if we start by considering the state of logic in continental Europe, as revealed by a study of publication between 1472 and 1530. As a brief glance at Risse’s Bibliographia Logica (8) will show, European presses produced an extremely large number of logic texts during this period. Some of them were editions of medieval authors alone, and some of them were editions of medieval authors combined with a contemporary commentary. Many more contained only contemporary writing, whether this took the form of a general introduction to logic, or a discussion of a particular topic such as terms or insolubles. Virtually none were anonymous, and virtually all were highly structured, with topics following one another in an orderly sequence. Even those few texts which were a compendium of shorter treatises were united either by author, as were the editions of Heytesbury’s works, or by theme, as was the 1517 edition of Strode, Ferrybridge, Heytesbury and others on the topic of consequences. The texts were devoted purely to logic, and natural science crept in only in predictable places, such as commentaries on the Analytica Posteriora, the discussion of incipit and desinit in works on exponibles, or editions of such earlier writers as Heytesbury and Menghus Blanchellus Faventinus. In the assessment of these texts, knowledge of the manuscript tradition is vital only when one wishes to know how original the writers were, or how accurately medieval texts were reproduced. On the whole the texts are self-explanatory, and simply by reading them one can get a good idea of how they might have been used in teaching.

    The situation in England could not have been more different. In the first place, the number of logic books published was extremely small, even if one bears in mind that some may have perished without trace. Including the two Libelli Sophistarum, only seven separate works seem to have appeared, and of these only two are by named authors, both medieval. The first author is Antonius Andreas whose commentary on the ars vetus appeared at St. Albans in 1483, and the second is Walter Burleigh, whose commentary on the Analytica Posteriora appeared at Oxford in 1517. Apart from the Libelli Sophistarum, the remaining works are a Logica which appeared at Oxford in 1483 [StC 16693]; the Opusculum insolubilium which appeared at Oxford in about 1517 [StC 18833] and in London in about 1527 [StC 18833a]; 9 and the Libellulus secundarum intentionum which appeared in London in 1498 [STC 15572], in about 1505 [STC 15573], and in 1527 [STC 15574] as well as in Paris before 150ο.(10) I do not know the provenance of the first two works, but the third is an edition of the medieval tract which starts "Bene fundatum preexigit debitum fundamentum”, and which is found incomplete in both Gonville and Caius 182/215 (p. 70) and Corpus Christi 378 (1o5 r 107 r).(11)" (pp. 135-136)

    (2) A Short Title Catalogue of Books Printed in England, Scotland &- Ireland and of English Books Printed Abroad 1475-1640. First Compiled by A. W. Pollard & G. R. Redgrave. Second Edition, Revised and Enlarged, begun by W. A. Jackson & F. S. Ferguson. Completed by Katharine F. Pantzer. Volume 2, I-Z, London 1976.

    (7) For the Logica Cantabrigiensis see L. M. De Rijk, 'Logica Cantabrigiensis -- A Fifteenth Century Cambridge Manual of Logic', in: Revue Internationale de philosophie (Grabmann), 29e année, 113 (1975), 297-315. For the Logica Oxoniensis, see L. M. de Rijk, 'Logica Oxoniensis. An Attempt to Reconstruct a Fifteenth Century Oxford Manual of Logic', in: Medioevo, III (1977), 121-164.

    (8) W. Risse, Bibliographia Logica. Band 1. 1473-1800, Hildesheim 1965.

  22. ———. 1979. "A Note on an Early Printed Logic Text in Edinburgh University Library." Innes Review no. 30:77-79.