William Heytesbury (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeWilliam HeytesburyFirst published Mon Jan 6, 2003; substantive revision Wed Mar 21, 2007William Heytesbury (before 1313–1372/3) was a fellow ofOxford's Merton College from 1330, where, with RichardKilvington, Richard Swineshead, Thomas Bradwardine and John Dumbleton,the Mertonian “Calculators”, he worked with logical puzzles,applying supposition theory to the logical exposition of problematicstatements (sophismata), attending in particular to scopeproblems and compounded and divided sense. He is particularly notedfor applying his methods to puzzles about motion and thecontinuum. His work curiously anticipates nineteenth-centurymathematical analysis of the continuum, and he is well know fordeveloping the Mean Speed Theorem concerning the distance covered overa period of time under uniform acceleration. His work had someinfluence on the deveopment of early modern science, though he cannotbe said to have been doing empirical science, his work being muchcloser to mathematical analysis.1. Biography2. Works and Influence3. Doctrine 3.1 Insolubles 3.2 On Knowing and Doubting 3.3 Heytesbury's Use of the Theory of Supposition 3.4 Compounded and Divided Sense 3.5 Heytesbury's Treatment of Continuous Magnitudes 3.5.1 Beginning and Ceasing 3.5.2 Maxima and Minima of Capacities 3.5.3 Kinematics and the Mean Speed Theorem 3.5.4 Heytesbury and the Physical Sciences BibliographyOther Internet ResourcesRelated Entries1. Biography William Heytesbury first appears to us as a fellow at Merton Collegein Oxford in 1330. He must have been born before 1313, probably inWiltshire, Salisbury Diocese. He served as the first bursar in hiscollege in 1338–39, and although he was named a fellow of the newQueen's College in 1340, he soon returned to Merton. He hadbecome a Doctor of Theology by 1348. He was chancellor of theUniversity in 1371–72, and may also have served an earlier term inthe same office, if the record do not reflect a mere tenure protempore, in 1553–54. William died just as he finished hischancellorship, in the winter of 1372/73.2. Works and Influence All of Heytesbury's works were probably written during his regency inthe Arts at Merton, so approximately 1331–1339. His most notedwork, the Rules for Solving Sophismata [Regulae solvendisophismata], dated to 1335 by one manuscript, consists of sixchapters, (1) On insoluble sentences (concerning self-referentialparadoxes) (2) On knowing and doubting (concerning reference inintensional contexts), (3) On relative terms (concerning thesupposition of relative pronouns), (4) On beginning and ceasing, (5)On maxima and minima (concerning the limits of capacities), and (6) Onthe three categories (concerning velocity and acceleration in thethree categories of place, quantity and quality). A work supplementingthe Rules, On the Proofs of Conclusions from the Treatiseof Rules for Resolving Syllogisms [De probationibusconclusionum tractatus regularum solvendi sophismata],which may not be by William himself, appears in early printed editionsand manuscripts. The Regulae was an advanced work in logic,and William also wrote a number of more elementary works, focusingchiefly on logical and semantical issues rather than paradoxesinvolving the continuum, including a collection of 32Sophismata, and a second collection of 39 sophismspurportedly proving that the respondent is a donkey (tu estasinus), entitled Sophismata Asinina. He produced anumber of short works at the more elementary level as well, includingOn the Compound and Divided Senses [De sensus compositoet diviso], and Concerning the Truth and Falsehood ofPremises [De veritate et falsitate propositionis]. Allof these works appear in the 1494 edition of his works by Locatellus.In addition, some works found only in manuscript include a treatise onconsequences, another on obligations, one on future contingents, and abeginner's book of definitions and divisions in naturalphilosophy. (For the mss. for all these, see Weisheipl1969. Additional works on insolubilia cited in Weisheipl 1969have been shown by Spade 1989 to be falsely attributed toHeytesbury.) Heytesbury was one of the second generation of the Oxford school of“Calculators,” building on Richard Kilvington'sSophismata (before 1325), and Thomas Bradwardine'sInsolubilia and Tractatus de Proportionibus(1328). His major work, the Regulae, had considerableinfluence. It is used in the anonymous Tractatus de SexInconvenientibus, and John Dumbleton's SummaNaturalium. In England the tradition of the Oxford School seemsto have faded out by 1400, but it became quite popular inItaly. Heytesbury's Regulae and Sophismatainfluenced Peter of Mantua (d. 1400) in his De Instanti andLogica, Paul of Venice (d. 1429) in his SummaNaturalium and Sophismata, Gaetano of Thiene(1387–1465), who commented on the works (probably1422–30), Paul of Pergula (d. 1456), who also commented onHeytesbury's works, and a score of more minor authors. Heytesbury'sworks became part of the curriculum by statute at Padua in 1487, andeditions of his works were printed at Pavia in 1481, and Padua in 1491and 1494. In the early 16th century his works were used atParis by John Major (d. 1540) and the members of his school. But withthe Humanist reaction against medieval logical developments,Heytesbury's work ceased to be used in the schools and descended intoobscurity.3. Doctrine Heytesbury's work rotates around sophismata. Asophisma is a statement which one can plausibly argue bothto be true and to be false, given certain explicit backgroundassumptions (the casus) which are, for the sake of theargument, not open to challenge. The resolution of these argumentsand determination of the real state of affairs forces one to dealwith logical matters such as the analysis of the meaning of thestatement in question, and the application of logical rules tospecific cases. Modern philosophers certainly discusssophismata even if they don't have the name. Forinstance, Bertrand Russell inaugurated a discussion entirely inHeytesbury's spirit of the sentence, ‘the present King ofFrance is bald’, given that there is at present no King ofFrance. These sophismata occurred for medieval thinkers inthe context of formal debates, in which the aim of the questioner(opponens) was to trap the answerer (respondens) incontradiction or absurdity, while the answerer had to answer eachquestion put, accepting all valid inferences from any admission hehad already made. These debates were a training device in dialecticand logic going back to the time of Plato and Aristotle, andAristotle's Topics and Sophistical Refutationmakes reference to the 4th-century Greek version of suchdebates. A sophistical refutation (a refutation of the sort a Sophistmight make) depends on some fallacy that the questioner counts on theanswerer not spotting. Every sophisma, of course, would seemto involve at least one sophistical proof, since it cannot be bothtrue and false in the conditions specified. In what follows I will give some relatively simple examples ofparticular sophismata and classes of sophismata,and Heytesbury's treatment of them, and some of Heytesbury rulesfor handling sophismata where the rules are illustrative, orof interest on their own. In most cases I will not work out thesophisma in detail. The reader should go to the texts to seejust how complex the cases considered can get, and how sure-footedHeytesbury is in his handling of them. The sophismata dealtwith by Heytesbury in his rules were not the most complex discussed(as a comparison to Kilvington will show), but it seems they wereintended for higher level students. A large part of the success ofHeytesbury's work rests on its having hit the right level ofdifficulty, as well as its determined effort to bring some order towhat inevitably became a chaos of arguments, and to offer the studentsome firm rules to guide him in the chances of the disputation.3.1 Insolubles The opening chapter of the work for which Heytesbury is most known,the Rules for Solving Sophismata, deals with‘insoluble’ sentences, i.e., self-referential sentences such as‘what I am now uttering is false’. (The original for theseis Eubulides's paradox of the liar.) Heytesbury seems to havethought insolubles to be indeed insoluble. For one thing, at somepoint even in what he takes to be the best solution one must maintaina strongly counter-intuitive proposition, which can only be justifiedwhen one sees that every other solution involves propositions thatare even more counter-intuitive. Consider the following case: Someonesays, “what I am now saying is false,” and says onlythat. Now given the conventions that would normally define what thewords here signify (what its words ‘commonly pretend tosignify’), the statement, if it is true, must be false, and ifit is false, it must be true. As Heytesbury sees it, this means thatit is contradictory to assume that the person indeed utters thesentence, and that he utters it with the signification specified inthe conventions that give meaning to his words. The best way out ofthe contradiction, Heytesbury thinks, is to hold that the words donot signify what they normally should by the language'sconventions. This is, as he confesses, a difficult solution of thecase to maintain, for one might ask where the words get theirsignification if not from the conventions of the language? If thelanguage itself does not have conventions setting its own conventionsaside in such cases as this, where do we get off announcing that thewords cannot have the meaning assigned to them? Do we want to claimthat the fellow somehow secretly assigns a different meaning to themas he speaks? But surely that may not be the case. The only reason to accept this resolution of the difficulty,Heytesbury thinks, is that every other resolution is worse. The otherpositions he considers, supposing that the sentence doessignify as the conventions of the language say, are (1) a truesentence is not merely one that signifies things are as in fact theyare, but also does not falsify itself, and so the insoluble is false,as is its contradictory, (2) that insolubles are not really sentencesat all, though they do signify, and so are neither true nor false, and(3) that even though an insoluble is a sentence which is true orfalse, it is not true, and it is not false. (This last was no doubtinspired by a line taken on the question of future contingents raisedin Aristotle's On Intrepretation 9, holding thatalthough it is true or false that the sea-battle will take placetomorrow, if the general has not decided whether to commit his forcesor withdraw, then it is not true, and it is not false, that it willtake place.) Heytesbury thinks that the answerer can be caught in flatout contradictions if he follows any of these lines, and so the onlyline left for him is to deny that the sentence signifies in accordwith the usual conventions, and defend the implausibilities thatarise. But these implausibilities are not the worst of it, for, asHeytesbury sees, a case can be constructed specifically against hisposition as follows: A person utters the sentence ‘this sentencesignifies that things are not as it signifies by the conventionsactually establishing its signification.’ He has no reply tooffer to that insoluble. Heytesbury's treatment ofinsolubilia is reasonably straightforward, and convincingenough to have served as a starting point for the considerations ofmost later medieval authors on the subject.3.2 On Knowing and Doubting The second chapter of the Regulae deals withsophismata based on the notions of knowing and doubting. Ineach case it is argued on the assumptions given both that one knowssomething, and that one is in doubt whether it is the case, orignorant that it is the case. So, assume that one does not knowwhether the proposition that has just been uttered is ‘Godexists’, which one knows must be true, or ‘a man is adonkey’, which one knows must be false. Assume also that one doesknow that it is one or the other of these. Finally, assume that thesentence is in fact ‘God exists’. Then one knows that thatsentence (‘God exists’) must be true, but one is in doubtwhether that sentence (the one just uttered) is true. Or, in adifferent case, assume that you saw Socrates yesterday, and know thatthe man you saw yesterday was Socrates, and you see Socrates today,but don't realize that it is Socrates, but mistake him forPlato. Then you know that this (the one you saw yesterday) isSocrates, but you do not know that this (the same person, seen today)is Socrates. (These are versions, again, of one of Eubulides'sparadoxes, this time, the paradox of the veiled person.) Heytesbury treats these cases by making a distinction. In the firstcase, let A be the offending sentence. Then ‘A, I knowto be true’ does not mean the same thing as ‘I knowA to be true’. The first means that I know ofA that it is true (the ‘divided sense’), thesecond that I know ‘A is true’(the‘compounded sense’). Neither of the sentences ‘Iknow A to be true’ and ‘A, I know to betrue’ follows from the other. I might know that ‘Godexists’ is true, but not know of sentence A, which says‘God exists’, that it is true, if I don't know whatsentence A says. Or I might know of sentence A thatit is true, on the basis of reliable authority, but not know whatA says, and doubt that God exists. Similarly, one might thinkSocrates dead, and still, Socrates one might know to be alive. Onedoes not know that Socrates is alive, but one knows of Socrates (whomone takes to be Plato) that he is alive. In the one case (thecompounded sense) the sentence taken as a whole (the dictum)is what one knows to be true, in the second (the divided sense) whatis known is the subject, a part of the sentence, and the predicate isknown of it. (Note that Heytesbury works out his solution here withoutassuming a distinction between the sentence and the propositionexpressed by it. He generally takes a nominalist line, and will haveno truck with propositions expressed by sentences. It should also benoted that the Latin propositio is best translated as‘sentence,’ or even ‘sentence-token,’ where itdoes not mean instead ‘premise.’ To express‘proposition’ in the modern sense, one would have had tospeak, perhaps, of significatio.) In the treatise On theCompounded and Divided Senses this sort of case falls under the8th way in which that distinction occurs, in connectionwith verbs indicating mental acts and acts of will. (Heytesbury takesthese cases to be very like the 1st way in which he saysthe distinction occurs, involving modality de dicto andde re.) There the simple case first examined is ‘Youknow one or the other of these to be true, therefore one or the otherof these you know to be true’, which is clearly invalid, for Imay know that the sea-battle either will or will not occur, and notknow which. Here the divided sense (one of them I know to betrue) is concluded illegitimately from the compounded sense (I knowone-or-the-other-of-them to be true). Similar false inferences are‘the man appears to be a donkey, therefore it appears that theman is a donkey’, and ‘you understand some man is going tocome here, therefore of some man you understand that he is going tocome here’. 3.3 Heytesbury's Use of the Theory of Supposition Heytesbury presupposes the theory of supposition, developed in12th and 13th centuries, in hisdiscussions. The supposition of a term in a sentence token is,roughly speaking, the reference of that term as determined by itsmeaning in general and its place within the sentence. ‘Materialsupposition’ occurs when a term is used for itself , as‘man’ is in ‘man is a noun.’ ‘Simplesupposition’ occurs when a word is used for a universal (forHeytesbury, no doubt a ‘mental word’ or concept), as in‘man is a species’. ‘Personal supposition’ occurswhen the word is used to refer to things falling under itssignification, as in ‘a man runs’. Personal supposition isdivided into ‘discrete’ and ‘common’ supposition,the former occurring when the term is used to refer to a definite,particular individual , as in ‘that man is running’, andthe second when it is used to refer to whatever falls under thesignification of the term. Common supposition is divided into‘determinate supposition’, in which one refers toeverything falling under the term's signification disjunctively,as in ‘some man runs’ = ‘this man or that man or theother man . . . runs’, and ‘confused supposition’, inwhich the term refers to everything falling under its signification,but not disjunctively. Confused supposition includes‘distributive supposition’, in which one can descendlogically from the sentence using the term conjunctively, as in‘every man runs’ = ‘this man and that man and theother man … runs’, and also ‘merely confused’ or‘non-distributive’ supposition, in which such a descentcannot be made, as for instance in ‘every tinker is a man’,which is not equivalent to ‘every tinker is this man and thatman …’. On the other hand, there is a certain kind ofdescent possible in the last example, for ‘every tinker isman’ = ‘every tinker is this man or that man or the otherman …’. This case is one of merely confused mobilesupposition, and where not even this sort of thing can be done, wehave merely confused immobile supposition. These notions about supposition are applied by Heytesbury inconjunction with two other devices. First of all, he recognizes thatin addition to categorical terms, which have supposition, there arealso often ‘syncategorematic’ terms in a sentence, whichinfluence the supposition of the terms around them, but may have nosupposition of their own. The only function of the word‘every’ in a sentence, for instance, is todistribute the supposition of the term immediately followingit, that is, it produces distributive supposition in the term, as in‘every man is running’. The most interestingsyncategorematic words for Heytesbury are generally those whichproduce confused supposition in the terms which follow them. Sincesyncategorematic terms only affect terms that follow them, moving a termconfused by a syncategorematic word before that word will negate theconfusing effect of the word. The second device is the notion thatsome terms are exponible, which will be explained below.3.4 Compounded and Divided Sense In The Compounded and Divided Senses, Heytesbury lays out a14th-century elaboration of the Aristotelian discussion ofthe fallacies of composition and division marked by the development oftechnical devices involving word order within Latin to distinguish thecontrasting compounded and divided senses identified in eachinstance. He uses the distinctions developed here on almost every pageof his more advanced works. William begins by considering casesinvolving modal terms, most often the term ‘possible’. Ingeneral, a divided sense occurs if the modal comes in the midst of aproposition, so that the subject precedes it. So we might say“you can be here and in Rome,” meaning that you have acapacity, which can be exercised at different times, to be in one orthe other of those places, and this is presumably true. It does notfollow that “it can be that you are here and at Rome,”marking the compounded sense by placing the modal term before theentire proposition, for that means that “you are both here andat Rome” can be true at one and the same time, and this, ofcourse, is false (at least for those of us writing inWisconsin). Similarly, “a white thing can be black” may betrue, but “it can be that a white thing is black”cannot. (It should be noted that the divided sense answers quite wellto Aristotle's treatment of modalities in the PriorAnalytics, whereas the compounded sense corresponds nicely to thetreatment of modalities by Theophrastus.) More mysteriously,“you can traverse this distance” is taken to be true evenwhen “it can be that you traverse this distance” isnot. Here the problem is that the compounded sense requires that thesentence be made true at one time, in an instant, and one cannottraverse a distance in an instant. It takes time. But Heytesburythinks that sometimes one can argue from a divided to a compoundedsense, as in “You can be a bishop, therefore it can be that youare a bishop.” Problems arise only when the statement inquestion cannot be made true in an instant. (In general, sometimes onesense follows from the other, and sometimes it doesn't, and Heytesburytries to come up with rules where he can to decide which is the case.)Many of Williams' sophismata most confusing to theuninitiated depend, as here, on compounded and divided senses appliedto times, that is, they present problems in tense logic. Insuch cases the Latin is often artificial enough so that one must relyquite determinedly on the explicit rules for its interpretation withinlogic, ignoring linguistic intuition, to follow what is intended. The distinction made by word order is an artificial one, of course,and drew undeserved negative comment from Renaissance humanistsconcerned with good Latinity. Heytesbury is trying to find a waywithin a natural language to make statements logically unambiguous,so that rules of inference can developed and applied to themsyntactically, a reasonable aim, of course, within logic. It did notoccur to medieval logicians to create an entirely artificial languagefor this purpose, as we have done, so they tried to develop aspecialized form of Latin for the task. Having said this, we canalso observe that Heytesbury's specifications about the meaningsassociated with word order are often fairly natural in English (hisnative tongue), and some of them may also have been natural inmedieval Latin, a living, spoken tongue, at least as it was spoken inEngland. In the 2nd way, one can have a compounded and dividedsense when a term producing merely confused supposition in a commonterm immediately following it is used. For instance, “eternally(after any future instant) there will be some man (compounded sense),therefore some man will be eternally (after any future instant)(divided sense)” does not follow, for the supposition of‘man’ is confused by the term ‘eternally’ in theantecedent, but stands in determinate supposition in theconsequent. (To see the point, assume for the sake of argument that noman is immortal.) Again, consider “body A begins totouch some point of body B.” here ‘begins’confuses the supposition of ‘point’, producing a compoundedsense, so that one cannot infer from this that “some point ofbody B body A begins to touch,” where‘point’ now has determinate supposition. This is importantif the points of body B form an infinite set with noassignable first member, so that the limit points of body Bare assumed not to belong to it. For us this distinction hangs onquantifier scope, so: ∀x(x is a time after thepresent → ∃y(y is a man &y exists at x)) will have a compounded sense, and ∃y(y is a man & ∀x(x is a time after thepresent → y exists at x)) gives the divided sense. In the 3rd way the two senses arise because relativepronouns produce a compounded sense joined to their antecedents. Soit does not follow that “every animal that can bray is a donkey,therefore every animal can bray,” or “therefore, everyanimal is a donkey.” A more complicated example:“Immediately after the present instant there will be someinstant which immediately after the present instant willbe” has a compounded sense, and since the antecedent of“which” has confused supposition (due to the‘immediately’ phrase preceding it), the sentence is true.“Immediately after the present instant there will be someinstant and it immediately after the presentinstant will be” is false, for now the “it” is not arelative, and does not inherit its antecedent'ssupposition. Thus it has determinate supposition, and there is nodeterminate instant that will be immediately after the presentinstant. In the 4th way, it turns out that there are some termsthat can be taken either categorically or syncategorematically. Oneparticularly troublesome case here for a modern English speaker isthe word ‘infinite’, which can be taken to refer to anactual infinite (used categorically) or a merely potential infinite(used syncategorematically). So from the true “infinitely manyare finitely many” it does not follow that “finitely manyare infinitely many.” In the first statement“infinite” is syncategorematic, rather like“every” or “some”, and so the sentence is takento mean “one quantity, and another quantity, and so onindefinitely, is finite.” (Heytesbury would not havethought an actually infinite number of cases of finite quantities ispossible, since he is a good Aristotelian. But however many casesthere are, more could be produced. Perhaps there are only a finitenumber of cases at any given time, but more can always be producedlater.) In the second statement “infinite” is categorical,and so it means that some finite number of quantities are actuallyinfinite quantities. The 5th way in which the distinction occurs involves theword ‘and’, and, as Heytesbury notes, this is easy tosee. So from “Socrates can carry stone A, and Socratescan carry stone B” it does not follow that“Socrates can carry stones A and B,”i.e., at the same time. Here Heytesbury has no device to mark thedistinction syntactically. So “Five and three are eight”has a compounded sense, and “five is eight” does notfollow, but “Socrates and Plato are at Plataea” has adivided sense, and “Socrates is at Plataea” doesfollow. The 6th way depends on the disjunction‘or’. Consider a distributive “or”: “Plato orSocrates runs.” From this it follows that “Plato runs orSocrates runs.” Now consider the same pattern of argument with“or” in a compounded sense: “every proposition or itscontradictory is true.” Clearly this does not follow. “Youdiffer from a donkey, therefore you differ from a man or from adonkey” follows, and Heytesbury proposes the rule that if‘or’ occurs after a term that produces distributed orconfused supposition, then an argument from an inferior (narrower)term to its superior (wider) term with the same supposition isvalid. In the example, ‘differ from’ contains an implicitnegation, and so distributes the supposition of donkey. (“Youdiffer from a donkey” = “You are not adonkey” = “you are not this donkey and you are not thatdonkey …”) Now a disjunctive term is superior to (widerthan) any part of itself, so “donkey or man” is superior to“donkey”. So the inference in question does follow. The 7th covers a number of cases involving the phrase‘it will be the case that,’ or ‘it was the casethat’ followed by a present tense. For instance, it does notfollow that, “it will be the case that you are every man in thishouse, therefore you will be every man in this house.” Thereason is that “you will be every man in this house” coversall future time (divided sense) and indicates that no one else willever be in the house, while “it will be the case that…” (compounded sense) indicates only that there will be atime at which the statement following it is true. Again, “itwill be case that Socrates is as big as Plato” (compoundedsense) indicates that there will be a determinate moment at which thetwo will be of the same size, whereas “Socrates will be as bigas Plato” (divided sense) indicates that Socrates will at sometime be as big as Plato is now, though Plato may have grown evenlarger by then. The child might ask, “will I ever be as old asyou are, Daddy?” The answer, hopefully, is that he will, butthis can only be true in the divided sense, of course.3.5 Heytesbury's treatment of continuous magnitudes3.5.1 Beginning and Ceasing The last three sections of the Regulae deal with issuesinvolving the continuum, and nowadays might even be regarded asmathematical in content. In the 4th chapter Heytesburydeals with the words ‘begins’ and ‘ceases’ in away that uncovers certain paradoxical properties of the temporalcontinuum. He works out the puzzles here by treating these words as‘exponibles’. Their ‘exposition’ amount to aspecification of the underlying logical structure of thesentence. (Despite the fact that the exposition of the termis sometimes referred to, the entire sentence in which the termoccurs always turns out to have a misleading grammatical form.) So‘Socrates begins to be white’ is exposited as‘Socrates was not white immediately before now, and now iswhite’, or as ‘Socrates is not white now, and is whiteimmediately after now’. The first thing to note here is that the time continuum givesHeytesbury two possible readings of ‘Socrates begins to bewhite’. A single, disjunctive exposition is not offered. Whynot? Perhaps it is because William remains true to Aristotelianlogic, and does not recognize ‘hypotheticals’, i.e.sentences compounded of other sentences using sentential connectives,as well-formed statements. A well-formed sentence must becategorical, and so have a single subject and a single predicate,however complex these two terms may become through the use ofexponible and syncategorematic terms in them. Or, it may be that eventhough he does recognize hypotheticals as sentences (as opposed to,say, to inference schemata and fragments of inference schemata), hestill is convinced that the sentence at issue must be truly acategorical one, though perhaps in a way it is ambiguous (Heytesburywould not have been uncomfortable with making it unambiguous in a wayno ordinary speaker would have anticipated, for this is akin toworking with real definitions to specify what is meant, which wouldhave been considered legitimate, even if most people would know onlythe nominal definition.) One might take his project, and that oflater medieval logic in general, to be a matter of specifying how theintroduction of complexity into the individual terms in a categoricalstatement affects the sense of the whole statement, so that one candeal logically with the complex terms that must be introduced toapply Aristotelian logic to more interesting stretches ofdiscourse. The second thing to note about the exposition of ‘Socratesbegins to be white’ is that the phrase ‘immediatelyafter’ or ‘immediately before’ occurs in both readingsof it. This phrase is further exposited, so that ‘Socrates iswhite immediately after now’ comes to ‘for every momentafter now, there is some moment before that moment and after now, atwhich Socrates is white’: ∀x[x is a moment after now → ∃y((y is before x and after now) & (Socrates is white at y))]. Heytesbury thinks that something can, in principle, be whiteimmediately after now for as short a time after now as onewishes. The exposition of the ‘immediately’ phrases givesHeytesbury the opportunity to present sophismata hanging onthe scope of the implicit quantifiers (which he himself views as amatter of divided and compounded senses). In particular, Heytesburydoes not think (i) Some instant will be immediately after the present instant. ∃y(y is a moment after now & ∀x(x is a moment after now → y is before x)) can ever be admitted, nor does he think it follows from the trueproposition that (ii) Immediately after the present instant some instant will be. ∀x(x is a moment after now → ∃y(y is moment after now & y is before x)), for the first has a divided sense (the existential quantifier is themain quantifier in our notation), so that one or another definiteinstant has the property indicated, and the second a compounded sense(the existential quantifier falls under the scope of the universalquantifier in our notation), so that it talks about the collection ofinstants with merely confused supposition, in such a way that alogical descent to the statement that some definite individualinstant has the property is not permitted. The property belongs tothe whole set of instants after now. Put in terms of supposition, thesyncategorematic phrase ‘immediately after the presentmoment’ renders the supposition of the term that follows itmerely confused. Heytesbury proposes a number of sophisms involving‘begins’ and ‘ceases’ with mathematicalcontent. Assume an object that Socrates now has fully in view,divided into proportional parts, so that the first part is the righthalf, the second the right half of the remaining portion, the thirdthe right half of what remains after the first two parts arededucted, and so on ad infinitum. Now assume that the objectbegins to be occluded by a second object, approaching from theleft. So now Socrates sees every proportional part of it, butimmediately after now he will not see every proportionalpart. Heytesbury grants here that (iii) now there begins to-be-occluded-from-Socrates's-sight (=O) some proportional-part-of-the-object (= P), ¬∃w(Ow) & ∀x(x is a moment afternow → ∃w∃y((y is before xand after now) & (Ow at y)))but denies that (iv)some proportional part of the object now begins to be occludedfrom Socrates sight.¬∃w(Ow) & ∃w∀x(x is a moment after now → ∃y((y is before x and after now) & (Ow at y))) The reason is that ‘begins’ immobilizes the terms thatfollow it. Again, suppose that Socrates is one foot long, and Plato two feetlong, and that Plato and Socrates are both increasing in length at auniform rate, so that at the final moment of an hour's growththey would both be three feet long, except that they cease to existat the very instant in which this would occur. It is argued thatSocrates will be of such a size as Plato will be, and is not now ofsuch a size as Plato will be, and so it would seem that he begins tobe of such a size as Plato will be. But he does not begin to be ofsuch a size as Plato will be, nor will he ever begin to be of such asize as Plato will be. In this rather involved case Heytesbury pointsout that ‘Socrates will be of such a size as Plato willbe’, though true in the case at hand, does not imply that thereis any instant in which Socrates and Plato are of the samesize. Whatever size Plato will come to be, Socrates will come to beof the same size, but not at the same instant, but rather alater instant. So there is no instant at which Plato and Socratesare of the same size, and immediately before which they were not (noris there any instant at which they are not of the same size, andimmediately after will be), and so Plato and Socrates do not begin tobe the same size at any time.3.5.2 Maxima and Minima of Capacities In the fifth chapter of the Regulae, on maxima and minima,Heytesbury considers the limits of capacities. He assumes here thatevery active capacity is measured against the resistance (the passivecapacity) which it can overcome. Thus an active capacity to liftweights will be measured by the weight (a resistance) that it enablesone to lift. Now we will naturally assign as one's capacity tolift weights the greatest weight one can lift, the limit ofone's capacity. In some cases, there will be no greatest passivecapacity by which the active capacity can be measured, but then (aslong as the active capacity has some limit) there will be a leastpassive capacity that the active capacity cannot overcome. Thus anygiven active capacity will divide the range of passive capacities onwhich it is measured into two sets, every element of one of which isgreater than every element of the other, and the limit of thecapacity will be that unique capacity which is at the boundarybetween the two sets, either the greatest in the first set, or theleast in the second. The situation is parallel to that with beginningand ceasing, where it turned out that two different expositions wereneeded for “A begins to be F,” oneassuming a first instant at which A is F, the othera last instant at which A is not. Heytesbury tries to laydown some rules as to which of these options to choose in anyparticular case, and he does the same with capacities, trying tospecify when a capacity's limit is the greatest capacity it canact upon (or the least by which it can be acted upon), and when it isthe least it cannot act upon (or the greatest by which it cannot beacted upon). Heytesbury is also interested in determining necessary andsufficient conditions for the existence of a limit to a capacity inthe first place. He constructs several cases in which no such limitin fact exists, and in the course of the exploration, evolves thefollowing rule (stated here for active capacities). A limit willexist if and only if the active capacity is of a sort to beapplicable to a continuous range of passive capacities of the samesort (say, weights), and: (1) Each capacity in the range on which themeasured capacity is assumed to act either can or cannot be actedupon by the active capacity being measured (the man's strengthcan or cannot lift the weight), but not both, (2) There is somecapacity in the range on which it is measured which it can act uponand some other which it cannot act upon, (3) if it can act upon agiven passive capacity in the range, it can act on any capacity lessthan that, and if it cannot act upon a given passive capacity in therange, it cannot act on any greater than that. Heytesbury constructscases violating each of these conditions to show theirnecessity. These rules are strongly reminiscent of Dedekind'sPostulate. Richard Dedekind (Stetigkeit und IrrationaleZahlen, 1872) characterized a continuous linear order as one inthe elements of the ordered set are related by a two-place relation,<, such that, if a, b and c are inK, then: (1) if a ≠ b, then a <b or b < a, (2) if a <b and b < c then a <c, (3) if a < b then a ≠ b, (4) if a <b, then (∃x)(x ∈ K and a < x < b) (i.e., K is dense under <), (5)K contains a denumerably infinite subclass R suchthat between any two elements of K there is an element ofR (this guarantees linearity), and finally, (6) if K′ and K″ are non-empty sets inK, such that every element of K is in one or theother, and every element of K′ is < every element of K″, then there is one and onlyelement x ∈ K (and so in K′ or K″, but not both) such if y ∈ K′ then y < x, if y ∈ K″ then x <y. The two sets K′ and K″ form a Dedekindcut. Dedekind's postulate is not satisfied in the Rationals, forthe square root of 2, for instance, can be made a boundary between K′ and K″, and (1)-(5) and theassumptions of (6) will be satisfied, without any such x ∈ K as (6) describes existing. It is,however, satisfied in Real numbers, and so on a linear continuum suchas Heytesbury envisioned. Dedekind used his postulate to constructReal Numbers from Dedekind cuts in Rationals, advancing the projectof constructing all numbers from Natural Numbers governed by thePeano postulates. Heytesbury has nothing like that in mind, ofcourse, but he does seem to state the postulate, and hiscommentators, at least, were aware that it is not satisfied inRationals.3.5.3 Kinematics and the Mean Speed Theorem In the sixth chapter of the Regulae, ‘On the threecategories’, i.e on place, quantity and quality, in which alonemotion can occur, William works with kinematic problems. Here inparticular, but in the previous chapters as well, he is quite clearthat he is working secundum imaginationem, that is, he isconsidering cases which can be described and talked about, but whichcould never occur, as he thinks, in reality. It is better to take hiswork and that of the other Calculators to be mathematical thanphysical. Heytesbury divides local motions into uniform and difformmotions, that is motions which involve the same rate of change in thesame direction throughout, and those which do not. A rolling wheelmoves with a difform motion because different parts of it move withdifferent motions at any given time, while an accelerating objectmoves with a difform motion for a different reason—it is difformover time. He is concerned, for one thing, to describe the speed ofan entire motion when it is not uniform from time to time, or overthe entire moving body. By convention, he decides to assign as thevelocity of a body the velocity of the fastest moving part, and thenconstructs a case in which, despite the fact that every part of anobject moves continually faster over the period of an hour, theobject itself continually slows. What he imagines here is that awheel spins faster and faster, but as it spins the outer rim iscontinually worn away (rather like the wheel with which one sharpensa knife), and is worn away rapidly enough so that the fastest movingpart of the wheel, which is the outer edge, of course, is alwaysmoving slower than the outer edge a moment before was. He nextconstructs an object which moves constantly at a uniform speed eventhough every point in the object slows down—a line with theendpoint missing is used, and the rate of slowing converges on zeroas one approaches the end of the line, so that the speed of the lineas a whole has to be assigned as the lowest speed a point on the linedoes not move at, there being no greatest speed at which a point onthe line does move at. After a few more paradoxical cases, even more imaginative,Heytesbury moves on to consider difform motions. The most interestingcase, of course, is uniformly difform motion, and Heytesbury showsthe mean—speed theorem, i.e., that a uniformly accelerated bodywill, over a given period of time, traverse a distance equal to thedistance it would traverse if it were moved continuously in the sameperiod at its mean velocity (one half the sum of the initial andfinal velocities) during that period. In the Deprobationibus the conclusion is drawn that a uniformlyaccelerated body will, in the second equal time interval, traversethree times the distance it does in the first, though this is only inthe consideration of a particular case, and no general proof isoffered. Domingo de Soto observed the applicability of the mean-speedtheorem to free fall in 1555. In addition to the Regulae, Heytesbury produced two othercollections of Sophismata, evidently intended for lowerlevel students, for many of them are considerably more elementarythan those of the Regulae, though the ingenuity of thesemedieval debaters is often revealed in the way in which an apparentlysimple sophisma can become interesting with the introductionof the sorts of themes involving continua and kinematics that we havebriefly reviewed. The second collection, the Sophismata Asininaendeavors to prove in every case that the respondent is adonkey. The last two sophismata in the first collection areof some interest physically. One argues that if anything iscondensed, something else is necessarily rarified, otherwise theuniverse, which contains no vacuum, will shrink. The other arguessimilarly that if anything is heated, something else must becooled. For a discussion of it, see Clagett 1941.3.5.4 Heytesbury and the Physical Sciences There has been some discussion of the meaning of the work ofHeytesbury and the other Calculators for the development of thephysical sciences. Here it must first be noted that Heytesbury didnot think of himself as doing natural science. His interest is alwayslogical in the broad sense, for he is always interested in whatfollows or does not follow from the case at hand, and, as we havenoted, he works often secundum imaginationem, with anexplicit disinterest whether the case can actually occur, as long asit is not self-contradictory. One might, as I have indicated, takehis interests to be mathematical, but, again, he might nothave done so. He shows occasional interest in working outcalculations, but his mathematics is for the most part puremathematics, not mathematics applied to physics. He is working at theborderline between mathematics and logic. Nonetheless, the work ofHeytesbury and the Calculators may have had some influence on thedevelopment of science, and contributed to the Scientific Revolution,in several ways. For one thing, they applied quantitative measures tocapacities and qualities in their work, and even if this was in noway observational, and involved no attempt to actually developworkable systems of measurement, they expanded the conceptualrepertoire beyond the Aristotelian assumption that nothing could bemeasured other than space, time and motion. One has to conceive ofmeasuring heat and force before one sets about doing it. So theselogicians may have made a critical contribution to the intellectualmilieu by assuring, through their exercises in logic, that universitygraduates became accustomed to the notion of measuring heat, force,and the like. For a second point, the treatment of difform motions byHeytesbury and his colleagues may have been of considerableimportance. There is no direct evidence that Galileo knew of or usedtheir results, but it is apparent from the presence of this work inthe logic curriculum in Italy that these results were in the air. Thenotion that a movement might be difform, yet regular and calculablebecause it was uniformly difform, was simply part of the intellectualequipment of a university graduate. Often one must dream before onecan act, and if the Calculators dreamed and did not act, that doesnot mean their dreams were irrelevant to the acts of others. It should also be noted that Heytesbury is working within anAristotelian tradition. His treatment of the continuum observes theAristotelian (ultimately, Anaxagorean) specifications that acontinuum is infinitely divisible potentially, though not actually,and many of the points he makes about this situation are rooted in adiscussion in Aristotle. So Aristotle's Physics VI 5and VIII 8 raise the issue whether there is a first instant at whicha change has taken place, or a last minute at which it has not, oneissue discussed in Heytesbury's treatment of ‘begins’and ‘ceases’. There were medieval thinkers (for instance,Henry Harclay and Walter Chatton), who at the beginning of thefourteenth century, tried denying the infinite divisibility ofcontinua, and the Calculators' paradoxes and their resolutionswere at least partly in response to the criticisms of theAristotelian view by these heterodox thinkers. (On this, see Murdoch1982.)BibliographyWilliam Heytesbury's WorksWilliam Heytesbury: 1984, On Maxima and Minima: Chapter 5 ofRules for Solving Sophismata, with an anonymous fourteenth-centurydiscussion, translated with introduction and study by JohnLongeway. Dordrecht: D. Reidel Publishing Company.William Heytesbury: 1994, Sophismata asinina. Uneintroduction aux disputes logiques du Moyen Age.Présentation, édition, critique et analyse par FabiennePironet. Paris: J. Vrin.William Heytesbury, Sophismata, Latin text transcribedby Fabienne Pironet, en route to a critical edition, at her website.William Heytesbury: 1979, William of Heytesbury on“Insoluble” Sentences, translated with notes by PaulSpade. Toronto: Pontifical Institute of Medieval Studies.William of Heytesbury: 1988, “The Compounded and DividedSenses,” and “The Verbs ‘Know’ and‘Doubt’,” translated Norman Kretzmann and EleonoreStump, in The Cambridge Translations of Medieval PhilosophicalTexts, Vol. 1: Logic and Philosophy ofLanguage. Cambridge: Cambridge University Press. Pages413-479.William of Heytesbury: 1983, “On Knowing and BeingUncertain,” translated by Ivan Boh. Available in mimeograph fromThe Translation Clearing House (see Other Internet Resources). William Heytesbury: 1959. “On the three categories.”Selections translated by E.A. Moody in Marshall Claggett, TheScience of Mechanics in the Middle Ages (Madison, Wisconsin:University of Wisconsin Press), 235-237, 270-277; reprinted in EdwardGrant, A Source Book in Medieval Science (Cambridge,Massachusetts: Harvard University Press, 1974), 237-243.William Heytesbury: 1494, Hentisberi de sensu composito etdiviso, Regulae solvendi sophismata, etc. Venice: BonetusLocatellus. Includes commentaries by Gaetano of Thienne. See Wilson(1960) for other editions and Mss. Includes the Sophismata at77ra-70vb. See Wilson (1960) 154-163 for a list of the thirty-twosophismata here.William Heytesbury: 1483. De probationibus conclusionumtractatus regularum solvendi sophismata. Pavia: 1483. It is notclear that this work is by Heytesbury.William's LifeWeisheipl, James A.: 1968, “Ockham and some Mertonians,”Medieval Studies 30, 151-175. For William's life, andhis place in his School.Weisheipl, James A: 1969, “Repertorium Mertonense,”Medieval Studies 31, 174-224. Lists William's works andmanuscripts, with those of the other members of his School.Spade, Paul Vincent: 1989, “The manuscripts of WilliamHeytestbury's Regulae solvendi sophismata. Conclusions,notes and descriptions,” Medioevo 15, 272-313. Secondary LiteratureBiard, Joël: 1983, “La signification d'objetsimaginaires dans quelques texts anglais du XIVe siècle(Guillaume Heytesbury, Henry Hopton),” in The Rise of BritishLogic, edited by Osmund Lewry. Toronto: Pontifical Institute ofMediaeval Studies. 265-283.Biard, Joël: 1989, ‘Les sophismes du savoir: Albert de Saxeentre Jean Buridan et Guillaume Heytesbury,” Vivarium27, 36-50. Boh, Ivan: 1984, “Epistemic and alethic iteration in latermedieval logic,” Philosophia Naturalis 21,492-506. Discusses the propositions ‘if A knows that P then Aknows that he knows that P,’ and ‘if it is necessary that P,then it is necessary that it be necessary that P’, as discussedby Heytesbury, Ralph Strode, and subsequent Italian thinkers.Boh, Ivan: 1989, “Frachantinian's debt toHeytesbury,” in Of Scholars, Savants and theirTexts. Studies in Philosophy and Religious Thought.,edited by Ruth Link-Salinger. New York, Bern and Franfurt am Main:Lang. 35-45. Buzzetti, Dino: 1992, “Linguaggio e ontologia nei commenti diautore bolognese al De tribus praedicamentis di William deHeytesbury,” in L'insengamento della logica a Bolognanel XIV secolo (Studi e memorie per la storiadell'Università di Bologna, Nuova serie, 8), edited byDino Buzzetti, Maurizio Ferriani, and Andrea Tabarroni. Bologna: Pressol'Istituto per la Storia dell'Università.579-604.Clagett, Marshall: 1941, Giovanni Marliani and Late MedievalPhysics. New York. See Chapter II, 34-58 for a discussion of thelast case in Heytesbury's Sophismata.Coleman, Janet: 1975, “Jean de Ripa and the OxfordCalculators,” Mediaeval Studies 37, 130-189.Federici Vescovini, Graziella: 1983, “L'influence desRegulae solvendi sophismata de GuillaumeHeytesbury. L'expositio de tribus praedicamentis deMagister Mesinus,” in The Rise of British Logic, editedby Osmund Lewry. Toronto: Pontifical Institute of MediaevalStudies. 361-379.Kretzmann, Norman: 1976, “Incipit/Desinit,” inMotion and Time, Space and Matter, edited by PeterK. Machamer and Robert G. Turnbull. Cleveland , Ohio: Ohio StateUnitersity Press. 101-136. An analysis of some discussions of sophismsinvolving ‘begins’ and ‘ceases’.Kretzmann, Norman: 1982, “Syncategoremata, exponibilia,sophismata,” in The Cambridge History of Later MedievalPhilosophy, edited by Norman Kretzmann, Anthony Kenny, and JanPinborg. Cambridge: Cambridge University Press. 211-245.Lecq, Ria van der: 1983, “William Heytesbury on‘necessity’,” in The Rise of British Logic,edited by Osmund Lewry. Toronto: Pontifical Institute of MediaevalStudies. 249-263.Murdoch, John: 1982, “Infinity and continuity,” inThe Cambridge History of Later Medieval Philosophy, edited byNorman Kretzmann, Anthony Kenny, and Jan Pinborg. Cambridge:Cambridge University Press. 564-591.Spade, Paul: 1975, The Mediaeval Liar: A Catalogue of theInsolubilia-Literature. Toronto: Pontifical Institute ofMediaeval Studies.Spade, Paul: 1976, “William Heytesbury's position on‘insolubles’: one possible source,” Vivarium14: 114-120.Sylla, Edith D.: 1971, “Medieval quantifications ofqualities: the “Merton School”,” Archive for theHistory of the Exact Sciences 8, 9-39. This, and Sylla'sother articles, treats Heytesbury within the context of the MertonSchool. Points out how the Calculators extended the practice ofquantification beyond spatial and temporal dimensions, to qualitiessuch as heat. Sylla's interest is chiefly with the scientificcontent of these speculations.Sylla, Edith D.: 1973, “Medieval concepts of the latitude offorms: The “Oxford Calculators”,” Archivesd'histoire doctrinale et littéraire du moyen âge40, 223-283. Discusses the metaphysical consequences found in theassumption that a form such as heat can take on differentintensities.Sylla, Edith D.: 1982, “The Oxford Calculators,” inThe Cambridge History of Later Medieval Philosophy, edited byNorman Kretzmann, Anthony Kenny, and Jan Pinborg. Cabridge: CambridgeUniversity Press. Pages 545-563. A general review, discussing inparticular the nature of a sophisma and teaching context,i.e., the disputations in which they occurred.Sylla, Edith D.: 1997, “Transmission of the new physics ofthe fourteenth century from England to the continent,” in Lanouvelle physique du XIVe siècle, Biblioteca de NunciusStudi e Testi, 24, edited by Stefano Caroti and PierreSouffrin. Florence: Olschki. 65-109. Wilson, Curtis: 1960, William Heytesbury: Medieval Logic andthe Rise of Modern Physics. Madison, Wisconsin: The Universityof Wisconsin Press. Contains a thorough account of the contents ofthe last three chapters of the Regulae, and of the logicaldoctrines contained therein.Yrjonsuuri, Mikko: 1994, “Obligationes:14th-century logic of disputational duties,” ActaPhilosophica Fennica 55, 7-176. An extensive study of theliterature of obligations, touching particularly on Heytesbury, DunsScotus, Roger Swineshead, Richard Kilvington and Walter Burley.Other Internet ResourcesMedieval Logic and Philosophy, maintained by Paul Spade (Indiana University)Guillaume Heytesbury -- Sophismata, transcribed Latin text edited by Fabienne Pironet. Her site also contains a bibliography for Heytesbury.[Please contact the author with additional suggestions.]Related Entries insolubles [= insolubilia] | Kilvington, Richard | Mertonian "calculators" | Paul of Venice | sophismata [= sophisms] | terms, properties of: medieval theories of Copyright © 2007 byJohn Longeway<longeway@uwp.edu> |
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