Abstract
Occupation with the moment of change in the Middle Ages starts out from Aristotle but progresses from there, as is typical of philosophy of the period. Not only does a lot of systematization take place in relation to the moment of change, but also considerable further development. The level of argumentation in the 13th and 14th century is not regained until the 20th century.
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Notes
Curtis Wilson: William Heytesbury, Madison (Wisconsin) 1956. Norman Kretzmann: Incipit/Desinit in: Machamer /Turnbull (eds.): Motion and Time, Space and Matter, Columbus (Ohio) 1976.1 have tried to work with the original texts as far as possible. This was facilitated by a lot of excellent editing work in recent years. Wherever the texts were edited and accessible I found Kretzmann’s and especially Wilson’s summaries to be reliable. A very helpful article for putting the medieval texts in a broader perspective is Knuuttilas ‘Remarks...’. A good short introduction to the medieval treatment of beginnings and endings from a formal point of view is found in Peter Ohrstrom: Temporal Logic — From Ancient Ideas to Artificial Intelligence, Dordrecht 1995. Another interesting introductory study is J.E. Murdoch’s ‘Propositional Analysis in Fourteenth Century Natural Philosophy’ (Synthese 40, 1979, p.117–146). Knuttila/Lehtinen ‘Change and Contradiction: A fourteenth century controversy’ (Synthese 40–1979 — p.189–207) is occupied with a ‘counter-current’, not with the ‘mainstream’ of medieval texts on the moment of change. I refer to it in ch. I, 3.4.
Kretzmann and Wilson differ on this point. Cf. Kretzmann: Incipit/Desinit, p. 103–105.
Kretzmann: Incipit/Desinit, p. 105.
Averroes’ remarks on the moment of change are in his commentary on Aristotle’s Physics VI, sections 40–46. A detailed discussion of them is provided by Knuuttila in: Remarks..., pp.254–256. Text: Aristotelis Opera cum Averrois Commentariis, Venetiis apud Junctas 1562- 1574, reprint Frankfurt 1962.
Cf. Knuuttila: Remarks..., p.257. The relevant remarks are in: Albertus Magnus [Albert the Great] Physicorum libri VIII (in: Opera omnia, ed. A.Borgnet Vives., Paris 1890), commentary on book VI, tractatus 2, ch.4.
Ibd. The relevant text is Aquinas’ commentary on Physics VI (Torino 1954), lectio 7, 822 and lectio 8, 838.
Cf. Knuuttila/Lehtinen: Change..., pp.l91f.
Walter Burleigh [also: Burley]: De primo et ultimo instanti, ed. by Herman and Charlotte Shapiro, Archiv für Geschichte der Philosophie 47 (1966), p. 157–173.
Concerning John of Holland and Wiliam Heytesbury cf. Wilson: William Heytesbury, pp.34ff.
William of Ockham: Summa Logicae part I, 75 and especially part II, 19.
Text in Nielsen op. cit.
These two examples are counted as the most frequently treated sophisms containing ‘incipit’ and desinit’ by Andrea Tabarroni. They appear as early as in Peter of Spain or in Matthew of Orleans. Cf. A. Tabarroni: ‘incipit’ and ‘desinit’ in a thirteenth century sophismata-collection, Cahiers de l’institut du moyen-age grec et latin [CIMAGL] 59 (1989), pp.61–111 (p.78). More sophisms on the topic are edited in the same volume of CIMAGL by Alain de Libera (Le sophisma anonyme ‘Sor desinit esse non desinendo esse’ du Cod. Parisianus 16135, pp.l 13–120) and by Sten Ebbsen (’Three 13th century sophismata about beginning and ceasing’, pp. 121–180). Cf. also Alain de Libera: La problematique de I’instant du changement au 13me siecle: contribution a l’histoire des ‘sophismata physicalia’, in: S.Caroti: Studies in Medieval Natural Philosophy, Firenze 1989. An impressive formal reconstruction of some sophisms of this kind is found in Ohrstrom, p.52–64.
The Sophismata of Richard Kilvington, Introduction, translation and commentary by Norman and Barbara Kretzmann, Cambridge 1990. Examples for the kind of sophismata in Kilvington are: Socrates is whiter than Plato begins to be white (No 1), Socrates is infinitely whiter than Plato begins to be white (No 2), Socrates begins to be whiter than Plato begins to be white (No 3), Socrates will begin to be as white as he himself will be white. Cf. also Norman Kretzmann: Socrates is whiter than Plato begins to be white, Nous 11 (1977), pp.7–15.
It is important to note in this context that most of the peculiar features of the Greek verb aspect do not translate into Latin.
An important exception is Walter Burleigh’s ‘De primo et ultimo instanti’. This text is not formulated in terms of the analysis-of-language level about ‘incipit’ and ‘desinit’ statements but directly about first and last instants. In this way the motivation is very clear and there is room for direct proofs with dense time, which are very good. Burleigh gives an account on the analysis-of-language level in his ‘De puritate artis Logicae’ (ed. by Ph. Boehner, Louvain 1951) in the chapter ‘incipit et desinit’. Lauge О. Nielsen correctly recognizes that not only the form but also the contents of the two texts considerably differ in detail (L.O. Nielsen: Thomas Bradwardine’s treatise on ‘incipit’ and ‘desinit’, edition and introduction, Cahiers de l’institut du Moyen-Age grec et latin 42 (1982) [Copenhague], p. 1–83; cf. p. 17.)
Knuuttila: Remarks..., p.254; Sorabji: Time..., p.419 and esp. Kretzmann: Incipit/Desinit, pp.l33f. (footnote 32). According to Kretzmann the most important passage in Aristotle for this context is Cat.6, 5al5–37.
A tractatus exponibilium, sometimes attributed to Peter of Spain, differs in an interesting respect. Its author characterizes res permanentes as objects”quarum totum esse acquiritur simul in instanti“, i.e. which acquire their whole being at a single instant. He characterizes res successivae as such objects”quarum esse acquiritur successive et pars post partem“, i.e. which acquire their being part by part. His examples are unusual, too: for rather odd reasons ‘white’ and ‘black’ are declared to be res successivae. Kretzmann thinks that the attribution to Peter of Spain is incorrect (Kretzmann: Incipit/Desinit, p. 133, footnote 26 and p. 134, footnote 39). Text: The Summulae Logicales of Peter of Spain, ed. by Joseph P. Mullally, Notre Dame/Indiana 1945, p. 114–117.
Wilson: Heytesbury, pp.32f. A revival of this seems to be the classification into ‘endurers’ and ‘non-endurers’ as made in Harold W. Noonan: Personal Identity, New York 1991.
In what follows I will use this plural. It seems to me correct because ‘res’ is feminine. Wilson, during English text, uses ‘res permanentia’ (cf. e.g. p.34), obviously taking ‘res’ to be neuter. I think he is wrong there.
Walter Burleigh: De Puritate Artis Logicae (Tractatus Brevior), Section ‘De incipit et Desinit’ (quoted according to the edition by Ph. Boehner, Franciscan Institute Publications, St. Bonaventure/Louvain 1951, p.59.
This example is found in the Summulae Logicales (p. 114/115 Mullally-edition) wrongly attributed to Peter of Spain, in Peter of Spain’s ‘Syncategoreumata’ (ed. by L.M. de Rijk, transl. by J.Spruyt, Leiden 1992, p.250/251; cf. also Kretzmann: Incipit/Desinit, p.122) as well as in Peter of Mantua (Wilson: Heytesbury, p.37).
Cf. ch.I 2.2.1.
William of Sherwood: Tractatus Syncategorematum, transl. by Norman Kretzmann as Treatise on Syncategorematic Words, Minneapolis 1968, p. 109. Latin text: William of Sherwood, ‘Syncategoremata’, ed. by J.R. O’Donnell, in: Medieval Studies vol. 3 (1941), pp.46- 93.
E.g. in Walter Burleigh: De Puritate..., p.61; Bradwardine, p.77.
William of Sherwood: Tractatus..., p. 109.
Burleigh: De Puritate..., p.61; William of Sherwood: Tractatus..., p. 110.
Peter of Spain: Syncategoreumata, p.257/258. Cf. also Kretzmann: Incipit/Desinit, p. 124. Summulae Logicales, p. 116/117.
William of Sherwood: Tractatus..., p. 109.
One wonders whether there might not also be something ‘in’ which Socrates exists. Spinoza’s bold answer a few centuries later was:”Yes. God/the universe/Substance“. As is clearly seen, this may be a big leap for a metaphysician but it is only a small step for a logician.
Apart from these rather fundamental problems, there is the rather more theory-immanent problem that it is difficult to see why ‘whiteness’ or ‘being well’ should be ‘res permanentes’. One can hardly say that ‘all their parts exist at the same time’. For it is difficult to see what, for instance, the parts of ‘being well’ might be. The problem could be alleviated by regarding the definition of ‘res successiva’ as the crucial one and regarding as ‘res permanens’ simply what is not covered by the definition of ‘res successiva’, no matter if this is the case as its parts exist simultaneously or rather because it lacks parts altogether. This seems to be the line taken by Burleigh in ‘De primo et ultimo instanti’ where an example for a ‘res permanens’ is the truth of the statement”Sortes currit“(p. 164). Burleigh’s reason is that the truth of a statement, unlike the succession of words in it, is a quality which is wholly there at once:”veritas est qualitas propositionis qu[a]e habet esse tota simul“. This clearly means no more than that there are no consecutive parts of the truth of a statement, but does not imply that the truth of a statement has any parts.
Doing so amounts to thinking in the manner of predicate logic. However, one should be aware of the fact that this habit is an exact mirror-image of the medieval logicians’ habit of thinking. Cum grano salts one can say that medieval logic turns everything into nouns (even words for colours); predicate logic turns everything into adjectives (since Quine even proper names).
This is stated in order to facilitate the definition of the State-prefix in what follows. It would be possible, but more complicated, to work with constant target and starting positions.
‘Referring to’ here indicates the relation which holds, e.g., between the property of being white and the predicate”...is-white“. F and P are meta-language variables.
Mind that F might stand for the predicate”...is a time at which Socrates exists“. This is, however, no existence-predicate; for it is possible to define it in a Quinean manner by saying that it is satisfied by a time iff there is something at this time which socratizes.
The example is rendered in Wilson: Heytesbury (p.34).
I have taken the expression ‘in the course of’ in this context from Sorabji: Time...p.408 (footnote 9). A similar view to that of John of Holland seems to be expressed in Burleigh’s ‘De primo et ultimo instanti’ (p. 173, point (8)). The point Burleigh makes here is rather complicated, but it is clear enough that Burleigh presupposes that there are instants of motion (“de re successiva bene tenet quod si haberet esse in instanti, habet esse in quolibet instanti ilius temporis in quo est primo [...] si motus, vel quelibet res successiva, habet esse in instanti, instans esset pars temporis.”)
Cf. the summaries in Wilson: Heytesbury, p.33–44, esp. his evaluation on p.44 and Kretzmann: Incipit/Desinit, p.lll as well as Sorabji: Time..., p.419.
There are some (as a rule, early) authors who work with discrete time, especially when dealing with sophisms. This becomes clear because they use phrases like ‘paenultimum instans’ which make sense only for discrete time. For example, this phrase appears in an anonymous English 13th century sophisma edited by Ebbesen (cf. p. 123) and in Henry of Ghent (cf.ibid. p. 126; text also in M. Braakhuis: ‘De 13de Eeuwse Tractaten over Syncategorematische Termen, vols. 1–2, Leiden 1979). The phrase also appears in William of Sherwood (Tractatus, p. 114); cf. ibid. Kretzmann’s footnote 12. The most detailed reconstruction of treating the moment of change with a discrete model is in 0hrstr0m loc. cit. 0hrstr0m mentions Richard Lavenham as an author in whose work a growing awareness of continuous time can be studied (op.cit. p.56). Dense time is also explicitly argued for in connection with the moment of change in Peter of Auvergne’s sophisma VII (edited in Ebbesen loc. cit, cf. pp,168f).
Cf. p. 166 of the Shapiro edition. Burleigh gives explicit rules for the assignment of limiting instants since, as mentioned, his treatise is not formulated on the analysis-of-language- level.
Cf. Shapiro edition, p. 163.
Cf. Burleigh: De puritate..., p.62 line 8–11.
Kretzmann conjectures that motivation for not allowing last instants of any states whatsoever is that a putative last instant t cannot be identified as a last instant at t already, as one can only say after t that t was in fact the last instant of the state in question. If, however, t is the first instant of a state, it can already be identified as such at t (Kretzmann: Incipit/Desinit, pp. 112f). I agree with Sorabji that this epistemological motivation is rather weak (cf. Sorabji: Time..., p.419). I think much stronger motivation could be inferred from what I take to be Aristotle’s view on substantial change: perishing is a process towards the end of existence of an object while this object still exists and of which there is no last instant. It terminates in a first instant of an object’s non-existence. Nielsen holds a similar view (cf. Nielsen, p.22). It is a little puzzling that Kretzmann and Sorabji discuss the point by referring to Peter of Spain. According to the critical edition of his ‘Syncategoreumata’ Peter seems to take the opposite view that there are in fact last instants of ‘res permanentes’ (cf. 5 on pp.252f).
Sorabji writes that this motivation is made explicit in Walter Burleigh’s ‘De primo et ultimo instand’, which he takes to be the best medieval text on the moment of change. He is sceptical whether other authors’ opinions are as clearly motivated as Walter’s or whether they just follow a more or less Aristotelian tradition. Sorabji: Time..., p.419. It is true that it is hard to see what else might have motivated those philosophers who took time to be discrete (for example William of Sherwood) to assign limiting instants in the way they did. Of course there then arises the question (to which I have no answer) of why someone who is acquainted with the ‘Physics’ would ever take time to be discrete in the first place.
Shapiro edition, p. 166.
It is important to notice that i) to iv) are implications and not equivalences. They can be turned into equivalences by asssuming the existence of a limiting instant in dense time, or by assuming that time is continuous (which is a stronger claim than that time is dense). Details in appendix C.
Cf. Wilson: Heytesbury, pp.39f and the Mulally edition of ‘Summulae logicales’.
Burleigh: De puritate..., pp.59ff.
Cf. John Le Page: Syncategoremata, partially edited in Braakhuis op. cit., pp.l98f. Cf. on this topic Tabarroni op. cit. p.75.
Edited in Ebbesen op. cit. Cf. p. 158:”nunc autem tempus vitae Socratis est terminatum ad dua instantia, primum et ultimum; est ergo dare ultimum instans vitae Socratis, et in illo instanti vivit...“(Socrates’ lifetime is limited by two instants, a first and a last one; so there is a last instant of Socrates’ life, and at it he is still alive).
Peter of Spain: Syncategoreumata, pp.253–55 (cf. also Wilson: Heytesbury, pp.38f). William of Sherwood: Tractatus..., p.110.
Cf. Tabarroni, p.75.
This is also exactly Gabon’s characterization of a ‘momentary event’. Cf. ch.II,l.
Cf.Wilson: Heytesbury, p.33, Burleigh: De primo et ultimo instanti, p. 164, 167, 169. Burleigh: De puritate...p.62 (In ‘De puritate...’ instantaneous ‘res permanentes’ become an exception to the rule as they have a last instant of existence).
Burleigh: De primo et ultimo instanti, p. 165; cf. also Wilson: Heytesbury, pp.33–35. Kilvington must have held a similar view. In sophisma No 1 he plausibly renders a process of becoming white by ‘remotio prasenti’ plus ‘positio futuris’ (he does not allow himself to be confused by the fact that white is a ‘res permanens’).
Cf. Wilson loc. cit.
For the differences between Burleigh I and Burleigh II compare also Nielsen op.cit. pp.l7f. By keeping Burleigh I and Burleigh II apart I do not want to claim that ‘De puritate...’ is the earlier work. In fact, according to Nielsen’s view, it must be the later work. However, I would claim that the view of ‘De primo et ultimo instanti’ is more progessive and more differentiated. Perhaps Burleigh even chose a simpler version for his elementary logic textbook than for his specialists’ treatise.
Guillelmi de Ockham Summa Logicae, St.Bonaventure, N.Y. 1974, pp.310f (Pars II, Cap. 19).
Sorabji: Time..., p.418/419. Ockham’s earlier treatment of ‘incipit’ and ‘desinit’ in Summa 1,75 is no better than 11,19 in this respect, since it is not occupied with assigning limiting instants at all, but with the question to what kind of ‘suppositio’ incipit- and desinit- statements belong. I shall not consider this point. Details are found in Nielsen’s introduction to Bradwardine. Nielsen convincingly argues that Ockham changed his view on the supposition question between writing 1,75 and 11,19 because of Bradwardine’s explicit criticism (found on pp.74f). Interestingly, Ockham’s very simple suggestion is the revival of some earlier authors’ views, which is called the (outdated) ‘expositio communis’ by John Le Page (Cf. John Le Page: Syncategoramata in: Braakhuis, pp.198. Cf. also Tabarroni, p.75). The opposite, equally over-simplified view that there are neither first nor last instants whatsoever is found in Matthew of Orleans (cf. Tabarroni, p.77).
For a discussion of this point I am grateful to Simo Knuuttila who, if I remember correctly, agreed that rest was usually regarded as a ‘res successiva’ as well as motion. Kretzmann thinks so too (’Incipit/Desinit’ cf. p.114).
While I think that Kretzmann is wrong in claiming that Aristotle favoured a ‘Neutral Instant Analysis’ for the s-change between rest and motion, I agree with him that most medieval logicians did (cf. Kretzmann: Incipit/Desinit, p. 114).
Phys.VIII 264a27.
Cf. Wilson: Heytesbury, p.34.
E.g. Walter Burleigh in ‘De primo et ultimo instanti’, cf. esp. p.163. Cf. also Peter of Spain: Tractatus syncategorematum, translated in: Kretzmann: Incipit/Desinit, p. 123:”...permanent things are intrinsically limited while successive things are not limited intrinsically but limited by permanent things — for example a change is limited by a quantity or a quality.“(res permanentes sunt in se terminate, sed res successive non habent in se terminos ut motus qui terminatur ad res permanentes, ut alteratio ad qualitatem, augmentum et diminutio ad quantitatem, critical edition, pp.250f). Peter’s detailed analysis of ‘res permanentes’ follows further down in his text; as we have seen, it is a little more complicated than he states in these introductory remarks.
Wilson: Heytesbury, p.38.
Cf. esp. William of Sherwood: Tractatus..., p. 110.
As I will show in ch. 11,1 Antony Galton is a most radical contemporary advocate of reading the present tense of ‘begins’ atemporally: he combines the idea that a begins to be F in the course of the history of the world with the idea that it does not do so at any time.
Ockham: Summa..., pars II, Cap. 19 (p.311).
The awareness that sentences are true with respect to times is, incidentally, once again a remarkable anticipation of Prior’s ideas.
According to Wilson: Heytesbury, p.42, William Heytesbury is criticised for this by Peter of Mantua and Gaetano de Thiene. The argument in which their criticism consists is also found in a text of Johannes Venator (died c.1427) rendered by Kretzmann in ‘Incipit/Desinit’ p. 128–130 (Logica, Tractatus III, Cap.6). Another indication that ‘incipit’ should be read temporally is found in Peter of Spain:” ‘incipit’ et ‘desinit’ sunt presentis temporis“(Syncategoreumata, p.254/5, cf. also Kretzmann op.cit. p. 125). In Walter Burleigh’s ‘De Puritate Artis Logicae (Tractatus Brevior)’ we find the opinion that both analyses are equally correct alongside the opinion that one has to decide on one analysis according to ‘res permanens’ or ‘res successiva’:”Sortes incipit esse albus’ potest dupliciter exponi...uno modo per negationem praesentis et positionem praeteriti... alio modo per positionem praesentis et negationem futuri“(p.60).”Sed ‘incipit’ in successivis et permanentibus aliter et aliter debet exponi, secundum quod propositiones sunt verae in permanentibus per positionem praesentis et negationem praeteriti. Sed in successivis debet exponi per negationem praesentis et positionem futuri.“(p.61). Since the second opinion is very well motivated in the text, I assume that the first opnion is just meant as a working hypothesis in the sense of”According to what has been said so far, one might think that...“.
Cf. William of Sherwood: Tractatus..., p.109/110 and Wilson: Heytesbury, pp.38f. I do not agree with Kretzmann (footnote 12, p. 109 of his translation). It is important to note that the point at stake is not at all influenced by the fact that William works with discrete time while most other logicians later on do not.
Annotation for experts in tense logic: A Priorian notation for (la1) would be ‘p л P—ip’, for (la’) ‘p л —iPp’ or ‘p л H-ip’; for (3b’) ‘—iq л Fq’ and for (3b’) ‘—iq л Gq’. A notation for (3b’) in the manner of v.Wright would be ‘-iqTq’. All formulas would have to be interpreted for t while p stands for ‘a is white’ and q stands for ‘a is in the course of moving’. ‘pTq’ is true at t iff p is true at t and there is a t’ > t, such that q is true at t’. This is the semantics of the T- operator as v.Wright introduces it in ch. 2 of ‘Norm and Action’ and in ‘And Then’ (p and q are so-called ‘generic propositions’ which are false and true relative to times).
Ockham: Summa... Pars II, Cap. 19 (p.311).
In my view, Ockham is correctly referring to 222a20/21.
Ockham loc. cit.
E.g. Bradwardine’s treatise on ‘incipit’ and ‘desinit’ and Kilvington’s ‘sophismata’.
Ockham, Summa... 11,19.
Wilson: Heytesbury, p.42.
Kilvington comes very close to it in sophisma 16 at the beginning point (c). In fact he seems to use Paul’s definiens. But the connection with the word ‘immediate’ is not explicit yet.
Paulus Venetus: Logica, p.107. Perreiah translates the sentence (p.203):” ‘before A you were and there was no instant before A unless between that and a you were’ [...] ‘no instant will be after В unless between that and В you will be’”. Wilson, who (p.42) points out Paul’s merits for our topic, translates:” ‘prior to a you existed and there was no instant prior to a such that between that instant and a you did not exist’ [...] ‘after instant b you will exist, and there will be no instant after b such that between that and b you will not exist’”.
For this term also cf. the discussion of Kienzle in ch. 11,1.
If this were the case, it would be incomprehensible why Paul analyses ‘immediate’ in this sense directly after having dealt at length with examples involving oscillation on the preceding pages.
It is easy to see that ImmP-P(t) implies P-P(t) and that ImmF-P(t) implies -F-P(t).
The following quotations are all from pp.60–62 of Walter Burleigh’s ‘De puritate...’
On the assumption that a limiting instant exists. This would be entailed by presupposing continuous time.
Explicitly so in William of Sherwood’s Tractatus..., p. 110. This is especially interesting since William takes time to be discrete, so that one might expect him to prefer a solution similar to Strang instead (cf. ch. 1,1.3.2. (b)).
Cf. Simo Knuuttila/Anja Inkeri Lehtinen: Change and Contradiction: A Fourteenth Century Controversy, Synthese 40 (1979), pp. 189–207 and Knuuttila: Remarks...
Knuuttila/Lehtinen: p. 190. See esp. the quotation from Carracioli in footnote 5, p.200.
Knuuttila/Lehtinen: Change..., pp.l95f.
I follow Knuuttila/Lehtinen: Change..., p. 194. cf. also the quote from Carracioli at the very beginning of Knuuttila: Remarks..., p.245.
Knuuttila/Lehtinen: Change..., p. 194:”In this solution a real contradiction is accepted in nature, because the termini of instantaneous change cannot, according to it, belong to different instants of time.“
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Strobach, N. (1998). The Moment of Change in the Middle Ages. In: The Moment of Change. The New Synthese Historical Library, vol 45. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9127-0_4
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