Four grades of necessity in Buridan

I’m currently reading through Paloma Pérez-Ilzarbe and María Cerezo’s History of Logic and Semantics: Studies on the Aristotelian and Terminist Traditions, a collection of papers in honor of Angel d’Ors, and learning all sorts of interesting things. In Calvin Normore’s paper, “Ex Impossibili Quodlibet Sequitur (Angel d’Ors)”, he looks at Buridan’s criterion for a good consequence and how it relates to the titular Parvipontanean thesis. Lots of interesting things in there especially in relation to 13th-century discussions of the principle, which I am more familiar with than the 14th-century ones. But what caught my eye was something that reminded me of a paper I’ve got on “possible impossibilities” which has been in draft format around 85% done for the last…um…five years (Some day I will finish it. It’s a good paper!) — namely, four different degrees of necessity that Buridan distinguishes. The discussion occurs in the Treatise on Demonstrations, &sec; 8.6.3 in Klima’s translation, in the context of different types of per se propositions. Because per se propositions have to be necessary, and “there are diverse grades of necessity”, there are therefore also diverse grades of perseity:

The first grade of necessity occurs when it is not possible by any power to falsify the proposition while its signification remains the same, nor [is it possible] for things to be otherwise than it signifies.

Another grade occurs when it is impossible either to falsify it or for things to be otherwise by natural powers, although it is possible supernaturally or miraculously, as in “The heavens are moving”, “The heavens are spherical”, and “[Any] place is filled.”

The third grade occurs with the assumption of the constancy of the subject, as in “A lunar eclipse takes place because of the interposition of the earth between the sun and the moon”, “Socrates is a man”, and “Socrates is risible”. These are said to be necessary in this way because it is necessary for Socrates, whenever he is, to be a risible man, and it is necessary, whenever there is a lunar eclipse, that it take place because of the interposition of the earth between the sun and the moon.

There is yet a fourth mode, which involves restriction. For just as ‘possible’ is sometimes predicated broadly, in relation to the present, past, and future, and sometimes restrictively, in relation to the present or the future, in accordance with what is said at the end of On the Heavens — that no force or power can be brought to bear on the past, i.e., on that which is done, but only on that which is or will be (for we say that everything that has been necessarily has been, and cannot not have been) — the same goes for ‘necessary’ and ‘impossible’, which are also predicated either with restriction or broadly (p. 733).

I find this discussion fascinating. First, the distinction between natural and supernatural necessities is quite relevant in connection with positio impossibili and the way this genre of obligatio is used in connection with theological reasoning. Second, the third grade sounds an awful lot like contemporary “analytic truths”. Third, it’s not entirely clear to me what the difference between the first and the third grade is; can anyone suggest an example of something that is necessary according to the third grade but not the first? Fourth, with the discussion of the fourth grade, Buridan has almost everything he needs to run Diodorus’s Master Argument — the different ways to define ‘necessary’, the necessity of the past — all he needs now is to ask whether there is something possible which neither is nor will be true!

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Spotlight on Thomas Bricot

I’m moving office right now, which involves packing up all of my books (my dept. admin expressed doubt when I said seven crates wouldn’t be enough; I’ve now filled up that many twice, and still have about 1-2 more crates’ worth left), and packing up all my books involves looking at all of them, and being reminded of the fact of just how many texts on medieval logic there are that have hardly been touched at all when it comes to modern commentary and analysis.

One of these is Thomas Bricot’s Tractatus Insolubilium, edited by E. J. Ashworth and published by Ingenium in 1986. The book has always interested me in part because it is so short — always a bonus when your Latin skills are never quite what you wish them to be — and I thought today I’d spend a bit of time poking around to see just what there is (or is not) that has been written about this book so far.

First, a bit about Bricot himself, because his name is certainly not one of the better known. He was born in Amiens, France in the middle of the 15th century, and he obtained his BA, MA, and doctorate from the university of Paris in 1478, 1479, and 1490, respectively. After 1490 he held various ecclesiastical and academic posts, in Amiens and Paris, and he died in Paris on April 10, 1516, so we’ve just missed one of his centenaries! More details about his life can be found in his entry in Thomas Sullivan’s Parisian Licentiates in Theology, A.D. 1373-1500. A Biographical Register; Sullivan calls Bricot “a leading figure in Parisian philosophical studies” (p. 115), and Ashworth notes that

Bricot’s work enjoyed considerable success in Paris in the last two decades of the fifteenth century as one can see from the number of editions printed there, as well as in other French centres (p. xiii)

and indeed, much of the references that I found to Bricot modernly are in the context of incunabula (cf., e.g., the Glasgow Incunabula Project). Ashworth’s edition of the Tractatus is based on printed editions only, as no manuscripts of the text are known (or were known in the 1980s). The earliest of these printed editions is from 1491, but the treatise was almost certainly composed earlier, in the 1480s, when Bricot’s focus was on philosophical rather than theological matters. (See more about his works here; a digitized version of the 1498 edition of the treatise on insolubles is available online here).

Given that Ashworth edited the text, it is no surprised that she is also the primary producer of modern commentary on his work:

  • Ashworth, E.J. 1972. “The Treatment of Semantic Paradoxes from 1400 to 1700”, Notre Dame Journal of Formal Logic 13, no. 1: 34-52.
  • Ashworth, E.J. 1974. Language and Logic in the Post-Medieval Period (D. Reidel), especially chapter 2.
  • Ashworth, E.J. 1977. “Thomas Bricot (d. 1516) and the Liar Paradox”, Journal of the History of Philosophy 15, no. 3: 267-280.
  • Ashworth, E.J. 1978. “Theories of the Proposition: Some Early Sixteenth Century Discussions”, Franciscan Studies 38: 81-121.
  • Ashworth, E.J. 1994. “Obligationes Treatises: A Catalogue of Manuscripts, Editions and Studies”, Bulletin de Philosophie Médiévale 36: 116-147.
  • Ashworth, E.J. 2016. “The Post-Medieval Period”, in the Cambridge Companion to Medieval Logic, ed. C. Dutilh Novaes & S. Read (Cambridge University Press): 166-193.

However, recently Hanke has been studying Bricot’s semantics, especially with respect to the views of one of Bricot’s students, John Mair, in his 2014 article, “The Bricot-Mair Dispute: Scholastic Prolegomena to Non-Compositional Semantics”, History and Philosophy of Logic 35, no. 2. (Interestingly, both Bricot and Mair were satirized by Rabelais (cf. Cambridge History of Renaissance Philosophy, p. 794; the CHRP discusses Bricot’s Aristotelian natural philosophy but not his logic). Finally, Bricot is discussed by Lagerlund in his chapter on “Trends in Logic and Logical Theory” in the Routledge Companion to Sixteenth Century Philosophy (2017).

So, there’s a short spotlight on the life of Thomas Bricot and modern discussions of his logic, for anyone who would like to investigate this rather underinvestigated author!

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Iterated epistemic modalities

There is a lot of very interesting medieval work on epistemic modalities, usually found in treatises De scire et dubitare, and whenever I present on this material at conferences, contemporary logicians always want to know whether they considered iterated modalities.

The answer is, it’s not always clear. Certainly the frameworks that they construct are more than apt for iterating, and there are some contexts in which you can find explicit denials of positive introspection (K\phi\rightarrow KK\phi) and multi-agent epistemic statements (e.g., K_aK_b\phi). But I have certainly never found any explicit discussion of unrestricted iteration, or even any clear examples of triple iteration.

Which is why this sophism caught my eye earlier today when reading (D’Ors 2015):

Sortes scit an Plato sciat an Sortes sciat an Plato sciat aliquid de eo.

This sophism is number three in a list of sophisms that “were taken as the ‘topical sophisms’ in connection to the syncategorem ‘an‘ (p. 142), and unfortunately the rest of the paper focuses on the first two, and doesn’t discuss this one further. There isn’t even a translation given, and I have to admit, I’m not entirely confident how to translate it myself; but it certainly does look like it should read something like:

Socrates knows whether Plato knows whether Socrates knows whether Plato knows something of him.

I haven’t been able to find any explicit discussion of this sophism by modern commentators; the closest I’ve found, other than its reference in (D’Ors 2015), is in (Streveler 1993), where it appears in the list of sophisms in the Magister Abstractionum(p. 145). I don’t have access to that text, but the sophism also occurs in Richard the Sophister:

Item, dubitet Plato utrum Sortes sciat aliquid de eo et similiter Sortes dubitet utrum Plato sciat aliquid de eo et proponatur: SORTES SCIT AN PLATO SCIAT AN SORTES SCIAT AN PLATO SCIAT ALIQUID DE EO. Et probatur sic: Sortem scire an Plato sciat aliquid de eo, scilicet de Sorte, est quoddam falsum; et Sortes scit quod nullum falsum scitur, ergo Sortes scit Platonem non scire Sortem // scire an Plato sciat aliquid de eo; ergo // Sortes scit an Plato sciat an Plato sciat etc.

Sed contra: Sortes neque scit Platonem scire neque non scire aliquid de eo; non ergo Sortes scit an Plato sciat an Sortes sciat an Plato sciat aliquid de eo.

I will not attempt a rendering of this at this point; instead, let’s look at a simplified version of the sophism that appears in the Abstractiones of Herveus the Sophister:

Item. In rei veratete Sor scit multa de Platone, Plato multa de Sorte, sed uterque dubitat, utrum reliquus sciat aliquid de se; inde sic: SOR SCIT, AN PLATO SCIAT ALIQUID DE EO. Probatio. Sor scire, an Plato sciat aliquid de eo, est falsum, et Sor scit, quod nullum falsum scitur; ergo Sor scit Platonem non scire hoc; ergo scit Sor, an Plato sciat, etc. Vel sic: Sor scit, an Plato sciat aliquid de eo, quoniam scit se ipsum scire; similiter scit Platonem scire aliquid de Plato; ergo Sor scit, an Plato sciat aliquid de eo. Contra. Ergo Sor scit Platonem scire vel non scire aliquid de eo. Quod falsum est: dubitat enim ipsum scire et ipsum non scire, etc.

Let’s give a stab at translating this:

The truth of things is that Socrates knows many things concerning Plato and Plato many things concerning Socrates, but one of them is uncertain whether the other knows something about him, hence thus: SOCRATES KNOWS WHETHER PLATO KNOWS SOMETHING ABOUT HIM. Proof: That Socrates knows whether Plato knows something about him is false, and Socrates knows that nothing false is known, therefore Socrates knows that Plato does not know this, therefore Socrates knows whether Plato knows, etc. Or so: Socrates knows whether Plato knows something about him, because he [Socrates] knows that he knows something about himself, similarly, he knows that Plato knows something about Plato, therefore Socrates knows whether Plato knows something about him. Contra: Therefore Socrates knows that Plato knows or does not know something about him, which is false: For he is doubtful that he himself knows and that he himself does not know.

I have to say, I’m not entirely sure what to make of this, but maybe I’ll come back to Richard’s version at some point and see if working through that one sheds any light!


  • D’Ors, Angel. 2015. “Tu scis an de mentiente sit falsum Sortem esse illum: On the Syncategorem ‘an‘”, in P. Pérez-Ilzarbe & M. Cerezo, eds., History of Logic and Semantics: Studies on the Aristotelian and Terminist Traditions (Brill): 128-151.
  • Streveler, Paul A. 1993. “A Comparative Analysis of the Treatment of Sophisms in MSS Digby 2 and Royal 12 of the Magister Abstractionum“, in S. Read, ed., Sophisms in Medieval Logic and Grammar. Nijhoff International Philosophy Series, vol 48. (Springer).
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Nugatoriness (Part 2)

This post follows up on one from a few weeks ago, where I started investigating the word nugatoria in the context of medieval logic. What I discovered (thanks google!) when writing that post is that the Ars Emmerana has an entire (albeit small) section on locutio nugatoria! That text is the focus of this post.

The anonymous Ars Emmerana (AE) is edited in de Rijk, Logica Modernorum vol. II, part 2. Concerning its date, de Rijk says both that the text “seems to have come into existence about the same time as the Ars Burana (third quarter of the twelfth century)” (LM II, 1, p. 400) and that “the Ars Emmerana…is older than the Ars Burana” (p. 403). Iwakuma suggests that the author of the AE was a Parvipontanus (Iwakuma, p. 332), and also suggests that the final portion of the text de Rijk edits is not part of the AE. We mention this here, because it is precisely that final page that we are interested in, whether the section De Nugatoriis properly belongs with the AE or not.


Every locution is worthless in which from the same parts of the oration there is a useless repetition of words, as in: ‘Socrates is a white man man.’

Similarly, every locution is worthless in which two predications are combined in which by the signification of one the signification of the other is understood, as in, ‘A man and an animal are Socrates and a man are.’

Similarly, every locution is worthless in which the same speech is combined to itself only by an interposed copulative conjunction, as in ‘A man and a man are’, unless the speech were a numeric or pronominal demonstrative.

[It is objected: ‘Men are, therefore a man and a man are’. An instance: ‘Now spirits live; therefore now a spirit lives’.]

But because ‘two’ is a numeric noun, for this reason this locution is admissable (competens): ‘Two and two are four’. Similarly, [because] ‘that’ [is] a demonstrative pronoun; for this reason, this locution is admissable: ‘That [thing] and that [thing] are white’.

Similarly, every locution is worthless in which from the same part of the locution two speeches are put forth the significations of which are always the same, as in ‘A human who can laugh reads’.

Similarly, whenever two speeches are brought together from the same part of the oration the significations of which are never the same, that locution is worthless, as in ‘A donkey who can laugh runs’ and ‘A braying man walks’.

Similarly, whenever this speech ‘which’ (qui) is put between such speeches the significations of which are either always the same or never the same, the locution is worthless, as in: ‘A man which is braying’, ‘A donkey which is able to laugh’.

Similarly, a locution is worthless whenever either the whole of a part or the part of a whole are enumerated [together], as in, ‘France and Paris’, and ‘Paris and France’.

Similarly, a locution is worthless on account of an inept demonstration, as when I say “That man runs”, indicating an ass, or vice versa.

[And it must be known that ‘to speak nonsense’ and ‘to advance a worthless locution’ are not the same, because it is possible that something is not nonsense but nevertheless corresponds to a locution put forwards as worthless. For when someone says, indicating an ass, ‘That man reads’, he speaks nonsense, yet nevertheless he does not put forward a worthless locution, because only that speech is worthless which never can be put forward except worthlessly.]


I have used “locution” for locutio, “oration” for oratio, and “speech” for dictio. I recognize that this is not the most immediately fluent of translations, but it helps reinforce the spoken nature of the material under discussion — it is repetitive speech, not repetitive words, that is nugatoria.

In reference to our previous post on this topic, it is interesting to note that nugatoria here has absolutely nothing to do with truth or falsity, per se.


  • Anonymous, Ars Emmerana, in Logica Modernorum, vol. II, part 2, edited by L.M. de Rijk, Van Gorcum & Comp., 1967: 143-174.
  • de Rijk, L.M., Logica Modernorum, vol. II, part 1, Van Gorcum & Comp., 1967.
  • Iwakuma, Yukio, “Influence”, in Cambridge Companion to Abelard, edited by Jeffrey E. Brower and Kevin Guilfoy, Cambride University Press, 2004, 305-335.
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The Logicians Who Say ‘Ne’

I’m currently in Stockholm for the 3rd Nordic Logic Summer School where I’ve been giving a introduction to logic in the Middle Ages. I brought along the Big Four (the textbooks of Bacon, Sherwood, Lambert, and Peter), as well as Kretzmann’s translation of Sherwood’s treatise on syncategorematic terms. The latter is a treatise that I’ve been wanting to read cover to cover for some time now, and flipping through it this week has only served to increase this desire.

There is one short chapter which regularly catches my eye because its title is the only one that Kretzmann doesn’t even try to translate: “the particle ‘ne'” (Sherwood/Kretzmann, p. 156).

This word, Sherwood notes, can be used either to turn an assertion (e.g., Socrates currit ‘Socrates runs’) into a question (e.g., Curritne Socrates ‘Is Socrates running?’) or “prohibitively”, that is, as a negation. (Note that English ‘no’ can also have a function of turning an assertion into a question, e.g., “Socrates runs” vs. “Socrates runs, no?” The latter sentence can be understood to be elliptical for something like “Socrates runs, does he not?” or “Doesn’t Socrates run?”).

It is the prohibitive use of ne that Sherwood focuses on, and he distinguishes two ways: ne can effect a prohibition by itself, as in the imperative ne currans (‘Do not run’), or “in such a way that what is prohibited is itself ordered together with something preceding it” (p. 156), as in volo ne curras (‘I want you not to run’).

The sophism sentence that Sherwood focuses his discussion of ‘ne’ around is:

Tu vis ne tibi concludatur, et [tu] caves ne tibi concludatur (p. 156).

The Latin structure is much more nicely parallel than the English; Kretzmann translates this as:

You want not to be confined, and you are wary lest you be confined (p. 156)

and the problem arises in that you can draw the apparent conclusion that you want/desire and are wary of one and the same thing (with the implication that wanting and being wary of are incompatible attitudes to have to one thing.

The first way of solving the sophism that Sherwood offers is one in which the two nes are not the same; the first, “some say”, is equivalent to ut non ‘that not’, and the second is equivalent to ut ‘that’; the latter carries with it a negation that the other doesn’t have, and thus these words only look to be the same but are not. In reply, Sherwood points out that ne always seems to carry with it negation. Furthermore, the second ne can be plausibly interpreted as ‘that not’, “for you are wary on this account, that you not be confined (propter hoc ut non tibi concludatur)” (p. 156).

A better way to solve the sophism is to note that ne can be used transitively and intransitively. If ne is used transitively, then the second sentence is understood as “you are wary of this lest you are confined”, while if it is intransitively, it is understood as “you are wary on this account, lest you are confined”. The first interpretation is false; the second is true. The correct conclusion to draw is that “you want and are wary on account of one and the same thing” (p. 156).

And that is the full and complete account of the Logicians Who Say Ne, as told by Guillelmus de Shyreswode.

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Nugatoriness (part 1)

Last week I was in Bonn for the Time and Modality workshop in Bonn, where I gave a talk based on my paper, “The Logic of Where and While in the 13th and 14th Centuries”.

One of the definitions I quoted comes from the anonymous Ars Burana (c1200):

In temporali et causali: si antecedens est falsum et consequens verum, nugatoria est [AB, p. 191].

I translated nugatoria as “worthless”, and the question came up as to why a temporal proposition (something of the form “X while Y”) is worthless when X is false but Y is true. Is it nugatoria because it is invalid? Or is it invalid because it is nugatoria? Are these terms equivalent? Nugatoriness (I can coin a word if I want to) is not something you commonly see in later 13th century and 14th century logical treatises — indeed, I’m not sure I’ve come across it before. But the question made me think, and I wanted to see what else AB had to say about being nugatoria.

This definition of the truth and falsity of temporal and causal propositions comes in the section De propositione ypothetica eiusque speciebus, with the species considered being conditionals, temporals, locals, causals, copulatives, disjunctives, and adjunctives. Of these seven, no truth conditions are given for local propositions; truth and falsity conditions are given for conditionals; only truth conditions for copulatives, disjunctives, and adjunctives; and only the conditions of being nugatory are given for causals and temporals. This appears, as far as I can tell, to be the only occurrence of nugatoria in this treatise.

However, in searching for hints about nugatoriness in AB, I found a couple of other 13th-century examples of the term; these will make up a hopefully future part 2 post!


  • Anonymous, Ars Burana, in Logica Modernorum, vol. II, part II, edited by L.M. de Rijk, Van Gorcum & Comp., 1967: 175-213.
  • Uckelman, Sara L., “The Logic of Where and While in the 13th and 14th Centuries”, in Advances in Modal Logic vol. 11, edited by Lev Beklemishev, Stéphane Demri, András Máté, College Publications: 2016, 535-550.
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Bulthuis on Burley on propositions

Over at the blog of the APA today, there’s an excellent interview with Nathaniel Bulthuis, whose research on Walter Burley’s views on the nature of propositions is germane to the interests of medieval logic.

Among the many quite interesting things he says in the interview is this:

Medieval philosophy – especially philosophy in the 13th and 14th centuries – rivals anything we have today in terms of its technical sophistication.

Whether you agree or disagree with this, pop on over to the APA blog to read the full interview!

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