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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Why medieval logicians aren’t ‘medieval’

‘Medieval’ is so often used as a pejorative, particularly in contemporary culture. One only need to hang around with #twitterstorians and #medievaltwitter to hear the collective teeth grinding whenever ‘medieval’ gets used by, e.g., modern journalists in this way.

Well, it’s time to grind my teeth a bit, but hopefully in a way that is at least a bit productive.

One of the thing that surprised me most when I began working in medieval logic was how interesting all of it was and how little anyone knew about it; why did it take until the 1950s that people started looking seriously at medieval logical texts? Why are their developments still not routinely integrated into modern logical pedagogy and research? The more I read about the history of logic and the history of philosophy, the more I saw that the consequences of the Renaissance and Humanist backlash against “Scholastic philosophy” were still perpetuating on to today. Scholastics weren’t interested in anything other than wrangling about angels on pinheads. Their motives are outdated, their methods are too abstruse. They don’t have anything to tell us today.

And thus we get 500 years of ignorance of medieval logic.

Unfortunately, this negative view of Scholasticism and its methods still abounds. Recently I was reading Steven Nadler’s “Who was the first modern philosopher?”, and in the midst of what is an otherwise very interesting and well-written piece, I came across this:

By contrast, it is more difficult to find such affinity between contemporary thought and medieval Scholastic theories and methods.

Huh, I thought, That’s surprising. I am routinely astounded by the amount of affinity between contemporary logical thought and medieval Scholastic logical theories and methods. I wonder what Nadler is referring to, or what he is classifying as “Scholastic theories and methods”?. A little bit later on, I read:

In terms of its method, the New Philosophy (for this is how contemporaries often referred to it) was generally characterized by a reliance on reason and empirical evidence rather than devotion to religious or ancient authority to determine what is and what is not philosophically and scientifically acceptable.

Huh, I thought, What about the after effects of the 1277 condemnations, wherein the Arts Masters began to investigate Aristotelian mechanics in a wholly secular fashion, without any reference to God and God’s capacities?. And then I read:

The modern philosophers, unlike their Scholastic predecessors and less progressive contemporaries, regarded terrestrial and celestial phenomena as constituted by one and the same kind of matter and governed by a single set of laws.

Huh, I thought again, What about that treatise I translated and analysed in my dissertation, where the anonymous author went to great pains to explain how Aristotelian syllogistics can cover both created and divine matter, to show how the so-called paralogisms of the trinity were in fact simply invalid. That sure sounds like someone regarding terrestrial and celestial phenomena as being governed by the same set of rules and reasons (even if maybe metaphyiscally they are distinct).

Now, Nadler hasn’t set out in his piece to give an account of the character of medieval philosophy; he’s trying to answer a question about “modern” philosophy. Nevertheless, throwaway remarks about how “medieval” medieval philosophy was — which is what the comments above amount to — are problematic. They are problematic because they range from misleading to outright wrong — or, if this is a correct view of medieval philosophy, then the appropriate conclusion to draw is that logic from the late 11th to the end of the 14th centuries was not “medieval”.

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Scholastic learning in late 15th C Italy: One very small snapshot

Today I picked up an ILL book which I’d requested for reasons entirely unrelated to philosophy/logic:

Cesare Cenci, Documentazione di vita assisana 1300-1530, volume II: 1449-1530 (Grottaferrata: Collegii S. Bonaventurae ad Claras Aquas, 1975).

Apparently, no one in Manchester is greatly enthused by Assisi, since virtually all of the pages were still uncut. I spent the better part of an hour this afternoon cutting them, which means I glanced at almost every page.

Capital letters catch my eye, and when I saw a Boetius, my first thought was “Huh, that’s a bit unusual in 15th C Italy, someone’s parents were clearly influenced by antiquity”, until I read the next phrase, de consolatione. Oh! It is the antique Boethius!

And then I saw super sententias elsewhere in the treatise, and sententie Iohannis Damasceni, and then I was reading from the beginning because here was a huge list of books, and I wanted to know the context.

The context is a donation of books from one Çacharias de Tergesto, a member of the Friars Minor and a warden of Istria, along with Almerichus de Mugla, Angelus de Castagnano, Antonius de Mugla, and Pasqualinus de Ubaldinis, to the Basilica of San Francesco d’Assisi in 1467. Because it is so interesting — and I am sure someone who is working in the transmission of Scholasticism in late 15th C Italy will find this relevant — I am reproducing the list of texts below; items in [square brackets] are notes added by the editor, providing further information identifying the texts:

in primis quatuor volumina cum tabulis conventus [s. Mariae] Venetiarum,: primum volumen est secundus et tertius d. Bonaventure super sententias, secundus Alexandri et quartus Ricardi. Item, 13 quinterni in pergameno, in quibus sunt aliqua Augustini et Gregorii. Item, magister sententiarum completus et sententie Iohannis Damasceni; suus. Item, quadragesimale dictum et pelegrino. Item, conpendium theologice veritatis [Hugonis de Argentina], in pergameno, cum tabulis coopertus. Item, tabula martiniana [Martini de Troppau] decreti et decretalium. Item, Tulius de officiis, cum tabulis et corio rubeo, cuius principium: quanuam te, Marce; finis vero: rebus misterii. Item, Persius [satyrae], cum tabulis et corio albo ad ligaturas; finis vero: liber, amen. Item, quadragesimale quodam in papiro, sine tabulis, cuius principium: voca, operarios; finis vero: ego dico sitio. Item, sermones predicabiles, in papiro, sine tabulis, cuius principium: fecit Deus duo; finis vero: tu es sacerdos. Item, Boetius de consolatione, in pergameno, sine tabulis. Item, Tulius de amicitia et paradoxe, in papiro, sine tabulis. (Charter no. 1)

(As it turned out, I was cutting pages while catching up with a friend who is writing her PhD thesis on parchment making practices, so we paused a moment to squee over the fact that some of these were noted as on parchment and others as on paper.

So, not exactly logic, but still, really cool!

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