Syllogism Mnemonics

The other day a colleague of mine asked if I had anything I could send him regarding the medieval syllogism mnemonics. I told him there was some info in the textbook I’m writing, but it’s rather idiosyncratic to the way I present the syllogistic, and that instead I’d write up something specific for him. When he protested, I told him it’d be a useful thing to have on the Medieval Logic blog, which we hope to update every Thursday. It’s this sort of thing precisely that blogging is good for! I may not need this collected information now, but I’m sure I’ll use it sometime in the future.

Syllogisms are characterised by their figure — the relative arrangement of the terms — and their mood — the triple of copulae that connect the terms.

Figure 1 Figure 2 Figure 3
P__M M__P P__M
M__S M__S S__M
P__S P__S P__S

(where P is the predicate of the conclusion (the major term), S is the subject of the conclusion (the minor term), and M is the middle term).

William of Sherwood in his Introduction to Logic (c1240, [1]) provides his students with a mnemonic verse to remember which figure is which:

The difference of the figures is retained in this verse: Sub pre prima bis pre secunda tertia bis sub (66).

That is, in the fist figure, the middle term is first the subject and then the predicate; in the second, it is the predicate twice; in the third, it is the subject twice. (This same verse occurs in Roger Bacon [5, ∥ 290], and Lambert of Auxerre, their contemporary, has the same content in a much more verbose form [5, 139].)

Following this, he then says:

The moods and their reductions, on the other hand, are retained in these verses:

Barbara celarent darii ferio baralipton
Celantes dabitis fapesmo frisesomorum
Cesare camestres festino baroco
Darapti felapton disamis datisi bocardo ferison
(66)

Kretzmann notes that this is “the oldest known surviving version of these famous mnemonic verses, and Sherwood may have been the inventor of them…However, there were earlier attempts at the same sort of device in the thirteenth century, and the word ‘Festino’ appears ‘in a MS dating at the latest from 1200′” [citing Bochenski] (66-67), though later in the same footnote he compares Sherwood’s verses to those in Peter of Spain and concludes that the differences between the two “strongly suggest at least one earlier version on which both men were drawing” (67).

Sherwood explains the mnemonic names as follows:

In these lines ‘a’ signifies a universal affirmative proposition, ‘e’ a universal negative, ‘i’ a particular affirmative, ‘o’ a particular negative, ‘s’ simple conversion [conversio simplex], ‘p’ conversion by limitation [conversio per accidens], ‘m’ transposition of the premisses, and ‘b’ and ‘r’ when they are in the same word signify reduction per impossibile. The first two lines are devoted to the first figure, the four words of the third line to the second figure, and all the other words to the third figure (67).

(Note that, conveniently, ‘a’ and ‘i’ are the first two vowels in Latin affirmo ‘I affirm’, and ‘e’ and ‘o’ are the two vowels in Latin nego ‘I deny’.) While the names that Sherwood introduces are typical/traditional, the explanation that he gives of the names is not. First, he doesn’t note that the first letter of the name indicates to which of the four basic syllogisms, Barbara, Celarent, Darii, and Ferio, the others can be reduced to. Second, his identification of ‘b’ and ‘r’ as the cues for reductio per impossibile are atypical (it’s not much of a mnemonic; why not, e.g., ‘r’ alone?)

Other roughly contemporary texts offer a slightly different version of the verses and explanation of the names. Roger Bacon in his Art and Science of Logic [5] has a verse for remembering the types of conversions (Maloney notes that similar verses also occur in the 12th-century Logica Ut Dicit and the Logic ‘cum sit nostra’la):

Simpliciter feci convertitur, eva per acci,
Acto per contra. Sic fit conversio tota (∥ 279)

and he notes that “‘E’ signifies a universal negative, ‘I’ a particular affirmative, ‘A’ a universal affirmative, and ‘O’ a particular negative” (∥ 279). The list of mnemonic names, identical to Sherwood’s, appears in ∥ 295, and is explained in ∥ 296; he corrects Sherwood’s apparent mistake and identifies ‘c’ as the code for reductio per impossibile. Lambert also has the same list of names (143), except that he names the second mood of the second figure “Campestres”; the addition of the ‘p’ is not warranted.

Peter of Spain in his Summaries of Logic, written shortly after Sherwood’s, [2] says:

Barbara Celarent Darii Ferio Baralipton
Celantes Dabitis Fapesmo Frisesomorum
Cesare Cambestres Festino Barocho Darapti
Felapto Disamis Datisi Bocardo Ferison

In these four verses are nineteen words representing the nineteen moods of the three figures, so that by the first word we understand the first mood of the first figure, by the second word the second mood, and so on for the others…

…By the vowel A we understand the universal affirmative, by E the universal negative, by I the particular affirmative, and by O the particular negative…

…It must be understood that all the moods indicated by a word beginning with ‘B’ are to be reduced to the first mood of the first figure, and ll the moods signified by a word beginning with C to the second mood, those beginning with D to the third, and those with F to the fourth. Also, wherever an S is put in these words, it signifies that the proposition understood by the immediately preceding vowel is to be converted simply. And by P it signifies that the proposition is to be converted accidentally [converti per accidens]. Wherever M is put, it signifies that a tranposition in premisses is to be done…Where C is put, however, it signifies that the mood understood by that word is to be confirmed by impossibility (191, 193).

Peter of Spain’s identification of “C” (by which we should understand NOT the “C” which begins the word) with reductio per impossibile is orthodox.

Copenhaver et al. also have a footnote, noting that “since the verses scan as Latin hexameters, Latin was almost certainly the language in which they were first written” (contra Prantl who “contends that the first mnemonic lines were Greek and formulated by Psellus (b. 1020)” [3, 519]), and that Sherwood’s verses “appear in the Dialectica monacensis and the Logica ‘cum sit nostra’, treatises dated before or around 1200 by de Rijk…note that de Rijk thinks that BARBARA CELARENT is a later interpolation in the Dialectica monacensis” (191). They also mention related material in the Ars burana and Ars emmerana, also edited by de Rijk, and some mnemonic verses found in a 9th C text [3]. These verses, appearing in Codex Sti. Galli 64, “though they do not contain the technical words, Barbara, etc., or their equivalent, yet served the purpose for which those words were afterwards invented” (519). (Because the verses are long and can be read in full, for free, on JSTOR, we do not quote them here.) Turner notes that

it would, of course, be idle to look for evidences of an original contribution to the science of logic. It belongs to an age in which originality was not a dominant characteristic of teachers of logic. It simply sums up what was to be found in the treatises of Apuleius, Martianus Capella, Cassiodorus, and Isidore. Its terminology does not vary essentially from that which was current in the schools of the ninth and tenth centuries (525).

In the few cases where the terminology used is not standard, this is due to the need to pick terms that fit the desired meter.

Let us return to the matter of the verses and material in the Dialectica monacensis, the Logica ‘cum sit nostra’, and the Artes burana et emmerana. De Rijk notes that “I have not been able to find these verses in twelfth century treatises. It should be noted, however, that the famous verses BARBARA, CELARENT had a few forerunners in two twelfth century tracts on syllogism” (401), namely the Ars emmerana and the Ars burana, which we will discuss below.

The Logica ‘cum sit nostra’ contains chapter on the syllogisms which in de Rijk’s edition [4] contains:

Barbara Celarent Darii Ferio
Baralipton Celantes Dabitis Fapesmo Frisesomorum
Cesare Campestres Festimo Baroco Drapti
Felaptdon Disamis Datisi Bocardo Ferison.

A notat universalem affirmativam, E notat universalem negativam, I particularem affirmativam, O particularem negativam…

…Consequenter dicendum est de reductione. Et sciendum quod omnes modi debent reduci in quatuor primos modos prime figure, quia omnes modi incipientes per B debent reduci in Barbara, per C in Celarent, per D in Darii, per F in Ferio.

Et reducuntur per tria, scilicet per conversionem, per transpositionem, et per impossibile. Unde sciendum quod S denotat conversionem simplicem, P conversionem per accidens, M transpositionem propositionum, C reductionem per impossibile. Unde versus:

S simplx, P per acc.
M transpos., C notat impossibile (436).

In the Dialectia Monacensis, we have the following:

Hec omnia facilius possunt haberi per hos versus:

Barbara * Celarent * Darii * Ferio * Baralipton *
Celantes * Dabitis * Fapesmo * Frisesmomorum

In hiis versibus sunt novem dictiones novem modis prime figure deservientes, prima primo et secunda secundo, et sic deinceps. Horum autuem versuum triplex est utilitas. Prima est quia scitur quales et quante debeant esse propositiones in qualibet materia; et hoc per vocales istarum dictionem: per A intelligitur universalis affirmativa, per E universalis negativa, per I particularis affirmativa, per O particularis negativa…Secunda utilitas est quia scitur qui sillogismi in quos habeant reduci; et hoc per initiales litteras istarum dictionum. Tertia utilitas est quia scitur per quid unusquisque sillogismus reducator: hec enim littera S ubicumque inventiur est signum simplicis conversionis; B vero significant conversionem per accidens; M vero significat transpositionem premissarum, idest quod de maiore fiat minor et econverso (494).

Note the use of “B” to stand for accidental conversion; this cannot be correct since (a) it leaves Fapesmo and Baralipton wholly unexplained and (b) Dabitis cannot be proven if the major premise is accidentally converted.

A separate verse is given for the second and third figure syllogisms:

Et sciendum quod omnia que dicta sunt de secunda et tertia figura, faciliter possunt haberi per hos versus:

Cesare * Camestres * Festino * Baroco *
Darapti * Felapton * Disamis * Datisi * Bocardo * Feriso *

In hiis versibus sunt decem dictiones. Inter quas prime quatuor deserviunt secunde figure et alie que secuntur deserviunt tertie figure, ita quod prima primo, et ita deinceps (497-498).

Of these verses, de Rijk notes “something peculiar must be noticed” (413) about their placement; the verse concerning the first figure appears after the discussion of the first figure, but the verses discussing the second and third figure occur together after the discussion of the third figure. He says that “this peculiar arrangement of the mnemonic verses strongly suggests that they were interpolated in the original treatise” (413) and that there was insufficient margin space to add the verses at the end of the discussion of the second figure, which is why they were saved for a later space.

What about the early hints towards the later mnemonics that de Rijk found in the Ars Emmerana and the Ars Burana? We have no verses in AE, but there is a clear attempt to encode the relevant information of a mood in terms of letters:

Notandum quod universalis affirmative designantur his quatuor literis E I O U; universales negative his: L M N R; particularis affirmative tribus: A S T; particulares negative tribus: B C D.

Secundum hoc he novem voces designent novem modos prime figure:

VIO * NON * EST * LAC * VIA * MEL * VAS * ERB * ARC

he quatuor voces designant quatuor modos secunde figure:

REN * ERM * RAC * OBD

sex modi tertie figure his sex designantur:

EVA * NEC * AUT * ESA * DUC * MAC (173).

Here, only the quality and the quantity of the propositions involved in the syllogisms are mentioned.

The mnemonic material in the Ars Burana is similar:

He littere E I O V significant universales affirmativas; et he littere L M N R significant universales negativas; et he A S T significant particulares affirmativas; et he B C D significant particulares negativas (199).

With this convention established, the author goes on to say that:

De modis igitur prime figure talis assignatur versus:

VIO NON EST LAC VIA MEL VAS ERB ARC (200).

Each of these syllogisms is then exemplified; VIO is Barbara, NON is Celarent, etc. A similar verse is given for the second figure: “REN ERM RAC OBD” (203), and the third: “EVA NEC AVT ESA DVC NAC” (205).

So much for the period up through the 13th century. What about afterwards? I have fewer 14th-century treatises on my shelves, but Buridan has a chapter on syllogisms in his Summulae de Dialectica [7]. In that chapter, the mnemonic names are taken completely for granted; Buridan refers to Disamis and Bocardo before he even introduces the concepts of figure and mood, thus assuming that these are already familiar to the reader (310). The mnemonic verse itself doesn’t appear for another 10 pages or so, where he introduces the verse (320), explains how to form a syllogism from the names given (321), and then explains “the verse insofar as it indicates the reduction of imperfect to perfect syllogisms” (321). His explanation follows that of Peter, Bacon, Lambert, etc.

The final person I’d like to mention in this post is Paul of Venice, if for no other reason than that I’m in the process of editing and translating his chapter on syllogisms from the Logica Magna, written at the very end of the 14th century. Paul also assumes familiarity with the names in that he uses them before he explains them:

Having seen what is required for a syllogism in the first figure, it remains to show the moods which are syllogized in the same [figure]. And they are six, namely: Barbara, Celarent, Darii, Ferio, Fapesmo, Frisesomorum.

(Note that he omits Dabitis and Celantes). He says:

Whence it must be noted that in whatever mode, some of these four vowels, namely a, e, i, o, are put in. ‘a’ denotes the universal affirmative, implicitly or explicitly. ‘e’ [denotes] the universal negative, implicitly or explicitly, ‘i’ the particular, indefinite, or singular affirmative, implicitly or explicitly. ‘o’ denotes the particular, indefinite, or singular negative, implicitly or explicitly.

(Paul differs from earlier writers in explicitly including indefinites and singulars in his syllogistic, and he assimilates them to the particulars, rather than to the universals, as some other commentators did.)

There is one way in which Paul’s discussion of the mnemonics is utterly unlike anything that I have seen preceding him. He says:

It is considered whether in a mood there are four such letters by the multiplication of one, or not…However, if this is so [that you have four letters rather than three, e.g., Barbarai], it will happen that you know the conclusion is able to follow according to the third or fourth denotation. Hence in Barbara is ‘a’ triplicated, and a simple solitary ‘i’ for that reason may be denoted by such mood that the premisses are implicitly or explicitly universals according to a two-fold precession of that term ‘a’, that is, because of their position. The conclusion which is able to be inferred either implicitly or explicitly is denoted by the third vowel, but because there is a final ‘i’, for this reason it may be denoted by the particular, indefinite, or affirmative itself, it is possible to infer either explicitly or implicitly, just as is stated in the example.

What’s going on in this section is not entirely clear, though it became clearer when we worked through my draft translation in the St Andrews medieval logic/Latin reading group a year or two ago. What we think is going on is that Paul is explicitly noting the validity of “Barbari”, a syllogism which has otherwise not been mentioned at all in any of the mnemonics above.

It is not until Paul introduces Fapesmo, the first non-perfect syllogism in hist list above, that he explains the other letters in the names:

Whereby any mood beginning with ‘B’ is able to be reduced to Barbara, everyone beginning with ‘C’ to Celarent, and everyone beginning with ‘D’ to Darii. And everyone beginning with ‘F’ to Ferio: But, as this is able to done, you ought to note that whenever ‘S’ is put down in these words, it signifies that the proposition indicated by the vowel immediately preceding ought to be converted simply. Wherever a ‘P’ is put down it signifies that the proposition indicated before the ‘P’ ought to be converted per accidens, and wherever an ‘M’, it is denoted that a transposition of the premises ought to be made, that is, to make the minor of the major and conversely. However, where ‘C’ is put down, it is denoted that this mood or a syllogism formed in the same should be reduced per impossibile.

He then gives this verse:

Simpliciter verti universaliter, ‘S’, ‘P’ vero per accidens, ‘M’ vult transponi, ‘C’ per impossibile duci.

I had intended in this post to also write about the verses relating to the square of opposition, especially the modal one; but I think that this point, that’s best saved for another post! One other aspect that I haven’t mentioned at all here is the mnemonic tradition in commentaries on the Prior Analytics, and that is because I have no idea what such a tradition is, or if it even existed. Again, a topic for another post! One final topic for another post, if only because I am running out of time, is how long these syllogism mnemonics persisted as pedagogical tools; in particular, I’d like to look at these and other 16th-century logic textbooks to see if the syllogism mnemonics show up in them.


Notes

[1] William of Sherwood, Introduction to Logic, translated with an introduction and notes by Norman Kretzmann (University of MN Press: 1966).

[2] Peter of Spain, Summaries of Logic, text, translation, introduction, and notes by Brian P. Copenhaver with Calvin Normore and Terence Parsons (OUP: 2014).

[3] William Turner, “Mnemonic Verses in a Ninth Century MS.: A Contribution to the History of Logic”, The Philosophical Review Vol. 16, No. 5 (Sep., 1907), pp. 519-526, JSTOR link

[4] L. M. de Rijk, ed., Logica Modernorum: A Contribution to the History of Early Terminist Logic vol. II, parts one & two (Van Gorcum, 1967).

[5] Roger Bacon, Art and Science of Logic, trans. Thomas S. Maloney, (PIMS 2009).

[6] Lambert of Auxerre, Logica or Summa Lamberti, T. S. Maloney, trans., (University of Notre Dame, 2015).

[7] John Buridan, Summulae de Dialectica, trans. with an introduction by Gyula Klima (Yale University Press, 2001).

[8] Paul of Venice, Logica Magna (Venice, 1499).

Advertisements
This entry was posted in Uncategorized and tagged , , , , , , , , , , , . Bookmark the permalink.

5 Responses to Syllogism Mnemonics

  1. jcasey12 says:

    Great work. I have a question not related to syllogisms I’m embarrassed not to know the answer to: what’s the first appearance of A, E, I, O as names of the four categorical propositions?

    Like

    • Good question — probably about the same time as the mnemonics, though it would be earlier. Certainly it’s a Latin innovation — the letters are explained as being the (first two) vowels you find in ‘affirmo’ and ‘nego’.

      Like

  2. jcasey12 says:

    Sorry, one more question, this time related to what you’ve written. In the chart, the first figure looks like what would be figure 4 and the other two seem switched according to the first mnemonic: “That is, in the fist figure, the middle term is first the subject and then the predicate; in the second, it is the predicate twice; in the third, it is the subject twice. ” Have I missed something obvious?

    Like

    • It’s a standard convention that the predicate is written first, and the subject second — reverse to the order that the words occur in a spoken English sentence.

      Like

      • jcasey12 says:

        Thanks for your replies. If I could bother you with a follow up. Could you explain (or point me to an explanation of) the “standard convention.” It seems odd it would exist because the Latin word order is the same as the English:

        Omne animal est substantia;
        Omnis homo est animal;
        ergo omnis homo est substantia

        It’s pretty much like that way for the rest of the text.

        Many thanks–

        Like

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.