TAPE 6: ABELARD (CONT.); ABBREVIATIO MONTANA

To follow this lecture you will need Readings, or A. Hyman and J.J. Walsh (eds.), Philosophy in the Middle Ages (Indianapolis: Hackett) and "Abbreviatio Montana" from N. Kratzmann and E. Stump (eds.), Logic and the Philosophy of Language (Cambridge: Cambridge University Press), p. 39 ff.

Abelard on Porphyry (cont.)

Paragraphs 23-8 have clarified various details. Now we return to the philosophical questions. Read paragraph 29 to the end of the passage in italics. This paragraph asks how a universal word can have meaning or significance. Significance is not the same as appellation, but it seems to presuppose that the word has appellation. In medieval linguistic theory it was supposed that words were "invented" and imposed by their inventor upon things which are the word's appellata; the inventor gave the word a significance that picks out the things that can be called by it. So when in the garden of Eden Adam named the animals, he called the donkey "donkey", giving the word such a meaning as would pick out other donkeys later - having four feet, long warm ears, etc. So it would seem that a word without appellation, without any sort of thing that can be called by it, can not have meaning. So what sort of thing can be named by a universal word? Not a universal thing, because paragraphs 9-22 have shown that there is no universal thing. Can it apply to some particular thing? Read the rest of the paragraph. (In the last few lines substitute "understanding" for "meaning" and for "idea" - the Latin text has intellectus: "Since it constitutes an understanding of no thing...But it seems that there cannot be an understanding . . . Every understanding is made either . . .)

Read paragraph 30. A universal word does have appellation. Its significance, although it does not arise from the different individuals, does pertain to them, namely by being their common likeness. This needs to be explained. Read paragraph 31. Compare paragraph 3, p.57, RH, lines 3-9.

Read paragraph 32 down to 3 lines above the bottom of p.61. Some comments "on the letter"; In the third line, "as we noted above", i.e. in paragraph 10-16. "Physics" means nature. In line 5, insert "a" before "man" each time, and underline "being a man". Socrates agrees with Plato in being a man. "Being" something, or "to be" something, is not a thing. In lines 7 and 8 the phrases in italics are quoted from Aristotle, who says that substances agree in not being in any other subject, in not undergoing contraries simultaneously, etc. These "not beings" in which Aristotle says substances agree are not things. (The point of the reference to Aristotle is to get his authority behind the way of speaking according to which substances agree in being or not being something that is not some thing.) So Socrates and Plato agree in being men, and "being a man" is not a thing. Abelard will call "being a something" a status: a vague and not particularly suitable word, but he needs some term to refer to these non-things in which things agree.

Read the rest of paragraph 32. Some comments "on the letter"; Near the bottom of p.61, "It seems, however, that we must avoid", i.e. because it seems an implausible thing to say. This is an objection, which is answered in the second line of p.62, at "But...". The last word on p.61, "since", should be "when", and the parenthesis closed just before "since" should extend to the end of the sentence. On p.62, line 4, "we appeal to no essence", i.e. to no thing. "Essence" is used here as equivalent to substance or thing. In the next line, "cause of" means "reason for". In the fourth last line of this paragraph put "He does not wish to appear in court" in quotation marks. His not wanting to appear is the cause or reason why he was whipped, but his not wanting is not an essence or thing. In the next line, in the phrase "those things themselves", "things" has a weak sense; "thing" here does not mean substance or thing, but characteristic. The status of man consists of those characteristics in men, the common likeness of which is the meaning of the word "man". So re-read that passage.

He is saying that the word "man" has meaning because it was imposed to pick out individuals who agree in having certain characteristics. "Having certain characteristics" is not a universal thing in which they all share, but it can all the same be the reason for imposing the word, just as not wanting to appear in court can be the reason why a man is whipped. Individuals can agree in having such characteristics, just as according to Aristotle individual substances agree in not being in any subject, in not accepting contraries simultaneously, etc. Individuals can agree in things that are not things.

Read paragraphs 33, 34 and 35. ("And let us first distinguish generally the nature of all understandings" I count as paragraph 34.) By "understanding", intellectus, Abelard does not here mean the faculty or power of understanding, but the act. (Except 4 lines down, where "understanding" does mean the faculty.) When we think of man, what do we understand? Our understanding is not, he says, the form of man (according to Boethius that is what it is - see Readings, pp. 17-18 (McKeon, pp. 97-8) - the same form as the man has exists in our mind). Rather, the understanding is an action of the mind directed to a mental image we produce. We can have an understanding, a grasp of a mental image, of something that has ceased to exist and can no longer be sensed, or perhaps never existed.

Read paragraphs 36 and 37. Paragraph 36 puts an objection, 37 answers it. There are some who identify the mental grasp with the image, whereas Abelard distinguishes the act of understanding from the image to which it is directed. (In paragraph 36 line 2, "they call the building of the tower which I conceive", i.e. they say that my conception or image of the fabric of the tower is the same as the understanding of the tower.) Abelard's answer concedes that in a sense Aristotle is right, but it is better to distinguish between the act of understanding and the image by means of which we understand the thing. The image can't be the understanding, because the image is inevitably unlike the thing in some respects (the image is not square and lofty), but we can still understand the thing even in those respects: by directing our mind upon the image of the tower we can understand that it is square and lofty even though the image is not. The drift of the argument is that even though being a man is not a thing, we can form an image of it (just as we can imagine imaginary towers), and through that image understand characteristics that men really have even if our mental image does not itself have these characteristics.

Read paragraph 38. This completes the discussion began at paragraph 34. Now we move to the question raised in paragraph 35. Read paragraph 39. This is clear enough. Notice in the second last sentence the distinction between signifying and naming. Read paragraph 40. The reference to Boethius seems to be to the passage in Readings, p.16 (McKeon, p. 94), paragraph 10. Abelard says that this is a "sophistical", i.e. fallacious argument: it is part of the aporetic part of Boethius' discussion. It is designed to put the reader into a situation from which there seems no way out. In paragraph 11 Boethius shows the way out, following "the reasons of other writers", viz. of Alexander. So, as Abelard says, you can't quote as Boethius' own view anything in paragraph 10. "We can moreover"; Anyway, in Abelard's theory there is something for the understanding to refer to, viz. the thing (if present to sense) or the image, proper or common.

Next comes a digression on Plato's theory of forms or ideas, in the neo-Platonist version which locates the Forms in the Cosmic Intelligence or in the mind of God. Read paragraphs 41-46. Is it possible that universal words have meaning by relating to God's ideas? Abelard does not reject this possibility. Near the end of paragraph 45 the "third thing" allows for the possibility, and in paragraph 46 I think "common conception" probably refers to it also. He seems to say that the status, being a man, in which all men agree, is enough to give universal words meaning by providing a common cause or reason for imposing or inventing the word "man", but maybe they also can have meaning by referring to the idea in God's mind by which God created all these individuals who agree in being men. But we can only have opinion of such Forms, since we do not have sense perception of them.

Read now paragraphs 47-53. The theory here is close to what Boethius says about abstraction, Readings, pp.16-17 (McKeon, pp. 95-7). Two small comments on p.64, line 15, "state" is status, "being a man". In the first line of paragraph 51, providence means foresight, foreknowledge.

Now Abelard can answer Porphyry's questions. I will leave you to read for yourselves paragraphs 54 to the end.

Status and dictum

The original aspect of Abelard's theory of universals is his account of how they can have meaning even though there are in reality no universal things to which they can refer. Their appellation is a set of many individual things which agree in something - but not in some thing, like Plato's idea (though Abelard is prepared to allow that universals may refer to God's ideas); the many individuals agree in status, i.e. in being whatever they are, in being men for example. Being a man is not a thing, but agreement in this non-thing can be the cause or reason of imposition.

Elsewhere in his logic, when discussing modal propositions (i.e. propositions of the form "It is necessary that (or contingent that or impossible that) Socrates is a man"), Abelard uses the term dictum to refer to what a proposition asserts. E.g. the dictum corresponding to the proposition "Socrates is a man" is the noun-phrase, "that Socrates is a man". E.g. we could say "That Socrates is a man is contingent", meaning that the statement "Socrates is a man" is contingent. The dictum could also be expressed, "Socrates' being a man"; so "Socrates' being a man is contingent". The dictum is what we might call a fact: the fact that Socrates is a man. But, Abelard argues, dicta (or, as we might say, facts) are not things. "If there is a rose, then there is a flower" is necessarily true, and would be necessarily true even if all roses and flowers had ceased to exist. It is necessarily true by virtue of a relation between the dicta, that there is a rose, that there is a flower. Even if God were the only entity in the universe, it would still be true and necessary that if there is a rose then there is a flower. So the dicta can't be things additional to God and to whatever other things he has created. Now a status is a dictum with the subject omitted. Take the dictum, "Socrates' being a man", omit "Socrates", and you have the status, "being a man". Like the dictum this is a noun phrase, but it does not refer to any thing. So Abelard recognises, besides words and things, dicta and status which are neither words nor things. In this he resembles the Stoics, who recognised lekta (the Greek equivalent of the Latin dictum), and also incomplete lekta (the equivalent of Abelard's status), as distinct from words and things. See Long and Sedley, The Hellenistic Philosophers, vol. 1., pp. 199-201. I don't know whether Abelard had a Stoic source, or whether he just thought of similar solutions to similar problems. For more on Abelard see M. Tweedale, Abailard on Universals.

John of Salisbury on the teaching of logic

After working through Abelard's comments on the first page of Porphyry's Introduction, in which Abelard surveys and refutes theories offered to answer the questions Porphyry says are too difficult and leaves aside, and offers his own, you may have some sympathy with some of John of Salisbury's remarks on the teaching of logic at that time. The Metalogicon (another made-up Greek title!) is a defence of logic (or of the artes sermocinales). But it also criticises the logicians of the time. "The ancients believed that Porphyry should be studied as an introduction to Aristotle. And right they were, provided that Porphyry's book is properly treated. For it should not be taught in such a way as to confuse and obfuscate those being instructed, or monopolize all their time . . . Because of its introductory nature Porphyry's work is entitled the Isagoge ["Introduction"]. But its very name is contradicted by those who become so engrossed in it that they leave no time for the principal essentials, on which the whole significance of the introductory work depends" (p. 111). "Nevertheless, at present, all are here [on Porphyry] declaiming on the nature of universals, and attempting to explain, contrary to the intention of the author [i.e. of Porphyry, who says that these are questions he will pass over] - contrary to the intention of the author what is really a most profound question, and a matter for more advanced studies" (p. 112). "They load insupportable burdens on the frail shoulders of their students . . . they pay no attention to proper order in teaching . . . Thus they . . . read the whole art into its title. With them, Porphyry practically teaches beforehand the contents of the Topics, the Analytics and the Elenchi [other logical treatises of Aristotle]" (p. 118). "That which is written should be studied with sympathetic mildness, and not tortured on the rack . . . until it renders what it never received. . . . [they] force Porphyry to cough up the opinions of all philosophers, and will not rest content until his short treatise teaches everything that has ever been written" (p. 148). He writes with feeling. John values logic, but insists that it is useful mainly in so far as it serves other subjects. It is unprofitable to spend too long on logic before going on to the study of literature, of the quadrivium, of theology, or some other more advanced subject - in which logic will prove very useful. "It is evident that dialectic, the highly efficient and ever-ready servant of eloquence, is useful to anyone in proportion to the degree of knowledge he possesses . . . Dialectic, however, is not great, if, as our contemporaries treat it, it remains forever engrossed in itself, walking round about and surveying itself, ransacking its own depths and secrets: limiting itself to things that are of no use whatsoever in a domestic or military, commercial or religious, civil or ecclesiastical way, and that are appropriate only in school" (p. 94). "One who has the sad misfortune of being in want of the other disciplines, cannot possess the riches that are promised and provided by dialectic" (p. 95).

Nevertheless, let us spend a little longer on dialectic, and read a very summary account of the sort of thing John learnt from Abelard at Mont Sainte Genevieve. Turn to Readings, p. 66 (or to Kratzmann and Stump, p. 39).

THE ABBREVIATIO MONTANA

The Abbreviatio Montana was written not long after Abelard's time there (montana refers to the Mont St Geniveve). It is in the form of a catechism, questions and answers; presumably the student learnt it and was then asked the questions and had to supply the right answers or go back and con it again. It is a summary of Boethius On Categorical Syllogisms and On Hypothetical Syllogisms.

Read the translator's Introduction, Readings, p.66, and then the introduction to the work itself, p.67. Comment: Dia does not mean two, and dialectic means conversation, not necessarily between just two. But it is true that ancient dialectic, illustrated for example in Plato's Socratic dialogues, was argument between two, questioner and answerer.

Now read paragraph 2, "Sound". Notice the use of the technique of division. On the top of p.68, "what human beings have rationally established" refers to imposition. Now read "3 Names and Verbs" and "4 Expressions". Notice that a complete expression is what we call a sentence.

Square of opposition

Read sections 5 and 6. In your notebook write down an example each of a universal affirmative proposition, a universal negative, a particular affirmative and a particular negative. These traditionally are called A,E,I,O respectively. Whether a proposition is universal or particular is its quantity, whether it is affirmative or negative is its quality. A and E are universal in quantity, I and O are particular in quantity; A and I are affirmative in quality, E and O are negative in quality. Read now sections 7, 8, 8a, 8b, 8c. In your notebook sketch the following diagram, called the square of opposition (found in MSS of Boethius On Categorical Syllogisms). Draw a square, and at the corners, starting with the top left and going clockwise, write the letters A,E,O,I (understand that the four propositions so represented have the same subjects and predicates). The diagonals join the contradictories, the top side joins the contraries, the bottom side joins the sub-contraries, and the other two sides join the subalterns. (Verify this from the definitions given in the text.)

Topics

Notice in the text the terms "topic" and "rule". On topic see the translators' introduction, Readings, p.66 (or p. 39). There it is said that the topic is a rule that warrants or justifies an "if - then" statement. More accurately, the "topic" is a place or pigeon-hole in which such rules will be found. The Greek topos, Latin locus, means a place. The English "commonplace" comes from this image. It is as if the arguer has a filing cabinet containing various places or compartments in which he will find material for various standard lines of argument. These places or topics have names, "from contraries" and so on. In the file "from contraries" you will find the useful rule, "If two propositions are contraries then if one is true the other must be false". Such a rule will warrant or justify an "if - then" statement that is the application of the rule, Since "Every man is an animal" and "No man is an animal" are contraries, we are warranted in saying, "If "every man is an animal" is true, then "no man is an animal" is false"; after this statement we can put in brackets the phrase "from contraries" to show how it is justified. To repeat: the topic is a place where you will find a rule that justifies or warrants an "if - then" statement that is an application of that rule. The doctrine of Topics goes back to a book on Topics by Boethius, and beyond that to Aristotle's Topics.

Conversion

Re-read the introduction to section 8, and then read section 8d. Notice that universal negatives are convertible both simply and per accidens. For finite and non-finite terms see near the bottom of LH, p.69, and two paragraphs above that for the difference between singular and particular propositions. Conversion is a kind of inference or argument: No man is an animal, therefore no animal is a man, by simple conversion. In your notebook write down in the form of an inference the simple conversion of E and I propositions and the per accidens conversion of an A proposition.

Hypotheticals in medieval and modern logic

Read now the introduction to section 9. In modern logic "disjoined hypotheticals" are called disjunctives and "conjoined hypotheticals" are called just hypotheticals. (They are also called conditionals, implications and entailments.)

Read to the end of section 9(b)i. What is said in this section will become clearer as you read on. But notice at this point some differences between this Boethian treatment of hypotheticals and that of modern logic. Modern logicians would represent a hypothetical as: "if A then B", and it doesn't matter what internal structure the A and the B may have (provided they are statements). But in Boethius' examples, and in the present text, the antecedent and consequent are predications, "If Socrates is a man, then Socrates is an animal". As I remarked in tape 3, Aristotle's logic was concerned with statements of the form "S is P", "subject is/is not predicate", and this is also true of the hypothetical arguments of Boethius. (The Stoics, on the other hand, in their treatment of hypotheticals were not concerned with the internal structure of antecedent and consequent: "If the first then the second".) Boethius is working in a peripatetic tradition.

So that's one point to notice: antecedent and consequent are both subject-predicate propositions. The second point is that our text requires that there be some discoverable logical connection between antecedent and consequent. "If A then B" is being treated as equivalent to an inference, "A, therefore B", and they want to know what other premise goes with A to establish B. The other premise is drawn from a topic: a rule that warrants the hypothetical statement. In modern logic, on the other hand, hypotheticals are all treated as what are called "material implications". For modern logic, "If A then B" is simply the assertion that it is not the case that both A is true and B is false. This is satisfied even if both are false (because then, of course, it is not the case both that A is true and B false, since A is not true); and it doesn't matter whether there is any discoverable logical connection between A and B. Thus the statement "If Paul Keating is the world's greatest treasurer then I'm the monkey's uncle" is a perfectly good hypothetical statement, as far as modern logic is concerned: since he is not the world's greatest treasurer and I am not a monkey's uncle the hypothetical is true, although there is no connection between its two parts. (Medieval and modern logicians have taken the opposite sides in a debate that goes back to the logicians Philo and Diodorus in ancient times - the medievals agree with Diodorus, the moderns with Philo. See Long and Sedley, Hellenistic Philosophers, vol. 1, pp. 208-211; also, Benson Mates, Stoic Logic, pp. 42-51, 97-8, 109.)

Hypotheticals

So in the passage following highlight the rules and understand in each case how the rule is supposed to justify the quasi-inference from antecedent to consequent. (I say quasi inference because strictly speaking a conditional statement is not an argument. "A, therefore B," asserts both A and B and infers B from A; "If A then B" asserts neither, but asserts that if A is asserted then B can be inferred and asserted.) Let's return to the text.

Read section 9b(ii). This explains the points made in 9b(i). Highlight the rules mentioned. Notice that all these rules are to be found in the topic, place, or pigeon-hole, labelled "From the whole". Read again the first paragraph of this section, and in line 2 notice "predicate", in line 5, "animal", which is the predicate of the antecedent, and in line 8, the application, notice "animal"; the relationship lies in the predicate of the antecedent, animal, because it is the predicate of the antecedent which is the whole in the application of the rule. Similarly in the second and third paragraphs "the whole" in the application is again "animal" - the antecedent removes, i.e. denies, "animal" of some subject. ("S is not P" removes P from S.)

In the fourth, fifth and sixth paragraphs the whole mentioned in the application of the rule is the subject of the antecedent. In the next three paragraphs look for two terms in the warranted hypothetical and in the application of the warranting rule.

Now read section 9b(iii). An integral whole is like a house in relation to its parts - roof, walls, floor. A universal whole is a class in relation to its members: a genus in relation to its species, or a species in relation to individuals. Keep reading, highlighting the rules, to the end of section 9 (on p.80). (The last argument on p.77 seems to be fallacious.) If you were a conscientious medieval student you would now learn all the rules by heart, and could say what rules are found in which topic.

Categorical syllogisms

This brings us to the categorical syllogism. Read to the end of p.80. In your notebook write out on three lines (one line for each statement) the syllogism at the end of 10b(i) first paragraph: Every man is an animal; but every risible thing [i.e. thing capable of laughing] is a man; therefore every risible thing is an animal. In a categorical syllogism there are three categorical (i.e. subject-predicate) propositions, namely two premises and one conclusion. There are three terms, each of which appears twice, namely "man", "animal", "risible thing". One of these, namely "man", does not appear in the conclusion, but appears in both of the premises: it is called the "middle term", i.e. the term that acts as go-between to establish the relationship between the other terms, which are called the extremes. Which figure the syllogism belongs to depends on where the middle term occurs in the premises, i.e. as subject or as predicate. If the middle term is the subject of one premise and the predicate of the other the syllogism belongs to the first figure, if the middle term is the predicate of both premises the syllogism belongs to the second figure, if it is the subject of both premises the syllogism belongs to the third figure. The premise stated first was called the proposition (in a more specialised sense of that word than it usually had), the second premise was called the assumptio, translated here as "addition". So the two premises are respectively the proposition and the addition. In the first figure the major term or extreme is the one predicated of the middle term, i.e. the one that appears in the conclusion as either subject or predicate, and in one of the premises as predicate. Now re-read section 10 from the beginning to the end of 10b(iii).

In talking about figures we ignored signs of quantity and quality. If we attend to those signs we are considering the mood of the syllogism. Read section 10b(iv) to the middle of RH side of p.81. "Mood" translates Latin modus, in other contexts usually translated mode; modus is as vague a word in Latin as mode is in English. If we use A,E,I,O to refer respectively to universal affirmative, universal negative, particular affirmative and particular negative, we can specify mood by a sequence of those letters. E.g. "every man is an animal, but every risible thing is a man, therefore every risible thing is an animal" is the syllogism "AAA in Figure 1".

Re-read the last paragraph of LH side p.80. Let me explain what this means. The syllogism we were just talking about corresponds to the composite hypothetical statement, "If every man is an animal and every risible thing is a man, then every risible thing is an animal". It also corresponds to another hypothetical statement: "If every man is an animal, then, if every risible thing is a man, every risible thing is an animal". The first hypothetical is of the form, "If A and B, then C"; the second is of the form "If A, then if B then C". Recall that the rules found in the topics are warrants or justifications for hypothetical statements. Now go back to 10b(iv) (cont.) in the middle of RH side of p.81. Let's number these paragraphs as we go. Paragraph 1 begins "In the first figure, for example". So read from here to the end of paragraph 4. The rules given are the warrant or justification for the hypotheticals corresponding to the syllogism AAA figure 1. In paragraph 2 the order of the premises is reversed - what was the proposition becomes the addition, and vice versa.

The next syllogism is mood EAE in figure 1. Read paragraphs 5-8. Paragraph 7 corresponds to paragraph 4: the hypothetical is of the form "If A, then if B then C". Paragraph 8 corresponds to paragraph 2: the order of the premises is reversed. Read now the similar treatment in paragraphs 9-11 of the third mood, AII in figure 1, and in paragraph 13 of the fourth mood, EIO in figure 1.

Conventions

Read paragraph 14. For the meaning of "major extreme" see 10b(ii): it is the term that is predicated of the middle in one of the premises; it appears as either subject or predicate of the conclusion and as predicate of the premise in which it appears. Later the convention was changed to define the major term as the one that is the predicate of the conclusion; it also became conventional to order the premises so that the first premise stated is the one in which the major term appears; and then it became customary to distinguish between the first figure and a fourth figure (both of which are lumped together as first figure in this text). The first figure is that in which the middle term is the subject of the first premise and the predicate of the second, stating the premises in the now conventional order that puts the predicate of the conclusion in the first premise, and the fourth figure is that in which the middle is predicate of the first premise and subject of the second in the conventional order. The syllogisms our text lists as the 5th-9th moods of the first figure are the moods of the fourth figure, in the later terminology.

Read paragraphs 14-18 and 20 (19 is too difficult - we'll skip it). In your notes write out the schemata for the nine syllogisms so far examined, using letters instead of the words used in the examples. For paragraph 1, write down, "Every M is A, (next line) Every R is M, (next line), therefore every R is A". And so on for the examples in the other paragraphs.

Reduction

Read paragraph 21. In your notes, at the margin on the L.H. side of your page, write out mood 8 (referring to the example in paragraph 18), using a separate line for each proposition, and using M for middle, S for subject of conclusion, P for predicate of conclusion. That is: "Every M (man is the middle term) is S (animal is the subject of the conclusion), (new line) No P (stone is predicate of the conclusion) is M, (new line) therefore some S is not P". Now on the same three lines, well over to the R.H. side of your page, write out mood 4 (example in paragraph 13), using M, S, P in the same way. Put the headings "mood 8" and "mood 4" above them. Now, to the right of mood 8, on the top line, with an arrow drawn to it from the second premise of mood 8, write "No M is P (by simple conversion)"; under that, on the second line, with an arrow from the first line of mood 8, write "Some S is M (by conversion per accidens)", and under that write the same conclusion as in mood 8, "Therefore some S is not P". Look back to p. 48, section 8d, to remind yourself about conversion. Now compare what you have just written, which is mood 8 transformed by conversion and transposition of premises, with mood 4 on the right. They should be the same. What you have just done is the process medieval logicians called "reduction" (or in this text "resolution"). The mood 8 syllogism has been reduced, i.e. led back, to, or transformed or resolved into, the mood 4 syllogism by a valid reasoning process (conversion and transposition of premises). In the second half of paragraph 21 you have a recipe for resolving the 9th mood into the 4th: write it out, following these instructions. Reduction shows that if some syllogism is valid (mood 4 in these cases) then some other syllogism must be valid.

Major term

Read the introductory paragraphs to sections 10b (v) and (vi). In the first it is said that the minor extreme is the subject of the conclusion (highlight that) and in the second that the major extreme is the predicate of the conclusion (highlight that). This was the convention later adopted for the first figure also, which led to the distinguishing of first and fourth figures [see above]. Notice at the end of these paragraphs generalisations about the sorts of conclusions that can validly be drawn in these figures. In figure 2 the conclusion must be negative, in fig. 3 it must be particular.

Now read section 10b(v) on syllogisms of the second figure. In the last paragraph there is a proof per impossibile, i.e. a reductio ad impossibile, proof of a thesis by showing that to suppose its contradictory leads to something necessarily false, a contradiction. Write out the following; write each step on a separate line, and number the lines. 1. Suppose Every M is A. 2. Suppose Some B is not A. Don't number the next line, but write: To be proved: that on suppositions 1 and 2, Some B is not M. Next line. 3. Suppose that "some B is not M" is false. 4. Therefore Every B is M (from 3, contradictories). 5. But Every M is A (supposition 1). 6. Therefore every B is A (from 4, 5, by AAA figure 1). 7. But Some B is not A (supposition 2). 8. Therefore Every B is A AND some B is not A (from 6, 7, conjunction). 9. But 8 is false (contradictory). 10. Therefore, on suppositions 1 and 2, 3 is false. 11 Therefore, on suppositions 1 and 2, Some B is not M. Re-read the paragraph and check that that's what it says.

Now read 10b(vi) on syllogisms of the third figure. See if you can work out for yourself the resolution of the mood 5 per impossibile, and try reducing some of the others by conversion of propositions and transposition of premises. Each reduction should make use of some first figure syllogism; while you try conversions, etc. keep your eye on the list of first figure moods.

Later medieval teachers of logic used a mnemonic verse to sum up the list of valid syllogisms and the method of resolution or reduction. Their summary was as follows:

Barbara, Celarent, Darii, Ferio (fig. 1)
Cesare, Camestres, Festino, Baroco (fig. 2)
Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison (fig. 3)
Bramantip, Camenes, Dimaris, Fesapo, Ferison (fig. 4)

The initial letters (B, C, D, F) link the fig. 2,3,4 syllogisms with the fig. 1 syllogism to which they can be resolved; thus Bramantip is provable using Barbara. The vowels indicate the quality and quantity of the three propositions (Barbara consists of three universal affirmative propositions). "S" shows that in reduction the proposition marked by the preceding vowel is converted simply, "p" that it is converted per accidens. "M" shows that the premises have to be transposed. "C" indicates that a reductio ad absurdum must be used. See Anton Dumitriu, A History of Logic (Tunbridge Wells, 1977), vol. 2, p. 113.

In retrospect on the Abbreviatio Montana a few comments. First, notice how prominent the topical rules are. In some cases they were enlightening (I think that is true of the rules justifying hypothetical propositions), in other cases (this is true of the syllogisms) they just restate in more abstract words the argument which they are supposed to justify. Second, no where does the author explain the vital notion of validity: i.e. that an argument is valid if and only if no argument of that type can have its premises true but its conclusion false. He has given you a list of valid arguments, but nowhere explained that they are valid or what this means. Third, his list is dogmatic; he does not show that these are valid or that the others (the other possible moods of the various figures of syllogism) are invalid. Aristotle's Prior Analytics explains what validity is, invalidates all the invalid moods of the various figures, and reduces all the valid moods of figures 2 and 3 to uninvalidated syllogisms of figure 1. (The Abbreviatio Montana shows you how to reduce some valid syllogisms of figures 2 and 3, but not all of them.) Fourth, there is no discussion of modal syllogisms (i.e. syllogisms in which the premises and conclusion are preceded by "It is necessary that" or "It is possible that" or "it is impossible that"; modal syllogisms had been treated at length in Aristotle's Prior Analytics. Boethius had translated the Prior Analytics, but his translation was not being copied and taught. The author of the Abbreviatio knows Boethius's books on categorical and hypothetical syllogisms, but his only reference to the Prior Analytics is in the obscure paragraph on p. 83.

For all that, the Abbreviatio Montana is a handy summary of logic. We will take John of Salisbury's advice and not spend too much more time on logic - in fact this is the only logic text we will read. But in understanding and evaluating the other philosophical texts we will make some use of what I hope you will have learnt from this text.

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