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 | |  John Buridan | |
John Buridan (c. 1300–1361) was the foremost logician of the later Middle Ages and in his hands the theory of the syllogism was reworked and developed well beyond anything seen before in the history of logic. His two most important logical works are the Treatise on Consequence and theSummulae de Dialectica. The presentation here is primarily based on the Treatise (for further discussion, see Lagerlund 2000, Chapter 5, and Zupko 2003, Chapters 5–6).In the Treatise, Buridan bases his discussion of the syllogism on a philosophical semantics that views syllogistic inference as a special case of the much more comprehensive theory of consequences. Like his immediate predecessors, he was for the most part uninterested in assertoric syllogisms and moves on quickly to temporal, oblique, variation, and modal syllogisms, though this does not prevent him from making some original contributions to the theory of the assertoric syllogism.According to Buridan, a syllogism is a formal consequence, and so syllogistic becomes a branch of the theory of formal consequence. As consequences, syllogisms are distinguished by their conjunctive antecedent and single-sentence consequent, and furthermore by their three terms — though this last condition is not necessary since Buridan also treats of syllogisms with more than three terms.Buridan treats of the three famous figures and notes that a conclusion can be either direct or indirect. In an indirect conclusion, the minor term is predicated of the major instead of the other way around. Since the premises are part of a conjunction and together form the antecedent of a consequent, they can easily switch places, which means that Buridan can define the fourth figure as a first figure with transposed premises and an indirect conclusion. Hence, he does not need to discuss it independently of the first figure.For Buridan, a formal consequence holds by the principle of uniform substitution. It is valid for any uniform substitution of its categorematic terms. A syllogism is a special kind of formal consequence since it requires for its validity that terms be conjoined across sentences. How the principle of uniform substitution is supposed to work here is a bit tricky and forces him to bring into play his general semantics as well as the notion of distribution. To spell out the relation of their terms and hence the validity of the first figure syllogisms, he reformulates the traditionaldici de omni et nullo rules (see King 1985: 71): - اقتباس :
| (8:1) | Any two terms, which are called the same as a third by reason of the same thing for which that third term supposits, not collectively, are correctly called the same as each other. | | (8:2) | Any two terms, of which one is called the same as some third term of which the other is called not the same by reason of the same thing for which that third term supposits, are correctly called not the same as each other. |
One could say a great deal about these rules, but the term that does most of the heavy lifting is ‘supposits'. Supposition is a theory of reference and it is the coreferentiality of terms in the different sentences in a syllogism that is the decisive factor in determining whether the principle of uniform substitution is satisfied or not. It is at this point that he introduces the theory of distribution.The rules governing the distribution of terms in a sentence are given as part of his account of common personal supposition. A term is distributed in a sentence if it is taken to refer to everything it signifies, such as if the term is in the scope of a universal quantifier. To indicate when a term is distributed he gives five rules, according to which universals distribute subjects, negatives distribute predicates, and no other terms are distributed. If we stay within the square of opposition (Buridan's theory of distribution, and hence his syllogistic, has wide application, extending far beyond the traditional A, E, I, and O sentences), this implies that universal affirmative sentences have their subject terms distributed, universal negatives have both terms distributed, particular affirmatives have neither term distributed, and particular negatives have only their predicate terms distributed. (For an influential criticism of this theory of distribution see Geach 1962, and King 1985 for a reply.)With his theory of distribution in place, Buridan turns to the syllogisms, and we see now that in order for a combination of premises to be acceptable, the middle terms must be distributed — otherwise we will not have a formally acceptable consequence. Buridan approaches the problem in combinatorial fashion. Given the four sentences and two possible positions for each we get 16 possible combinations. Some of these can be ruled out immediately based on the rules for distribution. A combination with only negative premises will not work at all; hence EE, EO, OE, and OO must be rejected. II has both middle terms undistributed and can thus be rejected. In the first figure, IA, OA, and OI have an undistributed middle. The other eight are accepted. In the second figure, we see that AA, AI, and IA must be rejected because of an undistributed middle. The other eight are accepted. In the third figure, IO and OI have an undistributed middle but the remaining nine combinations are accepted.At first glance, there are some surprises in Buridan's presentation of assertoric syllogistic. In the second figure he accepts indirect conclusions for IEO (Tifesno) and OAO (Robaco), and in the third figure indirect conclusions for AOO (Carbodo), AEO (Lapfeton), and IEO (Rifeson). He also accepts syllogisms concluding to what he calls an “uncommon idiom for negatives,” that is, when the predicate term precedes the negation, as in ‘Some B A is not’ (Quaedam B A est non). Such sentences make no sense in English, but Buridan treats them as equivalent to sentences where the predicate term quantified, as in ‘Some B is not some A’. He writes the sentences in this way because otherwise they would violate his rules for distribution and scope. Syllogisms concluding to an “uncommon idiom for negatives” add another three valid forms in the first figure and two in the second. If we tally this up and include all of the indirect conclusions, we get 33 valid moods, as opposed to 19 in Aristotelian syllogistic. If we then add the supplementary subalternate conclusions, we get 24 valid moods in traditional syllogistic but 38 in Buridan's systematization.Buridan is quite right to accept these additional 14 moods. They are valid. But his result is not as dramatic as it seems since the middle terms are either in the subject or predicate position. In the second and third figure an indirect conclusion becomes equivalent to transposing the premises. Hence Buridan's Tifesno, Robaco, Carbodo, Lapfeton, and Rifeson reduce to Festino, Baroco, Bocardo, Felapton, and Ferison, respectively. This is also obvious if we look at the names of these syllogisms, which suggest that Buridan has only reshuffled the letters of the names of the standard Aristotelian syllogisms. After having done all this over just a few pages — as mentioned above, Buridan is rather uninterested in assertoric syllogistic — he turns to the temporal, oblique, and modal syllogistic. Of these, it is modal syllogistic to which he devotes the most time.A temporal syllogism consists of sentences whose copulas involve temporal ampliation. In such sentences, the supposition of the subject term is extended to include past and future things as well as present things. The syllogistic for sentences involving oblique terms is important for Buridan's general theory of consequence, since this is where we find rules governing the behavior of oblique terms in distributive contexts. His investigation is extraordinarily detailed and extremely rigorous, qualities all the more impressive when we consider that he did not have the representational tools of modern symbolic logic.The syllogistic for composite modal sentences is straightforward and Buridan uses only a couple of pages in the Treatise to sketch its basic structure. The theory of the syllogism for divided modal sentences is given a much more thorough treatment. For Buridan, a modal copula always ampliates its subject term to stand not only for present, past, and future things but also possible things, unless the supposition of the subject term is explicitly restricted to what is actual. On this basis, he can give an exhaustive account of the logical relations between quantified divided modal sentences, which he presents in the octagon of opposition. Slightly simplified, and assuming that the complete octagon can be formed by some trivial equivalences holding between the modalities, it can be depicted as in Figure 1: - اقتباس :
Figure 1 Together with the octagon he also uses some consequences to prove the valid syllogisms. He first states the valid conversion rules: - اقتباس :
- (8:3) Every B is possibly A ⊃ Some A is possibly B
(8:4) Some B is possibly A ≡ Some A is possibly B
(8:5) Every B is necessarily not A ≡ Every A is necessarily not B
(8:6) (Quantity) B is contingently A ≡ (Quantity) B is contingently not A All these are valid assuming their subject terms are ampliated. He also employs the following consequences: - اقتباس :
- (8:7) Every B is necessarily not A ⊃ Every B is not A
(8:8) (Quantity) which is B is necessarily A ⊃ (Quantity) B is A
(8:9) (Quantity) B is A ⊃ Some B is possibly A
(8:10) (Quantity) B is necessarily (quality) A ⊃ (Quantity) B is possibly (quality) A
(8:11) (Quantity) B is contingently (quality) A ⊃ (Quantity) B is possibly (quality) A By stating these consequences and the octagon of opposition, Buridan has presented a virtually exhaustive syntactical account of modal logic, and, together with his semantics of supposition and distribution, constructed a powerful logic unmatched by anything presented in the history of logic before him.Buridan uses four methods to prove the valid syllogistic moods. All first figure moods are proved by the rules of class inclusion, that is, the dici de omni and dici de nullo rules. The second and third figure moods are proved using three different methods: either by conversion — that is, by (8:3), (8:4), (8:5) or (8:6) — by reductio ad impossibile, or by expository syllogism. Proof by impossibility is used on a few occasions, but Buridan's approach here differs in no way from Aristotle’s. This fourth way is frequently used to prove the valid third figure moods. Since the number of possible combinations of premises and conclusions in modal syllogistic is quite extensive, he limits himself to discussing those moods whose assertoric counterparts are valid, but even so he manages to discuss a large number of valid and invalid syllogisms (see Hughes 1989, Lagerlund 2000, Thom 2003, Klima 2008, and Dutilh-Novaes 2008).There has been considerable scholarly discussion of Buridan's modal syllogistic. It has been asked in particular whether it corresponds to any modern system of modal logic. The most popular answer is S5 (King 1985). Some have argued that Buridan must have been thinking in terms of some kind of possible worlds model (Hughes 1989 and Knuuttila 1993). Such comparisons of course reflect the extent to which these scholars have been impressed by Buridan's modal logic, which was without equal until the late twentieth century.(For an example of just how powerful his general logic was see Parsons 2014.) | |
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