Dici de omni

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Dici de omni from Worcester 13 - Dici de omni est quando nihil est sumere sub subiecto de quo non dicatur praedicatum

The dici de omni et de nullo (called by some writers, especially neo-Scholastic writers, the dictum de omni et de nullo) is a principle upon which all syllogistic inference is supposedly based. William of Ockham formulates it[1] as "it is denoted that nothing is subsumed under the subject without the predicate being predicated of it", adding the words 'it is denoted' in order to avoid the principle applying only to true propositions[2]. The dici de nullo applies when it is denoted or signified that nothing is to fall under the subject without the predicate being denied of it.

Aristotle's formulation

The principle is supposed to derive from Aristotle, Prior Analytics book I c. 1, 24 b 26-30. He says (English translation A.J. Jenkinson, Oxford 1928) "We say that one term is predicated of all of another, whenever no instance of the subject can be found of which the other term cannot be asserted: 'to be predicated of none' must be understood in the same way".

Giles of Rome[3] also argues that the principle proves the form of the syllogism. Somewhat later, J.N. Keynes (Formal Logic, London 1906, p.301) says Aristotle intended it as an axiom upon which all syllogistic logic is based. Lukasiewicz, citing Keynes, objects that it is nowhere formulated in the Prior Analytics as an axiom or principle, and that Aristotle's words are merely an explanation of the words 'to be predicated of all' or 'to be predicated of none'[4].

Other formulations

There are two distinct formulations of the principle. The first dates back at least to the early 13th century, and is found in writers such as Aquinas, pseudo-Aquinas, Ockham, Buridan and others. The early 13th century author William of Sherwood gives it as 'nothing is subsumed [sumere] under the subject of which the predicate is not asserted[5]. Aquinas has a similar version in his lectures on the Posterior Analytics book I chapter 9[6]. (See also the formulation by the anonymous author of Summa Totius Logicae, once attributed to Aquinas[7]). Ockham's version is mentioned in the introduction above. In his Summulae de dialectica[8], a commentary on Peter of Spain's treatise on logic, Buridan gives a similar version, saying that nothing is to fall under the subject term without the predicate also being predicated (or denied) of it. In his Questions on the Prior Analytics[9], he is more specific, saying "whenever the predicate is predicated of the subject affirmatively and universally, it is designated through such a proposition that this predicate is truly predicated of all that of which the subject is truly predicated. Both Ockham and Buridan mention the proposition 'designating' or 'denoting' that nothing falls under the subject etc., to preempt the objection that the principle applies otherwise only to true propositions.

The second formulation dates back at least to the neo-Thomist John Poinsot (John of St Thomas, 1589-1644), who says "whatever is universally predicated of some subject, is predicated of everything that is contained under that subject, and whatever is denied of some subject, is denied of everything that is contained under that subject"[10]. The key difference is the idea of 'predication of a subject' which is not in the standard 13th century formulation. It is obscure, but is closer to Aristotle's version. All later versions of the principle appear to derive from this second one. Thomas Reid[11] gives it as "what is affirmed of a whole genus may be affirmed of all the species and individuals belonging to that genus; and that what is denied of the whole genus may be denied of all its species and individuals".

John Stuart Mill gives it as "whatever can be affirmed (or denied) of a class may be affirmed (or denied) of everything included in the class", and says that this is called dictum de omni by logicians. He adds that the maxim is more suited to a system of [realist] metaphysics that has now (i.e. in the early 19th century) been finally abandoned, and depends on the idea that universals are a peculiar kind of substance, rather than a collection of individuals. It seems clear, however, that his objection applies only to the neo-scholastic and later version of the principle[12].

Britannica 1911 says "that which is affirmed or denied of any whole may be affirmed or denied of anything contained within (or 'any part of') that whole"[13], which is close to Aristotle's version. Lukaziewicz quotes it as quidquid de omnibus valet, valet etiam de quibusdam et de singulis, adding testily that this 'obscure principle' is not found in Aristotle.

Applications

Syllogisms in the first figure hold from the principle in a direct and obvious way. For example, 'every B is A', signifies that the predicate 'A' is truly predicated of everything of which 'B' is truly predicated. Thus if it is true that every C is B, it follows that 'B' is truly predicated of any C, and so follows from dici de omni that 'A' is also truly predicated of any C, i.e. every C is A. Thus the syllogism 'Barbara' is validated through dici de omni. Aristotle says that when a syllogism holds in this manifest and obvious way by the principle, it is a perfect syllogism.

Syllogisms of the second and third figure do not hold from the principle in such a direct and obvious way. For example, in the second figure syllogism 'no B is A, every C is A, therefore no C is B', 'A' is denied or removed from all B in the first or major premiss, but in the second or minor premiss there is nothing subsumed under 'B', which would have to happen if the argument were validated through dici de nullo in a direct and obvious way. Similarly, in the third figure syllogism 'every C is A, every C is B, therefore some B is A', in the major premiss 'A' is predicated of every C, but in the minor 'C' is not taken to be predicated of some other, which is what would have to happen if the argument were validated directly through dici de omni.

However, syllogisms of the second and third figure hold through dici de omni or de nullo, but in an opaque or indirect way, by reduction to the first figure. For example, in the second figure syllogism 'no B is A, every C is A, therefore no C is B', the major proposition converts to 'no A is B', given a first figure syllogism which can be validated by dici de nullo. Similarly, in the third figure syllogism 'every C is A, every C is B, therefore some B is A', the minor premiss 'every C is B' converts to 'some B is C', and the syllogism is valid under dici de omni[14].

See also

  • Supposition
  • Kneale & Kneale 1962, p.79, chapter 'The Syllogism', where the principle is called Dictum de omni and given as 'What is predicated of any whole is predicate of any part of that whole', mentioning also the tag Nota notae est nota rei ipsius ('The mark of a mark is a mark of the thing itself').
  • Distribution of terms

Links

Notes

  1. ↑ Summa Logicae III-1. ii
  2. ↑ Est autem dici de omni quando nihil est sumere sub subiecto, quin de eo dicatur praedicatum. Quod est sic intelligendum: non quod praedicatum conveniat cuilibet de quo dicitur subiectum, -- tunc enim non esset dici de omni nisi in propositionibus veris --, sed sufficit quod per talem propositionem denotetur quod nihil sit sumere sub subiecto, quin de eo dicatur praedicatum
  3. ↑ In Rhetoricam Aristotelis, in Jan Pinborg, Logik und Semantik im Mittelalter (Stuttgart-Bad Cannstatt: Fromann-Holzboog, 1972), pp. 200-203
  4. ↑ Lukaziewicz 1951 p. 47
  5. ↑ Kretzmann's translation, see Kretzmann 1966
  6. ↑ Tunc enim dicitur aliquid de omni, ut habetur in libro priorum, quando nihil est sumere sub subiecto, de quo praedicatum non dicatur [1]
  7. ↑ Est autem dici de omni, quando nihil est sumere sub subjecto, de quo non dicatur praedicatum; dici vero de nullo est, quando nihil est sumere sub subjecto, a quo non removeatur praedicatum. Not yet in the Logic Museum
  8. ↑ Book 5 c.1
  9. ↑ Book I Question 5
  10. ↑ Quidquid universaliter dicitur de aliquo subjecto, dicitur de omni quod sub tali subjecto continetur: quidquid negatur de aliquo subjecto, negatur et de omni contento sub tali subjecto. (Logica. Pars I., Lib.3, c.x)
  11. ↑ Aristotle's Logic [2]
  12. ↑ System of Logic (Mill) II. ii. 2
  13. ↑ [3]
  14. ↑ Buridan, op. cit.
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