Monday, September 20, 2010

Indivisibility and Unrepeatability: Subjective Parts

Before I attempt an answer to Paasch's question there is a preliminary notion that needs to be clarified, as it is crucial to Scotus' account of individuation, and it is also one of those medieval ideas that are obscure to us schooled in the thought of the modern predicate calculus.

In book II, d3 section 48 (Vatican edition) he says that we must not ask what it is by which such a division is formally incompatible to an individual (since it is formally incompatible by incompatibility), but rather what it is by which, "as by a proximate and intrinsic foundation", the incompatibility is in it. What is it in this stone by which, as by a proximate and intrinsic foundation, it is absolutely incompatible with it to be divided into subjective parts?

A 'subjective part' is a term that would have been familiar to any scholastic logician. Peter of Spain explains it [1], together with the notion of 'integral part', in his Treatise on Logic, a standard textbook of the time. "The [term] 'whole universal', as taken in this way, is anything superior and substantial, taken in respect of its inferior. For example, animal to man, man to Socrates. A 'subjective part' is said of any inferior, taken under the whole universal. […] The 'whole integral' is is a composite of parts having quantity, and its parts are called 'integral parts'. An integral part is what, taken with the other parts, gives the quantity of the whole'.

Walter Burley explains it [2] by definition as well as by example, saying that an individual is a subjective part of a species, because the species is directly predicated of the individual, and for the same reason the species is a subjective part of the genus, because the genus is directly predicated of the species, and this is the difference between an integral part and a subjective part, because the whole is directly predicated of a subjective part, but is not directly predicated of an integral part, but only indirectly. And so 'a hand is a man' is false, or 'a head is a man'.

The distinction probably comes from Porphyry's introduction to Aristotle's Categories, where he says that that individual is a part of the species, and the species by the genus, so that genus is a sort of whole, the individual is a part, and species both whole and a part.

Thus (returning to Scotus) we have the sense in which an Socrates is indivisible or 'individual'. The genus 'animal' can be divided into subjective parts such as man, giraffe, grasshopper and so on. The species 'man' can be divided into subjective parts such as Socrates, Plato and so on. But these cannot be divided in the same way. Socrates is not like a species having some member x of which we can truly say that x is Socrates. This conception of division is fundamentally different from modern logic. We predicate 'animal' or 'man' of some x (e.g. Socrates). But the subject is always an individual. We have man(Socrates) and animal(Socrates). We do not and cannot have animal(man), for as Geach notes, modern predicate logic assumes a fundamental distinction between the relation of class-inclusion (man to animal) and the relation between individual and class (Socrates to man, Socrates to animal). Thus we cannot even understand Scotus' conception of individuation unless we drop an idea that is part of the language of thought for modern analytic philosophers. The idea may be wrong and misguided, even incoherent (as Geach cogently argues). But we cannot even begin to 'get inside the head' of the medieval logician unless

This should also clarify the distinction between the modern notion of individuation as 'unrepeatability' and notion of it as indivisibility. Repeatability is another idea of modern predicate calculus. A predicate is repeatable when it be 'instantiated' more than once. An instance of F is an x such that Fx, another instance is a y such that Fy, and x <> y. Note that the instance must be an individual x or y, not another predicate. We can have man(Socrates) and man(Plato), but not animal(man), animal(giraffe), or at least not when the function-argument notation is understood as in standard predicate logic. But divisibility, as Scotus and other medieval logicians understand it, is fundamentally different. Animal is divisible because it is (as it were) instantiated by man or giraffe. 'Man' is repeatable because it can be instantiated, in exactly the same sense, by Socrates and Plato. But Socrates cannot be further instantiated.

Hence there is not really a problem of individuation for modern logic. The argument to a propositional function is guaranteed to be individual because anything other than a sign for an individual in the argument place (or a variable representing it) will make the expression ill-formed. Scotus, by contrast, has to explain why the Porphyrian tree comes to an abrupt halt with individuals such as Socrates and Plato, because he is assuming that the relation between Socrates and his species is fundamentally the same as that between the species and its genus. If the relation between Socrates and man is essentially like the relation between man and animal, why is it that we can't repeat Socrates into subjective parts, the way that we can repeat animal into subjective parts such as man and giraffe.

So, you can either argue that the problem of individuation is a silly problem that can't even be stated in modern logic. Or you can take it seriously, as I will try to do in a later post. More later.

[1] Totum universale, ut sic sumitur, est quodlibet superius et substantiale, sumptum ad suum inferius, ut animal ad hominem, et homo ad Socratem. Pars subiectiva dicitur quodlibet inferius, sub toto universali sumtum . . .Totum integrale est, quod est compositum ex partibus, quantitatem habentibus, et pars eius dicitur pars integralis. Pars integralis est, quae cum aliis partibus reddit quantitatem totius.

[2] Expositio super librum Porphyrii: De genere quidem et specie. Recapitulans dicit quod intelligendum est quod individuum est pars subiectiva speciei, quia species predicatur de individuo in recto; et propter eandem causam species est pars subieciva generis, quia genus predicatur in recto de specie; et hec est differentia inter partem integralem et partem subiectivam, quia de parte subiectiva vere predicatur suum totum in recto, sed de parte integrali non vere predicatur totum in recto, sed in obliquo. Manus enim et caput sunt partes integrales hominis, quia integrant hominem. Et ita hec est falsa: ‘manus est homo’, vel ‘caput est homo’. Text from here.

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