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What’s in a Name?
Towards a Nominalistic Computer Science – Without Sets or Other Abstract Objects
A paper by Adrian Larner
This paper was given at the De Montfort University Computer Science Summer Symposium 1994. I include it here not only for any intrinsic interest it might have but also because it explains to some extent the philosophy behind my approach to database theory and other disciplines.
Abstract
Computer Science, no doubt because of its mathematical roots, is steeped in Platonism: it posits the existence of such abstract objects as sets, classes, and types. Commonly, such an abstract object is introduced as the unique bearer of a common name that, intuitively, designates each of its “members” or “occurrences”. It is argued in this paper that the designation of things by common names can – with philosophical propriety and practical simplicity – be better explained and formalised nominalistically: without resort to extra, abstract objects.
1 INTRODUCTION
See William Kneale and Martha Kneale:
The Development of Logic, ClarendonPress, 1962.
Abelard expressed his position as:
Singuli homines discreti ab invicem ...
in eo ... conveniunt quod homines sunt;
non dico in homine, cum res nulla sit homo nisi discreta,
sed in esse hominem.
Esse autem hominem non est homo nec res aliqua.
(Individual persons,
distinct from one another ... have this in common:
being a person.
I say not [merely] “person” –
because nothing is merely “person” –
but “being a person”.
However, “being a person” is not a person,
nor is it any thing at all.)
The aim of this paper is to describe how names are related to the things that bear them, the things that they name, or designate. This term is not used in any special, technical sense. To say that the name, “Chicago”, designates the city, Chicago, is merely to say that that city has that name.
The scope of this enquiry includes both proper names of things, like “Chicago”, and common names of things (count nouns), like “city”; but it excludes names of stuff (mass nouns), like “gold” and “information”.
Controversy over names, and other general terms (universals) goes back a long way, certainly as far as Plato, whose theory was that things shared a common name by dint of participating in, or being like, an ideal form of which that name was the proper name. His opponents, the nominalists, argued that names were nothing, or almost nothing: flatus vocis (the mere breath of the voice), according to the mediaeval scholastic, Roscelin.
A mediating position was proposed by his contemporary,
Abelard:
things of a common name have some likeness, other than mere possession of the name;
but such a “likeness” is not itself a thing.
Notice that the extreme nominalist position does not distinguish
genuine commonality from non-systematic ambiguity, e.g. the same (proper)
given name designating each of two persons,
or the word “bank” designating here a riverside,
there a financial institution.
(See note, left.)
Most mathematicians and computer scientists are Platonists (the position is rarely defended). Modern nominalism is inspired by Nelson Goodman: see his Problems and Projects, Hackett Publishing Co, 1972, Section IV. The approach taken in this paper owes more to Geach, v.i.
The Platonist/nominalist controversy, however, is not of merely historical interest: the problems of our ancient and mediaeval ancestors have returned in new forms to haunt Twentieth Century philosophy, and computer science. (See note, left.)
2 PREDICATES AND SETS
See Willard Van Orman Quine:
Set Theory and its Logic, Harvard University Press, 1969
There is a very obvious function from common names to monadic predicates. In general, if the name, “D”, designates a thing, r, then we may form the predicate, “... is a D”, and this will hold true of r, i.e. the proposition, “r is a D” will be true. If “dog” designates Rover, then Rover is a dog.
But there are lots of them about –
predicates (as well as dogs) –
and they are what Frege called “ungesättigt”, unsaturated:
they have a “hole” in them, technically a place,
into which we may drop a name (such as “Rover”),
to form a proposition.
For reasons that are difficult to understand,
mathematicians do not like the unsaturated.
We would be inclined to interpret a relation,
such as “... is north of ...”, as a two-place predicate,
i.e. as true or false of things taken pairwise,
accordingly as the one was or was not north of the other.
The mathematician insists that such a relation is a set of ordered pairs of things.
But the mathematician is not in a philosophical state of grace.
(See note, left.)
REMARK: Perhaps Russell should have been warned (but, alas, he rather despised the mediaeval “scholastic” philosophers). Recall (above) Abelard’s Esse autem hominem non est homo nec res aliqua. (“Being a person” is not a person, nor is it any thing at all.)
(The original audience for this paper were expected to be acquainted with Russell’s “Paradox”. The following account is added for those that may not be. It may be skipped without serious loss to understanding of the paper.)
It was not ever thus: Lord Russell was innocent when he unintentionally proved that we could not replace predicates by something saturated: His temptation was to get rid of all those predicates, leaving “... is a member of ...” as the sole unsaturated component of the system; for each predicate, such as “... is a dog”, there was to be a thing, the set of dogs, such that to be a dog was to be a member of that thing. Russell’s knockdown proof of the incoherence of this notion of set membership is quaintly termed a “paradox”, but let us not be misled by that. The intractable predicate that he found – “... is not a member of itself” – arose directly from the incoherent notion that he was trying to formalise. (See REMARK, left, and added section, below.)
We can see what went wrong. The notion of set was intended to capture two quite different concepts: the application of a predicate and the aggregation of the things of which that predicate held true. Plato’s theory had met a similar end, the “Third Man” argument. If John and James are each called “man” only because they participate in the ideal MAN, who is also called “man”, then James and MAN can each be called “man” only because they participate in yet another, even more ideal, man. You just cannot keep a good third man down.
RUSSELL’S “PARADOX”
Russell’s so-called paradox arises as follows. In order to eliminate any predicate, such as “... is a person”, or – schematically (i.e. generally) “P(...)” – we introduce (despite Abelard’s warning!) a thing, p, (the set or class of persons, or, speaking more generally, of so-and-so’s; and we introduce the membership relation, “... is a member of ...”, or “... Î ...”
The objective is to eliminate all the other “unsaturated” predicates (“... is a person”, “... is a so-and-so”). But to do this we need to make a postulate: the Axiom Schema of Abstraction, which says that to be a so-and-so is to be a member of the class of so-and-so’s; formally:
- $p "x x Î p « P(x)
I.e. There exists something (p, the class of so-and-so’s), such that each thing, x, is a so-and-so – P(x) – if and only if it is a member of p, the class of so-and-so’s.
Granted this postulate, we may – as Russell intended – eliminate any predicate, “P(...)”, by substituting “... Îp”. The best laid plans, however, ...
Consider the predicate “x Ï x” (i.e. “NOT x Î x”; admittedly contrived but otherwise proper – at least, no less proper than “Δ itself).
This predicate – of non-self-membership – is true of each thing that (like most examples that come to mind) is not a member of itself (so of each dog, because dogs are not classes and have no class-members; and of the class of dogs, which – not being itself a dog – is not a member of the class of dogs). But this predicate of non-self-membership is false of some things: the class of classes, for instance, or the class of abstract objects.
Now recall that the schematic predicate, “P(...)”, in the Axiom Schema of Abstraction is postulated to give a true axiom no matter what actual predicate is substituted for it. So we substitute our predicate of non-self-membership, say “Q(...)”, where Q(x) is defined as (i.e. is a mere abbreviation for) x Ï x. We get the axiom:
- $p "x x Î p « Q(x)
I.e. There exists something (p, the class of non-self-members), such that each thing, x, is a non-self-member – Q(x) – if and only if it is a member of p, the class of non-self-members.
Granted the existence of this thing, the class of non-self-members, let us give it the name “q” and the axiom now gives us (after substitution of “x Ï x” for “Q(x)”):
- "x x Î q « x Ï x
I.e. The class of non-self-members, q, is such that each thing, x, is a member of q if and only if it is not a member of itself (x).
And now for the sting: the axiom says that the predicate, x Î q « x Ï x, holds true of each thing (x). And therefore it holds true specifically of q, the class of non-self-members:
- q Î q « q Ï q
I.e. The class of non-self-members is a member of the class of non-self-members if and only if the class of non-self-members is not a member of the class of non-self-members.
And this is plainly false: it is a contradiction (but not a paradox!) The Axiom Schema of Abstraction is false: it is false because it is founded on an incoherent notion, class membership. One simply cannot explain the common feature of similar things, such as persons, in terms of another thing: certainly not another (ideal, Platonistic) thing (the “third man”) but not any other kind of thing either: as Abelard said, “Being a person” is not a person, nor is it any thing at all.
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Copyright © 1994, 2001 Adrian Larner. The author asserts all moral rights.
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