[FOM] a series of grammatical confusions
slaterbh@cyllene.uwa.edu.au
slaterbh at cyllene.uwa.edu.au
Sat Feb 18 09:29:34 EST 2006
Gabriel Stolzenberg has asked for details of my March 2006 AJP paper
'Grammar and Sets'. Below are the arguments for the conclusions Nik
Weaver has singled out in a recent posting. But I have a far wider
ranging paper 'Logic and Grammar' coming out in Ratio shortly, showing
the 'series of grammatical confusions' extends well beyond these (indeed
there are many more just about sets in the AJP paper itself). I have
also completed a third paper pretty much on this topic, but centering on
what has been (strangely) called 'The Foundations of Mathematics' from
the end of the 19th century. That might be of particular interest to
FOM, of course. Electronic copies of these two further papers are
available upon request.
II. Collective Nouns, and Plural Descriptions
But what of the pair set itself? Is even that an object? Writers like
Lewis, Maddy, and now Potter, have seen the difficulty with identifying
singleton sets, or, at least, singleton sets whose members are
urelements. For where, or what is the singleton set of an apple? Maddy
has been led, even, to suppose it is the same as the apple itself. But
the solution is more easily found by looking first at non-singleton sets,
like pairs.
With two apples there is a single object in the vicinity, namely the
mereological sum of the two apples. But that clearly cannot be the pair of
apples, since that whole might be carved up in more than one way, and in
some ways with more than two components. The expectation, as a
consequence, has been that one must look elsewhere for another object to
be the pair of apples. But this supposed other object is a grammatical
mirage.
There are, in the area, at least three types of expression, with quite
different grammars, and without a close inspection it is far too easy to
run them all, or even just some of them together. Thus there are
collective nouns like ‘shoal’, ‘herd’, ‘pack’, ‘tribe’, collective nouns
like ‘pair’, ‘triplet’, ‘quartet’, ‘dozen’, and plural expressions like
‘the fish’, ‘the beasts’,‘the cards’, ‘the savages’. It is from
collective nouns of the first sort that Set Theory draws the idea of
collections of objects, but the distinctive thing about those natural
language terms is that they are species specific, to one degree or
another, so that they each describe certain mereological sums under a
certain aspect. That is to say, the principle of division of the whole
mereological sum is provided through the further count noun they are
normally associated with—‘shoal’ with ‘fish’, ‘herd’ with ‘beasts’,
‘pack’ with ‘cards’, ‘tribe’ with ‘savages’, etc. The fact that such
collections are mereological sums is also shown by the fact that shoals,
herds, packs, tribes, and the like, are located and can move around in
physical space, just like their members.
There is not the same to be said with regard to collective nouns of the
second sort, and not only because a complete description of the intended
set must be given—the pair is maybe of fish, the triplet of beasts, etc.
For these collective nouns can also be used when no physical objects are
involved, and so when there are no mereological sums of them. They
therefore only indicate the number of some things, and we have to
remember the general grammar of ‘y is one of a number of Ss’. This is not
of the form ‘y isin s’, with a singular term in place of ‘s’. A
specification of it, for instance, would be ‘y is one of 2 Ss’, which
relates ‘y’ to a plural term, and the original is just (En)(y is one of n
Ss).
John Burgess [PM 2004: 197 – 9] thinks there are two senses of ‘is one
of’, so that one might say, for instance, ‘y is a member of the set of
apples’ but ‘y is amongst the apples’, even though the expressions are
equivalent. That would allow a singular ‘s’ to occupy ‘y isin s’, while a
plural ‘xx’ occupies ‘y isamong xx’. Certainly ‘y is a member of
a/the/that pair of apples’is the same as ‘y is amongst 2/the 2/those 2
apples’ but ‘is one of’ could be used in both cases, and in the same
sense. For the equivalences show that ‘a/the/that pair of apples’ still
refer to the same things as ‘2/the 2/those 2 apples’— the former merely
refer to them (sic) in a different manner, namely collectively, i.e.,
taking them as a unit. One must take care about what is added to the bare
‘y is one of some apples’: clearly ‘y is one of 2 apples’ may speak about
the same apples, but is more specific about their number, and ‘y is one
of a pair of apples’ likewise. But the latter does not invoke a further
object, ‘a pair’, in addition to the two apples, it merely introduces a
certain numerical measure of the apples, by taking the two as a unit. ‘A
pair of apples’, in other words, differs from ‘2 apples’ simply in
changing ‘2 times 1 apple’ into ‘1 times 2 apples’. We can count with
such units, by counting in pairs, but we are then not counting some thing
other than apples; we are merely not counting the apples singly, i.e., one
by one. ‘There are 2 pairs of apples’ we say, but this is exactly
equivalent to ‘There are 4 (single) apples’, and ‘There is a quartet of
apples’. ‘There is a set of apples’ is equivalent to ‘(En)(there is an
n-tuple of apples)’. Words like ‘pair’, ‘quartet’, ‘16-tuple’ are thus
like ‘ounce’ and ‘pound’ in ‘There are/is 8 ounces/a pound of beef’: they
do not refer to further entities, but are instead the basis for
measurements of quantity.
Here are two pairs of alphabetic letters: a b c d. Notice that each of the
pairs is indicated without braces, as with ‘{a, b}’, since the latter,
i.e., the common ‘set-theoretic’ symbolism, does not represent sets as
merely numbers or quantities of objects but as independent objects,
distinct from their members. However, the four letters above, which were
taken as two pairs, also can be taken as a quartet, while the two ‘sets’
{a, b}, and {c, d} cannot be the same as the ‘set’ {a, b, c, d}. In fact
no two ‘sets’ can be the same as any one ‘set’, but 2 twos are exactly
the same as 1 four.
The reification of a unit of measure as a further separate object, maybe
arises through forgetting the difference between the two sorts of
collective term. For the mereological ‘tribe’ does have an objective
reference to an independent object, but ‘pair’ needs supplementing, and
then in ‘a pair of apples’, it merely qualifies the following
substantive. Maybe focussing on count terms and forgetting mass terms
also has something to do with the misconception. For the same matter of a
change of units even more clearly arises with fractions than with
multiples of individuals. In ‘There is a half of a loaf’ there is
obviously no reference to something other than bread: there is not, in
addition, reference to one of a range of mysterious, further objective
entities, ‘halves’, ‘quarters’, parts’, etc. There is merely a
specification of how much of a loaf there is, maybe as a prelude to
counting half-loaves, or totting up different parts of loaves to find an
equivalent sum of complete loaves, etc.
We can now tackle the question of what a ‘set-theoretic’ singleton is,
i.e., what ‘{y}’, or ‘{x: x = y}’ might represent. In the natural language
locution ‘a singleton S’, of course, ‘singleton’ just describes the S as
the only one of its kind, and does not refer to any other object. But
another way of representing being the sole S as ‘being one of’ something
might well be the prime source of the set-theoretic notion of
‘singleton’. For even if the number of things which are S is just 1 then
we can still say ‘y is one of those things which are S’, making *those
things which are S* what the sole S is one of. But ‘is one of’ is then
again succeeded by a plural term, not a singular one. If we read set
abstraction expressions as such plural terms, therefore, ‘{x: x = y}’
would be ‘those things which are y’, allowing ‘y isin {x: x = y}’ to be
‘y is one of the things which are y’. But then, identification of such
‘singletons’ with their single members, in the manner of Maddy, would
clearly be ungrammatical, since ‘y is those things which are y’ does not
make sense. Indeed the general identification of set abstracts with
plural terms could not be thoroughgoing, since not only would ‘y = {x: x
= y}’ be ungrammatical, so would ‘y = {x: Px}’ for any ‘P’, and therefore
also ‘{{x: Px}}’, and ‘{{x: Px}, {x: Qx}}’, because these latter would
have to be ‘{z: z = {x: Px}}’ and ‘{z: z = {x: Px} v z = {x: Qx}}’.
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