pc answers

["John E. Clifford" <pcliffje@crl.com>]

        Ok, what do you want to say? Let's take "Three men touched three dogs"
into logic without thinking too much about it.  That gives
        there are x,y,z,w,v,u, mutually distinct [actually a conjunction
of 15 non-identities] and for all x1, x1 is a relevant man just in case
x1 is x, y, or z and for all y1, y1 is a relevant dog just in case y1 is
w, v, or u [so far we have that there are three men and three dogs of
interest; now for the serious content, we have a choice among]
   1. for every relevant man z1 and every relevant dog w1, z1 touched w1
   2. for some relevant man z1 and every relevant dog w1, z1 touched w1
   3. for some relevant dog w1 and every releant man z1, z1 touched w1
   4. for every relevant man z1 and some relevant dog w1, z1 touched w1
   5. for every relevant dog w1 and some relevant man z1, z1 touched w1
   6. for some relevant man z1 and relevant dog w1, z1 touched w1.
        All of these can, of course, be expanded into disjunctions and
conjunctions involving the original terms, x,...,u and such formulae would
also provide relatively easy ways of saying things like "two of the men
touched two dogs each while the third did nothing, but all three dogs got
touched."  We can skip these niceties for now, I hope.
        Clearly, if 1 is true, the _le_..._le_ form is justified, every
man touches each of the dogs. But I think that the _le_ ... _le_ form is
also appropriate when 4 and 5 are simultaneously true: every man touches
at least one dog and every dog is touched by at least one man.  That is,
all the men were involved in the touching and so were all the dogs; only
the exact distribution of labor is left open. Notice that, like 1, this
form is symmetric (the order of the quantifiers, corresponding to the
order of the sumti, is irrelevant), so that the _se_ form will mean the
same thing.
        Since 2 entails 5 and 3 entails 4 (both are generally entailed by
1), we have to say that their conjunction is also appropriate for the
_le_..._le_, though less obviously so.  We have to work out that, once
again, every man touches at least one dog and every dog gets touched by at
least one man. We just get more information about the distribution of labor
(and belabored).  I suppose that mixtures, 2 and 4 or 3 and 5, would also
fit in to this pattern although they are not really symmetric.  But they
do all guarantee that everyone in both group gets involved.
        However, none of 2-5 taken alone seem appropriate for the double
_le_ form. They allow some man or some dog to escape participation in the
whole complex event and that seems not to fit with the sense of _le_.  It
does fit with _le'i_ (? the _le_ style mass), however -- assuming that the
the men and the dogs (as needed) have some reason to be together beyond
this complex event (I agree with xorxes that we ought not mass things
without some reason -- though keeping them in mind, under _le_, may be
reason enough).  Of course, any replacement of _le_ by _le'i_ will make a
true claim in the cases discussed above, but now we get to cases where
_le'i_ but not _le_ apply.  The rule seems to be that the particular
quantifier ("some") marks the _le'i_ here.  So, 2 and 5 are _le'i_ men and
_le_ dogs, 3 and 4 the reverse -- or, of course, all _le'i_.  That last
form is absolutely required for form 6, assuming that we can do this at
all.  Even if we allow that the two groups have some common ground for
massification, it seems that too little has been done to really count for
one mass touching the other.  We might be inclined to say only "the men
have started to touch the dogs" or some such thing indication a plan not
yet much executed.
        I'm not quite clear about what parity of reasoning does for the
intermediate cases, like "two of the three men touched the three dogs".
Clearly _le_ is justified when both men each touch all three and I think
it is when all are involved: each touches at least one dog and and each
dog is touched by at least one man (and the more precise version of that,
corresponding to 2 and 3).  Whether we want to bother with _le'i_ here is
less clear, especially with _le'i_re_le_ci_renma_ for some of the mixed
cases.
pc>|83