pc answers

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

Djer's additions about what is covered by the various quantifier prefixes
raises an interesting point but has it slightly off.  In the first place,
z1 is not defined to *be* the set {xyz} but to *range over* that set,
i.e., to take members of that set as values in genrating true sentences.
And the claim in 2 is just that for at least one (but maybe two or all
three) members, the replacement does give a true sentence.  For example,
it may be that it is true of man x that he touches all three dogs, w,v,
and u. And, if not of x, then maybe of y and, if not of y either, then
surely of z.  Notice that, "for some" ("at least one") includes the
possibility of "all," so that djer's list of possible all-dog-touchers
actually has seven members. But the members of this list are not the
substituends for the quantifier, which takes only individuals as
replacements.  The subsets come in only in the sense that, if all the
members of any one of those seven subsets gives a true sentence, then the
original sentence is true (another way to put this is that the original is
the disjunction of the corresponding claims about the three members of the
set).  The universal quantifier, w1 in the case of 2, means that all the
instances from the set must give trues for the original to be true (the
conjunction of the instances).  Expanding 2 out then gives first
 (w1)xTw1 or (w1)yTw1 or (w1)zTw1 (the disjunction of the z1 instances)
and then (xTw and xTv and xTu) or (yTw and yTv and yTu) or (zTw and zTv
and zTu) (expanding all of the universal w1's out)


        If we go back to basics, the nine simple sentences listed above,
each claiming one
man touched one dog, we can find 512 = 2^9 possible situations.  Of
these, exactly one,
the first in the usual ordering (as for truth tables) makes sentence 1
(all-all) true.  For
sentence 2 (and 3) (some-all), 169 situations makes each true (with some
uncalcuated
overlap of cases -- I've forgotten the rule calculating those).
(Actually, I calculated the
number of ways that this could fail to happen, e.g., where each man
touched two or fewer
dogs.  Since each man can do this in seven different ways -- every way
except touching
all three -- and they can do these independiently of how the other man
act, there are 7^3
= 343 ways.  The remaining cases, 169, must be the ones where the
sentence holds.)  By
the same kind of reasoning (seven ways to fulfill for each man -- all but
the case where
he touches no dog -- independently) 343 cases meet each of 4 and 5.
Again, the two sets
of cases overlap, significantly more than the 175 cases mathematically
required. Finally,
sentence 6 is verified in 511 cases, all but the last one in the standard
order.  The case
that satisfies 1 is included in both the 169 sets, the 169 set for 2 is
included in the 343 for
5 and similarly for 3 and 4.  Indeed, the overlaps of  2 and 3 are
included in the overlaps
of  4 and 5.  And so on.
pc>|83