On Feb 20, 2009, at 11:48 AM, Ed Barkmeyer wrote:
>> Laws of Quantifier movements:
>> 1. '(all x.P(x)) --> Q' equivalent-to 'exists x.(P(x)-->Q)',
>> provided x is
>> not free in Q
>
> I don't believe this axiom.
> I read it to say, for example,
> "if all the players are present, the match can begin"
> is equivalent to
> "there is at least one player whose presence (alone) will allow the
> match to begin" (01)
>
> The implication is correct only in one direction:
> exists x.(P(x) --> Q) implies (all x.P(x)) --> Q
> If there is a player whose presence allows the match to begin,
> then if all the players are present, the match can (surely) begin. (02)
I think the problem is that you are thinking of the conditional in
your example as signifying something causal (hence intensional)
instead of thinking of it as a bare, truth functional material
conditional. The equivalence in question is correct in the other
direction as well, at least if we reason classically (as you appear to
be doing here). (03)
To see this, suppose it is true that, if all the players are present,
the match can begin. Either the match can, in fact, begin, or it
can't. Suppose first that it can. Then it follows trivially (since a
material conditional is true if its consequent is) that if, say, the
captain of the team is present, the match can begin. Generalizing, it
follows that there is a player x such that, if x is present, the match
can begin. (04)
So suppose the match cannot in fact begin. Then, since, by
assumption, the conditional "if all the players are present, the match
can begin" is true, it must be that the antecedent is false, i.e., not
all of the players are present. Hence, some player is not present.
So for some player x, x is not present. Hence, trivially once again
(since a material conditional is true if its antecedent is false), for
some player x, if x is present, the match can begin. (05)
-chris (06)
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