From owner-cse191-sp08-list@LISTSERV.BUFFALO.EDU  Fri Feb  1 11:57:57 2008
Date:         Fri, 1 Feb 2008 10:25:11 -0500
From: "William J. Rapaport" <rapaport@cse.Buffalo.EDU>
Subject: 191: On the translation of "some dogs are pets"
To: CSE191-SP08-LIST@LISTSERV.BUFFALO.EDU

------------------------------------------------------------------------
Subject: 191: On the translation of "some dogs are pets"
------------------------------------------------------------------------

As I've indicated in lecture, and as the book states, a proposition of
the form:

	Some dogs are pets.

is represented in first-order predicate logic as:

	Ex[Dog(x) ^ Pet(x)]

i.e., as:  There is something that is both a dog and a pet.  It is NOT
represented as:  Ex[Dog(x) -> Pet(x)].

Why not?  Partly because the two FOPL propositions are not logically
equivalent.  One way to realize this is that the scopes are not
logically equivalent:

	(P ^ Q) is not equiv to (P -> Q)

They have different truth tables.  And binding them with an existential
quantifier doesn't change that status.

OK:  So if they're not equivalent, that tells us that only one of them
can be the correct representation, but why is it the conjunctive one and
not the conditional one?

Well, "some dogs are pets" is true
iff
there is at least one dog who is a pet.

And "Ex[Dog(x) ^ Pet(x)]" is true
iff
there is at least one thing that is a dog and is a pet.  

Therefore, these two propositions are equivalent, and the conjunctive
FOPL proposition is the correct translation of the English proposition.

Now "Ex[Dog(x) -> Pet(x)]" is also true IF there is a dog that is a
pet.  I.e., if there is a dog (call it "Fido") that is a pet, then
there is something (namely, Fido) that is such that if it is a dog
(which it is), then it's a pet (which it is).

But it is also true under a different circumstance:  It is also true about
my cat!  My cat, Bella, is such that, if she's a dog (which she isn't),
then she's a pet (which she is).  And that's true of her because
conditional propositions are true if their antecedent is false.

Now one of you in lecture asked about the related universally quantified
proposition:  Ax[Dog(x) -> Pet(x)].  Let me switch the example very
slightly to:  Ax[Dog(x) -> Animal(x)], so that we'll have a proposition
that everyone can agree is clearly true:  all dogs are animals.

Here's the puzzle:  This is also true about Bella the cat!  Why?
Because Bella is such that, if she's a dog, then she's an animal; and
that's true because conditionals with false antecedents are true.

But note that the reason that the existentially quantified **conditional**
is not the correct translation of "some dogs are pets" is not merely
that it's true about Bella the cat, but because "some dogs are pets" is
NOT true about Bella the cat (and that's because "some dogs are pets" is
true iff some dog is a pet; Bella the cat is irrelevant to whether some
dog is a pet).