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:

∃x[Dog(x) ∧ Pet(x)]

i.e., as: There is something that is both a dog and a pet. It is not represented as: ∃x[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 equivalent 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

And "∃x[Dog(x) ∧ Pet(x)]" is true

iff

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

Now "∃x[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.

What about the related universally quantified proposition: ∀x[Dog(x) → Pet(x)]? Let me switch the example very slightly to: ∀x[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).

For a different, more algorithmic (and perhaps clearer?) explanation, see CSE 191, Fall 2010, Lecture Notes for 22 Sep 2010