Second Order Predicate Logic vs. First Order

[ryan14]

Hey,

I'm studying Predicate Logic at the moment and I can't seem to wrap my head around the way that english sentences would convert into second order logic. What kind of sentence can be faithfully represented in PL2 but not in PL1? Sorry if this isn't the appropriate section; I'm actually in a Philosophy (of Math) class, so the sciences aren't really my strong suit.

Would "There exists an American philosopher" be one? Wiki mentions "There are no Albanian philosophers" but I don't see why this couldn't be translated into PL1 if you just made a predicate "is an Albanian philosopher." Or is that beyond the point?

Pretty much I'd like to see an example of what a PL2 sentence that couldn't be expressed in PL1 would look like.

Thanks,
Ryan

 

[JSuarez]

Try these ones:

"There are only a finite number of grains of sand"

This cannot be correctly formalized in a first order language, because you can't define the predicate "finite".

Or the Geach-Kaplan sentence:

"Some critics admire only each other"

You may see the explanation why this last sentence doesn't have a first-order formalization here:

http://books.google.pt/books?id=sdL...BA#v=onepage&q=geach kaplan sentence&f=false"

 

[ryan14]

Thanks for the fast reply and the link. Reading it now.

"There are only a finite number of grains of sand."
^ Can you explain why this cannot be translated into PL1?

Thanks again.

 

[JSuarez]

Because if you try to define "finite" in FOL, you'll end up with a predicate that is true in models that don't have more than finite, but prefixed, number of elements, or one that it's true also in infinite domains. This is a consequence of the compactness theorem for FOL, see here (cor. 22):

http://plato.stanford.edu/entries/logic-classical/#5"