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"

 

[CellCoree]

Hi Ryan,

First of all, it's great that you're studying Predicate Logic! It can definitely be a challenging topic, but it's also very important in many fields of science and philosophy.

To answer your question, the main difference between first and second order predicate logic is the ability to quantify over predicates themselves. In first order logic, we can only quantify over objects (i.e. "There exists an x such that..."). In second order logic, we can also quantify over properties or relations (i.e. "There exists a P such that...").

So, in response to your example, "There exists an American philosopher" could be expressed in both PL1 and PL2. However, let's take the example of "All dogs have four legs." In PL1, we could say "For all x, if x is a dog, then x has four legs." In PL2, we could say "For all P, if P is the property of being a dog, then there exists an x such that x has P and x has four legs." This allows us to explicitly refer to the property of being a dog, rather than just using the variable x to represent it.

Another example could be "Some philosophers believe in God." In PL1, we could say "There exists an x such that x is a philosopher and x believes in God." In PL2, we could say "There exists a P such that P is the property of being a philosopher and there exists an x such that x has P and x believes in God." Again, this allows us to refer to the property of being a philosopher, rather than just using the variable x to represent it.

I hope this helps to clarify the difference between PL1 and PL2. If you have any other questions, feel free to ask!

Best,