4
votes
2answers
223 views

Do isomorphic structures always satisfy the same second-order sentences?

I know that if two mathematical structures are isomorphic, then they satisfy the same first-order sentences. The converse is false. This is probably a completely obvious question, but is it true that ...
5
votes
2answers
132 views

Henkin vs. “Full” Semantics for Second-order Logic and Multi-Sorted First Order Interpretations

In this paper by Jeff Ketland, he notes: With Henkin semantics, the Completeness, Compactness and Löwenheim-Skolem Theorems all hold, because Henkin structures can be re-interpreted as many-sorted ...
5
votes
3answers
253 views

How are the full semantics of SOL and HOL specified?

In relation to this question about the "fundamental" character of possible logical systems, I realized that I just had an intuitive (and so, inadequate) understanding of the way logics higher than FOL ...
4
votes
1answer
410 views

advantage of first-order logic over second-order logic

As I look over the post that has the similar question, I began to wonder: The only reason I found is that first-order logic can prove validity of some second-order logic formula/sentences, as some of ...
2
votes
1answer
222 views

Independence results in first-order PA and second-order PA

There are statements $\varphi$ that are independent of first-order Peano Axioms. Are these statements also independent of second-order Peano Axioms? I'm reading Wikipedia articles around ...