In this case, that’s not true. The many-sorted logic, with axioms and all to emulate 2nd-order logic, has different properties than plain 1st-order logic (even though we may be emulating it in a plain 1st-order engine).
For example, in 2nd-order logic, we can quantify over any names we use. In 1st-order logic, this is not true: we can quantify over 1st-order entities, but we are unable to quantify over 2nd-order entities. So, we can have named entities (predicates and relations) which we are unable to quantify over.
One consequence of this is that, in 1st-order logic, we can never prove something non-tautological about a predicate or relation without first making some assumptions about that specific predicate or relation. Statements which share no predicates or relations are logically independent!
This limits the influence of concepts on one another, to some extent.
This sort of thing does not hold for 2nd-order logic, so its translation doesn’t hold for the translation of 2nd-order logic into 1st-order logic, either. (Basically, all statements in this many-sorted logic will use the membership predicate, which causes us to lose the guarantee of logical independence for other named items.)
So we have to be careful: an encoding of one thing into another thing doesn’t give us everything.
In this case, that’s not true. The many-sorted logic, with axioms and all to emulate 2nd-order logic, has different properties than plain 1st-order logic (even though we may be emulating it in a plain 1st-order engine).
For example, in 2nd-order logic, we can quantify over any names we use. In 1st-order logic, this is not true: we can quantify over 1st-order entities, but we are unable to quantify over 2nd-order entities. So, we can have named entities (predicates and relations) which we are unable to quantify over.
One consequence of this is that, in 1st-order logic, we can never prove something non-tautological about a predicate or relation without first making some assumptions about that specific predicate or relation. Statements which share no predicates or relations are logically independent!
This limits the influence of concepts on one another, to some extent.
This sort of thing does not hold for 2nd-order logic, so its translation doesn’t hold for the translation of 2nd-order logic into 1st-order logic, either. (Basically, all statements in this many-sorted logic will use the membership predicate, which causes us to lose the guarantee of logical independence for other named items.)
So we have to be careful: an encoding of one thing into another thing doesn’t give us everything.