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Full Idea
Second-order logic raises doubts because of its ontological commitment to the set-theoretic hierarchy, and the allegedly problematic epistemic status of the second-order consequence relation.
Gist of Idea
Second-order logic needs the sets, and its consequence has epistemological problems
Source
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §1)
A Reaction
The 'epistemic' problem is whether you can know the truths, given that the logic is incomplete, and so they cannot all be proved. Rossberg defends second-order logic against the second problem. A third problem is that it may be mathematics.
| 10751 | Second-order logic needs the sets, and its consequence has epistemological problems [Rossberg] |
| 10753 | Logical consequence is intuitively semantic, and captured by model theory [Rossberg] |
| 10752 | Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true [Rossberg] |
| 10754 | In proof-theory, logical form is shown by the logical constants [Rossberg] |
| 10758 | If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg] |
| 10756 | A model is a domain, and an interpretation assigning objects, predicates, relations etc. [Rossberg] |
| 10759 | There are at least seven possible systems of semantics for second-order logic [Rossberg] |
| 10757 | Henkin semantics has a second domain of predicates and relations (in upper case) [Rossberg] |
| 10755 | A deductive system is only incomplete with respect to a formal semantics [Rossberg] |
| 10761 | Completeness can always be achieved by cunning model-design [Rossberg] |