2004 | First-order Logic, 2nd-order, Completeness |
§1 | p.303 | 10751 | Second-order logic needs the sets, and its consequence has epistemological problems |
§2 | p.304 | 10752 | Γ|-S says S can be deduced from Γ; Γ|=S says a good model for Γ makes S true |
§2 | p.305 | 10753 | Logical consequence is intuitively semantic, and captured by model theory |
§2 | p.306 | 10754 | In proof-theory, logical form is shown by the logical constants |
§3 | p.306 | 10756 | A model is a domain, and an interpretation assigning objects, predicates, relations etc. |
§3 | p.306 | 10755 | A deductive system is only incomplete with respect to a formal semantics |
§3 | p.307 | 10757 | Henkin semantics has a second domain of predicates and relations (in upper case) |
§3 | p.307 | 10758 | If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model |
§3 | p.308 | 10759 | There are at least seven possible systems of semantics for second-order logic |
§4 | p.315 | 10760 | With possible worlds, S4 and S5 are sound and complete, but S1-S3 are not even sound |
§5 | p.317 | 10761 | Completeness can always be achieved by cunning model-design |