Timeline for Propositional Logic, First-Order Logic, and Higher-Order Logics
Current License: CC BY-SA 2.5
3 events
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| Feb 21, 2010 at 20:45 | comment | added | Haim | Although it's possible to reduce many problems in stronger logics (logics with generalized quantifiers, infinitary logics, etc) to the case of FOL, the reduction itself is far from being trivial. For example, it takes a lot of effort to prove that FOL logic+cofinality quantifiers has the compactness/completeness property, and the existence of many other basic model-theoretic properties for these logics (such as the Beth property and other interpolation properties) is yet unknown. ps. Shelah wrote a nice paper about this subject titled "Compact logics in ZFC". | |
| Feb 21, 2010 at 17:37 | comment | added | François G. Dorais | This is correct and this is one of the many serious difficulties with standard higher-order semantics. However, Henkin semantics reduce higher-order logic to first-order logic, which therefore has all of these desirable properties. | |
| Feb 21, 2010 at 16:32 | history | answered | Haim | CC BY-SA 2.5 |