FOIL axiomatized. (English) Zbl 1114.03014
Stud. Log. 84, No. 1, 1-22 (2006); correction ibid. 85, No. 2, 275 (2007).
Summary: In an earlier paper [“First-order intensional logic”, Ann. Pure Appl. Logic 127, No. 1–3, 171–193 (2004; Zbl 1061.03024)], I gave semantics and tableau rules for a simple first-order intensional logic, called FOIL, in which both objects and intensions are explicitly present and can be quantified over. Intensions, being non-rigid, are represented in FOIL as (partial) functions from states to objects. Scoping machinery, predicate abstraction, is present to disambiguate sentences like that asserting the necessary identity of the morning and the evening star, which is true in one sense and not true in another.
In this paper I address the problem of axiomatizing FOIL. I begin with an interesting sublogic with predicate abstraction and equality but no quantifiers. In [M. Fitting, “Modal logics between propositional and first-order”, J. Log. Comput. 12, No. 6, 1017–1026 (2002; Zbl 1017.03008)] this sublogic was shown to be undecidable if the underlying modal logic was at least K4, though it is decidable in other cases. The axiomatization given is shown to be complete for standard logics without a symmetry condition. The general situation is not known. After this an axiomatization for the full FOIL is given, which is straightforward after one makes a change in the point of view.
In this paper I address the problem of axiomatizing FOIL. I begin with an interesting sublogic with predicate abstraction and equality but no quantifiers. In [M. Fitting, “Modal logics between propositional and first-order”, J. Log. Comput. 12, No. 6, 1017–1026 (2002; Zbl 1017.03008)] this sublogic was shown to be undecidable if the underlying modal logic was at least K4, though it is decidable in other cases. The axiomatization given is shown to be complete for standard logics without a symmetry condition. The general situation is not known. After this an axiomatization for the full FOIL is given, which is straightforward after one makes a change in the point of view.
MSC:
| 03B45 | Modal logic (including the logic of norms) |
References:
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| [2] | Fitting, M. C., ’Modal logics between propositional and first-order’, Journal of Logic and Computation 12:1017–1026, 2002. · Zbl 1017.03008 · doi:10.1093/logcom/12.6.1017 |
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