Name
Abstract | ||
|---|---|---|
If is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely re- stricted when compared to a language where is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds se- mantics for predicates of sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate , we tackle both problems. Given a frame hW,Ri consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret at every world in such a way that p Aq holds at a world w 2 W if and only if A holds at every world v 2 W such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several 'paradoxes' (like Montague's Theorem, Godel's Second Incompleteness Theorem, McGee's Theorem on the !-inconsistency of certain truth theories etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sucient and necessary conditions for the existence of a suitable interpretation of at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic. We have presented parts of the paper in several talks; some of the results are pub- lished in (19) without proofs. We thank the audiences and the referee of the latter paper for numerous suggestions and hints. We are also indebted to Christopher von Bulow for his suggestions and for correcting many typographical errors. The work of Volker Halbach was sponsored by the Department of Logic and Philosophy at the University of California at Irvine and the research group Logic in Philosophy at the University of Constance. Philip Welch would like to acknowledge the support of the Institut fur Formale Logik, Vienna, where he held a Guest Professorship during the preparation of part of the paper. |
Year | DOI | Venue |
|---|---|---|
2003 | 10.1023/A:1023080715357 | J. Philosophical Logic |
Keywords | Field | DocType |
. possible worlds,modal logic,paradox,paradox.,necessity,possible worlds,binary relation,expressive power,standard model | Discrete mathematics,Unary operation,Computer science,Algorithm,Modal logic,Predicate (grammar),Functional predicate,Gödel's incompleteness theorems,Predicate (mathematical logic),Semantics,Transitive relation | Journal |
Volume | Issue | ISSN |
32 | 2 | 1573-0433 |
Citations | PageRank | References |
5 | 0.65 | 17 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Volker Halbach | 1 | 87 | 10.29 |
| Hannes Leitgeb | 2 | 115 | 19.26 |
| Philip Welch | 3 | 20 | 3.29 |