Abstract
This paper is an attempt to bring together two separated areas of research: classical mathematics and metamathematics on the one side, non-monotonic reasoning on the other. This is done by simulating nonmonotonic “logic” through antitonic theory extensions. In the first half, the specific extension procedure proposed here is motivated informally, partly in comparison with some well-known non-monotonic formalisms. Operators V and, more generally, Uδ are obtained which have some plausibility when viewed as giving nonmonotonic theory extensions. In the second half, these operators are treated from a mathematical and metamathematical point of view. Here an important role is played by Uδ -closed theories and Uδ -fixed points. The last section contains results on V-closed theories which are specific for V.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Alchourrón, C. E., P. Gärdenfors, and D. Makinson: 1985, ‘On the Logic of Theory Change: Partial Meet Contraction and Revision Functions’, The Journal of Symbolic Logic 50, 510-530.
Clark, K. L.: 1978, ‘Negation as Failure’, in Gallaire and Minker (eds.), Logic and Databases, Plenum, New York, pp. 293-322.
Feferman, S.: 1962, ‘Transfinite Recursive Progressions of Axiomatic Theories’, The Journal of Symbolic Logic 27, 259-316.
Hájek, P. and P. Pudlak: 1993, Metamathematics of First-Order Arithmetic, Springer, Berlin.
Halpern, J.: 1993, ‘A Critical Reexamination of Default Logic, Autoepistemic Logic and Only Knowing’, in Gottlob, Leitsch, and Mundici (eds.), Computational Logic and Proof Theory, Lecture Notes in Computer Science 713, Springer, Berlin.
Kent, C. F.: 1973, ‘The Relation of A to Prov⌌A⌍ in the Lindenbaum Sentence Algebra’, The Journal of Symbolic Logic 38, 295-298.
Marek, W. and M. Truszczynski: 1993, Nonmonotonic Logics; Context-Dependent Reasoning, Springer, Berlin.
McCarthy, T.: 1994, ‘Self-Reference and Incompleteness in a Non-monotonic Setting’, Journal of Philosophical Logic 23, 423-449.
McDermott, D. and J. Doyle: 1980, ‘Non-monotonic Logic I’, Artificial Intelligence 13, 41-72.
Moore, R. C.: 1985, ‘Semantical Considerations on Nonmonotonic Logic’, Artificial Intelligence 25, 75-94.
Niebergall, K.-G.: 1996, Zur Metamathematik nichtaxiomatisierbarer Theorien, dissertation, CIS, München.
Reiter, R.: 1980, ‘A Logic for Default Reasoning’, Artificial Intelligence 13, 81-132.
Rescher, N.: 1973, The Coherence Theory of Truth, Clarendon Press, Oxford.
Shapiro, S. (ed.): 1985, Intensional Mathematics, North-Holland, Amsterdam.
Smorynski, C.: 1985, Self-Reference and Modal Logic, Springer, Berlin.
Tarski, A.: 1956, Logic, Semantics, Metamathematics, Clarendon Press, Oxford.
Ursini, A.: 1978, ‘On the Set of ‘Meaningful’ Sentences of Arithmetic’, Studia Logica 37, 237-241.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Niebergall, KG. Nonmonotonicity in (the Metamathematics of) Arithmetic. Erkenntnis 50, 309–332 (1999). https://doi.org/10.1023/A:1005505417416
Issue Date:
DOI: https://doi.org/10.1023/A:1005505417416