## Abstract

Parametric logic is a framework that generalises classical first-order logic. A generalised notion of logical consequence—a form of preferential entailment based on a closed world assumption—is defined as a function of some parameters. A concept of possible knowledge base—the counterpart to the consistent theories of first-order logic—is introduced. The notion of compactness is weakened. The degree of weakening is quantified by a nonnull ordinal—the larger the ordinal, the more significant the weakening. For every possible knowledge base *T*, a hierarchy of sentences that are generalised logical consequences of *T* is built. The first layer of the hierarchies corresponds to sentences that can be obtained by a deductive inference, characterised by the compactness property. The second layer of the hierarchies corresponds to sentences that can be obtained by an inductive inference, characterised by the property of weak compactness quantified by 1. Weaker forms of compactness—quantified by nonnull ordinals—determine higher layers in the hierarchies, corresponding to more complex inferences. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory and from topology. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory—classification in the limit, with or without a bounded number of mind changes—and from topology—in reference to the Borel and the difference hierarchies. In this paper, we introduce the key model-theoretic aspects of Parametric logic, justify the concept of the knowledge base, define the hierarchies of generalised logical consequences and illustrate their relevance to Nonmonotonic reasoning. More specifically, we show that the degree of nonmonotonicity that is required to infer a sentence can be characterised by the least nonnull ordinal that quantifies the weakening of compactness used to locate the inferred sentence in the hierarchies.

## Keywords

Generalised logical consequence Hierarchies Closed world assumption Preferential entailment Nonmonotonicity## Preview

Unable to display preview. Download preview PDF.

## Notes

### Acknowledgements

The author would like to thank the anonymous reviewers for extremely detailed and most valuable comments that greatly helped improve the paper, and Kevin Kelly for having organised the stimulating workshop on the Logic of simplicity that led to this special issue of Studia Logica.

## References

- 1.Ambainis, A., R. Freivalds, and C. Smith, Inductive inference with procrastination: back to definitions,
*Fundamenta Informaticae*40:1–16, 1999.Google Scholar - 2.Ambainis, A., S. Jain, and A. Sharma, Ordinal mind change complexity of language identification,
*Theoretical Computer Science*220(2):323–343, 1999.CrossRefGoogle Scholar - 3.Bochman, A.,
*A Logical Theory of Nonmonotonic Reasoning and Belief Change*, Springer, 2001.Google Scholar - 4.Brewka, G., J. Dix, and K. Konolige,
*Nonmonotonic Reasoning: An overview*, Stanford University, CSLI lecture Notes 73, 1997.Google Scholar - 5.Chopra, S., and E. Martin, Generalized logical consequence: making room for induction in the logic of science,
*Journal of Philosophical Logic*31:245–280, 2002.CrossRefGoogle Scholar - 6.Dix, J., U. Furbach, and I. Niemelä,
*Nonmonotonic reasoning: towards efficient calculi and implementation*, in A. Voronkov, and A. Robinson (eds.),*Handbook of Automated Reasoning*, Elsevier, vol. 2, chap. 18, 2001, pp. 1121–1234.Google Scholar - 7.
- 8.Gold, E., Language identification in the limit,
*Information and Control*10:447–474, 1967.CrossRefGoogle Scholar - 9.Jain, S., D. Osherson, J. Royer, and A. Sharma,
*Systems That Learn: An Introduction to Learning theory, Second Edition*, The MIT Press, 1999.Google Scholar - 10.Kraus, S., D. Lehmann, and M. Magidor, Nonmonotonic reasoning, preferential models and cumulative logics,
*Artificial Intelligence*44:167–207, 1990.CrossRefGoogle Scholar - 11.Martin, E., Contextual hypotheses and semantics of logic programs,
*Theory and Practice of Logic Programming*12(6):843–887, 2012.CrossRefGoogle Scholar - 12.Martin, E., Logic programming as classical inference,
*Journal of Applied Logic*13:316–369, 2015.CrossRefGoogle Scholar - 13.Martin, E., P. Nguyen, A. Sharma, and F. Stephan,
*Learning in Logic with RichProlog*, Proceedings of the Eighteenth International Conference on Logic programming, Springer, LNCS 2401, 2002, pp. 239–254.Google Scholar - 14.Martin, E., A. Sharma, and F. Stephan, Unifying logic, topology and learning in parametric logic,
*Theoretical Computer Science*350(1):102–124, 2006.CrossRefGoogle Scholar - 15.Mendelson, E.,
*Introduction to Mathematical logic, 3rd edition*, Wadsworth and Brooks/Cole, 1987.Google Scholar - 16.Nienhuys-Cheng, S., and R. de Wolf,
*Foundations of Inductive Logic Programming*, Springer, LNAI 1228, 1997.Google Scholar - 17.Nute, D., Defeasible logic, in
*Handbook of Logics in Artificial Intelligence and Logic Programming*, vol. 3, Oxford University Press, 1994, pp. 353–395.Google Scholar - 18.
- 19.Reiter, R.,
*On closed-world data bases*, in J. Gallaire, and J. Minker, (eds.),*Logic and Data Bases*, Plenum Press, 1978, pp. 55–76.Google Scholar - 20.Reiter, R., A logic for default reasoning,
*Artificial Intelligence*13:81–132, 1980.CrossRefGoogle Scholar - 21.
- 22.Tarski, A., On the concept of logical consequence, in J. Corcoran, (ed.),
*Logic, Semantics, Metamathematics*, Second edition, Translations by J. Woodger, Hacket Publishing Company, 1983, pp. 409–420.Google Scholar