Abstract
Both the Beth definability theorem and Craig’s lemma (interpolation theorem from now on) deal with the issue of the entanglement of one language L 1 with another language L 2, that is to say, information transfer—or the lack of such transfer—between the two languages. The notion of splitting we study below looks into this issue. We briefly relate our own results in this area as well as the results of other researchers like Kourousias and Makinson, and Peppas, Chopra and Foo. Section 3 does contain one apparently new theorem.
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References
Alchourron C., Gärdenfors P., Makinson D. (1985) On the logic of theory change: Partial meet contraction and revision functions. The Journal of Symbolic Logic 50: 510–530
Beth E. W. (1953) On Padoa’s method in the theory of definition. Indagationes Mathematicae 15: 330–339
Cherniak C. (1986) Minimal rationality. MIT Press, Cambridge
Chopra S., Parikh R. (2000) Relevance sensitive belief structures. Annals of Mathematics and Artificial Intelligence 28: 259–285
Craig W. (1956) Review of E. W. Beth, On Padoa’s method in the theory of definition. The Journal of Symbolic Logic 21(2): 194–195
Craig W. (1957) Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory. The Journal of Symbolic Logic 22(3): 269–285
Craig W. (2008) The road to two theorems of logic. Synthese 164: 333–339
Grove A. (1988) Two modellings for theory change. Journal of Philosophical Logic 17: 157–170
Heyting A. (1966) In memoriam, Evert Willem Beth. Notre Dame Journal of Formal Logic 4: 289–295
Hodges W. (1993) Model theory. Cambridge University Press, Cambridge
Kourousias G., Makinson D. (2007) Parallel interpolation, splitting, and relevance in belief change. Journal of Symbolic Logic 72(3): 994–1002
Mark R. (1996) Belief revision and ordered theory presentations. In: Fuhrmann A., Rott H. (eds) Logic, action, information. De Gruyter, Berlin, pp 129–151
Makinson D. (2008) Sets, logic and maths for computing (undergraduate topics in computer science). Springer, Belin
Parikh, R. (1999). Beliefs, belief revision, and splitting languages. In L. Moss, J. Ginzburg, & M. de Rijke (Eds.), Proceedings of logic, language and computation, CSLI 1999 (pp. 266–278).
Peppas, P., Chopra, S., & Foo, N. (2004). Distance semantics for relevance-sensitive belief revision. In Proceedings of the international conference on knowledge representation and reasoning, KR2004.
Robinson A. (1956) A result on consistency and its application to the theory of definition. Indagationes Mathematicae 18: 47–58
Simon H. (1947) Administrative behaviour. Macmillan, New York
Tennant N. (2006) On the degeneracy of the full AGM-theory of theory-revision. Journal of Symbolic Logic 71(2): 661–676
van Benthem J. (2008) The many faces of interpolation. Synthese 164: 451–460
van Orman Quine W., Ullian J. (1978) The web of belief (2nd Ed.). Random house, New York
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Parikh, R. Beth definability, interpolation and language splitting. Synthese 179, 211–221 (2011). https://doi.org/10.1007/s11229-010-9778-3
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DOI: https://doi.org/10.1007/s11229-010-9778-3