Abstract
Logics for “generally” were introduced for handling assertions with vague notions,such as “generally”, “most”, “several”, etc., by generalized quantifiers, ultrafilter logic being an interesting case. Here, we show that ultrafilter logic can be faithfully embedded into a first-order theory of certain functions, called coherent. We also use generic functions (akin to Skolem functions) to enable elimination of the generalized quantifier. These devices permit using methods for classical first-order logic to reason about consequence in ultrafilter logic.
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References
Barwise, J. and S. Feferman, Model-Theoretic Logics, Springer-Verlag, New York, 1985.
Chang, C. L. and R. C. T. Lee, Symbolic Logic and Mechanical Theorem Proving, Academic Press, New York, 1973.
Carnielli, W.A. and A.M. Sette, “Default operators”, Abstr. Workshop on Logic, Language, Information and Computation, Recife, 1994.
Carnielli, W. A. and P. A. S., Veloso, “Ultrafilter Logic and Generic Reasoning”, G. Gottlob, A. Leitsch, and D. Mundici, (eds.) Computational Logic and Proof Theory (LNCS 1289): 34–53, Springer-Verlag, Berlin, 1997.
Chang, C. C. and H. J. Keisler, Model Theory North-Holland, Amsterdam, 1973.
Enderton, H. B., A Mathematical Introduction to Logic Academic Press; New York, 1972.
Fuhrmann, A., “Some Remarks on Ultrafilter Logic”, Studia Logica 73: 197–207, 2003 Special issue on Belief Revision,dedicated to S. O. Hansson’s 50th birthday; Olsson, E. J. (ed.).
Grácio, M. C. G., Lgicas Moduladas e Raciocínio sob Incerteza. D. Sc. dissertation, Unicamp, Campinas (1999).
Reiter, R., “A Logic for Default Reasoning”, Artif. Intellig. 13: 81–123, 1980.
Rentería, C. J., E. H. Haeusler, and P. A. S. Veloso, “NUL-Natural Deduction for Ultrafilter Logic”. Bulletin of the Section of Logic 32(4):191–199, 2003 abstract in Natural Deduction 2001, Rio de Janeiro, 2001.
Sette, A. M., W. A. Carnielli, and P. A. S. Veloso, “An Alternative View of Default Reasoning and its Logic”, in Haeusler, E. H. and Pereira, L. C. (eds.) Pratica: Proofs, Types and Categories: 127–158, PUC-Rio, Rio de Janeiro, 1999.
Shoenfield, J. R., Mathematical Logic, Addison-Wesley, Reading, 1967.
Turner, W., Logics for Artificial Intelligence, Ellis Horwood, Chichester, 1984.
Veloso, P. A. S., “On “almost all” and some Presuppositions”, Manuscrito XXII(2): 469–505, 1999 Special issue Logic, Language and Knowledge: essays in honour of Oswaldo Chateaubriand Filho; Pereira, L. C. P. D. and Wrigley, M. B.(eds.).
Veloso, P. A. S., “On Interpolation and Modularity for Ultrafilter Logic”, in In Abe, J. M. and Silva Filho, J. I. (eds.) Logic, Artificial Intelligence and Robotics - Proc. LAPTEC 2001: 270–278, IOS Press, Amsterdam, 2001.
Veloso, P. A. S., “On a Logic for “almost all” and “generic” Reasoning”, Manuscrito XXV(1): 191–271, 2002.
Veloso, P. A. S. and W. A. Carnielli, “Logics for Qualitative Reasoning”, CLE e-Prints 1, 2001 http://www.cle.unicamp.br/e-prints.
Veloso, S. R. M. and P. A. S. Veloso, ‘On Special Functions and Theorem Proving in Logics for ‘Generally’’, in Bittencourt, G. and Ramalho, G. L. (eds.) Advances in Artificial Intelligence: 16th Brazilian Symposium in Artificial Intelligence (LNAI 2507): 1–10, Springer-Verlag, Berlin, 2002.
Zadeh L. A., “Fuzzy Logic and Approximate Reasoning”, Synthése 30: 407–428, 1975.
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Veloso, P.A.S., Veloso, S.R.M. On Ultrafilter Logic and Special Functions. Stud Logica 78, 459–477 (2004). https://doi.org/10.1007/s11225-004-6045-y
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DOI: https://doi.org/10.1007/s11225-004-6045-y