[EMAIL PROTECTED] wrote: > Consider the following line of reasoning. Let p be the proposition > "Ronald was born in New York." From p, we can infer q: Ronald was > born in the United States. From q, we can infer r: It is possible > that Ronald was born in New Jersey. On the other hand, from p we can > infer s: It is not possible that Ronald was born in New Jersey. We > have arrived at a contradiction. What is wrong? Note: To answer the > question, familiarity with modal logic is not needed.
Dear Lofti, I guess it might be possible to find a solution in some modal-logic formalism. An alternative solution might be inspired on the belief revision and default reasoning research in the AI field. For instance, if you know q (Ronald was born in the US), you conclude r (it is possible that he was born in NJ), but this conclusion should be revised and might be retracted when new evidence arrives. In particular, after knowing p (he was born in NY) you should retract r. In any case, **the conclusions you draw must take into account all the available knowledge**. Therefore, if you only know q, then r holds, but if you know p and q then r does not hold. A typical example from the default-reasoning literature is that if you know that Tweety is a bird (q), you can assume that Tweety can fly (r). But if you also know that Tweety is a penguin (p), then you cannot deduce that he can fly (r). I think your paradox is formally identical to Tweety's classical example. In fact there is a subtle difference. In default reasoning the elementary proposition would be "Ronald was born in New Jersey", but in your formulation of the paradox, you defined r as "It is possible that Ronald was born in New Jersey", which should be understood as "If the only think we know is that Ronald was born in the US, it would be possible that he was born in NJ". In summary, the solution to you paradox might be that by "possible" you mean "compatible (i.e. not contradictory) with the available evidence". Therefore, "possible" would be context-sensitive, i.e., depending on the knowledge we have. For this reason, when you infer s you are using p as a part of the available evidence, and in this case you can not infer q. Does this solve your paradox? Best regards, Javier Díez ----------------------------------------------------------------- Francisco Javier Diez Phone: (+34) 91.398.71.61 Dpto. Inteligencia Artificial Fax: (+34) 91.398.88.95 UNED. c/Juan del Rosal, 16 E-mail: [EMAIL PROTECTED] 28040 Madrid. Spain http://www.ia.uned.es/~fjdiez _______________________________________________ uai mailing list uai@ENGR.ORST.EDU https://secure.engr.oregonstate.edu/mailman/listinfo/uai
