Perhaps, familiarity with modal logic is not needed.to answer the question. However, it is very helpful to use modal logic to clarify the confusion. Let us now formulate the reasoning in epistemic logic (a branch of modal logic) and see what happens.
Let s denote any statement, then in epistemic logic, the notation [i]s means that an agent i knows (or believes) s and <i>s = ~[i]~s means that s is epistemically possible for i. To contrast with your original arguement, let us define three propositions: p' : Ronald was born in New York q' : Ronald was born in the United States r' : Ronald was born in New Jersey. There may be two interpretations for the original arguement. First, you may mean p=p', q=q', and r=<i>r', so the three premises are 1. p --> q 2. q --> <i>r' 3. p --> [i]~r' In this interpretation, only the first premise holds. The second premise said that if Ronald was born in the United States, then agent i does not know Ronald was not born in New Jersey, which does not hold, since the antecedent does not specify any epistemic state of i. Similarly, the third premise does not hold, since p refers to an objective fact whereas [i]~r' refers to the subjective belief of i. The epistemic state of an agent may be independent of the objective fact. The second interpretation, maybe more close to your intention, is p=[i]p', q=[i]q', and r=<i>r', then the three premises are 1. [i]p' -->[i] q' 2. [i]q' --> <i>r' 3. [i]p' --> [i]~r' In such interpretation, the first premises holds if agent i has background knowledge about the basic geography of United States, i.e., [i](p' --> q') holds, then the first premises follows from the basic scheme K of modal logic. The third premises also holds for the same reason. However, the second premises does not hold since the antecedent only said that agent i knows Ronald was born in the United States, but it does not exclude the possiblity that agent i also knows Ronald was not born in New Jersey. In either interpretation, not all premises of your argument hold, so no contradiction can be derived based on the modal logic formulation of the problem. Hope that my humble opinion is helpful for clarifying the confusion. Best Wish. Churn-Jung Liau Institute of Information Science Academia Sinica, Taipei 115, Taiwan http://www.iis.sinica.edu.tw/pages/liaucj/eindex.html > -----Original Message----- > From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On > Behalf Of [EMAIL PROTECTED] > Sent: Tuesday, July 12, 2005 3:48 PM > To: uai@ENGR.ORST.EDU > Subject: [UAI] Is it a paradox? > > 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. > > -- > Lotfi A. Zadeh > Professor in the Graduate School, Computer Science Division > Department of Electrical Engineering and Computer Sciences > University of California > Berkeley, CA 94720 -1776 > Director, Berkeley Initiative in Soft Computing (BISC) > Tel.(office): (510) 642-4959 > _______________________________________________ > uai mailing list > uai@ENGR.ORST.EDU > https://secure.engr.oregonstate.edu/mailman/listinfo/uai > > _______________________________________________ > uai mailing list > uai@ENGR.ORST.EDU > https://secure.engr.oregonstate.edu/mailman/listinfo/uai _______________________________________________ uai mailing list uai@ENGR.ORST.EDU https://secure.engr.oregonstate.edu/mailman/listinfo/uai
