Abstract
Both I and Belnap, motivated the “Belnap-Dunn 4-valued Logic” by talk of the reasoner being simply “told true” (T), and simply “told false” (F), which leaves the options of being neither “told true” nor “told false” (N), and being both “told true” and “told false” (B). Belnap motivated these notions by consideration of unstructured databases that allow for negative information as well as positive information (even when they conflict). We now experience this on a daily basis with the Web. But the 4-valued logic is deductive in nature, and its matrix is discrete: there are just four values. In this paper I investigate embedding the 4-valued logic into a context of probability. Jøsang’s Subjective Logic introduced uncertainty to allow for degrees of belief, disbelief, and uncertainty. We extend this so as to allow for two kinds of uncertainty—that in which the reasoner has too little information (ignorance) and that in which the reasoner has too much information (conflicted). Jøsang’s “Opinion Triangle” becomes an “Opinion Tetrahedron” and the 4-values can be seen as its vertices. I make/prove various observations concerning the relation of non-classical “probability” to non-classical logic.
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This paper is dedicated to the memory of my dear friend Robert K. Meyer. Bob and I overlapped as Nuel’s students, and this formed the basis for our friendship of over forty-five years, for which I am most appreciative.
This is based on a talk given at the conference Logics of Consequence: A Celebration of Nuel Belnap’s Work in Philosophical Logic, April 3, 2009, University of Pittsburgh. Earlier versions were given at the Department of Logic and Philosophy of Science, Ghent University, December 9, 2008 and at the International Workshop on Truth Values at the Dresden University of Technology, May 31, 2008. This last presentation was “published” on the web: http://www.truthvalues2008.com/data/m_speakers.php?status=details&id=dunn.
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Dunn, J.M. Contradictory Information: Too Much of a Good Thing . J Philos Logic 39, 425–452 (2010). https://doi.org/10.1007/s10992-010-9134-6
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DOI: https://doi.org/10.1007/s10992-010-9134-6