Contradictory Information: Better Than Nothing? The Paradox of the Two Firefighters

Prominent philosophers (Bar-Hillel and Carnap, Popper, Floridi) have argued that contradictions contain either too much or too little information to be useful. We dispute this with what we call the “Paradox of the Two Firefighters.” Suppose you are awakened in your hotel room by a fire alarm. You open the door. You see three possible ways out: left, right, straight ahead. You see two firefighters. One says there is exactly one safe route and it is to your left. The other says there is exactly one safe route and it is to your right. While the two firemen are giving you contradictory information, they are also both giving you the perhaps useful information that there is a safe way out and it is not straight ahead. We give two analyses. The first uses the “Opinion Tetrahedron,” introduced by Dunn as a generalization of Audun Josang’s “Opinion Triangle.” The Opinion Tetrahedron in effect embeds the values of the “Belnap-Dunn 4-valued Logic” (Truth, Falsity, Neither, Both) into a context of subjective probability generalized to allow for degrees of belief, disbelief, and two kinds of uncertainty—that in which the reasoner has too little information (ignorance) and that in which the reasoner has too much information (conflict). Josang had only a single value for uncertainty. We also present an alternative solution, again based on subjective probability but of a more standard type. This solution builds upon “linear opinion pooling.” Kiefer had already developed apparatus for assessing risk using expert opinion, and this influences the second solution. Finally, we discuss how these solutions might apply to “Big Data” and the World Wide Web.