Abstract
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 Jøsang’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). Jøsang 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.
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Notes
- 1.
Graham and Richard Routley (later Sylvan) invented the word “dialethism” (now spelled by others “dialetheism,” closer to the Greek root for “dual truth”), expressing a stronger concept than “paraconsistency, a word introduced by Francisco Miró Quesada in 1976. See Priest et al. (1989, p. xx).
- 2.
Dunn (1976), p. 157: We are not claiming that there are sentences that are in fact both true and false. We are merely pointing out that there are situations where people suppose, assert, believe, etc. contradictory sentences to be true, and we therefore need a semantics that expresses the truth conditions of contradictions in terms of the truth values that the ingredient sentences would have to take in order for the contradictions to be true.
- 3.
Popper (1934, p. 50, 1959, p. 72) says: “The requirement of consistency plays a special rôle among the various requirements which a theoretical system, or an axiomatic system, must satisfy. ... In order to show the fundamental importance of this requirement it is not enough to mention the obvious fact that a self-contradictory system must be rejected because it is ‘false’. We frequently work with statements which, although actually false, nevertheless yield results which are adequate for certain purposes. ...(An example is Nernst’s approximation for the equilibrium equation of gases.) But the importance of the requirement of consistency will be appreciated if one realizes that a self-contradictory system is uninformative. It is so because any conclusion we please can be derived from it.”
- 4.
Note that Popper (1934, 1959) quoted in the previous footnote can be viewed as agreeing with Floridi that contradictions contain no information.
- 5.
JMD has given talks on “The Two Firefighters” first at the Logica conference (2012), and then at the University of Munich Center for Mathematical Philosophy, the Modal Logic Workshop on Consistency and Structure at the Center for Formal Epistemology of Carnegie Mellon University, the School of Cognitive Science at Jadavpur University in Kolkata (India), the Logic Group at the University of Alberta, and the Info-Metrics Institute at The American University. It was at this last that JMD and NMK decided to write this joint paper. We want to thank Aparajita Karmakar, David Makinson, John Winnie, and an anonymous referee for their insightful comments. But we do not mean to say that we have taken everything they said into account.
- 6.
A short list of “solutions” to the Russell Paradox includes beside Zermelo–Fraenkel set theory, NBG, MK, ML, etc., and theory of types (Russell’s own fix).
- 7.
We set things up so that everything starts off symmetrically (distributed) between the three choices. We have tried to throw a “veil of ignorance” over the situation so that you can bring no special knowledge to weigh the a priori odds of one choice over the other two. You can of course imagine that one of the firefighters is older and obviously more experienced than the other, and then we will add to the setup that they are twins, that they are wearing identical uniforms, with the same badges, can hand you the same certificates of training and of service, etc.
- 8.
At this point, we might eliminate O as an atomic proposition and define it as \( O=(R \& \lnot S \& \lnot L)\vee (\lnot R \& S \& \lnot L)\vee (\lnot R \& \lnot S \& L).\) Negation \(\lnot ,\)conjunction&, and disjunction \(\vee \) are defined in Dunn (1976) (generalizing their definitions by Jøsang 1997) to allow for conflict as well as ignorance). The formula for calculating the value of a conjunction is somewhat complicated (and disjunction inherits this since it is defined from conjunction and negation in the usual way using De Morgan’s Law). So it is probably wise for expository purposes not to use this definition of O here. We do point out that both conjunction and disjunction are associative and commutative so we can overlook grouping and order when forming them.
- 9.
It might make sense here to agree that \(p_{1}=0\), it is not the case that no escape is possible. In this case, you would assign probability 1/7 to each of the remaining possibilities and the resulting aggregate probabilities are 4/7. This is inessential to the argument.
- 10.
Alternatives to linear pooling include geometric and multiplicative approaches. These have the property that the support of the combined distribution is the intersection of the supports of the components. In this case, no escape. With linear pooling, the support is the union of the supports. NMK has found in risk management applications that differences in support for default rates for similar portfolios are common across experts. For the subjective approach to default prediction, see Kiefer (2007, 2011).
- 11.
Of course, the list of probabilities has to satisfy some properties if they are “coherent.” Usually, we consider probabilities of zero or one, certainties, as likely to be unrealistic, or perhaps approximations for very small or large probabilities. In fact, these extremes arise all the time but are hidden. They arise in the (subjective) specification of the relevant event space, the list of what can happen. But in our application, these are the probabilities of interest, and the ordinary rules should apply.
- 12.
Lohr (2013) argues that the term “Big Data” should be credited to John Mashey, the Chief Scientist at Silicon Graphics in the 1990s, when he gave a number of unpublished talks on the subject in the late 1990s—see, e.g., http://static.usenix.org/event/usenix99/invited_talks/mashey.pdf.
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Dunn, J.M., Kiefer, N.M. (2019). Contradictory Information: Better Than Nothing? The Paradox of the Two Firefighters. In: Başkent, C., Ferguson, T. (eds) Graham Priest on Dialetheism and Paraconsistency. Outstanding Contributions to Logic, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-25365-3_12
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