Contradictory Information Can Be Better than Nothing The Example of the Two Firemen

Transcription

1 Contradictory Information Can Be Better than Nothing The Example of the Two Firemen J. Michael Dunn School of Informatics and Computing, and Department of Philosophy Indiana University-Bloomington Workshop on the Philosophy of Information: The Value of Information Info-Metrics Institute, American University April 26, 2013

2 This is a kind of dual sequel to my paper Contradictory Information: Too Much of a Good Thing, J. of Philosophical Logic (2010), pp I presented this at previous workshop on philosophy of information. Today s Example of the Two Firemen can be seen as motivating the formal apparatus of the earlier paper. I will argue that contradictory information can sometimes be Better Than Nothing. Rephrasing to match the theme of this workshop: Contradictory information can sometimes have value.

3 Bar-Hillel and Carnap (1953, p. 229)) wrote: "It might perhaps, at fi rst, seem strange that a self-contradictory sentence, hence one which no ideal receiver would accept, is regarded as carrying with it the most inclusive information.... A self-contradictory sentence asserts too much; it is too informative to be true."

4 Floridi (2011) wants a strong semantic theory of information" that avoids what he labels the Bar-Hillel Carnap Paradox (BCP). He says contradictions contain zero information, giving among other reasons that "inconsistent information is obviously of no use to a decision maker." My story of the Two Firemen is intended as a counterexample to the claim of uselessness of inconsistent Information.

5 The Two Firemen 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 Scenario 1. You see two firemen. One says the only safe route is to your left. The other says to your right. Contradictory information!

6 Scenario 1

7 Scenario 2. You find no one to give directions. Incomplete information!

8 Scenario 2

9 Question: Which scenario would you prefer? Obvious answer: A rational agent would prefer to be in Scenario 1. Contradictory information in this case is better than no information at all.

10 A More Homey Example The essence of the example of the two firemen can be duplicated over and over again. An instance very familiar to me goes like this: My wife Sally and I are leaving the house. I reach in my pocket and cannot find my car keys. I tell Sally I think they are in a jacket pocket in the closet. She tells me they are on the piano. Again, this is all useful information, and I will use it in my search. But it is contradictory.

11 Let us focus though on the firemen, and examine several ways out (pun intended). First a few abbreviations: L: Left is a way out. R: Right is a way out. S: Straight is a way out. O: There is just one way out.

12 First Way Out: This is not an explicit contradiction, like L & ~L. It is only a partial contradiction, (L & O) and (R & O) are only contraries. First Response: But contraries are only a stronger form of contradiction. (L & O) & (R & O) entails L & ~L.

13 Second Way Out: The firemen made separate statements contradicting each other. There is no contradiction until I conjoin them. Reject adjunction rule (Jaśkowski Discussive Logic). Second Response: But it is important to consider the two statements together, and if that is not conjunction then I don t know what conjunction means. If I had reason to believe one more than the other, I might just disregard that other, but in the situation I described I have equal reason to believe each. Also, it is only by my considering the two statements that I see that the two firemen agree on O.

14 Third Way Out: We are not dealing with a conjunction of L & O and R & O, but rather a disjunction. One fireman says L & O. The other says R & O. But suppose the first fireman then says: We can t both be right, but I am certain one of us is right. This makes perfect sense, retreating to what in correspondence David Makinson has described as: using their disjunction as a fall-back reading or integration of the conflicting inputs.

15 So even if neither of the firemen speaks up, I could reason more or less the same way. Both seem experts who have up-to-date info about the fire, and this is all the information I have readily (and safely) available. So probably at least one of them is right, i.e., (L & O) v (R & O). L & O entails ~ S, R & O entails ~ S, therefore ~ S.

16 Makinson is of course famous for the AGM (Alchourron, Gardenfors, Makinson) notion of belief revision. The key idea is that if one believes P, and then learns that ~ P, that one retreats to a consistent set of beliefs that includes ~ P (and hence excludes P). Makinson s response to The Two Firemen is in the same spirit.

17 Third Response: Suppose though that there is what Carl Hewitt calls a pervasively inconsistent background. Then unless one is going to fix everything at once, there is no consistent set of beliefs to drop back to.

18 To be a bit precise about this let s suppose that you are a dialethist (like Graham Priest who supposes there are true contradictions) and you were sitting in your room reflecting on the truth and beauty of the Russell Paradox (how the set of all sets not members of themselves is both a member of itself, and not a member of itself). Or suppose that you are a mere computer scientist reflecting on how the World Wide Web has made you a cyborg by extending your system of belief, with search replacing recall.

19 I really do not disagree with Makinson. I just think we have to somehow extend belief revision to accommodate pervasively inconsistent environments. Relevance logic could be a key component. Note that all the entailments so far mentioned in this talk are relevant implications within Anderson-Belnap s system of first-degree relevant entailments. Real trouble would raise its head if we allowed the irrelevant implication sometimes called Explosion : A & ~ A entails B.

20 Let s leave this topic for another day and get back to our two firemen. Scenario 1. Same as Scenario 1 (two firemen) but you can clearly see that you cannot safely go straight ahead. There are only two options, left and right. It now might seem that there is no reason to have a preference between Scenarios 1 and 2.

21

22 But wait, they still agree that there is a way out. So I can risk running left or right as opposed to staying in my room and preparing for my death.

23 Now suppose that a third fireman shows up and points right saying that this is the only safe way to go. Now I have some reason to run right.

24

25 WWW: Better Than Nothing? Suppose you want to find an answer to a certain yes/no question on the WWW. Which of the following scenarios do you prefer? A. You google and get no (relevant) response. B. You google and get multiple conflicting responses.

26 Perhaps such investigations will sort things out so that you have, in your mind at least, a definite answer as to yes or no. But in a worse case, where you do not have the mind, or time, to do that, at least you might assign a value within the Opinion Tetrahedron, if only based on your gut reaction.

27 I, and I expect you, would prefer to be in scenario B. In this circumstance you can at least try to sort the situation out, e.g.:

28 1) You can count the relative number of opinions on either side. 2) You can somehow assess the credentials (authority/motives) of the sources on either side. 3) You can look at the arguments, if any, provided by the sources. 4) You can try to find cited facts, try to reproduce cited experiments. Etc.

29 It is worth emphasizing that the utility of contradictions is due not just to their content but also to their pragmatic context. Some of the tools described on the previous slide might be taken as logical fallacies. E.g., counting the number of sources can be interpreted as an argument from repetition -- Fifty Million Frenchmen Can t Be Wrong. Anatole France responded: If fifty million people say a foolish thing, it is still a foolish thing. But the aim here is not to prove P by the number of sources that say it, but rather to take a vote to determine the subjective likelihood that P. An improvement would be to check for duplications (one source merely repeating another source), and to have trust weightings based on reliability (expertise, honesty, lack of bias, etc.).

30 Finally a quick overview of the formal background from my previous talk:

31 Subjective Logic A. Jøsang, Artificial Reasoning with Subjective Logic, Proceedings of the Second Australian Workshop on Commonsense Reasoning, Perth A. Jøsang, An Algebra for Assessing Trust in Certification Chains, Proceedings of the NDSS 99 Network and Distributed Systems Security Symposium, The Internet Society, San Diego From Wikipedia, the free encyclopedia Subjective logic is a type of probabilistic logic that explicitly takes uncertainty and belief ownership into account. In general, subjective logic is suitable for modeling and analysing situations involving uncertainty and incomplete knowledge[1][2]. For example, it can be used for modeling trust networks and for analysing Bayesian networks. Arguments in subjective logic are subjective opinions about propositions. A binomial opinion applies to a single proposition, and can be represented as a Beta distribution. A multinomial opinion applies to a collection of propositions, and can be represented as a Dirichlet distribution. Through the correspondence between opinions and Beta/Dirichlet distributions, subjective logic provides an algebra for these functions. Opinions are also related to the belief functions of Dempster-Shafer belief theory. A fundamental aspect of the human condition is that nobody can ever determine with absolute certainty whether a proposition about the world is true or false. In addition, whenever the truth of a proposition is expressed, it is always done by an individual, and it can never be considered to represent a general and objective belief.

32 w = (0.7, 0.1, 0.2) Ternary coordinates

33 DM4 Belnap-Dunn 4-valued Logic T(rue) {t} {t, f} { } B(oth) N(either) Logical (truth) Order {f} F(alse) Approximation (knowledge, information) Order

34 Two Kinds of Uncertainty: Too little information, too much information Uncertain = N Disbelief = F Belief = T Uncertain = B

35 Add a line and visualize it as an Opinion Tetrahedron! N F T B

36 Establish coordinates by dropping altitudes from each vertex to the center of the opposite side and by convention assign each the length 1.0 (measuring from 0 at the base to 1 at the vertex). They intersect at A point (b, d, u, c) in the Opinion Tetrahedron is to be understood as follows: b = degree of belief, d = degree of disbelief, u = degree of uncertainty (ignorance), c = degree of contradiction. 0 b, d, u, c 1.

37 Thank you.

38 Some references A. R. Anderson, N.D. Belnap, and J. M. Dunn, Entailment: The Logic of Relevance and Necessity, vol. 2,Princeton University Press, 775 pp. Belnap, N.D. (1977): How a computer should think, in: G. Ryle (ed.) Contemporary Aspects of Philosophy. Oriel Press, Stockfield, Belnap, N.D. (1977): A useful four-valued logic, in: J.M. Dunn, G. Epstein (eds.), Modern Uses of Multiple-Valued Logic. Reidel, Dordrecht, J. M. Dunn (1967), "The Effective Equivalence of Certain Propositions About de Morgan Lattices," Journal of Symbolic Logic, 1967, pp J. M. Dunn (1976): "Intuitive Semantics for First-Degree Entailments and Coupled Trees," Philosophical Studies, 29, pp A. Jøsang (1997), Artificial Reasoning with Subjective Logic, Proceedings of the Second Australian Workshop on Commonsense Reasoning, Perth. A. Jøsang (1999), An Algebra for Assessing Trust in Certification Chains, Proceedings of the NDSS 99 Network and Distributed Systems Security Symposium, The Internet Society, San Diego.