On an Information-Theoretic Approach toBetter Questions and Answers
鈴木 聡 (Satoru SUZUKI)
駒澤大学総合教育研究部非常勤講師

Groenendijk and Stokhof [1] consider the goodness of an answer in terms of the partition
induced by a question. AQ is deﬁned as the set of all cells of the partition Q that are compatible
with an answer A: AQ := {q ∈ Q : q ∩ A ̸= ∅}. In order to compare the goodness of answers,
they use a qualitative ordering >Q among answers with relative to Q. Groenendijk and Stokhof
propose that when an answer A is incompatible with more cells of Q than an answer A′ , thatis, AQ ⊂ A′Q , A is considered to be a better answer to the question than A′ . When AQ = A′Q ,if A does not extra irrelevant information, that is, A′ ⊂ A, A is considered to be a betteranswer than A′ . Combining both constraints, Groenendijk and Stokhof deﬁne A’s being abetter answer to a question Q than A′ , in symbols, A >Q A′ as follows: A >Q A′ iﬀ (i)AQ ⊂ A′Q , or (ii) AQ = A′Q and A′ ⊂ A. However, Groenendijk and Stokhof’s qualitative
approach is too coarse-grained to achieve the goal, for we can use this approach only when
all cells of a partition are equiprobable. In order to overcome the limitations, van Rooij [4]proposes an information-theoretic approach based on conditional Shannon entropy. An answer
A is better than an answer A′ to a question Q iﬀ the information value of A is higher thanthat of A′ , or in case both of them are the same, the self-information of A is lower than that ofA′ : (1) A >Q A′ iﬀ (i) IVQ (A) > IVQ (A′ ), or (ii) IVQ (A) = IVQ (A′ ) and I(A) < I(A′ ).
Moreover, van Rooij considers the goodness of a question. He proposes that a question Q isbetter than a question Q′ with respect to a mutually exclusive and exhaustive set of hypotheses
H iﬀ the expected information value of Q is higher than that of Q′ , or if both of them are thesame, Q is less ﬁne-grained than Q′ : (2) Q >H Q′ iﬀ (i) EIVH (Q) > EIVH (Q′ ), or (ii)
EIVH (Q) = EIVH (Q′ ) and, for any q ∈ Q, there is such q ′ ∈ Q′ that q ⊆ q ′ , in symbols,Q′ ⊆∗ Q. Both IVQ and EIVH are deﬁned in terms of conditional Shannon entropy. IVQ (A)is deﬁned as the reduction of Shannon entropy of a question Q when an answer A is learnt:
IVQ (A) := H(Q) − HA (Q), where H(Q) is the Shannon entropy of Q, and HA (Q) is theShannon entropy of Q conditional on an answer A. So we can rewrite (1) as follows: (3)A >Q A′ iﬀ (i) HA (Q) < HA′ (Q), or (ii) HA (Q) = HA′ (Q) and I(A) < I(A′ ). EIVH (Q)is deﬁned as the average reduction of Shannon entropy of a set of hypotheses H when an answer
to a question Q is learnt: EIVH (Q) := H(H)−HQ (H), where HQ (H) is the Shannon entropyof H conditional on a question Q. So we can rewrite (2) as follows: (4) Q >H Q′ iﬀ (i)HQ (H) < HQ′ (H), or (ii) HQ (H) = HQ′ (H) and Q′ ⊆∗ Q. We [6] provided a qualitative1

approach that is as ﬁne-grained as the information-theoretic (conditional-Shannon-entropytheoretic) one, in terms of measurement theory, and proposed a new version of complete logic
of better questions and answers—Better-Question-and-Answer Logic (BQAL)—the model ofwhich is based on this qualitative approach. Generally, the standard model of economics isbased on global rationality that requires an optimising behavior. But according to Simon [5],cognitive and information-processing constrains on the capabilities of agents, together withthe complexity of their environment, render an optimising behavior an unattainable ideal.
Simon dismisses the idea that agents should exhibit global rationality and suggests that theyin fact exhibit bounded rationality that allows a satisﬁcing behavior. It is considered to be
not irrational but boundedly rational in questions and answers that we cannot always form apartition induced by a question. How can we deal with this case? This is the main problem inthis talk. Neither van Rooij [4] nor we [6] consider this case. Rényi [3] provides a clue for solving
this problem. He generalizes Shannon entropy to satisfy the weighted quasi-arithmetic mean
property. This type of entropy is called Rényi entropy. Moreover, he extends the coverage ofShannon entropy so that it can be applied when we cannot form a partition. We call this type ofentropy incomplete Shannon entropy. The aim of this talk is to furnish a solution to the main
problem above by proposing a new version of logic of better questions and answers—Logic-ofBetter-Questions-and-Answers (LBQA)—the model of which is based directly on incomplete
conditional Shannon entropy without depending on measurement theory. BQAL in [6] was aversion of propositional logic, whereas LBQA in this talk is a version of ﬁrst-order logic thelanguage of which is based on Halpern [2], which can express both probability and incomplete
conditional Shannon entropy statements in an object language. （使用言語：日本語）

参考文献[1] Groenendijk, J., Stokhof, M.: On the semantics of questions and the pragmatics of answers.
In: Landman, F., Veltman, F. (eds.) Varieties of Formal Semantics, pp. 143–170. Foris,Dordrecht (1984)[2] Halpern, J.Y.: Reasoning about Uncertainty. The MIT Press, Cambridge, Mass. (2003)[3] Rényi, A.: On measures of information and entropy. In: Proceedings of the fourth Berkeley
Symposium on Mathematics, Statistics and Probability 1960. pp. 547–561 (1961)[4] van Rooij, R.: Comparing questions and answers: A bit of logic, a bit of language, andsome bits of information. In: Sommaruga, G. (ed.) Formal Theories of Information: fromShannon to Semantic Information Theory and General Concepts of Information, LNCS,
vol. 5363, pp. 161–192. Springer-Verlag, Heidelberg (2009)[5] Simon, H.A.: Models of Bounded Rationality, vol. 1. The MIT Press, Cambridge, Mass.
(1982)[6] Suzuki, S.: Measurement-theoretic foundations of logic for better questions and answers.
In: Zeevat, H., Schmitz, H.C. (eds.) Bayesian Natural Language Semantics and Pragmatics,
Language, Cognition, and Mind, vol. 2, pp. 43–69. Springer-Verlag, Heidelberg (2015)2