Yalcin (2010) shows that Kratzer’s model (1991) does not validate some intuitively valid inferences and validates some intuitively invalid ones. In order to handle this problem, he adopts a model based directly on a probability measure. However, as Kratzer (2012) says, “Our semantic knowledge alone does not give us the precise quantitative notions of probability and desirability that mathematicians and scientists work with”, Yalcin’s model seems to be unnatural as a model for comparative epistemic modals. We (2013) proposed a new version of logic—modal-qualitative-probability logic ( \(\textsf{MQPL}\) )—that has the following four merits: (i) The model of \(\textsf{MQPL}\) reflects Kratzer’s intuition above in the sense that it is not based directly on probability measures, but based on qualitative-probability orderings that represent probability measures. (ii) \(\textsf{MQPL}\) does not cause Yalcin’s problem. (iii) The model has no limitation of the size of the domain. (iv) The model, which is based on qualitative-probability orderings that represent nonstandard-real-valued probability measures, can deal with the cases that Kolmogorov probability theory cannot. However, we (2013) did not prove the completeness of \(\textsf{MQPL}\) . Segerberg (1971)’s proof based on a probability measure model has been the standard proof of the completeness of logic of qualitative probability. This proof uses Scott (1964)’s representation theorem of measurement theory. The aim of this paper is (i) to give the reason why we adopt not a probability measure model but a qualitative-probability model as a model for a comparative epistemic modal, which essentially contributes to simplification of the proof of the completeness of \(\textsf{MQPL}\) , in terms of a representation theorem and arbitrary reference, (ii) to show that the proof system of \(\textsf{MQPL}\) is simpler than that of Segerberg’s \(\textsf{PK}\) , and (iii) to prove the completeness of \(\textsf{MQPL}\) . Because the construction of finite Boolean algebras to make it possible to apply Scott’s representation theorem covers a considerable part of Segerberg’s proof, whereas our proof does not require this construction, our proof has the merit that it is by far simpler than Segerberg’s proof.

Measurement-Theoretic Foundations of Logic of Epistemic Modals

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发表日期：

2026年

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Satoru Suzuki

摘要

Yalcin (2010) shows that Kratzer’s model (1991) does not validate some intuitively valid inferences and validates some intuitively invalid ones. In order to handle this problem, he adopts a model based directly on a probability measure. However, as Kratzer (2012) says, “Our semantic knowledge alone does not give us the precise quantitative notions of probability and desirability that mathematicians and scientists work with”, Yalcin’s model seems to be unnatural as a model for comparative epistemic modals. We (2013) proposed a new version of logic—modal-qualitative-probability logic ( \(\textsf{MQPL}\) )—that has the following four merits: (i) The model of \(\textsf{MQPL}\) reflects Kratzer’s intuition above in the sense that it is not based directly on probability measures, but based on qualitative-probability orderings that represent probability measures. (ii) \(\textsf{MQPL}\) does not cause Yalcin’s problem. (iii) The model has no limitation of the size of the domain. (iv) The model, which is based on qualitative-probability orderings that represent nonstandard-real-valued probability measures, can deal with the cases that Kolmogorov probability theory cannot. However, we (2013) did not prove the completeness of \(\textsf{MQPL}\) . Segerberg (1971)’s proof based on a probability measure model has been the standard proof of the completeness of logic of qualitative probability. This proof uses Scott (1964)’s representation theorem of measurement theory. The aim of this paper is (i) to give the reason why we adopt not a probability measure model but a qualitative-probability model as a model for a comparative epistemic modal, which essentially contributes to simplification of the proof of the completeness of \(\textsf{MQPL}\) , in terms of a representation theorem and arbitrary reference, (ii) to show that the proof system of \(\textsf{MQPL}\) is simpler than that of Segerberg’s \(\textsf{PK}\) , and (iii) to prove the completeness of \(\textsf{MQPL}\) . Because the construction of finite Boolean algebras to make it possible to apply Scott’s representation theorem covers a considerable part of Segerberg’s proof, whereas our proof does not require this construction, our proof has the merit that it is by far simpler than Segerberg’s proof.