Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being (...) true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic. (shrink)

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore-Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on (...) the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture. (shrink)

Che 1-pu. Han'guk kŭndae yurim ŭi kulchŏl : ilche hyŏmnyŏk Yurim ŭi hyŏngsŏng kwa chŏn'gae -- 1. Sŏron -- 2. 19-segi huban-Taehan Chegukki Yurim ŭi tonghyang -- 3. 1910-1920-yŏndae Kyŏnghagwŏn kwa ilche hyŏmnyŏk Yurim ŭi hwaltong -- 4. 1920-1930-yŏndae Yurim tanch'e ŭi chojik kwa hwaltong -- 5. 1930-1940-yŏndae Chosŏn Yudo Yŏnhaphoe wa 'Hwangdo yuhak' -- 6. Kyŏllon -- Che 2-pu. Han'guk kŭndae Yurim ŭi che yangsang -- 7. Hanmal, ilche ha Yang Pong-je ŭi pyŏnsin kwa hwaltong -- 8. 1920-yŏndae singminji (...) Chosŏn Yurim kwa Ilbon ŭi Yusima Sŏngdang -- 9. 20-segi ch'o ilche hyŏmnyŏk Yurim ŭi Kyŏnghagwŏn hwaltong : Yi Tae-yŏng, 1874-1950 ŭl chungsim ŭro -- 10. Ilche kangjŏmgi Song Chu-hŏn ŭi saengae wa hwaltong. (shrink)

1. Ikkŭnŭn mal -- 2. Han nara ŭi yet kŭlssi ro tora kaja : K'ang Yuwei, sŏye wa mihakchŏk chŏnhwan -- 3. Chosŏn ŭi yesul ŭn illyu ŭi pigŭk ŭl tamnŭnda : Yanagi Muneyosi, minye uĭ palgyŏn -- 4. Nŏhŭi nŭn T'ap ŭi himch'am ŭl poannŭn'ga : Ko Yu-sŏp, Han'guk hyŏndae mihak ŭi t'ansaeng -- 5. Naganŭn mal.