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Probabilities on Sentences in an Expressive Logic
, 2012
"... 1 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 higherorder logic are ideally suited for repre ..."
Abstract

Cited by 5 (2 self)
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1 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 higherorder logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truthvalues. 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 wishlist, 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 wishlist into technical requirements for a prior probability