Jon Williamson Draft of August 5 , 2005 Combining

There is a clear connection between probability and logic: both appear to tell us how we should reason. But how, exactly, are the two concepts related? The need for a coherent answer to this question has become increasingly urgent in the past few years, particularly in the field of artificial intelligence. There, both logical and probabilistic techniques are routinely applied in an attempt to solve complex problems such as parsing natural language and determining the way proteins fold. The hope is that some combination of logic and probability will produce better solutions. After all, both natural language and protein molecules have some structure that admits logical representation and reasoning; yet inherent uncertainties also demand the use of probabilistic methods: this structure is only partially known and does not in any case fully determine a solution—context or environment also play a role. Objective Bayesianism offers one answer to this question of the relationship between probability and logic. Objective Bayesians argue that an agent’s initial degrees of belief (her prior belief distribution) should be consistent with her background knowledge but should be non-committal in other respects—i.e., her degrees of belief should be far from the extremes of 0 and 1 unless such strong commitment is warranted by background knowledge. Probability theory is then used to draw conclusions from a prior assignment of beliefs. According to objective Bayesianism, probability generalises deductive logic: deductive logic tells us which conclusions are certain, given a set of premises, while probability tells us the extent to which one should believe a conclusion, given the knowledge of the premises (certain conclusions being awarded full degree of belief). Typically moreover, the premises objectively (i.e. uniquely) determine the degree to which one should believe a conclusion. The papers in this volume address objective Bayesianism and other proposals for combining probability and logic. The papers all stem from presentations made to the Second Workshop on Combining Probability and Logic (Progic2005), an interdisciplinary workshop held at the London School of Economics on 6th-8th July 2005. The objective Bayesian approach is currently widely applied in statistics. However, Bayesian statisticians are rarely explicitly objectivist. More often they tacitly follow an objectivist methodology by using non-committal or noninformative priors, as opposed to following the subjectivist approach of eliciting