I provide a method of measuring the inconsistency of a set of sentences from 1-consistency, corresponding to complete consistency, to 0-consistency, corresponding to the explicit presence of a contradiction. Using this notion to analyze the lottery paradox, one can see that the set of sentences capturing the paradox has a high degree of consistency (assuming, of course, a sufficiently large lottery). The measure of consistency, however, is not limited to paradoxes. I also provide results for general sets of sentences.

I present two measures of information for both consistentand inconsistent sets of sentences in a finite language ofpropositional logic. The measures of information are based onmeasures of inconsistency developed in Knight (2002).Relative information measures are then provided corresponding to thetwo information measures.

In this paper we present a probabilistic notion of entailment for finite sets of premises, which has classical entailment as a special case, and show that it is well defined; i.e., that the problem of whether a sentence is entailed by a set of premises is computable. Further we present a natural deductive system and prove that it is the strongest deductive system possible without referring to probabilities.