 In the late 19th and early 20th century Frege, followed by Russell and Whitehead, showed how deductive logic may be represented in the kind of rigorous formal system we now call quantified predicate logic. For the first time logicians had a fully formal deductive logic powerful enough to represent all valid deductive arguments that arise in mathematics and the sciences. In this logic the validity of deductive arguments depends only on the logical structure of the sentences involved. This development in deductive logic spurred some logicians to attempt to apply a similar approach to inductive reasoning. The idea was to extend the deductive entailment relation to a notion of probabilistic entailment for cases where premises provide less than conclusive support for conclusions. These partial entailments are expressed in terms of conditional probabilities, probabilities of the form \(P[C \pmid B] = r\) (read “the probability of C given B is r”), where P is a probability function, C is a conclusion sentence, B is a conjunction of premise sentences, and r is the probabilistic degree of support that premises B provide for conclusion C. Attempts to develop such a logic vary somewhat with regard to the ways in which they attempt to emulate the paradigm of formal deductive logic.