Conditional Deduction Under Uncertainty

Conditional deduction in binary logic basically consists of deriving new statements from an existing set of statements and conditional rules. Modus Ponens, which is the classical example of a conditional deduction rule, expresses a conditional relationship between an antecedent and a consequent. A generalisation of Modus Ponens to probabilities in the form of probabilistic conditional inference is also well known. This paper describes a method for conditional deduction with beliefs which is a generalisation of probabilistic conditional inference and Modus Ponens. Meaningful conditional deduction requires a degree of relevance between the antecedent and the consequent, and this relevance can be explicitly expressed and measured with our method. Our belief representation has the advantage that it is possible to represent partial ignorance regarding the truth of statements, and is therefore suitable to model typical real life situations. Conditional deduction with beliefs thereby allows partial ignorance to be included in the analysis and deduction of statements and hypothesis.