Redefining material implication with subjective logic

Material implication is traditionally denoted as (x→y), where x represents the antecedent and y the consequent of the logical relationship between the propositions x and y. Material implication is a truth functional connective, meaning that it is defined by a truth table. While truth functional connectives normally have a relatively clear interpretation in normal language, this is not the case for material implication. It could for example be expressed as: "if x is true, then y is true". However, this does not say anything about the case when x is false, which is problematic for the interpretation of the corresponding entries in the truth table. In this paper we introduce a probabilistic view of material implication and show that it is not closed under binary truth values, and that it in fact produces uncertainty in the form of a vacuous opinion. When seen in this light, it becomes clear that the traditional definition of material implication is based on the over-simplistic and misleading interpretation of complete uncertainty as binary logic TRUE. We redefine material implication with subjective logic to preserve the uncertainty that it unavoidably produces in specific cases. We then compare the new definition of material implication with conditional deduction, and show that they reflect the same mathematical equation rearranged in different forms.