Explicit and Implicit Belief in First Degree Entailment with Strict Implication

We introduce sans serif upper F upper D upper E Subscript c Superscript), an extension of FDE with strict implication and a classicality constant, and we show that it formalizes the distinction between explicit and implicit belief. In the style of Levesque’s formalization of these two concepts, explicit beliefs are modelled as sets of formulas closed under our extension of FDE, while implicit beliefs form, in a sense, the classical closure of explicit beliefs. We establish an embedding of Levesque’s logic of explicit and implicit belief into sans serif upper F upper D upper E Subscript c Superscript). This result shows that sans serif upper F upper D upper E Subscript c Superscript) is a viable generalization of Levesque’s logic lifting some of its limitations. Unlike a similar generalization introduced by Lakemeyer, sans serif upper F upper D upper E Subscript c Superscript) comes with an Australian-plan semantics and so it is an alternative potentially attractive to those who prefer the Australian plan over the American one.