Epistemic and doxastic logic with restrictions (2010)
The hierarchies of deductive systems LER – n and LDR – n are presented as extensions of the classical propositional calculation, with n ≥ 1. LER – n is the epistemic Logic with depth restrictions – n, LDR – n is the doxastic Logic with depth restrictions –N. The LER – 1 and LDR – 1 systems are the classic propositional calculation. The LER– (n + 1) system can be seen as the result of applying the rule: from X, + X is inferred, once to the theorems of the LER – n system, in addition, the validity of the + (X) axioms is restricted → Y) → (+ X → + Y) and + X → X in terms of the depth (complexity with respect to the operator +) of X and Y, and also include generalized versions with restrictions of the axes of positive introspection and negative. The LER system results from the meeting of the hierarchy systems, and can be seen as the S5 modal Logic system with various types of restrictions. Changing + X → X to + X → ~ + ~ X builds the LDR – n hierarchy and the LDR system; The latter can be seen as the KD45 modal Logic system with various types of restrictions. The systems are characterized with semantics of possible embedded worlds, with which certain limits are imposed on the problem of logical omniscience.