Epistemic and doxastic logic with restrictions

Are presented as extensions of classical propositional calculus hierarchies of deductive systems LDR–n and LER–n with n > 1. LER–n is the epistemic logic with restrictions, LDR–n is the doxastic logic with restrictions. The systems LER–1 and LDR–1 are the classical propositional calculus. System LER–(n + 1) can be seen as the result of applying the rule: if X is theorem of LER–n then +X is theorem of LER–(n + 1). Systems also restricts the validity of the axioms +(X → Y ) → (+X → +Y ) and +X → X, in terms of depth (complexity with respect to the operator +) of X and Y , and also includes restricted versions of the axioms of positive and negative introspection. LER system results from the union of LER–n systems, and can be seen as the S5 modal logic system with different types of restrictions. Changing +X → X by +X →～+～X are built LDR–n and the LDR systems. LDR can be seen as the KD45 modal logic system with different types of restrictions. The systems are characterized with a embedded worlds semantics, with which the ‘omniscience logical problem’ is limited