Reflection and Truth

(where an a f f i r m a t i v e sentential operator is informally characterized as one for which a p r i n c i p l e of d i s t r i b u t i o n over conjunction is not o b v i o u s l y implausible). Examples of such operators include the familiar modal operators ~ and 0, and also various epistemic and doxastic operators. I want to consider certain principles that may be held to govern certain of these operators. and 0 c o n f o r m to this principle in all 'normal' systems of modal logic, and it seems reasonable to suppose that a knowledge operator and a belief operator should as well, at least when it is 'ideally rational' belief that we attempt to c h a r a c t e r i z e by means of the belief operator. The d i s t r i b u t i o n principle above in fact follows from two other principles that together determine the class of normal systems of modal logic, and find frequent a p p l i c a t i o n to epistemic and doxastic operators. They are 2) A-nee: ~ ~ ~ ~ A~; and 3) D i s t ' ~ '" A(~ ~ t). ~ .A~ ~ A t. Together they express the deductive closure of A. And again, the beliefs and knowledge of ideally rational agents can plausibly be held to meet this standard. Also commonly invoked, for such agents, is a c o n s i s t e n c y condition, 4) Consis: A~Z ~ A ~ .