Doxastic logic

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The doxastic logic is a modal logic formalized which logical aspects of the faith or opining. The term "doxastic" comes from the ancient Greek doxa , "belief". In a doxastic logic, a new operator of the type B (x) is introduced, which intuitively means “It is believed that x is the case”. The set then means a set of opinions or beliefs. A belief is thus formalized as a modal operator: ${\displaystyle \mathbb {B} }$

${\displaystyle \mathbb {B} }$: { } ${\displaystyle b_{1},b_{2},...,b_{n}}$

Types of Belief

Raymond Smullyan defines the following types of reasoners ( hereinafter referred to as “agents” as is customary in knowledge logic) with regard to a set of beliefs . ${\displaystyle \mathbb {B} }$

• Correct reasoner: A correct agent never believes in a wrong belief.
• Incorrect Agent (inaccurate reasoner): A incorrect agent believes a false belief at least.
• Smug Agent (conceited reasoner): An agent is conceited, if he believes that he is never incorrect.
• Consistent Agent (consistent reasoner): A consistent Agent believes never a belief and its negation.