Theories of Knowledge and Ignorance

What does it mean to say that an agent only knows a particular fact,i.e., knowing that fact and not more than that? The problem of describing so-called minimal knowledge has been discussed in the literature since 1985, when Halpern and Moses published a paper on Knowledge and Ignorance.The present chapter reviews a considerable part of the most important proposals for only knowing and provides a number of generalizations over these proposals. The focus of this study is on a theoretical understanding of the subject, but several (possible) applications are indicated too. The proposals for solving the problem of minimal knowledge vary along some dimensions. Most of these proposals are restricted to a single agent, whereas a few deal with the multi-agent case. Also, most deal with the problem of minimal knowledge on the level of meta-language, by formulating inferential conditions, semantic constraints on verifying models, or rules for establishing belief sets; a few suggest an explicit operator for only knowing in the object language. Moreover, the majority of proposals employ specific modal systems which, e.g., point out whether the agent is (fully) introspective, i.e., knows that she knows (and knows that and what she does not know); the authors, however, suggest and discuss general modal approaches too. Finally, the advantages of `going partial' (using models that may leave the truth value of certain propositions undefined) are demonstrated.