Sentences, Propositions and Logical Omniscience, or What does Deduction tell us

Abstract: In deductive reasoning, if φ is deduced from some set Γ, then φ is already implicit in Γ. But then how do we learn anything from deduction? That we do not learn anything is the (unsatisfying) answer suggested by Socrates in Plato’s Meno. This problem is echoed in the problem of logical omniscience prominent in epistemic logic according to which an agent knows all the consequences of his/her knowledge. An absurd consequence is that someone who knows the axioms of Peano Arithmetic knows all its theorems. Since knowledge presumes belief, the lack of closure of (actual) beliefs under deduction remains an important issue. The post-Gettier literature has concentrated on the gap between justified true belief and knowledge, but has not concerned itself with what belief is. This question, or at least an analysis of sentences of the form Jack believes that frogs have ears, has been prominent in the philosophy of language. But even there, less attention has been paid to how we know what someone believes. This turns out to matter