中央研究院 資訊科學研究所

This work benefits from discussions over the years with Johan van Benthem, Adam Bjorndahl, Hanti Lin, and Konstantin Genin.  Epistemic logic studies the modal operator Kp, interpreted as “s knows that p”.  The standard, “minimalist”approach to epistemic logic (e.g., Williamson) views possible worlds semantics as a neutral forum in which to investigate principles that seem intuitively valid for knowledge.  The minimalist approach is subject to a number of familiar shortcomings: logical omniscience, the difficulty of finding plausible principles that can be stated entirely in terms of K, failure of the possible worlds semantics to explain the principles assumed, and failure to connect with real methodological or epistemological concerns.

In this talk, I present an alternative, explanatory semantics for inductive knowledge whose models encompass concrete, cognitive agents who modify their beliefs in light of new information.  The primitive vocabulary includes operators for time, belief, information, and methodological possibility.

Time and information are handled in a framework closely related to topological subspace semantics.  In that richer setting, inductive knowledge that p can be explicated explicitly as stable, true belief that p achieved by an agent who is capable of eliminating error with respect to p eventually.   I will present the semantics and discuss some striking consequences.

(I) The system S4 is not even close to being valid, but something like S4 is valid if the axioms are re-expressed in terms of methodological possibility of knowledge, rather than in terms of knowledge itself.  The similarly restricted version of S5 is not valid.

(II) A scientist can know that p while rejecting q, even though p is logically equivalent to q, as in the history of quantum mechanics.

(III) The scope of scientific knowability is very broad in the proposed semantics.  It suffices that the hypothesis is objective (independent of the scientist’s beliefs) and capable of being sharpened with true auxiliary hypotheses that make it sharply testable.

(IV)  It is a major theme in the philosophy of science that scientific knowledge allows for luck in finding true auxiliary hypotheses according to which a general theory stands up to test.   According to the proposed semantics, allowance for that kind of luck is necessary if deductive consequences of inductive knowledge are to be knowable.  Thus, there is a deep, logical motivation for what appears to be a mere, sociological contingency.

(V) The minimalist approach to epistemic logic implies that it is impossible to know one’s own Moore sentence: “p is true but I don’t know that p”.

In the proposed semantics, it is plausibly possible (if not desirable) to know one’s own Moore sentence---without failure of deductive cogency between the Moore sentence and its conjuncts!

(VI) Common knowledge is often assumed, but its origin is rarely explained.

In the proposed semantics, it is straightforwardly possible for an expert to spread common knowledge of a scientific law over a classroom filled with gullible pupils.  I call that the “high school science theorem”.