Loosely, are the following two statements equivalent?

(1) It is epistemically necessary that P.

(2) It is known that P.

If they are not, in what ways could they be different? I'm thinking in terms of natural language here, not under any specific logical system.

  • 2
    It seems a proposition can be epistemically necessary without being known, for example Fermat's Last Theorem was epistemically necessary 50 years ago despite it not being known.
    – George
    Aug 15 '13 at 20:06
  • I'd say that it was epistimically compelling rather than neccessary - there are examples when the epistimically neccessary turned out to mathematicians surprise to be wrong. Nov 18 '13 at 1:44

Thanks George, my original answer was wrong. Here's a less wrong, and this time negative answer.

(1) says that: for all agents a, a knows that P
(2) says that: for some agent a, a knows that P

The meaning of (1) follows from the fact that epistemically necessary propositions are such that they are true in all epistemically accessible worlds of all agents (this second "all" is the key, because each agent has her own epistemic accessibility relation). The meaning of (2) is straightforward: "P is known" simply means that there exists someone who knows P to be true.

In case it's not already obvious why (1) and (2) are not equivalent, let's fully unpack the meanings of the two sentences. Each agent α has an associated accessibility relation R(α). Using this, we observe that:

(1) says that: for all agents α, for all R(α)–accessible worlds w, P is true at w
(2) says that: for some agent α, for all R(α)–accessible worlds w, P is true at w

Based on these observations, we can present a counterexample to the equivalence of (1) & (2) by having a world where some agent α knows P, but some other agent β does not know P. This situation would show that P is known (by someone), but nevertheless not epistemically necessary (because at least one agent, viz. β, doesn't know P).

  • 1
    What is the motivation for the logical semantics of (2)? It seems P could happen to be true in the current world of evaluation without anyone "knowing" it to be true. For example, a certain lottery ticket might be a powerball winner, with nobody "knowing" it (having not looked at it yet).
    – George
    Aug 16 '13 at 0:27
  • Of course, you're absolutely correct! I don't know what I was thinking. I'll fix it. Aug 16 '13 at 0:45
  • 1
    Thanks for the clarification. For what system of logic is this the typical semantics for "epistemicaly necessary" and "knows"? Kripke's modal logic doesn't have agents, as far as I'm aware, so it can't be that one. Is this the typical semantics for modal logics with agents? Are these semantics in dispute in contemporary philosophy or are they taken as largely standard?
    – George
    Aug 16 '13 at 2:53
  • It's a multi-agent epistemic logic. It differs from the generic modal logic by the re-interpretation of the Box as the knowledge operator K indexed to agents so that K(a,p) reads "agent a knows that p", which within the Kripke semantics reduces to: p is true in all R(a)-accessible worlds. So yes, pretty typical, standard semantics for modal logics with agents. What is, however, in dispute is the various conditions on the accessibility relations, e.g., reflexivity, transitivity, symmetry, etc. If you haven't already, check out Hintikka's Knowledge and Belief (this is where it all began!). Aug 16 '13 at 3:43

"Epistemically necessary" is a technical term used in systems of logic and philosophy. "Known" is a term from natural language, which is sometimes also used technically. Therefore it isn't entirely coherent to ask if they mean the same thing in natural language. If "epistemically necessary" can be said to have a natural meaning, it's one derived from its technical sense.

It's probably accurate, however, to say that the concept of "epistemic necessity" is an attempt to formalize at least some aspects of the natural language concept of "known."

As to whether they are equivalents in a technical sense, that would depend on the particular system, and how it defines those terms within it.

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