Necessitation in Modal Logic

trushaev December 07, 2019 at 01:24 1775 views 1 comments

I'm confused as to why necessitation is presented as an axiom in certain axiom systems of modal logic (specifically S5 and related axiomatizations for Epistemic logics). My confusion, to be clear stems from not seeing the intuitive motivation for introducing such an axiom, rather than from any sort of confusion regarding the model theoretic semantics underlying these axiomitizations.

To clarify, necessitation states that p -> Kp for all wff p, where 'Ka' is interpretted as "the agent knows that a". The problem is that such an axiom entails complete omniscience about true statements, e.g. these axioms would entail that given that every human was born on some exact date, therefore I know the exact birthdate of every human.

Of course, there are other more intuitively acceptable axiom systems for epistemic logics, so I understand that a variety of notions of "knowledge" might appear in the literature on the topic, but my confusion largely stems from the fact that the most popular axiom systems for epistemic logic all seem to incorporate necessitation. What's even more strange is that in the informal/philosophical literature on the topic, the distributive law K(p -> q) -> (K(p) -> K(q)) is considered extremely controversial, but nobody seems to mind necessitation.

Does anyone have any insight as to why this is so, or could someone at least point me to literature that might motivate necessitation in some contexts?

Comments (1)

quickly December 16, 2019 at 05:08 #363515 0 likes

I have never seen necessitation presented as an axiom. Rather, it is usually presented as a deduction rule. It states that if [math]\phi[/math] is a theorem of a modal logic [math]\mathbb{K}[/math], then [math]\square\phi[/math] is a theorem of [math]\mathbb{K}[/math]. In other words, every theorem of the logic is necessary. In epistemic logics, the necessitation rule states that every agent knows every proposition derivable from no assumptions (i.e., every agent knows every tautology). This captures the idea that epistemic agents are, or ought to be, logically consistent.