Where p designates some proposition, M designates a model and w designates a world. Sorry if this question is too basic, but I think it is so basic that none of the basic textbooks and articles which I have consulted give a clear explanation. I understand that the first formula expresses that p is true under model M, in world p. But what does the second formula express?

A different example is from epistemic logic. What is the difference between: M, w |= ¬Kap and M|¬Kap, w |= ¬Kap

In the context of dynamic epistemic logic, the binary operator 'M | φ' takes a model M and a formula φ and 'updates' M, by removing from M all the worlds where φ is false ([Holliday], Lecture 13, Slide 7; [van Benthem], Chapter 15, Definition 15.2.1):

M | φ = {v ∈ |M| : M, v |= φ}.

In the context of dynamic epistemic logic, this represents the process of learning, and can be used to give explications of belief-revision and so on. Here are some references to the relevant literature:

van Benthem, J. (2010) Modal Logic for Open Minds, Stanford, CSLI Lecture Notes #199.
Holliday, W.H. (2012) Modal Reasoning, Lecture Course (Spring), UC Berkeley.

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