What is knowledge? How can we acquire knowledge? When can we say that we know something? Do we know that we know something? Those are the issues that puzzled Chinese philosophers about 2000 years ago, witness a famous dialogue below between Zhuangzi (approx.369–286 BC) from the Daoism School and Huizi (390–317 BC) from the School of Names: One day Zhuangzi and Huizi are strolling on Bridge Hao.
Zhuangzi: “Look how happy the fish are just swimming around in the river.”
Huizi: “How do you know they are happy? You are not a fish.”
Zhuangzi: “And you are not me. How do you know I don’t know the fish are happy?”
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Reference  studied epistemic logic from both internal and external perspectives.
In this book, possible states/worlds are denoted by variables w, v, s, t, but also sometimes x, y, as seems convenient in the statement of notions and proofs.
I have benefited from a discussion of this point with Johan van Benthem. His new book  contains a further analysis of this issue in terms of logical validities for various dynamic actions, and concrete illustrations in the context of games and protocol update.
In this book, we will not use the original privacy motivations for product update. In our later applications to qualitative and quantitative preference change, the different events are rather different “signals” that can apply to worlds, whose ordering tells us something about how agents are to evaluate their relative betterness or plausibility.
For a first programmatic discussion in this vein, see .
Say, with epistemic S5, the frame still has equivalence relations after DEL update.
Still, the reduction does not settle computational complexity: Translation via the axioms may increase formula length exponentially. Still, for the case of PAL, the complexity of satisfiability remains that of epistemic logic, viz. Pspace-complete (cf. ).
This again requires a reduction axiom for conditional common knowledge, which can be done as well. A good reference for such issues of language design in DEL is .
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Liu, F. (2011). Dynamic Epistemic Logic. In: Reasoning about Preference Dynamics. Synthese Library, vol 354. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1344-4_2
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