Comments on 'modal fixed point logic and changing models'

Abstract
This is indeed a very nice draft that I have read with great pleasure, and that has helped me to better understand the completeness proof for LCC. Modal fixed point logic allows for an illuminating new version (and a further extension) of that proof. But still. My main comment is that I think the perspective on substitutions in the draft paper is flawed. The general drift of the paper is that relativization, (predicate) substitution and product update are general operations on models, and that it is important to check whether given logical languages are closed under these operations. FO logic is closed under relativization, predicate substitution and product constructions (such as those involved in relative interpretation). The minimal modal logic is closed under relativization, which explains the reduction of epistemic logic (withhout common knowledge) + public announcement to epistemic logic simpliciter (as observed in Van Benthem, [2]). The reduction breaks down as soon as one adds common knowledge. The minimal modal logic is also closed under substitution, which explains the reduction of epistemic logic plus (publicly observable) factual change to epistemic logic simpliciter, via the following reduction axioms (I use !p := φ for the operation of publicly changing the truth value of p to φ): [!p := φ]p ↔ φ [!p := φ]q ↔ q (p and q syntactically different ) [!p := φ]¬ψ ↔ ¬[!p := φ]ψ [!p := φ](ψ1 ∧ ψ2) ↔ [!p := φ]ψ1 ∧ [!p := φ]ψ2 [!p := φ][i]ψ ↔ [i][!p := φ]ψ Unlike the case of relativisation, this can be extended to the case of epistemic logic with common knowledge, by means of: [!p := φ]CGψ ↔ CG[!p := φ]ψ We get the following..
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