In the preceding chapter, we have introduced a logical framework for static preference. Continuing on this platform, our main concern in this chapter is the dynamics of preference change. Our preferences are modified constantly through commands of moral authorities, suggestions from friends who give good advice, or just changes in our own evaluation of worlds and actions. Living in a society, our preference is also affected by what others like or dislike, as vividly described in the Chinese classic Record of Music 1:
- Belief Revision
- Preference Change
- Dynamic Logic
- Public Announcement
- Default Rule
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The is the first work on music in Chinese history. It is believed that the book was written about two thousands year ago, but the precise time and author are still controversial.
In addition to the arrows drawn, our betterness relations always have reflexive loops.
Reference  analyzes newly defined betterness relations in a set-theoretic format.
Just as with actions of public announcement in Chapter 2, the completeness theorem here does not give us an explicit valid principle for dealing with iterated modalities \(\langle\sharp \varphi\rangle\langle\sharp \psi\rangle \chi\). Indeed, there is no such principle, as the effect of two consecutive suggestions may be genuinely different from making just one suggestion. Consider a simpler principle \(\langle\sharp \varphi\rangle\langle\sharp \varphi\rangle \psi \leftrightarrow \langle\sharp \varphi\rangle\psi\), it holds for factual assertions φ. But it need not hold for non-factual φ which themselves refer to the ordering. After the first ♯φ action, we have changed the ordering, and the worlds where φ is now true need not be the same ones as those where φ was true before. Thus there is more structure to our dynamic logics than what we have brought to light so far.
Reference  already noted how relativization and redefinition make up the standard notion of “relative interpretation” between theories in logic when objects are kept fixed – while product update relates to more complex reductions forming new objects as tuples of old objects.
It is instructive to see the difference with \(\sharp \varphi(R)\) in the above PDL-style format.
Conservative upgrade is a radical command to produce, not φ, but \(Best(\varphi)\).
Such judgments occur in the literature on conditional obligation: see .
We do not consider dynamic actions that directly transform the strict betterness relation, since these are less intuitive, while also presenting some technical difficulties.
This “lexicographic” policy for belief revision was first suggested in .
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Liu, F. (2011). Preference over Worlds: Dynamic Logic. In: Reasoning about Preference Dynamics. Synthese Library, vol 354. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1344-4_4
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