Abstract
In this chapter, we will change the atmosphere of our topics and our logical methods a bit: Worlds will make place for objects, modal logic for first-order logic, and there will be differences in style as well. Eventually, however, all will fit back into one uniform paradigm for this book.
Keywords
- Optimality Theory
- Modal Logic
- Representation Theorem
- Priority Order
- Hybrid Logic
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Notes
- 1.
In Chapter 10, we will discuss cases where the syntactic view is really richer.
- 2.
Note that in optimality theory the optimal alternative is chosen unconsciously; we are thinking mostly of applications where conscious choices are made. Also, in optimality theory the application of the constraints to the alternatives lead to a clear and unambiguous result: either the constraint clearly is true of the alternative or it is not, and that is something that is not sensitive to change. We will loosen this condition and consider issues that arise when changes do occur.
- 3.
Unlike in Chapter 8 belief does not enter into this definition. This means that \({Pref}(x,y)\) can be read as x is superior to y, or under complete information x is preferable over y.
- 4.
This way of deriving an ordering from a priority sequence is the “leximin ordering” of [62].
- 5.
Note that, although we used n priorities in the proof to make the procedure easy to describe, in general \(^{2}log(n)+1\) priorities are sufficient for the purpose.
- 6.
References
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Grosof, B.N. 1991. Generalising prioritization. In Proceedings of the 2th international conference on principles of knowledge representation and reasoning (KR 91), eds. J. Allen, R. Fikes, and E. Sandewall, 289–300. San Fransisco, CA: Morgan Kaufmann.
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Liu, F. (2011). Preference from Priorities: Static Logic. In: Reasoning about Preference Dynamics. Synthese Library, vol 354. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1344-4_7
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