Abstract
The notion of preference arises from comparisons of alternatives. To formalize such comparisons and study them, there are two ways to go. We can represent preference in terms of qualitative binary relations, as we did in our modeling in Chapter 3, following a long logical tradition ([83, 100, 122] and [92]). Or we can introduce a utility or evaluation function which will assign values to the alternatives being compared. The latter quantitative method has been dominant in many research areas, e.g., game theory and social choice theory (cf. [74, 113, 114] and [63]).
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Notes
- 1.
This position was held by famous utilitarians such as Jeremy Bentham (1748–1832) and John Stuart Mill (1806–1876). It has even been claimed that this position is found millennia earlier in China with Mozi.
- 2.
A well-known result is that a preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function. In this chapter, we only consider representable preferences.
- 3.
We will discuss deontic applications in much greater detail in Chapter 11.
- 4.
In [10] the range is natural numbers up to a maximal element (Max). The values are normalized to Max. For me the distance between the numbers seems essential, so normalization is not an option. Similarly I like to be able to subtract unrestrictedly.
- 5.
Let us look at the relation between \(\mathcal{L}_{A}\) and \(\mathcal{L_E} \). From \(\mathcal{L}_{A}\) to \(\mathcal{L_E} \), we can define a translation: a formula of the form \(B^{m}\varphi\) is translated into \(K(q^{m}\rightarrow \varphi)\). This is to say that in the language \(\mathcal{L_E} \), we can express the same notions as [10] without introducing additional belief operators. This advantage leads to the much simpler completeness proof we will see. It becomes even more prominent when constructing reduction axioms for dynamics in the later sections. On the other hand, we can easily translate \(\mathcal{L_E} \) back into \(\mathcal{L}_A\): q m will be \(\neg B^m\bot\), which means that \(\mathcal{L}_{A} \) and \(\mathcal{L_E} \) are equivalent.
- 6.
The main reason why this argument can remain so simple, compared to earlier numerical systems, is our use of propositional constants for values of worlds.
- 7.
Actually, there are two different plausible interpretations here. One either thinks of the event as a command to change one’s evaluation of some current state, or as something which itself has a value that leads to a prescribed value change in the world where the event takes place.
- 8.
Other numerical rules are possible, but most of our later points would apply then as well.
- 9.
In practice, one will normally choose a small natural number, say, between 0 and 10, to denote the reliability or the relative force.
- 10.
The conditions for the epistemic relations ∼ are omitted, as they are routine.
- 11.
This also makes immediate sense in a multi-agent context, employing relative forces. One agent a may take the boss’s commands seriously, whereas agent b may not.
- 12.
Another relevant approach are the DEL models for trust developed in [109].
- 13.
This is closer to Veltman’s intuition that worlds are ordered by how many stated regularities they have satisfied in a longer discourse.
- 14.
As for a comparison with the dynamic semantics for defaults in [192], I suspect that the latter can be embedded into DEEL, but not vice versa.
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Liu, F. (2011). Intermezzo: A Quantitative Approach. In: Reasoning about Preference Dynamics. Synthese Library, vol 354. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1344-4_6
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