Subjective probability theory as well as the theory of DS belief functions take the objects of belief to be propositions. Possibility theory does so only indirectly, although possibility measures on a field of propositions [ltr]AA[/ltr] can also be defined without recourse to a possibility distribution on the underlying set [ltr]WW[/ltr] of possibilities.A possibility [ltr]ww[/ltr] in [ltr]WW[/ltr] is a complete and consistent description of the world relative to the expressive power of [ltr]WW[/ltr]. [ltr]WW[/ltr] may contain just two possibilities: according to [ltr]w[size=13]1w1[/ltr] tomorrow it will be sunny in Vienna, according to [ltr]
w2w2[/ltr] it will not. On the other end of the spectrum, [ltr]
WW[/ltr] may comprise all metaphysically possible, or even all logically possible worlds (for more see the entry on
possible worlds.)[/size]
Usually we are not certain which of the possibilities in [ltr]WW[/ltr] corresponds to the actual world. Otherwise these possibilities would not be genuine possibilities for us, and our degree of belief function would collapse into a truth value assignment. However, to say that we are not certain which possibility it is that corresponds to the actual world does not mean that all possibilities are on a par. Some of them will be really far-fetched, while others will seem to be more reasonable candidates for the actual possibility.This gives rise to the following consideration. We can partition the set of possibilities, that is, form sets of possibilities that are mutually exclusive and jointly exhaustive. Then we can orderthe cells of this partition according to their plausibility. The first cell in this ordering contains the possibilities that we take to be the most reasonable candidates for the actual possibility. The second cell contains the possibilities which we take to be the second most reasonable candidates. And so on.If you are still equipped with your possibility distribution from the preceding section you can use your degrees of possibility for the various possibilities to obtain such an ordered partition. Note, though, that an ordered partition — in contrast to your possibility distribution — contains no more than ordinal information. While your possibility distribution enables you to say how possible you take a particular possibility to be, an ordered partition only allows you to say that one possibility [ltr]w[size=13]1w1[/ltr] is more plausible than another possibility [ltr]
w2w2[/ltr]. In fact, an ordered partition does not even enable you to express that the difference between your plausibility for [ltr]
w1w1[/ltr] (say, tomorrow the temperature in Vienna will be between 70°F and 75°F) and for [ltr]
w2w2[/ltr] (say, tomorrow the temperature in Vienna will be between 75°F and 80°F) is smaller than the difference between your plausibility for [ltr]
w2w2[/ltr] and for the far-fetched [ltr]
w3w3[/ltr] (say, tomorrow the temperature in Vienna will be between 120°F and 125°F).[/size]
This takes us directly to ranking theory (Spohn 1988 and 1990 and, especially, 2012), which goes one step further. Rather than merely ordering the possibilities in [ltr]WW[/ltr], a pointwise ranking function [ltr]κ:W→N∪{∞}κ:W→N∪{∞}[/ltr] additionally assigns natural numbers from [ltr]N∪{∞}N∪{∞}[/ltr] to the cells of possibilities. These numbers represent the grades of disbelief you assign to the various (cells of) possibilities in [ltr]WW[/ltr]. The result is a numbered partition of [ltr]WW[/ltr],[ltr]κ[size=13]−1(0),κ−1(1),κ−1(2),…,κ−1(n)={w∈W:κ(w)=n},…κ−1(∞).κ−1(0),κ−1(1),κ−1(2),…,κ−1(n)={w∈W:κ(w)=n},…κ−1(∞).[/ltr][/size]
The first cell [ltr]κ[size=13]−1(0)κ−1(0)[/ltr] contains the possibilities which are not disbelieved (which does not mean that they are believed). The second cell [ltr]
κ−1(1)κ−1(1)[/ltr] is the set of possibilities which are disbelieved to degree 1. And so on. It is important to note that, except for [ltr]
κ−1(0)κ−1(0)[/ltr], the cells [ltr]
κ−1(n)κ−1(n)[/ltr] may be empty, and so would not appear at all in the corresponding ordered partition. [ltr]
κ−1(0)κ−1(0)[/ltr] must not be empty, though. The reason is that one cannot consistently disbelieve everything.[/size]
More precisely, a function [ltr]κ:W→N∪{∞}κ:W→N∪{∞}[/ltr] from a set of possibilities [ltr]WW[/ltr] into the set of natural numbers extended by [ltr]∞∞[/ltr], [ltr]N∪{∞}N∪{∞}[/ltr], is a pointwise ranking function just in case [ltr]κ(w)=0κ(w)=0[/ltr] for at least one [ltr]ww[/ltr] in [ltr]WW[/ltr], i.e., just in case [ltr]κ[size=13]−1(0)≠∅κ−1(0)≠∅[/ltr]. The latter requirement says that you should not disbelieve every possibility. It is justified, because you know for sure that one possibility is actual. A pointwise ranking function [ltr]
κ:W→N∪{∞}κ:W→N∪{∞}[/ltr] on [ltr]
WW[/ltr] induces a
ranking function [ltr]
ϱ:A→N∪{∞}ϱ:A→N∪{∞}[/ltr] on a field [ltr]
AA[/ltr] of propositions over [ltr]
WW[/ltr] by defining for each [ltr]
AA[/ltr] in [ltr]
AA[/ltr],[/size]
[ltr]ϱ(A)=min{κ(w):w∈A} (=∞ if A=∅).ϱ(A)=min{κ(w):w∈A} (=∞ if A=∅).[/ltr]
This entails that ranking functions [ltr]ϱ:A→N∪{∞}ϱ:A→N∪{∞}[/ltr] are (finitely) minimitive (and hence super-additive), i.e., for all [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr],[ltr]ϱ(A∪B)=min{ϱ(A),ϱ(B)}.(9)(9)ϱ(A∪B)=min{ϱ(A),ϱ(B)}.[/ltr]
As in the case of possibility theory, (finitely minimitive and non-conditional) ranking functions can be directly defined on a field [ltr]AA[/ltr] of propositions over a set of possibilities [ltr]WW[/ltr] as functions [ltr]ϱ:A→N∪{∞}ϱ:A→N∪{∞}[/ltr] such that for all [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr]:[ltr]=∞,[/ltr]
الثلاثاء مارس 15, 2016 11:13 am من طرف free men