Aggregating multiple fuzzy values for a given observation Hmmm... this question might be a bit low-brow. I'm no mathematician. 
Let's say that I have an unordered sequence of fuzzy values $T = (t_1 \ldots t_n)$, $T \in (0,1]^n$. Each value is a positive indication towards some observation (hence, the values are not probabilities in that a value of 0.1 does not indicate a value of 0.9 against, just that we have a weak positive affirmation of the observation).
To take an example, let's say that the observation is "whether or not there has been a hurricane in the US today": we have evidence from a weather forecast two days ago saying that there would be one today (0.1), there's been a high level of traffic on twitter with the keyword hurricane (0.2) and there's been breaking news reports about a hurricane in Florida (0.7).
I'm trying to formalise a means of aggregating these readings into a final score (the absolute value of the score makes no difference, but the relative value compared to other aggregations does). For what I'm doing, I want each fuzzy value to reduce the cumulative "uncertainty" about the observation. I also need to set an external $max\in(0,1]$ value on the aggregation result:
$F_0(T,max) = 0$;
$F_k(T,max) = (max - F_{k-1})t_k + F_{k-1}$; 
$F(T,max) = F_n(T,max)$.
Taking the hurricane example, let $max = 1$:
$F_0 = 0$;
$F_1 = (1 - 0) * 0.1 + 0$;
$F_2 = (1 - 0.1) * 0.2 + 0.1 = 0.28$;
$F_3 = (1 - 0.28) * 0.7 + 0.28 = 0.784$.
This aggregation is nice (for my scenario) because it rewards high fuzzy values better than 
multiple medium or low values, it's order indepedent, and the answer is in $(0,max]$.
I'm having a lot of trouble formalising this fairly simple idea... don't have the necessary know-how or lingo (probably clear by now). So I have the following questions:


*

*Has anyone seen a similar such aggregation somewhere?

*How should I call this sequence of values, where the order is not important, and duplicates should be maintained? An unordered sequence? A set with duplicates? An unordered n-tuple?

*How should I formulate this function? Should I formulate it for an unordered pair first, and then for an unordered tuple? Is there a neat way of demonstrating commutativity?


Any help or nudges in the right direction would be greatly appreciated.
 A: I'm at least acquainted with the idea of fuzzy logic, as distinct from probabilities. Having said that: while you are not looking for a probabilistic sort of formula, you have stumbled upon one.
Taking max = 1, your formula can be finessed to the form

Fk+1  = (1 − Fk) tk + Fk  = Fk + tk − Fktk

which is the same thing as the "probability" of at least one of event A or event B happening, where A and B are independent random events, occurring with probability Fk and tk respectively. — This is why your construction is order independent (and in fact can be proven so using this observation).
To say the same thing in different terms: if you take the "fuzzy AND" (logical conjunction) of two qualities with fuzzy strengths x and y to be the product xy, then the formula x + y − xy is the corresponding "fuzzy OR" which will satisfy de Morgan's laws, i.e. NOT(x AND y) = (NOT x) OR (NOT y), if you take NOT x = (1 − x). Of course, you may not want this if you don't want a small positive fuzzy value to represent a large fuzzy value for its negation.
It seems to me that whatever you choose max to be, you can re-scale it to get identical behaviour to the case max = 1. So the utility of this interpretation is unlikely to hinge on that particular parameter.
For your other questions:


*

*A collection in which number is important, but order is not, is a bag or multi-set (these are synonyms). Furthermore, that's all that it is; they're basically "set-like collections" which are equivalent to a histogram.

*I would call the function "fuzzy OR" or "fuzzy disjunction".

*If you want to make the fuzzy OR of a collection (a "bag") well-defined, it suffices to show commutativity (which makes it well-defined for bags of size at most two), and then show that it is associative — that is, that OR(x, OR(y,z)) = OR(OR(x,y), z) — which ensures that if it is well-defined for bags of size k, it is also well-defined for bags of size k + 1.
