Dear Lotfi, On Mon, 8 Sep 2003, Lotfi A. Zadeh wrote:
> It is true that in some instances ambiguity and imprecision > are close in meaning, but in most instances this is not the case. > Example l: Robert asked me to meet him at 8. Did he mean 8 am or 8 > pm? Here we have ambiguity but no imprecision. Example 2: Robert asked > me to meet him a few minutes before 8 am. Here we have imprecison but no > ambiguity. The Oxford English Dictionary gives the following under "ambiguous": "2. Of words or other significant indications: Admitting more than one interpretation, or explanation; of double meaning, or of several possible meanings; equivocal. (The commonest use.)" This is the meaning I have in mind. We might interpret "a few minutes before 8 am" as referring to an interval centered on 7.58 am. We might also interpret it as referring to an interval centered on 7.55 am. Since these are two distinct interpretations of the expression, both of which seem reasonable, the definition forces me to conclude that the expression is ambiguous. Since the possible interpretations are fairly close to each other, the ambiguity in this example will often not be of practical significance. In such cases we often choose to ignore it and speak as if there was no ambiguity. But since we are being technical here, and since one can think of examples where even this ambiguity could lead to crucial misunderstandings, I have to insist on being strict. We may have ambiguity without imprecision. We may also have ambiguity resulting from imprecision. > You suggest interpreting " approximately X" as a probability > distribution centering on X. This is a legitimate interpretation. But > how would you deal with Version 2 using this interpretation ? (Version > 2: If I choose option A ,I will get "approximately a." If I choose > optionB, I will get "approximately b" with probability " approximately > p," or "approximately c" with probability l - "approximately p," with > a lying between b and c. Which option should I choose? Assume a pdf is specified for each occurrence of the word "approximately". Clearly we can calculate the expectation under option A. For option B, use the symbol P for the probability which is "approximately p". The combined pdf is given by averaging the pdfs for approximately b and approximately c, weighted by P and (1-P) respectively. Since P is unknown, we obtain the true combined pdf by integrating over all possible values of P. We then integrate over the combined pdf to get the expectation for option B. Having computed an expected payoff for both options, we choose the option for which it is maximised. > I believe that in the course of trying to come up with an > answer, you will come to the realization that, contrary to conventional > wisdom, existing bivalent-logic-based decision theories break down when > imprecision is introduced. For the type of problem under discussion, where we are asked to select one of a finite and clearly defined set of options, I disagree: the nature of the problem is bivalent and a bivalent theory is well suited for handling it. > The reason, as I have pointed out in earlier > messages,is that such theories are focused on partiality of certainty, > but fail to provide tools for dealing with partiality of truth, > partiality of possibility and partiality of preference. What is not > recognized to the extent that it should, is that certainty, truth and > possibility are distinct concepts. I agree with these points. It may be possible to come up with relevant and well defined problems for which it is necessary to model partiality of truth, but that has not been the case for the examples discussed here recently. sincerely, Konrad
