From pcliffje@CRL.COM Sat Mar 6 22:45:13 2010 Date: Tue, 30 May 1995 13:32:50 -0700 From: "John E. Clifford" Subject: fuzzy logic To: Bob LeChevalier X-From-Space-Date: X-From-Space-Address: LOJBAN%CUVMB.BITNET@uga.cc.uga.edu Message-ID: Not too surprisingly, logicians have been involved with Loglan/Lojban from almost the start and so provisions have been made for extended logics from early on -- originally just many-valued or probability-valued bu t fuzzy soon after Zadeh's original paper. In one sense, this has been pretty easy, since the *language* of even the most extreme Wooky logics differ hardly at all from that of a standard logic and certainly not from such a language modified to be speaka ble in real-world contexts. In short, as xorxes has pointed out, lojban is equipped already to do fuzzy logic -- linguistically at least. A couple of distinction make the discussion clearer, however. Zadeh -- and some even madder epigones -- have developed at least three fuzziness theories: set theory, logic, and arithmetic. The set theory takes the range of the characteristic function of a set from the usual {0,1} to [0,1], from a set with two members to a closed real interval. But the underlying logic of this theory is two-valued: c-set(object)=r gives the right value or it does not. The fun comes in figuring out how the values for derivative sets comes from that for basic sets -- various kinds of intersections and unions and (worst, since it does not work at all regularly) subselections (red horses as opposed to things red and horses). Considering the range of possible members, each set could be seen to have a characteristic membership gradient (thank you, and) and one of the developments finding sets related to a given set but with different gradients, sharper ("very," "extremely," "perfectly" -- this lasted tended to be almost perpendicular) or flatter ("sorta," "somewhat," "more or less" and so on). In lojban these are the tanrus and the lujvos with _mutce_ and its compounds and opposites (and probably other words as well, if we need them). >From this, they moved (almost naturally) to taking the value of the characteristic function for the object as value of the sentence "object is a member of set" or even "set object." This undergirds a logic and has its own definitions of the ususal connectives (strange but no stranger than those in the probability interpretation and falling well within the framework laid down by the two-valued system, the least restrictive logic). Among the new connectives actually expressed in this system, most result in sentences which increase or decrease the truth value of underlying sentence in a fixed way, similar to -- but different from -- the effects of modifying the membership gradient on sets. The modificiations are, in both cases, mathematically defined in standard mathematics. The metalanguage of the logic is still two-valued, i.e., the truth value of a sentence either is or is not r, some specific real number. The fuzzyists have noted the problem that xorxes keeps pointing out, that officially the characteristic function value or the truth value is this very precise number. To meet this they have developed fuzzy numbers and the corresponding fuzzy arithmetic. I haven't fiddled with this much (it's not logic so I would be less adept at it) but it seems to function on my grandmother's principle, "Many a mickle maks a muckle," with a variety of "numbers" not unlike the range of items lojban has in quantifier set s: "many," "few," "several," and so on, including the "about n" sort. These then serve as values for fuzzy characteristic or truth functions in the latest versions of the earlier theories. The metatheory to this theory is a fuzzy set theory over ordinary numbers, which are more or less in given ranges. pc >|83