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Date: Sat, 6 Nov 93 01:21:34 -0500 From: Dan Schwartz <schwartz@iota.cs.fsu.edu> Message-id: <9311060621.AA01638@iota.cs.fsu.edu> To: interlingua@ISI.EDU Subject: higher-order KIF and CG's

Dear Fritz Lehman, This is in response to your request for "a few more examples of higher-order or mixed-order statements". Here follows a brief description of two such systems, both having two distinct semantic levels, and both being in fact a multivalent logic at the first level and a bivalent logic at the second level. I'll leave it to the KIF/CG aficionados to determine whether these can be expressed in their systems. First is a portion of my doctoral dissertation, completed in 1981, and finally published as [1] in 1987. It's aim was to formalize as much as I could of the logic associated with L.A. Zadeh's semantics of fuzzy sets. To illustrate, the language allows propositions like middle-aged(x) == ~old(x) & ~young(x) where middle-aged, old, and young are interpreted as fuzzy subsets of a universe of ages, so that, for an individual age, a, the truth value of say middle-aged(n-a) ('n-a' being the name of age a) is the degree of membership of age a in the fuzzy set associated with middle-aged. Then, as is customary in fuzzy sets theory, the connectives ~, &, and V are interpreted for instantiated formulas exactly as in Lukasiewicz logic, i.e., as 1-, min, and max. As a result, the connective == becomes what Zadeh referred to as "semantic equivalence" (which is closely related to something known in the literature on multivalent logics as "strong equivalence"). To wit, the relations on either side of the == are semantically equivalent if, under all possible instantiations, both sides receive the same truth value. Considering that two relations either are, or are not, semantically equivalent, this leads to a bivalent logic for talking about the properties of semantic equivalence. For example, p(x)==q(x) &-dot q(x)==r(x) ->-dot p(x)==r(x) expresses transitivity of semantic equivalence (the dotted connectives are second level). I was able to develop a semantically complete axiomatization of this system, using a rather obscure theorem from set theory. Semantic completeness can be seen to depend crucially on this theorem, to the extent that semantic completeness holds for theories with countably infinite languages, but fails for uncountable theories. The languages allow fuzzy relations of any arity, but do not include quantifiers. Second is a logic recently published as [2], which aims at capturing such reasoning as *Most* birds can fly. Tweety is a bird. ----------------------------------- It is *likely* that Tweety can fly. and *Usually*, if something is a bird it can fly. Tweety is a bird. --------------------------------------------- It is *likely* that Tweety can fly. Here the task was to devise a language which allows for expressing the implicit relationships between quantification (all, most, many, few, etc.), likelihood (certainly, likely, uncertain, unlikely, etc.) and usuality (always, usually, sometimes, seldom, etc.). All such modifiers are interpreted probabalistically. To illustrate, let M="most x", C="certainly', and L="likely". Then the first syllogism above would be expressed as M C (bird(x)->can_fly(x)) C bird(Tweety) ----------------------------- L can_fly(Tweety) and the first line, e.g., would be "true" if the probability of the proposition bird(x)->can_fly(x) falls within some specified range, say [2/3,1], where this is computed as the conditional probability that something can fly, given that it is a bird. (Similarly, negation, conjunction, and disjuction have probabalistic interpretations at this level.) Thus lower-level (unmodified) propositions have degrees of truth, given as probabilities, and upper-level propositions (using modifiers) are either true or false. >From the standpoint of the upper level, the three lines of the above syllogism are all atomic formulas, and logical combinations of such involving higher-leveled versions of not, or, and, implies, and iff are allowed. There is also an intermediate form of implies, introduced to avoid nesting of inferences at the lower level. For example, letting A="all x", the formula MC(p(x)->q(x))<--->A(Cp(x)-->Lq(x)) says For most x it is certain that p(x) implies q(x) iff for all x, if it is certain that p(x) then it is likely that q(x). The -> is interpreted as above (as conditional probability), the <---> is the classical bivalent iff, and the --> is the intermediate implies, also interpreted as conditional probability. The resulting logic validates all such syllogisms and formulas as above, and is aimed at devising a new approach to nonmonotonic reasoning. It has not yet been axiomatized. [1] D.G. Schwartz, Axioms for a theory of semantic equivalence. Fuzzy Sets and Systems, 21 (1987) 319--349. [2] D.G. Schwartz, Toward a logic for fuzzy syllogisms. Proceedings of the Second IEEE Conference on Fuzzy Systems, San Francisco, CA, March 31--April 1, 1993, pp. 71--75. ************************************************************************ Daniel G. Schwartz Office 904-644-5875 Dept. of Computer Science, MC 4019 CS Dept 904-644-2296 Florida State University Fax 904-644-0058 Tallahassee, FL 32306-4019 Home 904-385-7735 U.S.A. schwartz@cs.fsu.edu ************************************************************************