Presumption and prejudice in logical inference

Two nonstandard modes o f mference, conftrmatton and dental, have been shown by Bandler and Kohout to be vahd m fuzzy propositional and predicate logics. I f demal ts used m combmatton with modus ponens, the resultmg mference mode ("augmented modus ponens") ytelds more prectse bounds on the consequent of an implication than are usually called for m approximate reasoning. Slmdar results hold for augmented modus tollens constructed from conftrmatton and conventtonal fuzzy modus tollens Two simpler modes o f mference, presumption and prejudtce, are also vahd under the same assumptions as conftrmatton and dental PreJudice tmposes an upper bound on the truth value of the consequent of a fuzzy tmphcatton regardless of the truth value of the antecedent; presumption imposes a lower bound on the truth value of the antecedent regardless of that o f the consequent. Some of the consequences o f presumptton and prejudzce cast doubt on the suttabthty o f fuzzy propostttonal and predtcate logtcs for use m expert systems that are destgned to process real-world data. A logic based dtrectly on fuzzy sets is explored as an alternative. Fuzzy set logic supports fuzzy modus ponens and modus tollens but does not entad the more problematic modes of confirmation, demal, presumptton, and prejudwe However, some of the expressive power derivable from the dtverstty of fuzzy proposttzonal logws and thetr dertvattve fuzzy predicate logics ts lost. K E Y W O R D S : f u z z y logic, inference modes, propositional logic, predicate logic, set logic * Supported m part by a NASA/ASEE Summer Faculty FeUowshlp Address correspondence to Professor Thomas Whalen, Department of Dectswn Sciences, Georgia State Umversity, Atlanta, Georgia 30303 International Journal of Approxtmate Reasoning 1989, 3 359-382 © 1989 Elsevier Science Pubhshmg Co , Inc 655 Avenue of the Americas. New York, NY 10010 0888-613X/89/$3 50 359 360 Thomas Whalen and Brian Schott I N T R O D U C T I O N An elementary logical statement 1s the basic unit of any logic system, in the sense that it is the smallest logical unit to which a truth value can be assigned within the system One of the most fundamental ways of dividing the general field of logic is by the way statements are analyzed In propositional logic, elementary statements are treated as unanalyzed units; in predicate logic, an elementary statement asserts that a particular object has a particular attribute or that a particular set of objects stand in a particular relation to one another; and in set logic, an elementary statement asserts that an object belongs to a particular set. In the following section we discuss propositional logic, considering both the traditional binary truth value systems and multlvalent fuzzy systems. The latter discussion concentrates on multivalent logical operations that follow the axioms of continuous triangular norms (T-norms) and their dual triangular conorms (Schwelzer and Sklar [1, 2]) Drawing upon recent work on the nonstandard inference modes of confirmation and denial (Bandler and Kohout [3], Hall [4], Schwartz [5]), we derive "augmented" versions of modus ponens and modus tollens for multlvalent logic, using the method of "residuation" (Trillas and Valverde [6]) to derive both an upper bound and a lower bound for the inferred truth value; these bounds often coincide, yielding a unique answer Two additional modes of inference, presumption and prejudice, are derived using the same techniques; these modes entail a lower bound on the truth of the antecedent and an upper bound on the truth of the consequent from knowledge of the truth of the implication alone The third section presents a parallel development of similar results for the case of predicate logic It concludes with a simple demonstration of "knowledge base psychosis," in which multiple rules of fuzzy predicate logic interact to preclude the denial of any base value of the antecedent variable due to presumption, and to preclude the affirmation of any base value of the consequent variable due to prejudice In the fourth section we consider inferences derived from a more fundamental grounding in set logic We consider a single primary universe of discourse consisting of objects that are classified into fuzzy subsets on the basis of the values of several attributes; an inference rule in this sytem consists of the assertion that all objects belong to a consequent fuzzy subset C at least as strongly as they belong to an antecedent fuzzy subset A. In this environment, ordinary fuzzy modus ponens and modus tollens are well-defined according to the "standard strict" lmphcatlon operator and the compositional rule of inference, but confirmation, denial, presumption, and prejudice are not derivable. Thus, no a priori constraints are placed on the antecedent or the consequent Presumption and Prejudice m Logical Inference 361 We next illustrate these results with a simple example based on the fuzzy relation between fast driving and poor fuel economy. We conclude with a summary of the implications of these results and an outline of future research involving the concept of "usuahty" in fuzzy set logic P R O P O S I T I O N A L LOGIC An implication rule in propositional logic takes the simple form " i f (antecedent) then (consequent)," or " I f A then C " for short In standard twovalued logic, the only way that this lmphcatlon can be judged false is if A is true while C is false; thus, the implication is held to be true if C is true or if A is false In other words, the truth or falsity of the statement " I f A then C " is identical to the truth or falsity of the statement " C or Not A " Many theoretical treatments of material implication in multivalent propositional logic preserve this equivalence; in such a logic the degree of truth attached to " I f A then C " can be computed by finding the truth value of " C or Not A " Different systems of multlvalent logic arise from the choice of a truth function to represent " o r " and to a lesser extent from the representation of "Not " The most important class of such systems IS the class of S-lmphcations, in which the " o r " operator is defined by a T-conorm (Schwelzer and Sklar [1, 2], Bonissone [7]), the most important S-imphcatlons are Kleene-Dienes, probabihstic and Lukasiewlcz. Other systems of multivalent logic make a distinction between " I f A then C " and " C or Not A , " but every propositional tmphcation operator ~ defines a truth value Tr(A ~ C) = I[Tr(A), Tr(C)] that varies directly with Tr(C) (the truth of C) and inversely with Tr(A ) (the truth of A ). 1 The most important class of logics that do not postulate an identity between " I f A then C " or " C or Not A " is the class of R-implications, which define the modus ponens operation in terms of a T-norm and derive the lmphcation operator to fit; the most important R-imphcatlons are Brouwer (also known as "standard star"), quotient, and Lukaslewicz. (Note that the Lukaslewicz logic satisfies the definitions of both Slmphcation and R-implication, this versatihty is surely not unrelated to the popularity of the logic.) Several authors have performed more or less empirical comparisons among operators gleaned from the literature, with conflicting results depending on which particular real or simulated domain of application was used (Mizumoto [8, 9], Whalen and Schott [10]). On the theoretical side, other researchers start with a particular set of assumptions and infer an lmphcatlon operator that suits them (Sanchez [11], Smets and Magrez [12,13]). The safest overall conclusion The only major exception is Mamdam's operator, which is not a material ~mphcauon operator ("If-then") but rather a conjunctaon operator ("and") Inference m Mamdam's system is more properly treated as a generalized alternatwe syllogism than as generahzed modus ponens 362 Thomas Whalen and Bnan Schott from these studies ~s that different situations appear to require different implication operators. Modus ponens allows us to infer a lower bound for Tr(C), the truth value of the consequent, from the truth value of the imphcation Tr(A --, C) and the truth value of the antecedent Tr(A); this lower bound is equal to the truth value of " A and (ff A then C ) , " where " a n d " is defined by the modus ponens generating function mp (Tnllas and Valverde [6]) proper to the implication operator I that is used to define " i f then" where T r ( C ) _ m p [ T r ( A ) , Tr(A --, C)] mp(x, t ) = m f { y : I(x, y)>_t} If the antecedent and the lmphcatlon are both totally true [Tr(A) = Tr(A ~ C) = I], then the lower bound on the truth value of the consequent Is 1, or " t rue , " since mp(1, 1) = 1 under any modus ponens generating function mp. One is also the universal upper bound of truth values, so in this case the truth value of the consequent Is completely determined as "true " On the other hand, if the antecedent is totally false (truth value = 0), then the lower bound on the truth value of the consequent is only the universal lower bound zero, since mp(0, x) = 0 for any x This does not mean that the consequent is false, only that modus ponens places no restnctlons on the truth or falsehood of the consequent For truth values of the antecedent and the lmphcatlon between total truth and total falsehood, the lower bound for the truth of the consequent varies monotonically Modus tollens allows us to infer an upper bound for Tr(A), the truth value of the antecedent, from Tr(C) and Tr(A --* C) by computing the truth value of " ( I f A then C) but not C , " defining "but not" by the modus tollens generating function mt proper to the implication operator" Tr(A)_<mt[Tr(C), Tr(A ~ C)]