A Modern Syllogistic Method in Intuitionistic Fuzzy Logic with Realistic Tautology

The Modern Syllogistic Method (MSM) of propositional logic ferrets out from a set of premises all that can be concluded from it in the most compact form. The MSM combines the premises into a single function equated to 1 and then produces the complete product of this function. Two fuzzy versions of MSM are developed in Ordinary Fuzzy Logic (OFL) and in Intuitionistic Fuzzy Logic (IFL) with these logics augmented by the concept of Realistic Fuzzy Tautology (RFT) which is a variable whose truth exceeds 0.5. The paper formally proves each of the steps needed in the conversion of the ordinary MSM into a fuzzy one. The proofs rely mainly on the successful replacement of logic 1 (or ordinary tautology) by an RFT. An improved version of Blake-Tison algorithm for generating the complete product of a logical function is also presented and shown to be applicable to both crisp and fuzzy versions of the MSM. The fuzzy MSM methodology is illustrated by three specific examples, which delineate differences with the crisp MSM, address the question of validity values of consequences, tackle the problem of inconsistency when it arises, and demonstrate the utility of the concept of Realistic Fuzzy Tautology

Uogólnione schematy i reguły wnioskowania w logice rozmytej

In Chapter 1 we recall the most important notions in the theory of fuzzy sets. In Chapter 2 we give some basic facts regarding approximate reasoning. In the main part of the thesis we consider functional equations and inequalities mentioned above when one function is given - usually a t-norm T (or a semicopula or any other generalization of a classical conjunction). Therefore we show some solutions for chosen families of fuzzy im- plications. In Chapter 3 we focus on hypothetical syllogism - there are solutions for (CRI-GHS), (BK-GHS) and (HS), but also we present here some algebraic properties of a composition sup-T. Chapter 4 contains information regarding solutions of (CRI-GMP), (BK-GMP) and (MP). In Chapter 5 analogous facts for (CRI-GMT), (BK-GMT) and (MT) can be found. Solutions for (CRI-GRA), (BK-GRA) and (RA) have been described in Chapter 6. In Chapter 7 we shortly give some remarks concerning some other possible functional equa- tions that might be received when combinig rules of inference and different fuzzy relations. In last Chapter 8, we present a different method of reasoning - similarity based reasoning. Also we give some remarks for two main strategies in approximate reasoning - FITA (First Infer Then Aggregate) and FATI (First Aggregate Then Infer) with respect to some proven theorems

On the chaining syllogism in fuzzy logic

In this paper we deal with an extension of the fuzzy chaining syllogism in fuzzy logic, which was initiated by Zadeh [LA. Zadeh, Syllogistic reasoning in fuzzy logic and its application to reasoning with dispositions, IEEE Trans. on Systems, Man and Cybernetics, 15, 754-763, 1985]. We introduce the extended fuzzy chaining syllogism, and investigate it with three important classes of fuzzy implications against the validity of their deduction scheme in fuzzy logic. We show that the fuzzy chaining syllogism holds for an arbitrary fuzzy implication and a continuous extended triangular norm as long as the fuzzy syllogism holds. For an arbitrary triangular norm and an S- or QL- implication we prove that the generalised Kleene-Dienes implication constitutes a lower bound for the inference result of the chaining syllogism. Similarly for Zadeh's triangular norm (minimum). and an R-implication I we obtain this I as a lower bound for the conclusion of the chaining syllogism