Omitting Types Theorem for Fuzzy Logics

Petr Cintula,Denisa Diaconescu

doi:10.1109/tfuzz.2018.2856084

Abstract

In this paper, we generalize the omitting types theorem, an important result of classical model theory, for a wide class of fuzzy logics, containing the prominent logics of left-continuous t-norms and uninorms.

Highlights

M ODEL theory of fuzzy structures is a branch of mathematical fuzzy logic [1], [26], [27] which is recently getting significant attention [2]–[7].In classical model theory, a type is a collection of formulas illustrating how elements in a structure might behave
4) In classical logic, the isolated types over a complete theory are realized in all its models
This is clearly not the case in our general setting; again, an interesting question would be to explore in which fuzzy logics the claim holds

Summary

INTRODUCTION

M ODEL theory of fuzzy structures is a branch of mathematical fuzzy logic [1], [26], [27] which is recently getting significant attention [2]–[7]. Omitting types theorem can be investigated in different settings, e.g., for continuous logic or in the context of model theory of metric structures [12], [13]. Really “tailored” for the use in classical logic; a more complex notion is needed to obtain the result in our fuzzy setting (in Remark 5 we show that in the classical setting both notions coincide; one can observe how strong properties of classical logic are needed to prove this fact) There is another complication caused by the fact that model theory of fuzzy logics is mostly studied in the setting without crisp equality of elements (see the comments before Lemma 6 for more details).

LOGIC MTL

OMITTING TYPES THEOREM FOR A WIDE CLASS OF FUZZY LOGICS

CONCLUSION AND FUTURE WORK