Towards Fuzzy Partial Logic

In this paper, we discuss the possibility to develop fuzzy partial logics in which some formulas may have undefined truth values. The main idea is to consider semantics of these logics formed by algebras of truth values extended by a special value "*". This value may have several interpretations, such as "undefined", "meaningless", "non-applicable", etc. This approach requires extension of the original connectives to new ones that behave as the original ones if all of their arguments are defined. We also present a general method of defining the new connectives and outline axioms and deduction rules for these logics.