Consistency, Roots And Divergency Degrees Of Sets Of Propositions

In this article, three proposional logical systems-the classical,Lukasiewicz and L* proposional logical systems-are included.We study consistency,roots and divergency degrees of sets of propositions in these systems. We divide this article into three chapters.The question whether a theory(a proposional set) we are dealing with is consistent is one of the crucial question in any logical systems. is said to be consistent if -D( ) F or else it is inconsistent.We always pay attention to the consistent proposional sets.So,it is necessary for us to study the consistency of proposional sets to apart from the inconsistency proposional sets.In the first chapter of this article,we discuss the consistency of proposional sets in L* systems and mainly study two special and important proposional sets-maximal proposional sets and complete proposional sets.Some examples are given to further explain these two sets.At the same time, we distinguish the L* systems from the classical logical system on the above aspect.As is well known human reasoning is approximate rather than precise in riature.There is a natural question in any logical system:suppose that a theory is given, then which formulas can be deduced from by using deduction rules?Professer Wang has proposed the concept of roots of proposional sets firstly in [4].From the sense of roots,we see that all of -consequences can be obtained as long as the least -consequence(root) is known.Notice that [1] only proved that every finite proposional set has root in classical logical systems.we still don't know the existence of roots and its characteristics in L* system.In the second chapter of this article,we study and answer this question.At first,we give the defination of the dependent proposional sets,and get this conclusion on the dependent sets:any finite dependent set has roots,otherwise any infinite dependent set has no roots.Secondly,we define the reduct of the proposional sets.With the help of this concept,we extend the above conclusion to any proposional set.Moreover,the question of to what extent a fuzzy theory is consistent or inconsistent is also one of the crucial question in any fuzzy logical system.[9] Forturiately,the idea based on the concept of divergence degrees of theories can be used to solve this problem.Divergence degrees of theory change from 0 to 1, is said to be fully divergent if its divergence degree is 1. Obviously,every inconsistent theory is fully divergence.However,is fully divergence theories to be inconsistent?This is a unclear problem. In the third chapter of thisarticle ,this problem are to be solved in the three systems.