On Implications

Implications are essential concepts in the whole process of modern logic development.The naissance of material implication greatly produced many controversies,and the paradoxes of implication have always been the focus of those arguments.On the basis of different explanations and descriptions,all different kinds of implications were formed,and so did different modern logic systems.Authors of this paper will try historically to distinguish and clarify their true senses among all sorts of implications.Lin's Entailment Logic,LEL for short,was ever misconstrued by some people.Critics rebuked that conceptions of necessity and possibility could not be defined in LEL.Relationship of mutual entailment existed between necessity and possibility in notional calculus system Cn of LEL.Systems of LEL comprised paradoxes of logic. 1=0 could be proved in the axiom of the system N for natural numbers,but the first axiom of N is 1≠0,therefore,N is inconsistent.Another purpose of this paper is to clarify some facts and clear the air.