The Paradoxes Of Implication And The Problems Of Induction:a New Logic Pattern Based On The Unification Of Deduction And Induction

One of the main aims of research in logic is to analyze natural language arguments. If any logical system is to accomplish this aim, the forms of natural language arguments and the logical laws behind the forms must be depicted with sufficient accuracy. Whether classical logic is accurate enough to be of value in the analysis of natural language arguments has always been a topic of hot debate in academia. The "paradoxes of implication" and the "problem of induction" (i.e. paradoxes associated with induction) have arisen from these debates and have been subject to sustained discussion. Various solutions put forth each have their own deficiencies and are, thus, hard to accept. On the basis of an examination of previous research, this paper attempts to break out of the age-old paradigm of treating deduction and induction as separate research subjects. As a single solution to the problems of both, it presents a new logic pattern based on the unification of deduction and induction.This paper first introduces the motivations, goals, and guiding principles of this research project. It then reviews the paradoxes of implication and their relation to the problems of induction, as well as examining previously proposed solutions. In the process. the difference between natural language "If.... then..." statements and material implication in classical logic is clarified in detail. After which, the conclusion is drawn that the paradoxes of implication and the problems of induction arise due to this non-correspondence between the logical forms depicted by classical logic and the true logical forms of natural language arguments.Next, this paper puts forth that implication propositions concerning matters of fact (as opposed to relations of ideas) can be divided into two types:"observed" and "inferred". The meaning of an "observed" implication proposition is very close to that defined by material implication. However, the meaning of an "inferred" implication proposition is not. Rather, the objects denoted by an "inferred" implication proposition have not all been observed to satisfy the implication proposition. Thus, the grounds for claiming the truth of an "inferred" implication proposition can only be induction. It is negligence of the difference between these two types of implication propositions that has given rise to the "paradoxes of implication". On the basis of this division, three common characteristics of previous solutions to the paradoxes are summarized and asserted to be the reason why previous solutions have time and again proven deficient.Building on the foundation of the above research, the basic principles of inductive arguments and the problems associated with traditional approaches to induction are discussed and a new logic pattern is presented as a solution to the above problems. Grounded in induction, this new logic pattern differs from previous solutions to the "paradoxes of implication" on the three common characteristics discussed above. The practice in classical logic of depicting "inferred" implication propositions as universal statements linking abstract properties with other abstract properties is refuted and replaced by a relation of concrete objects with other concrete objects, which thereby unifies the traditional logical forms of deduction and induction. First, a simplified, easy-to-use’three-step pattern" is presented. Then, a more rigorous formal proof is given, for which only a single additional axiom need be added to the axiomatic system for classical propositional logic. This additional axiom is derived from the foundational principles of statistics.The assumptions of a "stochastic process" and "equal selection" are at the heart of this new axiom, prompting an in-depth discussion. According to basic statistical principles, if the objects of the premises and of the conclusion are selected in the same way from a stochastic process, then because of this special relation between the objects, it can be inferred with high probability that the properties of the objects will be the same. This is not to say that both groups of objects will have identical properties, but rather that the difference between the objects of the conclusion and those of the premises will not be greater than the difference between the objects of the premises themselves. Rather than trying to depict the basis for these already widely accepted assumptions, the new logic pattern presented in this paper delineates the logical form of inferences that start off with the assumption of a "stochastic process" and "equal selection".After the presentation of the pattern, this paper systematically elucidates this new logic pattern to be a unified solution to the problems of induction and the paradoxes of implication. It also briefly explores the practical value and application of the pattern.