On The Conceptual History Of Formal Implication

This paper is devoted to exploring the ideological context of formal implication in the history of western logic and its practical effect in philosophy of logic,trying to put forward new ideas in both historical and theoretical aspects.The former belongs to the study of the history of logical thought.Diodorus Cronus,B.Russell and C.S.Peirce's thoughts on formal implication are mainly discussed,thus three conclusions are drawn:(1)Although the initial claim of Diodorus' implication theory is the modal considerations in the semantic of conditional sentences,his unique view of proposition and modality makes his implication theory actually become a "special formal implication"varied in the domain of moments.(2)As the author of the concept of "formal implication",Russell's understanding in Principia Mathematica is not mature enough.After mastering Frege's logical quantifier theory,after making critical distinctions between proposition and propositional function,he eventually elaborates the complete connotation of formal implication.The conclusion is that the formal implication is a general relationship between the propositional functions that share individual variables.Each substitution of individual constant to individual variable expresses a "material implication" relationship.(3)Peirce develops the modern logic quantification theory through the work of logical algebra and existential graphs,and thus clearly reveals the connotation of formal implication.But unlike Russell,he also favors Diodorus' modal considerations and attempts to extend the scope of the quantification to the range of possibilities.At the end of this paper,the concepts of general formal implication and special formal implication are taken to examine the material implication paradoxes and the restricted quantifier theory,which highlights the unique meaning of the concept of"special formal implication".These belong to the study of application of the thought of formal implication,thus two main conclusions are drawn:(1)There emerges a unique approach to resolve the counterexamples when material implication paradoxes are analyzed from the perspective of formal implication.Sliced into the deep logical structure behind the daily language from the quantitative perspective,some conditional sentences' components do not necessarily express propositions but propositional functions,thus these conditionals' logical structure is a special formal implication quantified within a limited individual domain.(2)"The restricted quantifier theory" suggests that each occurrence of quantifier should be restricted by the scope of the domain,which may make the antecedent and consequent in the conditional sentence both restricted by different "restricted quantifiers".However,from the point of view of formal implication,such conditional sentence can still be reduced to general formal implication constrained by logical quantifier,so it is actually another kind of "special formal implication"(which is different from the Diodorean "special formal implication").