Formalizing the conditionals of Diodorus Cronus and Chrysippus: logic and potential applications in human thinking

Although formal logics are supposed to express some aspects of human thinking, none of them achieve the desired success. These failures sometimes cause - at conferences - voices of resignation claiming that logic is not a science whose object is human thinking. And yet Aristotle considered logic to be an organon, or tool for practicing other sciences. Accepting this point of view, the present paper starts from the assumption that the main incompatibilities of formal logics with our thinking are due to the fact that the implications defined in these logics are very distant in nature from the conditional we use in everyday thinking. Our paper presents two proposals to formalize conditional in such a way that both they and the inferences defined by them do not have properties alien to our thinking. To achieve this goal, we make the assumption that the content of sentences must play a fundamental role in the formal reconstruction of both implication and inference. An essential idea of our approach is that the truth of a conditional is grounded in the inference relation between antecedent and consequent, and more precisely, it should depend on the possibility or impossibility of deriving a contradiction from acceptance and/or rejection of the antecedent and consequent content – an approach characteristic for inferentialism. Since the essence of the paper is a conditional, propositional logic is a sufficient field of consideration.