Deductive completeness and conditionalization in systems of weak implication.
Maribel Díaz · Notre Dame Journal of Formal Logic · 1980
I wish to investigate the conditions under which certain systems of implication satisfy deductive completeness of the kind associated with the deduction theorem (in the sense of Curry and Feys [2]).The systems that I investigate have received considerable attention in the last two decades: the implication fragment of Relevance Logic, R-» (Church's Weak Implication); the implication fragment of strict implication, S4-»; the implication fragment of Anderson and Belnap's System of Entailment, E-»; and the system of Ticket Entailment, T-».None of these systems satisfy deductive completeness except under certain conditions which may be interpreted as the satisfaction of conditions of relevance (for R->), modality (for S4->), relevance and modality (for E-•), and inference ticket/inference distinctions (T->).Thus we might say that they are each deductively complete for an extended notion of deductive completeness.In Section 2, I formulate natural deduction systems NR-», NS4-», NE->, and NT-* which are deductively equivalent respectively to R-», S4-», E-*, and T-•.Each system involves only two rules, one of which is modus ponens and one a form of conditionalization.The conditionalization rule in each case is based on the deduction theorem of the corresponding axiom system.Furthermore, each system is the result of adding a further restriction to only the rule of conditionalization for the previous system.In this form we can see more clearly the relationship between the systems and intuitionistic implication (H-•); and what relevance, necessity, and ticket entailment amount to.Finally, in Section 3, I show how to formulate T-* in terms of a restriction on the rule for modus ponens, and how adding this restriction to modus ponens in E-», R->, S4-», and H-* affects these systems.Let S be a deductive system with an implication operator, z>, which satisfies the rule modus ponens (MP): A, A D B ^B. Curry and Feys [2] called such a system deductively complete if, whenever from a premise B, and possibly other premises, we can derive A, then we can derive B z> A from these other premises alone.It has been shown by Gentzen [3] that