Relevant entailment—semantics and formal systems

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Abstract
This work results from an attempt to give the vague notion of relevance a concrete semantical interpretation. The idea is that propositions may be divided into different “domains of relevance”. Each “domain” has its own “T” and “F” values, and propositions “belonging” to one domain can never entail propositions “belonging” to another, unconnected one.The semantics we have developed were found to correspond to an already known system, which we call here RMI⥲. Its axioms are the implication-negation axioms of the system RM ([1, Chapter 5]). However, as Meyer has shown, RM is not a conservative extension of RMI⥲, since RMI⥲ has the sharing-of-variables property ([5], and [1, pp. 148–149[), which the implication-negation fragment of RM has not.RMI⥲ has four advantages in comparison to its more famous sister R⥲ (the pure intentional fragment of the system R; see [1]):a) It has a very natural (from a relevance point of view) many-valued semantics, the simple form of which we describe here.b) RMI⥲, ⊢ A1 → [A2 → (… → (An → A) …)] iff there is a proof of A from the set {A1, …, An} that actually uses all the members of this set. In R⥲, this holds only if we talk about “sequences” instead of “sets”. This is somewhat less intuitive (see [1, pp. 394–395]).c) RMI⥲ is a maximal “natural” relevance logic, in the sense that every proper extension of it limits the number of “domains of relevance” (§III).

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