Abstract. It is proved that the relevance logic R (without sentential constants) has no structurally complete consistent axiomatic extension, except for classical propositional logic. In fact, no other such extension is even passively structurally complete.
In [60] N. Belnap presented an 8-element matrix for the relevant logic R with the following property: if in an implication A → B the formulas A and B do not have a common variable then there exists a valuation v such that v (A → B) does not belong to the set of designated elements of this matrix. A 6-element matrix of this kind can be found in: R. Routley, R.K. Meyer, V. Plumwood and R.T. Brady [82], Below we prove that the logics generated by these two matrices are the only maximal extensions of the relevant logic R which have the relevance property: if A → B is provable in such a logic then A and B have a common propositional variable.
The first attempts were given by A. Hofstadter and J. C. C. McKinsey in [7] and J. J~rgensen in [8]. It is worth noticing that the key problem of G. H. von Wright's book Norm and Action was giving a definition of entailment of norms. Cf. also N. Rescher [9], B. Chellas [4], E. Sosa [ll], [12], C. L. Hamblin [6]. Very interesting remarks concerning the notion of entailment of norms can be found in papers of 0. Weinberger (cf. 1141) and L. Gumanski (cf. [S]). ~7 1 ) . Cf. A. Ross [lo], C. L. Hamblin [6].
In Handbook of Philosophical Logic M. Dunn formulated a problem of describing pretabular extensions of relevant logics (cf. M. Dunn [1984], p. 211; M. Dunn, G. Restall [2002], p. 79). The main result of this paper is described in the title.
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