**Two problems still open in relevance logic: semantics for E mingle and semantics for Disjunctive Syllogism as a rule of proof**

This research project is based at the Universidad de León (Spain) and is funded by the Spanish Ministry of Science and Innovation (MCIN/AEI/ 10.13039/501100011033), grant no. PID2020-116502GB-I00. It runs from 2021 till 2024.

**Summary**

Consider the following notions:

- Logic B. Routley and Meyer’s basic logic B is the essential relevance logic and the fundamental logic in the development of Routley-Meyer semantics.
- E Mingle (EM). The logic EM is the result of adding the restricted “mingle” axiom, Mr, to Anderson and Belnap’s logic of entailment E (Mr is the thesis (A→B)→[(A→B)→(A→B)]).
- Disjunctive Syllogism (RDS). The rule RDS reads as follows: If A and ∼A∨B, then B.
- Disjunctive Syllogism as rule of proof (rDS). Understood as “rule of proof”, RDS is restricted to the ensuing form: If A and ∼A∨B are theorems, then B is a theorem.

The aim of this proposal is twofold. Relevance logic and Routley-Meyer semantics (RM-semantics) are the connecting thread of this twofold aim.

First aim of our proposal. In what still nowadays is the fundamental work on RM-semantics, Routley et al. refer to EM only once in order to state that Mr is an “intractable principle” not interpretable in RM-semantics (*Relevant logics and their rivals*, Ridgeview, 1982, p. 344). Then, it follows that EM is not interpretable in this semantics either. This may be the reason why 45 years after the publication of *Entailment* I, our ignorance about EM is such that it lacks a semantics whatsoever, despite the fact that Anderson and Belnap consider EM one of the main neighbors of E and that the other 7 main meighbors they quote are long ago very well-known logics (*Entailment* I, Princeton Univ. Press, 1975, p. 343).

Contrarily to the opinion of the creators of RM-semantics, we think that EM can be endowed with this type of semantics once it has been provided with an axiomatization free from Mr and similar axioms. The first aim of our proposal is then to define an RM-semantics for EM and its extensions.

Second aim of our proposal. The fact that relevance logics lack RDS is “the hardest thing to swallow concerning relevance logics”, as Dunn and Restall put it (Relevance Logic, Kluwer, 2002, p. 32). Anyway, most relevant logicians think that the admissibility of the rule makes more “palatable” its actual absence. Admissibility of RDS amounts to have it as a “rule of proof”, i.e., in the form rDS. However, as its is shown in Seki’s works, current RDS admissibilty proof techniques are applicable only in the case of logics with certain properties. The second aim of our proposal is then to free the introduction of rDS in logics interpretable in RM-semantics from the strict conditions the said proof techniques impose. In particular, we prove the following fact. Le S be a logic interpretable in RM-semantics and equivalent to or including the basic logic B. Then, the extension of S with rDS is given an RM-semantics.

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