The undecidability of entailment and relevant implication

In this paper we show that the propositional logics E of entailment, R of relevant implication and T of ticket entailment are undecidable. The decision problem is also shown to be unsolvable in an extensive class of related logics. The main tool used in establishing these results is an adaptation of the von Neumann coordinatization theorem for modular lattices.Interest in the decision problem for these systems dates from the late 1950s. The earliest result was obtained by Anderson and Belnap who proved that the first degree fragment of all these logics is decidable. Kripke [11] proved that the pure implicational fragments R→ and E→ of R and E are decidable. His methods were extended by Belnap and Wallace to the implication-negation fragments of these systems [3]; Kripke's methods also extend easily to include the implication-conjunction fragments of R and E. Meyer in his thesis [14] extended the result for R to include a primitive necessity operator. He also proved decidable the system R-mingle, an extension of R, and ortho-R (OR), the logic obtained from R by omitting the distribution axiom. Various weak relevant logics are also known to be decidable by model-theoretic proofs of the finite model property (see Fine [5]). Finally, S. Giambrone [7] has solved the decision problem for various logics obtained by the omission of the contraction axiom (A → ⦁ A → B) → ⦁ A → B, including R+ − W (R+ minus contraction). It is worth noting that even where positive results were obtained, the decision methods were usually of a complexity considerably greater than in the case of other nonclassical logics such as intuitionistic logic or modal logic, a fact which already indicates the difficulty of the decision problem.