On Sahlqvist formulas in relevant logic (Q1670491)

This is an interesting paper proving a Sahlqvist theorem for relevant logic similar to the one definable in modal logic. It is also proven that there are formulas in the relevant language which do not define elementary classes, thus showing that ``the problem of elementarity is not trivial'' (p. 674). The author remarks that the result in the paper ``improves our understanding of the first order properties of the Routley-Meyer frames which are definable in relevant logic'' (p. 674). It is important to note the following points. (1) The relevant language used includes $\circ $ (fusion) and $t$ (the Ackermann constant) in addition to $\rightarrow $ (implication), $\wedge $ (conjunction), $\vee $ (disjunction) and $\sim $ (negation). (2) It is used an unreduced Routley-Meyer semantics where validity is defined w.r.t. a set of designated points in the models (cf. [\textit{R. Routley} et al., Relevant logics and their rivals. Part I: The basic philosophical and semantical theory. Atascadero, California: Ridgeview Publishing Company (1982; Zbl 0579.03011)]). The particular class of models used corresponds to Sylvan and Plumwood's minimal De Morgan logic $\mathrm{B_M}$ (cf. [\textit{R. Sylvan} et al., Relevant logics and their rivals. Vol. II. A continuation of the work of Richard Sylvan, Robert Meyer, Val Plumwood and Ross Brady. Aldershot: Ashgate Publishing (2003; Zbl 1398.03011), ``Non-normal relevant logics'', p. 10--16]) expanded with $\circ $ and $t$. (We note that the semantical conditions $x\leq x^{\ast \ast }$ and $x^{\ast \ast }\leq x$ generally do not hold in this class of models, whence neither $\sim \sim A\rightarrow A$ nor $A\rightarrow \sim \sim A$ are validated, while the rule contraposition holds only in the form $A\rightarrow B/\sim B\rightarrow \sim A$.)