The Relevant Fragment of First Order Logic

Abstract

Under a proper translation, the languages of propositional (and quantified relevant logic) with an absurdity constant are characterized as the fragments of first order logic preserved under (world-object) relevant directed bisimulations. Furthermore, the properties of pointed models axiomatizable by sets of propositional relevant formulas have a purely algebraic characterization. Finally, a form of the interpolation property holds for the relevant fragment of first order logic.

Infinitary Propositional Relevant Languages With Absurdity

“…Finally, by Proposition 2.5 of [2], the pair pΘ, Φq is satisfiable in V S , which is a contradiction since by definition Θ says that at least one of φ P Φ must hold. % When |PROP| ě ω, inconsistency is expressible by a single formula in the extension…”

“…For all u and arbitrary e n such that u P B M4 1 and M 4 ( Ire n us, we have that M 4 ( Gruabs, and given that there is such a u, we have that aZ 1 b. But one of the formulas in Θ implies that there is also v P B M4 For clause (2) in Definition 4, suppose that i P t1, 2u and xZ i y, so there is e n (n P ω) in the sequence p˚q such that there is u P B M4 i , M 4 ( Ire n us and M 4 ( Gruxys. Now let R rU M 4 j s M 4 |K ybc for some b, c P U M4 j , i.e., R 4 ybc by Lemma 10.…”

“…For clause (2) in Definition 3, suppose that xZ pβ`1qi y pβ`1 ď αq, i.e., rel ďβ`1tp Mi pxq Ď rel ďβ`1 -tp Mj pyq, and R j ybc for some worlds b, c from M j . Where Fmla(L Ñ 8ω ) stands for the class of propositional relevant formulas of L Ñ 8ω , consider nrel ďβ -tp Mj pcq " tφ : M j , c .…”

“…There are certain differences in method worth mentioning, though. In [2], there was an appeal to the machinery of saturated models in order to establish a preservation theorem characterizing relevant formulas as a fragment of first order logic. This was, in fact, unnecessary for a much more direct proof through a simple application of the compactness theorem of first order logic was possible.…”

“…It would have simply required the introduction of the notion of a relevant directed n-bisimulation, a finite approximation of a relevant directed bisimulation. 2 This approach is so basic that it generalizes to logics having some minimal forms of compactness such as L ∞ω . That is the main motivation behind our introduction of relevant directed α-bisimulations in Definition 4.3.…”