On First-Order Pure Relevance Logics

Abstract We describe two ways of introducing a universal quantifier in the context of the maximal propositional relevance logic $$\textbf{RMI}_{{\mathop {\rightarrow }\limits ^{\lnot }}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>RMI</mml:mi> <mml:mover> <mml:mo>→</mml:mo> <mml:mo>¬</mml:mo> </mml:mover> </mml:msub> </mml:math> . One takes it to be an infinite version of a purely relevant counterpart of the additive conjunction that is usual to add to purely multiplicative relevance logics. The other takes it to be an infinite version of the multiplicative (or ‘intensional’) conjunction of relevance logics. We provide semantics and corresponding (strongly) sound and complete proof systems in both cases (separately), as well as to the logic that is obtained by combining them.