Logics of left variable inclusion and Płonka sums of matrices

Bonzio S.;Moraschini T.;PraBaldi M.

2021-01-01

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Abstract

The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic ⊢. We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic ⊢ is related to the construction of Płonka sums of the matrix models of ⊢. This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate them in the Leibniz hierarchy.