Finite and analytic Hilbert-style systems for Paraconsistent Weak Kleene and Bochvar-Kleene logics

Trends in Logic XXII, Cagliari, Italy (online presentation)

No finite Hilbert-style axiomatization was known so far for Paraconsistent Weak Kleene and Bochvar-Kleene logics. In this presentation, we offer finite and analytic Set-Set (that is, multiple-conclusion) systems for these logics, as well as we show finite traditional Hilbert-style systems for them, answering positively to the open question of whether these two logics are finitely axiomatizable. At the end, we point to future directions of research fundamented on two-dimensional notions of logic, proposing to capture in the same logical structure these two logics plus the associated (non-Tarskian) ts- and st-consequences.