Filter-induced entailment relations in paraconsistent Gödel logics

Abstract

We consider two expansions of Gödel logic $\mathsf{G}$ with two versions of paraconsistent negation. The first one is $\mathsf{G_{inv}}$ -- the expansion of $\mathsf{G}$ with an involuitive negation ${\sim_\mathsf{i}}$ defined via $v({\sim_\mathsf{i}}ϕ)=1-v(ϕ)$. The second one is $\mathsf{G}^2_{(\rightarrow,-\!<)}$ -- an expansion with a so-called strong negation $\neg$. This logic utilises two independent valuations on $[0,1]$ -- $v_1$ (support of truth or positive support) and $v_2$ (support of falsity or negative support) that are connected with $\neg$. Two valuations in $\mathsf{G}^2_{(\rightarrow,-\!<)}$ can be combined into one valuation $v$ on $[0,1]^{\Join}$ -- the twisted product of $[0,1]$ with itself -- with two components $v_1$ and $v_2$. The two logics are closely connected as ${\sim_\mathsf{i}}$ and $\neg$ allow for similar definitions of co-implication -- $ϕ-\!<χ:={\sim_\mathsf{i}}({\sim_\mathsf{i}}χ\rightarrow{\sim_\mathsf{i}}ϕ)$ and $ϕ-\!<χ:=\neg(\negχ\rightarrow\negϕ)$ -- but do not coincide since the set of values of $\mathsf{G}^2_{(\rightarrow,-\!<)}$ is not ordered linearly. Our main goal is to study different entailment relations in $\mathsf{G_{inv}}$ and $\mathsf{G}^2_{(\rightarrow,-\!<)}$ that are induced by filters on $[0,1]$ and $[0,1]^{\Join}$, respectively. In particular, we determine the exact number of such relations in both cases, establish whether any of them coincide with the entailment defined via the order on $[0,1]$ and $[0,1]^{\Join}$, and obtain their hierarchy. We also construct reductions of filter-induced entailment relations to the ones defined via the order.