Filter-induced entailment relations in paraconsistent Gödel logics

We consider two expansions of Gödel logic $G$ with two versions of paraconsistent negation. The first one is $G_{\mathrm{inv}}$ — the expansion of $G$ with an involuitive negation ${\sim }_i$ defined via $v\left({\sim }_i\varphi \right)=1-v\left(\varphi \right)$. The second one is

— an expansion with a so-called ‘strong negation’ ¬. 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 ¬. Two valuations in

can be combined into one valuation v on ${[0,1]}^{\bowtie }$ — the twisted product of $[0,1]$ with itself — with two components $v_1$ and $v_2$. The two logics are closely connected as ${\sim }_i$ and ¬ allow for similar definitions of co-implication —

and

— but do not coincide since the set of values of

is not ordered linearly.

Our main goal is to study different entailment relations in $G_{\mathrm{inv}}$ and

that are induced by filters on $[0,1]$ and ${[0,1]}^{\bowtie }$, 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 ${[0,1]}^{\bowtie }$, and obtain their hierarchy. We also construct reductions of filter-induced entailment relations to the ones defined via the order.