  We introduce a paraconsistent expansion of the G\"{o}del logic with a De
Morgan negation $\neg$ and modalities $\blacksquare$ and $\blacklozenge$. We
equip it with Kripke semantics on frames with two (possibly fuzzy) relations:
$R^+$ and $R^-$ (interpreted as the degree of trust in affirmations and denials
by a given source) and valuations $v_1$ and $v_2$ (positive and negative
support) ranging over $[0,1]$ and connected via $\neg$. We motivate the
semantics of $\blacksquare\phi$ (resp., $\blacklozenge\phi$) as infima
(suprema) of both positive and negative supports of $\phi$ in $R^+$- and
$R^-$-accessible states, respectively. We then prove several instructive
semantical properties of the logic. Finally, we devise a tableaux system for
branching fragment and establish the complexity of satisfiability and validity.
