Plenty of work has been done in intuitionistic logic, where we remove from classical logic the law of excluded middle: $$\vdash P \lor \lnot P$$. However, what if we instead removed the law of noncontradiction: $$\vdash \lnot (P \land \lnot P)$$ ? In this logic, it would be possible for a proposition to be both true and false. This could also be viewed as a trivalent logic with three truth values: $$\mathsf{T}$$, $$\mathsf{F}$$, and $$\mathsf{TF}$$.
It seems then that Russell's paradox, for example, is no longer a paradox. If we construct $$S = \{ x : x \notin x \}$$ in naive set theory, and ask if $$S \in S$$, there is no contradiction - simply a proof that $$S \in S$$ is both true and false (has truth value $$\mathsf{TF}$$). Since most paradoxes are constructed the same way, would such a contradictory logic be immune to paradoxes? Are there still any paradoxes that arise?
Edit: If we modify Russell's paradox to \begin{align*} S\ &=\ \big\{ x: x \notin x \text{ and } \lnot (x \in x \text{ and } x \notin x) \big\} \\ &=\ \{ x: \operatorname{TV}(x \in x) = \mathsf{F} \text{ and } \operatorname{TV}(x \in x) \neq \mathsf{TF} \} \end{align*} where $$\operatorname{TV}(\varphi)$$ denotes the truth value of $$\varphi$$, do we now run into a contradiction?