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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?

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The claim that (some) contradictions are both true and false is dialetheism. For an overview of dialethic logics, which allow true contradictions, see this very useful encyclopaedia article. As you will see, the story is a tangled one, far too tangled to outline again here.

On the particular question as to whether we can save naive set theory by adopting a dialethic logic, there is an excellent discussion in Luca Incurvati's new book Conceptions of Set (CUP 2020). For a blog post discussing the relevant chapter, see here. Short version: there's no version of dialethic logic which is both tolerably well motivated and yet can also give us a set theory with naive comprehension which is strong enough to be useful.

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