Should we think about superposition in a fuzzy or in a paraconsistent manner?

Suppose we define a predicate $L$ as "is alive." We then take Schrödinger’s cat, $c$, and put it in its box and wait for a time for it to be equally likely for it to be dead and alive. How should we think about the sentence $Lclandlnot Lc$? If superposition is fuzzy, then we should find that:

$$Lclandlnot Lc leftrightarrow frac{1}{2}$$

But if superposition is instead paraconsistent we find:

$$Lclandlnot Lc leftrightarrow top$$

Is one more appropriate to superposition, or is there another logic we should use?