what is the cardinality of finite sets in three-valued logic? In binary logic the member-of ($\in$) takes two values 0 or 1, so the power set
of a finite set of cardinal $n$ is $2^n$.
In three-valued-logic could we say that the member-of ($\in$) takes three values (True,
False, Undefined)? so the cardinal of power set of a finite set of cardinal $n$ is $3^n$?
 A: This is not a complete answer, and perhaps there are canonical definitions that solves the bellow problems, but this post is about how careful we need to be before we can answer such question

It is very sensitive to the definitions of cardinality, subset, and of powerset (Let $T$ be "true", $U$ be undefined, and $F$ be false).
If $A$ be such that $\lceil 0\in A\rceil=T$, $\lceil 1\in A\rceil=U$ and $\forall x, x\neq 0,1\implies \lceil x\in A\rceil=F$ (note, here we also need to be very careful about how we write the axiom of extensionality, as $\ne$ can be a bit different from what you would normally imagine, but for now I will ignore this problem).
What is the cardinality of $A$: $1$? $2$? or maybe something in between?

Is $\{1\}\subseteq A$? And is $\{0,1\}\subseteq A$? If the latter is yes, can we have $X\subseteq Y$ such that $|X|>|Y|$?

The definition of powerset depends a lot on how you define subset, but you can also consider "levels of subsets", e.g.

*

*$X$ is a "true subset" of $Y$ if it preserves all "true membership" and "undefined membership" from $X$. (if $a$ is really in $X$, it is really in $Y$, and if it is undefined in $X$, it is undefined in $Y$)

*$X$ is "weak subset" of $Y$ if it preserves all "true membership" from $X$

*$X$ is "strictly weak subset" of $Y$ if it preserves all "true membership" from $X$ and  the truth values have decreasing property ("undefined member" can become "not a member", but not vice versa)

*$X$ is "strong subset" of $Y$ if it preserves all "false membership" from $X$

*$X$ is "strictly strong subset" of $Y$ if it preserves all "false membership" from $X$ and the truth values have increasing property ("undefined member" can become a "true member", but not vice versa)

(Those 5 are just examples, maybe there are other more natural definitions you can consider).
From each such definition you can define different definition of powerset, and for each such definition of powerset, and for each definition of cardinality you will have different result.
