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If I have three values and these truth tables:

Or:

$$\begin{array}{c|c|c|} & A & \text{true} & \text{false} \\ \hline A & A & A & A \\ \hline \text{true} & A & \text{true} & \text{true} \\ \hline \text{false} & A & \text{true} & \text{false} \\ \hline \end{array}$$

And:

$$\begin{array}{c|c|c|} & A & \text{true} & \text{false} \\ \hline A & A & A & A \\ \hline \text{true} & A & \text{true} & \text{false} \\ \hline \text{false} & A & \text{false} & \text{false} \\ \hline \end{array}$$

does this have a name? The construction is such that when A enters into a logic operation, the result is A, otherwise normal logic applies.

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Try Bochvar's internal three-valued logic (also known as Kleene's weak three-valued logic)

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