Consider the following (complete and distributive) three-element lattice $\mathbf{A}_3= \langle \{1,a,0\}, \wedge, \vee, \Rightarrow, ^*,1, 0 \rangle$ where $1^*=0, a^*=1, 0=1^*$ and $x \Rightarrow y = (x^* \vee y)$ for any $x,y \in \{1,a,0\}$. So ($1\Rightarrow a) = a$ and $(1\Rightarrow 0)= 0 $ and for the rest of elements $\Rightarrow$ always receives value $1$. Also the set of designated values is $\{1,a\}$
Question 1. Is this a well-known logic? If yes could someone please give me some bibliography references?
Question 2. Does there exist some propositional formula that allows me to "distinguish" the element $a$ from $0$ ? For instance, I can distinguish 1 and 0 in the sense that if I set $v(p)=1$, then $v(p \wedge \neg \neg p )$ is in the set of designated elements, however, for $v(p)=0$, $v(p \wedge \neg \neg p )$ is not in the set of designated elements.