Given 2 True/False boolean propositions $\mathcal{P}_1\{x_1\}$ and $\mathcal{P}_2\{x_2\}$, where $x_i\in \{T,F\}$
there are 7 possible relations between $\mathcal{P}_1\{x_1\}$ and $\mathcal{P}_2\{x_2\} \\$
0 implications:
1) $\mathcal{P}_1$ bears no relation to $\mathcal{P}_2$ and by contraposition $\mathcal{P}_2$ bears no relation to $\mathcal{P}_1 \\$
1 implication:
2) $\mathcal{P}_1\{T\} \Rightarrow \mathcal{P}_2\{T\}$ and by contraposition $\mathcal{P}_2\{F\} \Rightarrow \mathcal{P}_1\{F\}$
3) $\mathcal{P}_1\{T\} \Rightarrow \mathcal{P}_2\{F\}$ and by contraposition $\mathcal{P}_2\{T\} \Rightarrow \mathcal{P}_1\{F\}$
4) $\mathcal{P}_1\{F\} \Rightarrow \mathcal{P}_2\{T\}$ and by contraposition $\mathcal{P}_2\{F\} \Rightarrow \mathcal{P}_1\{T\}$
5) $\mathcal{P}_1\{F\} \Rightarrow \mathcal{P}_2\{F\}$ and by contraposition $\mathcal{P}_2\{T\} \Rightarrow \mathcal{P}_1\{T\} \\$
2 implications:
6) $\mathcal{P}_1\{T\} \Longleftrightarrow \mathcal{P}_2\{T\}$ alternatively $x_1 = x_2$
7) $\mathcal{P}_1\{T\} \Longleftrightarrow \mathcal{P}_2\{T\}$ alternatively $x_1 \neq x_2 \\$
- Question: Are there any technical names for these groups of relations, and have I missed any relations? I'm pretty sure this is an exhaustive list.
I would guess that the 2 implications are called bijections.
I can't think of a name for the 1 implication
I think the 0 implications are called mutually exclusive.
- I've never studied propositional calculus
Thanks in advance for any help and pointers
Turns out the answer is no there is not.