Propositional logic: $B$ or $B$, $B$ and $B$

Question

Just an extremely basic question:

Is $B$ or $B$ equivalent to $B$? Is it also the case that $B$ and $B$ is equivalent to $B$?

Answer 1

Yes.

$$B \land B = B = B \lor B$$

You can verify this by considering two cases.

• Let $B=T$.
• Let $B=F$.

Answer 2

Alternative way of seeing this is through "propositions as types", which is a beautiful framework through which to view constructive logic. When you're more comfortable with logic-as-syntax, I recommend reading the Homotopy Type Theory book just for its intro to type theory at the beginning.

• If you have an object $b$ of type $B$, then you have an object of type $B \times B$: namely $(b, b)$. Conversely, if you have an object of type $B \times B$ - say that's $(a, b)$ - then you have an object of type $B$ (namely $a$).
• If you have an object $b$ of type $B$, then you have an object of type $B + B$: namely $\text{inl}_{B+B}(b)$ where $\text{inl}_{X+Y}$ indicates "the object obtained by the left-hand injection $X \to X+Y$". Conversely, if you have an object of type $B + B$, then either it's $\text{inl}_{B+B}(b)$ or $\text{inr}_{B+B}(b)$ for some $b$ (where $\text{inr}_{X+Y}$ indicates "the object obtained by the right-hand injection $Y \to X+Y$); in either case, we've got an object of type $B$ again, namely $b$.