On page 6 of ‘An invitation to applied category theory’ (isbn 9781108711821) it is stated:
The set $\mathbb{B}=\{true, false\}$... has an order $false \leq true$
It then goes on to say:
Thus $false \leq false$, $false \leq true$, and $true \leq true$ but $true \nleq false$. In other words $A \leq B$ if A implies B.
Are these subsequent definitions axiomatic or is there an explanation for them? Unfortunately searching for this has drawn a blank so far.
This is just an example of a partial order: by definition $A\leq B$ if $A$ implies $B$. So, there is good reason for this definition, since implication is an interesting concept, and the two elements $false\leq false$ and $true\leq true$ of the partial order are required by the axioms of partial orders, but there's nothing mathematically requiring the elements $false\leq true$ and $true\not\leq false$. Indeed, there is also the opposite partial order on $\mathbb B$, where $A\leq B$ if and only if $B$ implies $A$, as well as the discrete partial order, where $A\leq B$ if and only if $A=B$.