Solution
= Solution
Express existence of a <minimum> using only the given strict-order relation:
$$
\boxed{\varphi=\exists m\,\forall x\,\neg(x<m).}
$$
In the closed <interval> $[0,1]$, the witness is $m=0$. In the open <interval> $(0,1)$, every possible witness $m$ has the smaller member $m/2$, so the <first-order sentence> is false. These witnesses are explanations in the ambient <real numbers>, not additional symbols in the <first-order signature>.