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Two elements $u<v$ of a totally ordered set form an adjacent pair when no domain element lies strictly between them. Every pair of consecutive integers is adjacent, while a dense order has no adjacent pair. A gap in a subset of the real numbers matters here only through which points belong to the domain: $1,2$ are adjacent in $[0,1]\cup [2,3]$.

Express that the order is a dense order:In either real interval, $(x+y)/2$ lies strictly between any two distinct ordered elements and remains in the domain. In the integers, $x=0,y=1$ have no intervening integer, so this first-order sentence is false. The endpoints of a closed interval do not prevent a dense order.

Combine a minimum with an adjacent pair in a linear order:In $[0,1]\cup [2,3]$, choose $m=0,u=1,v=2$. The minimum and the adjacent pair in a linear order both exist. In $[0,1]$, the dense order property rules out such an adjacent pair. In $\mathbb{Z}$, adjacent pairs exist, but no minimum exists: for any $m$, the integer $m-1$ is smaller. Thus this first-order sentence has precisely the required three truth values. Neither the endpoints nor the numerical witnesses occur as constant symbols in the first-order sentence.