Can we expand "induction principle" to a partial order $(X, \leq)$?

Induction can be performed over any relation $R$ over a set $X$, provided the relation is well-founded: any subset $S \subseteq X$ must have a minimal element with respect to $R$. A minimal element of $S$ is an element $m \in S$ such that there is no $x \in S$ with $x R m$.

For total orders, being well-founded is equivalent to being well-ordered.

Note: Assuming the axiom of dependent choice (a weaker form of the axiom of choice), one can show that a relation is well-founded if and only if there is no infinite descending chain of elements in $X$ (with respect to the relation $R$).

Note 2: The Axiom of Choice is equivalent to stating that any set can be well-ordered. Thus, one could in principle do (transfinite) induction over any set.

The problem is that the Axiom of Choice is not constructive, which means that no one knows anything about what the well-ordering given by the axiom looks like. Therefore, it is in practice impossible to use that well-ordering.

We can't even do induction on a total order if it's not well-ordered. Like on $\Bbb Q$ or $\Bbb R$ with the standard order. So in general a partial order is out of the question.

One could impose a well-ordering-like requirement on the partial order (every non-empty subset of $X$ has a minimal element), and then it is called a well-founded partial order. In that case one can indeed induct on a partial order.