You can formalize inductive logic, but it is usually though to require the introduction of an ambient Bayesian probability theory. The reason first-order logic alone doesn't work is because first-order logic examines whether an argument is valid or invalid. An argument is valid if and only if the truth of the premises guarantees the truth of the conclusion. By contrast, an inductive argument provides probable support for the conclusion. To evaluate inductive arguments you need a way to evaluate the degree of support a set of premises provide for the conclusion. First-order logic makes no such provision. Hence to formalize inductive logic probability theory is used. Intuitively, what you want to know is how probable a state of affairs P is given the premises.
As a corollary, the argument form you cite (∃x: F(x) ∴ ∀x: F(x)) does not really capture the form of inductive reasoning. First, the existential quantification requires only one entity that is F to be satisfied. Almost no inductive arguments have that form, and usually involve a large number of observations. In fact, in the sciences, inductive arguments typically make testable hypotheses relative to background hypotheses from other scientific domains (This, by the way, is another reason first-order logic cannot really "formalize" inductive reasoning adequately). Hence the form you list here is at best a highly idealized way of thinking about inductive reasoning that is so divorced from inductive reasoning in practice as to be useless. Secondly, inductive arguments most often have the form of predictions about what our future observations will likely be, and don't require a universal quantification over a domain. For example, if you argue, "I've tasted thousands of lemons and every single one has been sour. Therefore, the very next one I taste will likely be sour." This is a perfectly cogent inductive argument that does not require a universal quantification over a domain.
Check out the following page if you want to know more: https://plato.stanford.edu/entries/logic-inductive/