 ::::Systems I can think of that are powerful enough to express that statement are Girard's System F, and Martin-Löf's Intuitionistic type theory. The first is clearly second-order, but not a logical system: there are no proof rules. The second is a logical system, but does not fit the usual definition of being n-th order, and while one can argue it is higher-order, it is not clear that you need that for the formalization of tertium exclusum. However, we can add the Law as an axiom schema to first-order propositional calculus, meaning that for every well-formed proposition P the disjunction P ∨ ¬P is an axiom – so there are infinitely many axioms, but all are first-order. And you can consider the statement, and try to prove it mathematically, not in the formal system, that for every P the disjunction P ∨ ¬P is a theorem that follows from the axioms. A logical system may or may not have that property. To formalize it does not require the use of higher-order constructs.   