 The notion of model-theoretic validity mimics the notion of universal validity, but is defined just with the help of the set-theoretic apparatus developed by Tarski (1935) for the characterization of semantic concepts such as satisfaction, definability, and truth. (See the entry on  Tarski's truth definitions.)  Given a Fregean language, a structure for the language is a set-theoretical object composed of a set-domain taken together with an assignment of extensions drawn from that domain to its non-logical constants. A structure is meant by most logicians to represent an assignment of meanings: its domain gives the range or “meaning” of the first-order variables (and induces ranges of the higher-order variables), and the extensions that the structure assigns to the non-logical constants are “meanings” that these expressions could take. Using the Tarskian apparatus, one defines for the formulae of the Fregean language the notion of truth in (or satisfaction by) a set-theoretic structure (with respect to an infinite sequence assigning an object of the domain to each variable). And finally, one defines a formula to be model-theoretically valid just in case it is true in all structures for its language (with respect to all infinite sequences). Let's abbreviate “\(F\) is true in all structures” as “MTValid\((F)\)”. The model-theoretic characterization makes it clear that “MTValid\((F)\)” is definable purely in terms of concepts of set theory. (The notion of model-theoretic validity for Fregean languages is explained in thorough detail in the entries on  classical logic  and  second-order and higher-order logic;  see also the entry on  model theory.)[9]