 Apparently, the first to use a version of universal validity and explicitly propose it as both necessary and sufficient for logical truth was Bolzano (see Bolzano 1837, §148; and Coffa 1991, pp. 33–4 for the claim of priority). The idea is also present in other mathematicians of the nineteenth century (see e.g. Jané 2006), and was common in Hilbert's school. Tarski (1936a, 1936b) was the first to indicate in a fully explicit way how the version of universal validity used by the mathematicians could itself be given a characterization in terms of concepts of standard mathematics, in the case of Fregean formalized languages with an algorithmic grammar. Essentially Tarski's characterization is widely used today in the form of what is known as the model-theoretic notion of validity, and it seems fair to say that it is usually accepted that this notion gives a reasonably good delineation of the set of logical truths for Fregean languages.