 A remarkable fact about logical truth is that many have thought it plausible that the set of logical truths of certain rich formalized languages is characterizable in terms of concepts of standard mathematics. In particular, on some views the set of logical truths of a language of that kind is always the set of sentences of the language derivable in a certain calculus. On other, more widespread views, the set of logical truths of a language of that kind can be identified with the set of sentences that are valid across a certain range of mathematical interpretations (where validity is something related to but different from the condition that all the sentences that are replacement instances of its form be true too; see below, section 2.3). One main achievement of early mathematical logic was precisely to show how to characterize notions of derivability and validity in terms of concepts of standard mathematics. (Sections 2.2 and 2.3 give a basic description of the standard mathematically characterized notions of derivability and validity for standard quantificational languages, with references to other entries.)