 The restriction to artificial formulae raises a number of questions about the exact value of the Fregean enterprise for the demarcation of logical truths in natural language; much of this value depends on how many and how important are perceived to be the notes stripped from the natural language expressions that are correlates of the standard logical expressions of formalized languages. But whatever one's view of the exact value of formalization, there is little doubt that it has been very illuminating for logical purposes. One reason is that it's sometimes relatively clear that the stripped notes are irrelevant to truth-conditional content (this is especially true of the use of natural language logical expressions for doing mathematics). Another of the reasons is that the fact that the grammar and meaning of the artificial formulae is so well delimited has permitted the development of proposed characterizations of logical truth that use only concepts of standard mathematics. This in turn has allowed the study of the characterized notions by means of standard mathematical techniques. The next two sections describe the two main approaches to characterization in broad  outline.[7]