 As mentioned in 1.2 above, one feature of the deductivist style is that paradigmatic mathematical proofs are expressed entirely in some appropriate formal language (for example, first-order predicate logic with identity). This allows the validity of a given proof to be easily, indeed mechanically, ascertained. But of course few, if any, of the proofs circulated and published by mathematicians have this form. What qualifies as a proof for working mathematicians ranges from the completely informal to the detailed and precise, with every (or almost every) gap filled in. However, even detailed and precise proofs are rarely expressed purely in the language of logic; rather, they are a mixture of ordinary language, mathematical, and logical symbols and terminology.