Editing Propositional formula (section)

=== Identity ===
Tarski asserts that the notion of IDENTITY (as distinguished from LOGICAL EQUIVALENCE) lies outside the propositional calculus; however, he notes that if a logic is to be of use for mathematics and the sciences it must contain a "theory" of IDENTITY.<ref>Tarski p.54-68. Suppes calls IDENTITY a "further rule of inference" and has a brief development around it; Robbin, Bender and Williamson, and Goodstein introduce the sign and its usage without comment or explanation. Hamilton p. 37 employs two signs ≠ and = with respect to the '''valuation''' of a formula in a formal calculus. Kleene p. 70 and Hamilton p. 52 place it in the predicate calculus, in particular with regards to the arithmetic of natural numbers.</ref> Some authors refer to "predicate logic with identity" to emphasize this extension. See more about this below.