This is an introduction to the philosophy and use of OBJ, emphasizing its operational semantics, with aspects of its history and its logical semantics. Release 2 of OBJ3 is described in detail, with many examples. OBJ is a wide spectrum first-order functional language that is rigorously based on (order sorted) equational logic and parameterized programming, supporting a declarative style that facilitates verification and allows OBJ to be used as a theorem prover.

It is often claimed that the essence of functional programming is the use of functions as values, i.e., of higher order functions, and many interesting examples have been given showing the power of this approach. Unfortunately, the logic of higher order functions is difficult, and in particular, higher order unification is undecidable. Moreover (and closely related), higher order expressions are notoriously difficult for humans to read and write correctly. However, this paper shows that typical higher order programming examples can be captured with just first order functions, by the systematic use of parameterized modules, in a style that we call parameterized programming. This has the advantages that correctness proofs can be done entirely within first order logic, and that interpreters and compilers can be simpler and more efficient. Moreover, it is natural to impose semantic requirements on modules, and hence on functions. A more subtle point is that higher order logic does not alwa...