Metalanguages

The IFF terminology is organized along metalevels. In general, the terminology introduced at any level uses the terminology in the same or higher metalevels. Object level ontologies use lower metalevel terminology and functionality. Lower metalevel namespaces and ontologies (meta-ontologies) use higher metalevel terminology and functionality. The namespaces and meta-ontologies in the natural part use metashell terminology and functionality. There is an infinity of IFF metalevels: 1, 2, 3, … n, … ‘meta’, ‘type’ and ‘iff’, with one metalanguage based at each metalevel.

IFF = ‹IFF₁ ⊃ IFF₂ ⊃ … ⊃ IFF_n ⊃ … ⊃ IFF_meta ⊃ IFF_type ⊃ IFF_iff ⊃ logic›.

Any metalanguage can be used by the meta-ontologies and ontologies at the same or lower levels. These metalanguages can be defined in terms of their top-down nesting. Proceeding in a top-down fashion, the metalanguage hierarchy starts with a logical shell logic, which enables a lisp-like first-order expression using connectives and restricted quantification. In addition to the logical shell, there is a hierarchy of metalanguages coordinated with the core kernel called the IFF metastack. Each metalanguage includes the ones above, with the logical shell as innermost. Any object-level language uses the IFF₁ metalanguage in its axiomatization. Thus, conceptually there are an infinite number of metalanguages. But we hasten to add, there are actually only the five metalanguages

IFF = ‹IFF_n ⊃ IFF_meta ⊃ IFF_type ⊃ IFF_iff ⊃ logic›,

since level n is a generic and parametric level. Of course, this makes IFF_n a parametric metalanguage. The IFF-META namespace serves as an interface between the lisp-like syntax of the logic metalanguage and the parametric IFF_n metalanguage. It does this by servicing the natural part of the IFF containing the generic metalevels. Furthermore, if (using a strict, minimalist category-theoretic philosophy), the IFF axiomatization is simplified by discarding the metashell along with the metastack links between the natural part and the metashell, then there is only one parametric metalanguage

IFF = ‹IFF_n›.

To understand this in the concrete, see the code examples for the IFF syntax.