The IFF Architecture

Overview

IFF Architecture

μ : metashell

ν : natural part

ο : object level

The IFF architecture (Figure 1) is a two dimensional structure consisting of levels (the vertical dimension) and namespaces (the horizontal dimension). Within each level, the terminology is partitioned into namespaces. The number of namespaces and the content may vary over time: new namespaces may be created or old namespaces may be deprecated, and new terminology and axiomatization within any particular namespace may change (new versions). In addition, within each level, various namespaces are collected together into meaningful composites called meta-ontologies. At any particular metalevel, these meta-ontologies cover all the namespaces at that level, but they may overlap. The number of meta-ontologies and the content of any meta-ontology may vary over time: new meta-ontologies may be created or old meta-ontologies may be deprecated, and new namespaces within any particular meta-ontology may change (new versions).

Modular Structure

The IFF architecture (Figure 1) is a two-dimensional structure with metalevels ranging along the vertical axis and namespaces ranging along the horizontal. It is vertically partitioned into the object level at the bottom, the supra-natural part or metashell at the top, and the vast intermediate natural part. The natural part contains the namespaces located at metalevels 1, 2, 3, … n, …; and the metashell contains the namespaces located at metalevels ‘meta’, ‘type’ and ‘iff’. The natural part is further divided horizontally into pure and applied aspects. The pure aspect itself is divided into core and structural components. The IFF core consists of the core component of the natural part plus the metashell. The IFF metastack, which is the kernel of the IFF core, consists of the kernel of the core component plus the kernel of the metashell.

Figure 1: IFF Architecture (exploded iconic version)

Metashell

The supra-natural part or metashell (Figure 2) consists of only three levels. The tiny (only five terms) IFF-IFF namespace, which is located on the iff level at the very top of the IFF architecture, provides the terminology for the fundamental ideas of abstract set and function. The medium-sized (∼ 450 terms) IFF-TYPE namespace, which is located on the type level just above the meta level, provides typing terminology for the IFF-META namespace. The core concepts of basic collections, metastack orders and fundamental relations are introduced on the type level. The very large-sized IFF-META namespace, which is located on the meta level just above the natural part of the IFF, services the natural part consisting of the generic metalevels. The IFF-IFF namespace represents a (directed) graph of abstract sets and functions, the IFF-TYPE namespace represents a finitely-complete category of abstract sets and functions, and the IFF-META namespace represents a topos of Cantorian featureless abstract sets and functions.

Natural Part

Along the horizontal dimension, the natural part (Figure 2) is partitioned into pure and applied aspects. The applied aspect contains meta-ontologies, such as the IFF institution theory meta-ontology (IFF-INS), the IFF first order logic meta-ontology (IFF-FOL) and the IFF ontology meta-ontology (IFF-ONT), providing the terminology and axiomatization needed for the logical and semiotic functionality in applications. The pure aspect, which is partitioned into core and structural components, represents a natural topos hierarchy. The pure aspect contains meta-ontologies, such as the IFF set theory meta-ontology (IFF-SET) in the core component and the IFF category theory meta-ontology (IFF-CAT) in the structure component, axiomatizing the set-theoretic and category-theoretic foundations needed elsewhere in the IFF. The natural part is vertically divided into an infinite number of metalevels 0 < n, with level 0 being the object level and level n being a generic metalevel. In the pure aspect of the natural part of the IFF, the axiomatization for any concept is located in one generic module. Such a module gives the axiomatization at level n (1 ≤ n < ∞) for that concept. The core of the IFF consists of the core component of the natural part plus the metashell. The kernel of the IFF core is called the IFF metastack. The connection (Figure 1) between the generic finite level axiomatizations and the metashell axiomatizations relies heavily upon the metastack partial orders.

Core

The IFF core axiomatization contains many terms, partitioned according to whether the term is a core concept, a diagram term, a (co)limit term or an exponent term. The IFF core axiomatizates finite limits of specific shapes at the type level and general finite limits at all lower levels. Finite limits are connected with Dedekind’s abstract definition of finiteness (Lawvere and Rosebrugh, 2003). Following the principle of conceptual warrant, terminology has been placed in the IFF core only when it is needed in the lower or more peripheral namespaces. All metalevel ontologies import and use, either directly or indirectly, the core meta-ontology. This includes the all meta-ontologies in the structural component of the IFF axiomatization. The IFF core axiomatization is in adjunctive form. The generic level of the core plays a central role in the IFF axiomatization — this is the most referenced namespace in the natural part of the IFF. The kernel and finite limits namespace axiomatizations of the core rely heavily upon the metastack partial orders for the basic collections, respectively. The finite colimits namespace is a categorical dual to the finite limits namespace. The axiomatization for exponents and curry begins at the meta level of the metashell. The collection of kernel namespaces in the IFF core, called the IFF metastack, represents a chain of toposes, which anchors the entire IFF architecture. The argument for the metastack structure is based upon the Cantorian expansion. Furthermore, the vertical structuring of the IFF metastack between metalevels n and n+1 (and between metalevel n and the IFF-META namespace of the IFF metashell) requires axiomatization for the metastack partial orders at metalevel n+2 (and the IFF-TYPE namespace of the IFF metashell).

Design Goals

The IFF is a descriptive category metatheory. A central design goal of the natural part is to represent the intuitions of the working category theorist, using their illustrations, called commutative diagrams, to represent their ideas. A central design goal of the metashell is to support the terminology in the natural part in conformance with the strong restrictions of the syntactic mechanism incorporated in the IFF grammar. There is a strong boundary between the object levels of the IFF and the metalevels. The object level represents “real world” concepts, whereas the metalevel provides the mechanism to structure these concepts. Although the metalevel contains metalanguages describing the object languages at the object level, the distinction between the object and meta levels is not exactly Tarskian, since the object level in the IFF contains only instance data. Traditional logical structuring of object level data is incorporated in the metalevels of the natural part, mostly at the lowest metalevel 1.

Design Principles

In general, the axiomatization of the IFF uses full first order expression, consisting of set, predicate (unary relation), (unary) function and (binary) relation symbols, with the usual logical connectives and quantifiers. The development of the IFF is governed overall by two design principles: conceptual warrant and categorical design. Both are limitations of the logical expression. Conceptual warrant seeks to limit the content of logical expression, by requiring us to justify any introduction of terminology (and attendent axiomatizations). These should be based upon need at lower metalevels and object level. Categorical design seeks to limit the form of logical expression to atomic expressions: set declarations (declarations that an element is in a set), equations (commutative diagrams, since they use only unary functions) or relational expressions (declarations that a pair is in a relationship).A major goal is to require that the natural part conform to the principle of categorical design. Hence, the natural part should be expressed atomically as set declarations, equations and relational expressions — but mostly equations. Since functions are unary, these equations correspond to the commutative diagrams of category theory.

Transition

The colors illustrating the levels in the current architecture (Figures 1 and 2) closely correspond to those describing the previous architecture; the most important change was expansion of the lower metalevel in the previous architecture into the natural part of the current architecture. In short, the old lower metalevel became the natural part, the old upper metalevel became the IFF-META namespace, the old top metalevel became the IFF-TYPE namespace, and the old Ur metalevel became the IFF-IFF namespace.

Figure 2: IFF Architecture (detailed version)

Logical Expression

An important design constraint of the IFF is that the natural and objective parts are fully conformable to the categorical design principle. This means that all axiomatization in the natural and objective parts are atomic, consisting of either set declarations, equations or relational expressions (relation declarations). Since the equations, which make up the largest part of the axiomatization of the natural and objective parts, consist in the composition of unary functions, they correspond to the commutative diagrams of category theory. In one sense, the atomic axiomatization of the IFF natural part is much closer in spirit to the intuitions of the working category-theorist, than the usual idea that the foundations of category theory has a first order expression. In the IFF, such first order expression only appears in the metashell. In other words, in the IFF traditional logical expression has been relegated to the metashell. The metashell is used by the various meta-ontologies in the pure aspect of the natural part: either the single meta-ontology IFF-SET in the core component; or the various meta-ontologies in the structural component, including the IFF-CAT meta-ontology, the IFF 2-category theory meta-ontology (IFF-2CAT), and the future IFF double category theory meta-ontology (IFF-DCAT). The metashell acts as a bootstrapping mechanism from which the natural part can be unfurled: first unfurl the core component; next unfurl the structure component; and last unfurl the applied aspect, including the IFF institution theory meta-ontology (IFF-INS).