Meaning, Opacity, and Function: An Observer-Relative Classification of Artifacts
Davide Bragetti, Carmelo Asaro, Giuseppe Francesco Italiano · IRIS Research product catalog (Sapienza University of Rome) · 2026
Whether an artifact is understood, merely apprehended, or functional depends on the observer confronting it rather than on the artifact alone. We take three binary predicates over a domain of artifacts – structured meaning M, residual opacity C, and functionality K, each read relative to an agent – and study the finite datum they induce: which of the seven regions they carve are inhabited for that agent. Two modelling choices are stipulated before anything is derived: the semantic predicates need not exclude one another, and denotation is confined to their overlap Θ = M ∩ C. The admissible data are then exactly the covers of the predicate set, so their number – 109 labelled, 34 up to relabelling – is the classical cover count of a three-element set and not a new enumeration. Three results follow. The types with empty symbolic carrier are exactly those omitting the two overlap cells, seventeen in number; since a query's denoting token must lie in that carrier – a requirement that is definitional – an agent of such a type poses no query whatever architecture it carries. The six-coordinate structural profile that a type fixes is realizable if and only if it satisfies four explicit Boolean conditions, whence 39 of the 64 profiles occur. And architectures inducing the same compositional closure may still differ in the generation depth they assign its elements and in the pairs on which their composition is defined, two being the least size of the functional set at which this can happen. The predicates are supplied model data throughout: identifying the intended assignment from external evidence lies outside the present formalism. --- VERSION 4 (2026-08-19) — a substantial scope-refined revision of version 3 (10.5281/zenodo.21853111, 80 pp). This version is 61 pp. WHAT LEFT THIS VERSION, AND WHERE IT REMAINS AVAILABLE. The generative layer – "The Generation of Functionality" (v3 section 6), including the conditional compositional-representability theorem and the conservative-extension result – has been deliberately moved out of the scope of this paper, pending its companion treatment. The gallery of fourteen illustrative circle configurations (v3 section 8.2), together with the three-circle unrealizability proposition and the gallery-coverage proposition, has likewise been moved out of scope. An interpretive appendix has been extracted to companion material; a scope sentence remains in the body. ALL OF THAT MATERIAL REMAINS AVAILABLE AND CITABLE IN VERSION 3, which is permanent and unchanged at 10.5281/zenodo.21853111. NO RETRACTION IS IMPLIED AND NONE IS INTENDED. No result was withdrawn as incorrect, and no surviving proof depended on any of the removed material: the reduced text builds with zero undefined references. The geometry contract the paper does rely on – the rendering-fidelity remark and the seven-region figure – is retained and relocated. WHAT IS STRENGTHENED IN THIS VERSION. A type-determined structural-profile subsection, characterising exactly which of the six-coordinate profiles are realizable (39 of 64). An explicit agent-architecture and compositional-closure subsection in the execution layer. A three-memories definition – storage, execution, linkage – with the corollary that linkage is not determined by storage. A fixed-point result for internal generation. Deferred proofs are inlined with their results rather than deferred to an appendix. The positioning and limitations sections are rewritten and expanded. CORRECTIONS. Four source-attribution corrections in the related-work discussion (Searle; Belnap; Floridi; Cattaneo and Ciucci). Each narrows what is attributed to an external source. None touches a theorem, a proof, a definition, or any count. Two of the four – Belnap and Cattaneo–Ciucci – rest on non-conforming copies of paywalled sources, and their re-verification against a conforming copy is still open; they are published here in the more conservative wording rather than left in the form shown to be inaccurate. They are non-result-bearing.