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Peirce semiotics
I said:
This frame can be aligned with Peirce’s classical semiotic triangle of things, signs, and symbols to define models and ontologies, the former mapping things to symbolic categories, the latter things to concepts. Ontological prisms are derived through simple translation.
Claude said:
Two technical concerns before engaging with the substantive move, because both affect what “simple translation” can mean.
On the Peirce attribution. Peirce’s own semiotic triad is object, sign (representamen), interpretant — not “things, signs, and symbols.” “Symbol” enters Peirce’s apparatus at a different level: it’s one of three sign-types in his icon/index/symbol trichotomy (a classification of how a sign relates to its object — by resemblance, by causal connection, or by convention), not a third vertex parallel to “thing” and “sign.” So “things, signs, and symbols” as a triangle is closer to Ogden–Richards (referent/thought/symbol) than to Peirce, and even there the middle term is thought or reference, not “sign.” This matters for your framework specifically because if “symbol” is doing double duty — once as a vertex of the triangle, once as a Peircean sign-subtype meaning “conventional sign” — the conventionality property I asked about last turn (whether logical dimension’s autonomy rests on convention) may already be smuggled in by the choice of triangle, rather than argued for.
On the directionality of “model.” In standard model theory, a model is an interpretation function that maps symbols to things — vocabulary to domain elements and relations, the satisfaction relation ⊨ running from language to structure. Your formulation has it the other way: model = “things to symbolic categories.” That reversal might be intentional — KR ontologies often do run the mapping nomologically from reality outward into category systems — but it’s worth flagging since it inverts the textbook direction, and the rest of the argument depends on which way the arrow points.
Setting both aside, the substantive content of the move is strong and has real precedent. What you’re describing — model as the things↔symbols correspondence, ontology as the things↔concepts correspondence — closely tracks Guarino and Giaretta’s classical distinction (1995) between a conceptualization (an intensional structure organizing a domain, prior to language) and an ontology (a logical theory whose axioms constrain the models of a logical language to approximate that conceptualization). In that lineage, “model” already has the technical job of relating symbols to a domain, and “ontology” already has the job of encoding a conceptualization — so your Model/Ontology split is recovering a known and well-tested distinction, not inventing an ad hoc one.
Now, the key claim: “ontological prisms are derived through simple translation.”
Here is the strongest reading I can construct. Model gives you the pair (Things, Symbols). Ontology gives you the pair (Things, Concepts). Both share the same first term. The prism’s three dimensions — extensional, intensional, logical — are just the union of these two pairs: Things, Concepts, Symbols. Nothing new has to be stipulated to get a third dimension; it falls out automatically once you have both a model and an ontology defined over the same domain.
On this reading, “translation” doesn’t name a primitive concept→symbol correspondence at all. It names a derived relation: for any given thing, Ontology supplies its concept and Model supplies its symbol, and the correspondence between that concept and that symbol is obtained by composing the two mappings through their shared anchor — schematically, Symbol = Model ∘ Ontology⁻¹. The translation is “simple” precisely because it requires no independent theory of how concepts encode into symbols; it’s a composite, not a primitive, relation.
If this is right, it actually answers the autonomy question from last turn in a way I hadn’t offered: logical dimension’s independence from intensional dimension rests on neither formal manipulability nor pure convention, but on what might be called referential parallelism — concepts and symbols are each defined independently relative to things, and only contingently aligned with each other through that shared reference. Two ontologies could conceivably share an extensional base and a logical encoding while diverging in their conceptualization, or share a conceptualization while diverging in notation — because nothing forces concept and symbol into direct correspondence except passing through Things.
The question this leaves open: is “simple translation” exactly this composition through the shared anchor — in which case the concept↔symbol relation is only ever as reliable as Model and Ontology individually are, and could fail to compose cleanly (a thing might be under-determined by its concept, or ambiguously symbolized) — or do you intend something stronger, a direct and lossless concept-to-symbol mapping that doesn’t route through Things at all, with “translation” naming that direct operation instead?
Signs & symbols
I said:
This frame can be aligned with a revisited Peirce’s semiotic triad of things, signs, and symbols to define models and ontologies, mapping symbolic categories and concepts to things, respectively.
Claude said:
Both corrections land, and they land cleanly together rather than independently.
On attribution. “Revisited” does the necessary work — it concedes up front that this isn’t Peirce’s actual triad (object/representamen/interpretant) but a deliberate adaptation borrowing his name and general trichotomous structure. That’s a defensible move once flagged, and it neutralizes the misattribution concern without requiring you to abandon the Peircean framing you clearly want for its philosophical weight.
On directionality. Reversing to “symbolic categories and concepts mapped to things” now puts both Model and Ontology in the standard model-theoretic direction — interpretation functions running from a representational vocabulary into a domain, the way the satisfaction relation ⊨ actually works. This is more than a cosmetic fix; it changes the formal shape of “translation” for the better.
Here’s the technical upgrade this produces. In the previous formulation, Model and Ontology had different domains and codomains pointing opposite ways (Things→Symbols, Things→Concepts), so deriving a concept↔symbol translation required composing one mapping with the inverse of the other — Symbol = Model ∘ Ontology⁻¹ — which only works cleanly if Ontology is invertible (injective: one concept per thing, no collapsing). That’s a strong and somewhat artificial requirement.
Now both mappings point the same direction into the same codomain: Model: Symbols → Things, Ontology: Concepts → Things. This is exactly the shape of a cospan, and the natural way to derive a concept↔symbol correspondence from two maps sharing a codomain is a pullback: the set of pairs (s, c) such that Model(s) = Ontology(c). No inverse required. Translation becomes “this symbol and this concept agree on what thing they pick out” rather than “running one mapping backward through the other.” That’s a more robust and more standard construction, and it tolerates exactly the kind of underdetermination (several symbols for one thing, several concepts for one thing) that the inverse-based version couldn’t handle gracefully.
One loose thread the rewording leaves dangling. The triad is still named with three terms — things, signs, and symbols — but the operationalized mappings only use two roles: symbolic categories and concepts. “Sign” appears in the triad’s name but not in either mapping. Three readings are available:
- “Sign” is just a synonym for “symbol” here, and the triad is really binary in function (things, and one mediating term called sign-or-symbol indifferently) — in which case naming both is redundant.
- “Sign” is the genus and “symbol” the species relevant to the logical dimension specifically (recall Peirce’s icon/index/symbol as one sign-type, the conventional one) — meaning “sign” names the broader category that “concept” and “symbolic category” are both instances of, with intensional and logical dimensions each picking out a different kind of sign.
- “Sign” was meant to map to “concept” (closer to the interpretant role) while “symbol” maps to “symbolic category” (the representamen role) — i.e., the triad’s three terms correspond one-to-one with three roles, and the “respectively” in your sentence is silently doing a things/signs/symbols ↔ —/concepts/symbolic-categories alignment that just wasn’t spelled out.
Which of these is intended? It matters for whether the prism still has exactly three independent dimensions, or whether “sign” is an unused fourth wheel that should be dropped from the triad’s name.