Things & labels
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 labelled with signs, respectively.
Claude said:
This revision does real structural work, and it resolves the dangling-“sign” question from last turn — though not via any of the three readings I offered. A fourth possibility has emerged that’s stronger than all of them.
What changed. “Sign” is no longer a mapping target standing alongside “concept” and “symbolic category.” It’s been relocated into the codomain: things are no longer bare arguments of the Model and Ontology mappings — they are things labelled with signs. So the compound entity being mapped to is not Thing but (Thing, Sign).
This means sign is not a fourth dimension, and not a genus for symbol, and not a synonym for either concept or symbol. It’s something more foundational: the minimal identifying layer that makes a thing available to be mapped to at all. A bare thing, on this reading, cannot be the target of Model or Ontology directly — it must first be picked out, individuated, indexed. Sign is that act of individuation, prior to and independent of whatever concept or symbolic category later gets attached to the same thing.
Why this is a good fit for your extensional dimension specifically. Recall the very first definition in this conversation: extensional knowledge is “facts as they can be observed.” Observation is never of a bare thing — it’s of a thing under some minimal identification: this one, here, now, distinct from that one. Sign, in this revised triad, is exactly that observational individuating function — not yet a concept (no interpretation), not yet a symbolic category (no formal/conventional encoding), just the bare act of telling one thing from another so it can be referred to at all. So “sign” turns out to live inside the extensional dimension as its enabling condition, rather than sitting beside intensional and logical as a competing fourth term. The vocabulary has four words — thing, sign, symbol, concept — but the architecture still has exactly three dimensions, because sign is subordinate to (constitutive of) extensional access rather than parallel to intensional or logical access.
The formal payoff. This also sharpens the pullback construction from last turn. Previously: Model: Symbols → Things, Ontology: Concepts → Things, with translation defined as agreement on which thing is hit. Now the shared codomain is richer — Model: Symbols → (Things, Signs), Ontology: Concepts → (Things, Signs) — so translation requires agreement not just on the thing but on the thing as identified by the same sign. This is a real strengthening, not just decoration: if Things include abstracta or unobservables that can’t be compared directly, Sign supplies the observable proxy that makes the equality in the pullback actually checkable. Signs are what make the pullback decidable in practice, not just well-defined in principle.
What’s still open. Is “thing labelled with sign” a primitive — i.e., an axiom that things are never encountered except already signed, so there is no bare-Things layer beneath it at all — or is it constructed, via some separate labelling function L: Things → Signs that you compose on top of an independently existing Things layer? The difference matters: on the primitive reading, sign is built into the very identity conditions of “thing” (no thing without a sign, full stop — closer to a Peircean view where there’s no object outside semiosis); on the constructed reading, Things retain an independent, presemiotic existence, and signs are simply how they happen to get accessed for the purposes of this framework. Which is intended?
From Things to facts
I said:
Labelling function necessary to observation
Claude said:
Good — that settles the modal status of L, but settling that it’s necessary opens a sharper question about what kind of necessity, and a formal simplification follows from taking “function” literally.
The formal simplification first. If L: Things → Signs is a genuine function — single-valued, as “labelling function” implies — then “a thing labelled with a sign” is not independent data alongside the thing. It’s just the thing, plus the (unique, already-determined) value L(t). Specifying both Thing and Sign in the codomain, as I did last turn, was redundant once L is fixed: a Sign carries no information beyond what L already assigns to its Thing. So the codomain of Model and Ontology can revert to plain Things, as in the “revisited Peirce” formulation two turns ago. What L actually contributes is not a richer codomain but a mediated equality test: the pullback comparing Model(s) and Ontology(c) doesn’t compare Things directly — it compares L(Model(s)) and L(Ontology(c)). Sign is the verification interface, not an extra dimension of the target. This is the precise version of what I gestured at loosely last turn (“signs make the pullback decidable”) — it’s not that signs enlarge the codomain, it’s that they’re the only thing the pullback is actually allowed to test, when Things themselves (especially abstracta) resist direct comparison.
This gives extensional dimension a clean boundary. If L is necessary for observation, then a Thing is extensionally available — observable, a candidate for extensional knowledge — exactly when L is defined on it. Things outside dom(L) are not falsified or excluded from existence; they simply never enter the extensional dimension at all. This is a quasi-Kantian boundary (a noumenal remainder unreachable by observation) but earned formally rather than asserted philosophically: it falls straight out of treating L as partial rather than total. Two properties of L now matter and you haven’t yet specified either:
- Totality — is L defined on all Things, or only some? Partiality would give you a principled, non-mysterious account of the unobservable: not “things beyond knowledge” in the mystical sense, but simply things outside the domain of a specific function.
- Injectivity — can two distinct Things share a Sign? If not injective, the extensional dimension has a resolution limit: Things that differ but are labelled identically become observationally indiscernible, a Leibniz’s-Law failure restricted specifically to the extensional dimension (they could still differ intensionally or logically, if Ontology or Model individuate more finely than L does).
On the necessity claim itself. You said L is “necessary to observation,” which is itself a modal statement — and this framework has a vocabulary purpose-built for exactly this kind of statement, since ontological modality was introduced earlier as the declarative qualifier of how strongly something holds. So the claim “L is necessary to observation” can itself be located inside the apparatus it’s describing: is this a metaphysical necessity (no process could count as observation at all without an individuating labelling step — true in every possible epistemic system, by what observation means) or an epistemic/nomological one (true of observation as it happens to be realized in the kinds of cognizers or systems this framework targets, but not a conceptual truth about observation per se)? The distinction matters for the prism as a whole: a metaphysical reading makes L part of the definition of the extensional dimension, undetachable from it even in principle; a nomological reading makes L a contingent feature of how extensional knowledge happens to get realized, which leaves room for an L-free mode of observation that the framework simply isn’t modeling.
Which is intended — and, separately, is L total or partial over Things?
I said:
“The world is the totality of facts, not of things” (Ludwig Wittgenstein)
Claude said:
This is proposition 1.1 of the Tractatus, following directly from proposition 1 — and it’s not a decorative epigraph. It cuts against the anchor that’s been quietly accumulating across the last several turns.
Tracking the drift. The framework’s very first formulation, in your opening message, defined extensional knowledge as “facts as they can be observed.” Then the semiotic-triangle excursion introduced “things, signs, symbols” as the triad’s vocabulary, and from that point on, Model and Ontology were both specified as mapping to Things — Things became the operative anchor of the extensional dimension, with Facts quietly dropped from the vocabulary. The Wittgenstein quote reasserts the original term and explicitly denies the substitution: the world — i.e., what extensional dimension is of — is the totality of facts, not of things. On this proposition, the last several turns have been analyzing the wrong primitive.
Why this isn’t merely terminological. Wittgenstein doesn’t deny that objects (Gegenstände) exist — he calls them the substance of the world (2.021) — but he insists they’re never encountered except already configured: an object can only occur within a state of affairs, never in isolation (2.0122), and what is the case — a fact — is the existence of a state of affairs (2), i.e., a combination of objects, not an object simpliciter. So Things, in your framework’s terms, may well be real, but they are not what gets observed. What gets observed is always already a configuration — a Fact. This means L, the labelling function you just declared necessary to observation, was mis-typed two turns ago. Its domain isn’t Things; it’s Facts (configurations of things). A bare Thing is exactly the kind of entity Wittgenstein says cannot be the direct target of any observational act.
This reframes, rather than answers, my totality/partiality question. Asking whether L is total or partial over Things was the wrong question — Things-in-isolation are never individually up for labelling at all, so totality/partiality doesn’t apply to them in the way I posed it. But the question resurfaces, correctly typed, one level up: is L total or partial over Facts? Wittgenstein actually gives you machinery for exactly this, and it’s sharper than mere partiality: the totality of facts determines not only what is the case but also what is not the case (1.12) — so unobserved/unobservable territory isn’t a gap in L’s domain (an epistemic shortfall), it’s negatively constituted by the very totality that fixes what does hold. Non-facts aren’t outside L’s reach; they’re excluded by the totality’s own boundary.
Consequence for Model and Ontology. If the world is facts, the pullback codomain I set up two turns ago needs retyping across the board: Model: Symbols → Facts, Ontology: Concepts → Facts, and translation is agreement (via L) on which fact — not which thing — is picked out. This is a real strengthening, because Facts are structured (a configuration of things-in-relation) where bare Things were not, which gives the pullback something compositional to actually check rather than bare numerical identity.
One tension worth naming explicitly. This sits awkwardly against the Aristotelian backbone from early in the conversation, where ontological modality was cast as the declarative counterpart to substance and accident — a framework that takes individual substances (things bearing properties) as primitive. Wittgenstein’s logical atomism is standardly read as a rival to exactly that picture: facts/states-of-affairs as primitive, with objects as their abstracted simples, not the reverse. You can have both lineages in the same architecture, but only if they’re assigned to different jobs rather than treated as interchangeable foundations.
So: is Facts-not-Things meant to replace Things as the anchor of extensional dimension outright — with Things demoted to simples that are only ever reached by analyzing facts, never observed directly — or do you want a two-tier extensional dimension, where Things-as-substance remains the Aristotelian metaphysical bedrock while Facts is specifically what L operates on?
I said:
Facts-not-Things meant to replace Things as the anchor of extensional dimension outright
Claude said:
Taking this as settled, the consequences propagate through every piece of formal apparatus built over the last several turns, and one of them can’t be quietly absorbed — it has to be resolved or accepted as a fault line.
Retyping the apparatus. L, which you declared necessary to observation, now has signature L: Facts → Signs (partial or total still open, see below). Model and Ontology both retype to Symbols → Facts and Concepts → Facts respectively, and the pullback comparing Model(s) and Ontology(c) goes through L(Model(s)) = L(Ontology(c)) over Facts, not Things. None of this is new work — it’s the mechanical consequence of the substitution you just confirmed, carried through the existing scaffolding.
What happens to “Things.” Outright replacement doesn’t eliminate Things from the architecture — Wittgenstein doesn’t eliminate objects either; he calls them the substance of the world (2.021) and insists every statement about a complex can be resolved into statements about its constituents (2.0201). So Things survive, but demoted: they are no longer the target of any direct mapping (Model, Ontology, or L), only reachable indirectly, via logical analysis of Facts into their constituents. The triad’s earlier name — “things, signs, symbols” — is now mistyped at its first term: it should read facts, signs, symbols, with Things relocated one level down as what facts decompose into, not what they’re built from in the sense of being directly mapped to.
The Aristotelian fault line, named precisely. This is the consequence that doesn’t resolve itself mechanically. Ontological modality was grounded several turns ago as the declarative counterpart to Aristotelian substance and accident — a metaphysics where individual substances, bearing properties, are primitive. Wittgensteinian logical atomism is standardly read as the rival position: facts (states of affairs) are primitive, objects are what analysis abstracts out of them, never given independently. By choosing outright replacement rather than the two-tier option, you’ve selected the version of the framework where these two lineages are in genuine tension rather than comfortably stacked.
There is one reconciliation available, and it’s worth putting on the table rather than just flagging the clash: read Aristotelian substance as mapping onto the Wittgensteinian object (which, again, Wittgenstein himself calls substance — 2.021 is not a coincidence of vocabulary), and read Aristotelian accident as mapping onto a fact — the configuration in which that object occurs. On this reading, accidents were never properties borne directly by a bare substance available to observation; they were always the substance’s participation in some state of affairs. Aristotle’s framework then describes the sub-factual layer correctly characterized, while Facts becomes the properly extensional, observable layer Wittgenstein insists on. The two lineages stop competing for the same job and instead describe adjacent levels — substance/accident below the line of observability, fact/non-fact at it.
If you don’t want this reconciliation, the alternative is to let the tension stand as a genuine fault line in the framework — which is a legitimate choice, but one that would mean ontological modality’s Aristotelian grounding and extensional dimension’s Wittgensteinian grounding are answering to two different and not fully compatible metaphysics.
Still open, now correctly typed: is L total or partial over Facts — and does Wittgenstein’s 1.12 (the totality of facts determines both what is the case and what is not) mean partiality isn’t even the right concept here, since what L doesn’t reach isn’t a gap but a negatively-determined boundary? And separately, the modal status question from two turns ago is still unanswered: is “L necessary to observation” metaphysical necessity (no observation without labelling, in any possible system) or nomological (true of how observation happens to work here, in this framework’s intended domain)?