When meaning becomes computable

Key points

• A set is a collection; a class is a collection with a membership condition; an instance is a member; a predicate is a statement pattern with slots; a constraint is a predicate every acceptable state of the world must satisfy. Everything a reasoner does is built from these five ideas.
• A graph records which things are connected and by what label. A formal ontology records, in addition, what the labels mean and what must be true of anything that carries them. A graph can be traversed; an ontology can be reasoned over. Storing data in a graph does not establish that the second operation is available.
• A toy model with three axioms and six facts lets a reader check every step by hand. One conclusion follows (a synthetic agent is a therapeutic agent). One tempting conclusion does not (a second agent is not a therapeutic agent), and the essay shows exactly which two added axioms make it follow.
• Under the open-world assumption of OWL, absence of a statement is not a negative statement. Under the closed-world assumption of a database, it is. Much confusion about what an ontology "knows" comes from reading one under the rules of the other.
• Description logics make this precise, and they are not academic. SNOMED CT distributes its concept definitions as OWL axioms in the OWL 2 EL profile, a deliberately restricted logic chosen so that classifying hundreds of thousands of concepts stays tractable.

What a computer is being asked to do

The first essay in this publication asked what it takes for two people to mean the same thing (Why medicine needs a shared language). The second traced the long effort to classify (From classification to clinical understanding). This essay asks a narrower and colder question. Suppose we have agreed on a concept and written it down in a form a machine can read. What, exactly, can the machine then do with it, and what does the machine still not know?

The question matters because "computable" is routinely overclaimed. A slide shows circles connected by arrows; a database is renamed a Knowledge graph; a system is said to "understand" relationships. None of that tells the reader whether the system can draw a single conclusion that was not typed in. The only honest way to answer is to build something small enough to inspect completely, state every assumption, and watch what follows.

Everything in the worked example below is synthetic. The agents, nuclides, and target are invented for teaching; the identifiers are local to this page and correspond to no product, terminology code, or regulatory category.

Five ideas: sets, classes, instances, predicates, constraints

Formal representation is built from a small number of ideas, and they need separating before use.

A set is a collection of things, defined by its members and nothing else. A class is a set with a membership condition: "the things that are radiopharmaceuticals" is a class because a condition decides membership, and a new thing that satisfies the condition is included without anyone editing a list. If everything satisfying one class's condition also satisfies another's, the first is a subclass of the second. Every therapeutic radionuclide is a radionuclide.

An instance (an individual) is a particular member of a class. The vial on the bench is an instance; "radiopharmaceutical" is a class. Confusing the two produces errors that are hard to see: "radiopharmaceutical is a product category" is about the class, "this vial is a radiopharmaceutical" is about an instance.

A predicate is a statement pattern with slots. "x is a radiopharmaceutical" has one slot; "x is labeled with y" has two. Fill the slots and you get a fact: "Agent B is labeled with Nuclide L." In OWL, one-slot predicates are classes and two-slot predicates are properties; in a graph, a two-slot predicate is a labeled edge. The lexicon entry on Relationship gives the short form: a relationship has a type and a direction.

A constraint (an axiom) is a predicate that every acceptable state of the world must satisfy. "Every therapeutic radionuclide is a radionuclide" constrains classes; "anything labeled with something is a radiopharmaceutical" constrains a property. The set of constraints is what "ontology," in its computational sense, mostly refers to. Facts about instances are data; constraints are the theory that gives the data its meaning.

Two further ideas follow. An interpretation is one possible world: an assignment of things to classes and property slots that makes every fact true. A model of an ontology is an interpretation that also satisfies every constraint. A statement is entailed if it is true in every model. That definition, which descends from Tarski's account of logical consequence, is what "the computer inferred it" means when the phrase is used carefully. It does not mean the computer guessed well. It means there is no acceptable world in which the statement is false.

A graph and an ontology are different objects

A Knowledge graph is, at minimum, a set of nodes and labeled, directed edges. In the Resource Description Framework (RDF), the W3C's data model for such graphs, each edge is a triple: subject, predicate, object (W3C, RDF 1.1 Concepts, 2014). A graph database stores millions of triples and answers questions by traversal: start at a node, follow edges of a given label, collect what you reach.

Traversal is powerful and it is not reasoning. Consider three triples: Agent B is labeled with Nuclide L; Nuclide L has type Therapeutic Radionuclide; Agent A and Agent B both bind Target P. A traversal from Agent B reaches Therapeutic Radionuclide in two hops. A traversal from Agent A reaches Agent B in two hops via Target P. The graph treats both paths the same way. It has no idea that the first is the kind of pattern that could define a class, or that the second says nothing about whether Agent A is used for therapy. A query for "agents connected to a therapeutic agent within two hops" returns Agent A, and the result looks authoritative. The graph did what it was asked; nothing in it could say that it was asked the wrong thing.

Hogan and colleagues, surveying the term, hold that a graph becomes a knowledge graph by the addition of schema, identity, and context, and that the schema may range from a lightweight vocabulary to a formal ontology with a logical semantics (Hogan et al., 2021). The word "may" carries the whole distinction; the monograph on Knowledge graphs develops it. A graph with a formal ontology attached can answer "what follows?" A graph without one can only answer "what is connected?"

A formal ontology in the sense of the Web Ontology Language (OWL) adds to the graph a set of axioms with a mathematically defined meaning (W3C, OWL 2 Overview, 2012): which classes are subclasses of which, which are equivalent to which combinations of other classes and properties, which exclude one another, what the domains and ranges of properties are. A reasoner computes the entailments: it classifies (finds every subclass relation that follows), realizes (finds every class each individual belongs to), and checks consistency (whether any model exists at all). The lexicon entry on Reasoning gives the short definition; what follows is a worked case.

A toy model, stated in full

Here is the entire model, in plain words, in description-logic notation, and in the Manchester syntax that OWL editors display. Every conclusion below can be checked against these lines alone.

In plain words: there are radiopharmaceuticals and radionuclides. Therapeutic radionuclides and diagnostic radionuclides are both radionuclides. A radiopharmaceutical is labeled with one or more radionuclides. We define a therapeutic agent as a radiopharmaceutical labeled with at least one therapeutic radionuclide; this is an equivalence, not merely a necessary condition. Then the facts: Agent B is a radiopharmaceutical labeled with Nuclide L, a therapeutic radionuclide. Agent A is a radiopharmaceutical labeled with Nuclide F, a diagnostic radionuclide.

In notation (Description logic symbols: ⊑ is "is a subclass of," ≡ is "is equivalent to," ⊓ is "and," ∃ is "has some," ∀ is "has only," ⊥ is the empty class, ¬ is "not"):

Classes      Radiopharmaceutical, Radionuclide, TherapeuticRadionuclide,
             DiagnosticRadionuclide, TherapeuticAgent
Property     labeledWith   (domain Radiopharmaceutical, range Radionuclide)

Axioms (the terminology, or TBox)
A1   TherapeuticRadionuclide  ⊑  Radionuclide
A2   DiagnosticRadionuclide   ⊑  Radionuclide
A3   TherapeuticAgent  ≡  Radiopharmaceutical  ⊓  ∃labeledWith.TherapeuticRadionuclide

Facts (the assertions, or ABox)
F1   Radiopharmaceutical(agentB)
F2   labeledWith(agentB, nuclideL)
F3   TherapeuticRadionuclide(nuclideL)
F4   Radiopharmaceutical(agentA)
F5   labeledWith(agentA, nuclideF)
F6   DiagnosticRadionuclide(nuclideF)

The same axiom A3 in Manchester syntax, as an OWL editor would show it:

Class: TherapeuticAgent
    EquivalentTo: Radiopharmaceutical and (labeledWith some TherapeuticRadionuclide)

The split into a TBox (terminological axioms about classes) and an ABox (assertions about individuals) is standard in the description-logic literature (Baader et al., 2003). It is the first thing to look for when someone claims a system "has an ontology": are there any TBox axioms at all, or only labeled data?

What follows

Entailment E1: Agent B is a therapeutic agent. Take any model of the axioms. In it, agentB is in Radiopharmaceutical (F1), has a labeledWith link to nuclideL (F2), and nuclideL is in TherapeuticRadionuclide (F3). So agentB is in the class "Radiopharmaceutical and has some labeledWith link to a TherapeuticRadionuclide," which by A3 is exactly TherapeuticAgent. This holds in every model. Entailed.

Nothing in the data said "Agent B is a therapeutic agent." The conclusion is new and, given the axioms, certain. A reasoner performing realization would add it. This is what the title's "computable" refers to, and it is worth seeing how modest and how exact it is.

Entailment E2, at the class level, with no individuals at all. Add NuclideClassL ⊑ TherapeuticRadionuclide and ClassLAgent ≡ Radiopharmaceutical ⊓ ∃labeledWith.NuclideClassL. Now ClassLAgent ⊑ TherapeuticAgent is entailed: anything labeled with some NuclideClassL is labeled with some TherapeuticRadionuclide, so it meets the definition in A3. This is classification, the computation of the subclass hierarchy from definitions, and it needs no instance data. It is the operation SNOMED CT depends on, discussed below.

What does not follow

Non-entailment N1: it does not follow that Agent A is not a therapeutic agent. The temptation is strong: Agent A is labeled with a diagnostic nuclide, so surely it is not therapeutic. But look for a model in which agentA is in TherapeuticAgent. Nothing in A1 to A3 says that a diagnostic radionuclide cannot also be a therapeutic one. Take a model in which nuclideF is in both classes. Then agentA is a radiopharmaceutical labeled with some therapeutic radionuclide, and A3 places it in TherapeuticAgent. Every axiom and fact is satisfied. Since a model exists in which "agentA is not a therapeutic agent" is false, the statement is not entailed.

The gap is missing disjointness. Humans reading the class names assume "diagnostic" and "therapeutic" exclude each other; the logic assumes nothing it was not told. Rector, Brandt, and Schneider documented the opposite failure in SNOMED CT, where modeling choices about part-of relations let a reasoner classify a foot-related finding under a pelvis-related one: entailments that follow correctly from axioms nobody meant (Rector et al., 2011). Our case is the mirror image, an intended entailment that fails because an axiom nobody thought to write is absent.

So add it:

A4   TherapeuticRadionuclide  ⊓  DiagnosticRadionuclide  ⊑  ⊥

Plain words: nothing is both. Now does "agentA is not a therapeutic agent" follow?

Non-entailment N2: still no. By A3, if agentA is a therapeutic agent it has some labeledWith link to a therapeutic radionuclide. That cannot be nuclideF, which A4 and F6 now exclude. But nothing says nuclideF is the only thing agentA is labeled with. Construct a model with an extra individual nuclideX in TherapeuticRadionuclide and labeledWith(agentA, nuclideX). Every axiom and fact is satisfied. The negative is still not entailed.

This is the open-world assumption, the single most important difference between an OWL ontology and a database. In a relational database, under what Reiter named the closed-world assumption, a fact not in the table is false (Reiter, 1978): "is Agent A labeled with anything other than Nuclide F?" returns no rows, and "no rows" is read as "no." In OWL, the absence of a second labeledWith fact means only that we have not said. OWL was designed for the web, where any document is incomplete by construction, and the choice was deliberate (Horrocks et al., 2003; W3C, OWL 2 Primer, 2012).

To make the negative follow, the world must be closed for this individual explicitly:

A5   {agentA}  ⊑  ∀labeledWith.DiagnosticRadionuclide

Plain words: everything Agent A is labeled with is a diagnostic radionuclide. Suppose agentA were a therapeutic agent. By A3 it is labeled with some therapeutic radionuclide n; by A5, n is diagnostic; by A4, n is in the empty class. No such model exists, so ¬TherapeuticAgent(agentA) is entailed, given A1 to A5 and F1 to F6. Remove any of A3, A4, or A5 and it is not.

Figure 1. The toy model as a graph, with the inference and the two non-inferences (inline SVG, synthetic example). Solid arrows are asserted facts. The dashed arrow is entailment E1, derived from axiom A3. The arrow marked with a question is non-entailment N1/N2: the graph contains the path from Agent A to a diagnostic nuclide, but "not a therapeutic agent" does not follow until A4 (disjointness) and A5 (closure) are added. All names are local teaching labels.

Alt text: a diagram with two rows. In the top row, agentB is linked by labeledWith to nuclideL, which has type TherapeuticRadionuclide; a dashed arrow from agentB to the class TherapeuticAgent is labeled "entailed by A3." In the bottom row, agentA is linked by labeledWith to nuclideF, which has type DiagnosticRadionuclide; a dotted arrow from agentA to "not TherapeuticAgent" carries a question mark and the note that it is not entailed until axioms A4 and A5 are added. The three axioms are printed in the middle.

Reading the result

Three lessons sit inside this small exercise.

First, the positive conclusion was cheap and certain; the negative was expensive and conditional. That asymmetry belongs to the open world, not to this example. OWL is good at saying what something is, given a definition, and reluctant to say what something is not, because "is not" requires knowing that the description of the world is complete. Clinical reasoning is full of negatives ("no evidence of," "not a candidate for"), and representing them well is a known difficulty (Rector, 1999). A system that stores clinical facts in OWL and answers "any contraindication?" with "none found" has silently switched from open-world logic to closed-world querying, and the reader should ask which rules were in force when the answer was produced.

Second, the axioms that made the negative follow were axioms about meaning, not data. A4 says the two nuclide classes exclude each other; A5 says our description of Agent A is complete. Neither is a fact about any nuclide; both are commitments by the modeler. This is the sense in which an ontology is a theory and not a database. The theory can be wrong (a nuclide might be used both ways, in which case A4 was a mistake), and the reasoner will faithfully derive the consequences of the mistake.

Third, the reasoner performed exactly three operations: realization (E1), classification (E2), and consistency checking (the step that proved the negative after A5). Everything a description-logic reasoner does reduces to these. When a product claims "reasoning," the question is which of the three it performs and over what axioms.

From a toy to a logic family

Description logics are a family of formal languages, studied since the 1980s, each defined by which constructors it allows (intersection, existential restriction, universal restriction, negation, number restrictions, nominals, and so on) and each with known computational properties (Baader et al., 2003; Baader et al., 2017). The trade is always the same: more constructors give more expressive definitions and slower reasoning, sometimes exponentially slower, sometimes undecidable. OWL's designers chose an expressive logic, SHOIN for the first OWL and SROIQ for OWL 2, as the basis of the full language (Horrocks et al., 2003; W3C, OWL 2 Overview, 2012).

Our toy model used intersection, existential restriction, equivalence, subclass, and (in A4) disjointness. Those constructors belong to a small logic called EL, extended as EL++ by Baader, Brandt, and Lutz, who showed that classification in it is polynomial even for very large ontologies (Baader et al., 2005). OWL 2 standardized a corresponding profile, OWL 2 EL (W3C, OWL 2 Profiles, 2012). The profile permits existential restriction ("has some") and forbids universal restriction ("has only"), permits intersection and forbids union, permits disjointness and forbids general negation.

That is why A5 was marked "outside OWL 2 EL" in the figure. The closure axiom that made the negative follow uses ∀, a constructor the profile excludes precisely because it makes reasoning harder. An ontology inside EL can be classified quickly and can say what things are; it has given up some ability to say what they are not. This is the explicit price of tractability, and a modeler who needs the negative must leave the profile or obtain it by other means, for instance closed-world querying over the classified result, with the switch of assumptions made explicit.

Reasoners exist for each level. HermiT, Pellet, and FaCT++ handle full OWL 2. ELK is specialized for the EL profile and was built to classify ontologies the size of SNOMED CT in seconds rather than hours (Kazakov et al., 2014). Running the toy model through any of them should reproduce E1 and E2, decline N1 and N2, and report the contradiction that yields the negative once A4 and A5 are present. The draft has been checked by hand; the source check should include running it.

SNOMED CT as the working case

SNOMED CT is the largest clinical terminology in use and, less visibly, a description-logic ontology. Its predecessor SNOMED RT was designed in the 1990s with a description-logic foundation so that concept definitions could be classified by machine (Spackman et al., 1997). The current logical model consists of concepts, descriptions (the human-readable terms attached to them), and relationships, of which "is a" relationships form the subsumption hierarchy and attribute relationships carry the definitions (SNOMED International, logical model). A fully defined concept has, in effect, an equivalence axiom like our A3; a primitive concept has only subclass axioms, like our A1.

SNOMED International distributes those definitions as OWL axioms within the OWL 2 EL profile, through its OWL axiom reference set, and documents the practice in its OWL guide (SNOMED International, OWL Guide). The inferred "is a" hierarchy that users browse is not hand-maintained; it is computed by classifying the stated axioms with an EL reasoner. The choice of EL is the choice described above: hundreds of thousands of concepts, classified with each release, in a logic whose worst case is polynomial.

Two consequences follow for anyone building on SNOMED CT, including NucLex. First, the profile's limits are SNOMED's limits: a definition that needs "only," "not," or a count cannot be stated, and the modeler must find an EL approximation or accept a primitive concept. The lexicon entry on SNOMED CT and the monograph on SNOMED CT and the NucLex niche take this up. Second, the open-world asymmetry applies in full. SNOMED can tell a reasoner that a procedure is a kind of imaging procedure; it was not built to say what a procedure is not, and systems that need exclusions build them outside the terminology.

For this publication's project the lesson is concrete. A proposed NucLex extension adding radiopharmaceutical and procedure concepts in the style of the toy model would inherit SNOMED's logic and reasoner, gaining classification and realization for free, and would inherit the limits: every disjointness stated by hand, every negative obtained explicitly. This describes what the formalism offers, not anything NucLex has built. No NucLex ontology, extension, or reasoning service exists in this release.

What remains outside the representation

The central question was double, and the second half deserves a direct answer.

Outside the representation is everything the axioms do not say. In the toy model that included whether diagnostic and therapeutic nuclides exclude each other (until A4), whether our description of Agent A was complete (until A5), what "therapeutic" means beyond "labeled with a therapeutic radionuclide" (always), and why anyone cares (always). The reasoner never asked and could not have. A representation is exactly as deep as its axioms, and the depth was chosen by people.

Outside it, too, is the act of interpretation that connects a symbol to the world. The reasoner established that agentB is in TherapeuticAgent. It did not establish that any vial on any bench is a therapeutic agent, because the link between the symbol and the vial is not a logical fact; it is an act of reference performed by whoever entered the data, under conditions the logic does not record. The monograph on Description logics discusses this boundary. The essay on A patient story across time and modality shows what happens when that act is performed many times, by many people, over years.

And outside it is whether the axioms are right. The logic guarantees that conclusions follow from premises and nothing about the premises. If A4 is wrong, the derived negative about Agent A is wrong with the full authority of a formal proof. The remedy is not less logic. It is the recognition that an ontology is a set of claims made by people, with provenance, subject to review, and revisable, which is where the later sections of this publication go.

Meaning becomes computable, then, in a precise and limited sense. A definition written as an axiom lets a machine decide membership and subsumption with certainty, under stated assumptions, for the concepts the axioms cover. The machine does not know what the concepts are for, does not know what it has not been told, and does not know whether what it has been told is true. Those three remain the work of the people who write the axioms. That is not a disappointment. It is the division of labor the formalism was designed to make possible.

Limitations

This is an AI-assisted draft that has not been source-checked or reviewed by a domain expert or an editor. Its claims should be read accordingly.

The toy model is wholly synthetic. Agent A, Agent B, Nuclide L, Nuclide F, and Target P are teaching labels corresponding to no product, radionuclide, molecular target, or terminology code. "Therapeutic agent" is defined by a single criterion for the exercise and is not offered as a definition for clinical, regulatory, or terminological use. Whether any real radionuclide should be modeled as belonging to exclusive diagnostic and therapeutic classes is a modeling question the essay does not answer; several real nuclides are used for both purposes, which is one reason A4 is presented as a modeler's commitment rather than a fact.

Each entailment and non-entailment was checked by hand against the direct semantics of OWL 2 and the standard definitions of model and entailment. The model has not been run through a reasoner as part of this draft; doing so is a required step of the source check, and the Manchester-syntax fragment is meant to make that straightforward.

The account of SNOMED CT's use of OWL 2 EL follows SNOMED International's published logical model and OWL guide as the writer understands them. The source check must confirm the current release format and the name and status of the OWL axiom reference set against the edition in force. The characterization of the EL profile (existential restriction permitted, universal restriction excluded, union excluded, disjointness permitted) should be verified against the W3C profiles document.

The essay compresses the history and theory of description logics severely; readers needing formal definitions and complexity results should consult the Description Logic Handbook and the 2017 introduction by Baader and colleagues. The essay describes what the formalism makes possible and no implemented NucLex ontology, extension, reasoning service, or terminology endpoint. None exists in this release.

Sources and further reading

• W3C OWL Working Group. 2012. OWL 2 Web Ontology Language Document Overview (Second Edition). W3C Recommendation, 11 December 2012. https://www.w3.org/TR/owl-overview/
• W3C OWL Working Group. 2012. OWL 2 Web Ontology Language Profiles (Second Edition). W3C Recommendation, 11 December 2012. https://www.w3.org/TR/owl2-profiles/
• W3C OWL Working Group. 2012. OWL 2 Web Ontology Language Primer (Second Edition). W3C Recommendation, 11 December 2012. https://www.w3.org/TR/owl2-primer/
• W3C RDF Working Group. 2014. RDF 1.1 Concepts and Abstract Syntax. W3C Recommendation, 25 February 2014. https://www.w3.org/TR/rdf11-concepts/
• Baader F, Calvanese D, McGuinness DL, Nardi D, Patel-Schneider PF, eds. 2003. The Description Logic Handbook: Theory, Implementation and Applications. Cambridge University Press. 2nd ed. 2007.
• Baader F, Horrocks I, Lutz C, Sattler U. 2017. An Introduction to Description Logic. Cambridge University Press.
• Baader F, Brandt S, Lutz C. 2005. Pushing the EL envelope. In: Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI 2005). Morgan Kaufmann.
• Kazakov Y, Krötzsch M, Simančík F. 2014. The incredible ELK: from polynomial procedures to efficient reasoning with EL ontologies. Journal of Automated Reasoning 53(1):1-61.
• Horrocks I, Patel-Schneider PF, van Harmelen F. 2003. From SHIQ and RDF to OWL: the making of a Web Ontology Language. Journal of Web Semantics 1(1):7-26.
• Brachman RJ, Levesque HJ. 2004. Knowledge Representation and Reasoning. Morgan Kaufmann.
• Reiter R. 1978. On closed world data bases. In: Gallaire H, Minker J, eds. Logic and Data Bases. Plenum Press.
• Hogan A, Blomqvist E, Cochez M, et al. 2021. Knowledge graphs. ACM Computing Surveys 54(4): Article 71.
• SNOMED International. SNOMED CT logical model. SNOMED CT Starter Guide. https://docs.snomed.org/snomed-ct-practical-guides/snomed-ct-starter-guide/5-snomed-ct-logical-model (access checked 10 October 2026).
• SNOMED International. SNOMED CT OWL Guide. https://confluence.ihtsdotools.org/display/DOCOWL (to be access-checked at source check).
• Spackman KA, Campbell KE, Côté RA. 1997. SNOMED RT: a reference terminology for health care. Proceedings of the AMIA Annual Fall Symposium: 640-644.
• Rector AL, Brandt S, Schneider T. 2011. Getting the foot out of the pelvis: modeling problems affecting use of SNOMED CT hierarchies in practical applications. Journal of the American Medical Informatics Association 18(4):432-440.
• Rector AL. 1999. Clinical terminology: why is it so hard? Methods of Information in Medicine 38(4-5):239-252.