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Table 8.4 Axioms, Inference Rules, Theorems: A Theory
AXIOMS INFERENCE RULES THEOREMS
Class(Thing) And-introduction: Given P, Q, it is valid to infer P / Q. If P / Q are true, then so is P ( Q.
Class(Person) Or-introduction: Given P, it is valid to infer P V Q. If X is a member of Class(Parent), then X is a member of Class(Person).
Class(Parent) And-elimination: Given P / Q, it is valid to infer P. If X is a member of Class(Child), then X is a member of Class(Person).
Class(Child) Excluded middle: P V i P (i.e., either something is true or its negation is true) If X is a member of Class(Child), then NameOf(X, Y) and Y is a String.
If SubClass(X, Y), then X is a subset of Y. This also means that if A is a member of Class(X), then A is a member of Class(Y). If Person (JohnSmith), then j ParentOf(John Smith, JohnSmith).
If X is a member of Class (Parent) and Y is a member of Class(Child), then J (X =Y).
Term versus Concept: Thesaurus versus Ontology
To help us understand what an ontology is and isn't, let's try to elaborate one of the distinctions we made in the last chapter: that between a term and a concept.12 One way to illustrate this distinction is to differentiate between a thesaurus and an ontology (specifically, a high-end ontology or logical theory, i.e., on the upper right in the Ontology Spectrum of Figure 7.6).
12For further discussion of the distinction between terms and concepts, refer to (ISO 704, 2000).
^209^ Table 8.5 Ontology Example
Classes (general things) Metal working machinery, equipment, and supplies;
metal-cutting machinery; metal-turning equipment; metal-milling equipment; milling insert; turning insert, etc.
Instances (particular things) An instance of metal-cutting machinery is the “OKK
KCV 600 15L
Vertical Spindle Direction, 1530x640x640mm 60.24"x25.20"x25.20 X-Y-Z Travels Coordinates,
30 Magazine Capacity, 50 Spindle Taper, 20kg 44 lbs Max Tool Weight, 1500 kg 3307 lbs Max Loadable Weight on Table, 27,600 lbs Machine Weight, CNC Vertical Machining Center" (http://www.okkcorp .com/kcvseries.html)
A kind of metal working machinery is metal cutting machinery.
A kind of metal cutting machinery is milling insert.
Geometry, material, length, operation, ISO-code, etc.
1; 2; 3; “2.5", “inches"; “85-degree-diamond";
“231716"; “boring"; “drilling"; etc.
If milling-insert(X) & operation(Y) & material(Z)=HG_Steel & performs(X, Y, Z), then has-geometry(X, 85-degree-diamond).
[Meaning: If you need to do milling on high-grade steel, then you need to use a milling insert (blade) that has an 85-degree diamond shape.]
Figure 8.8 displays the triangle of signification or triangle of meaning. It attempts to display in an abbreviated form the three components (the angles) of the meaning of natural languages like English. The first component, at the lower left, is the terms, that is, the symbols (the labels for the concepts) or the words of English and the rules for combining these into phrases and sentences (the syntax of English). In themselves, they have no meaning until they are associated with the other components, such as other angles of "Concepts" and "Real-World Referents."
For example, if asked for the meaning of the term "LKDF34AQ," you would be at a loss, as there is no meaning for it. If asked, however, for the meaning of "automobile," you would know what is meant because there is an associated thing in the world (the real-world referent that has four tires, an engine, is manufactured by Ford or Honda, gets particular miles to the gallon, and so on) and there is a concept in our human mental model that stands for (or "represents")
Relations: subclass-of, (kind_of), instance-of, part-of, has-geometry, performs, used-on, etc.
that real thing in the world. That is why there is a dotted line between Term and Real-World Referent in Figure 8.8; there is no direct link. Humans need a concept to mediate between a term and the thing in the world the term refers to.
A thesaurus generally works with the left-hand side of the triangle (the terms and concepts), while an ontology in general works more with the right-hand side of the triangle (the concepts and referents), as depicted in Figure 8.9.
Recall from the previous chapter that a thesaurus is developed primarily as a classification space over a domain, a set of domains, or even over the entire world, such as Roget's 1916 thesaurus—for the purpose of conceptual navigation, search, and information retrieval. Therefore, the semantics of the classification space can remain relatively weak, characterizing the simple semantic relations among conceptual labels (terms), and so structured mostly taxonom-ically by broader-than and narrower-than relations. All you really need to know about a term node in a thesaurus is that it is semantically distinct from other nodes (hence, removing ambiguity), and it is broader than or narrower than certain other terms. No complicated notion of the meaning has to be captured and represented.