Abstract
The purpose of this paper is to present the higher order formalization of RDF and OWL with setting up ontological meta-modeling criteria through the discussion of Russell’s Ramified Type Theory, which was developed in order to solve Russell Paradox appeared at the last stage in the history of set theory. This paper briefly summarize some of set theories, and reviews the RDF and OWL Semantics with higher order classes from the view of Russell’s Principia Mathematica. Then, a set of criteria is proposed for ontological meta-modeling. Several examples of meta-modeling, including sound ones and unsound ones, are discussed and some of solutions are demonstrated according to the meta-modeling criteria proposed.
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Notes
- 1.
- 2.
A class notion in set theories is different from one in ontology descriptions.
- 3.
The universe of natural numbers is factually defined as sets of sets that include the empty set as number zero and powersets of sets as number successors.
- 4.
In the letter, it is stated that “The system \(\varOmega \) of all numbers is an inconsistent, absolutely infinite multiplicity.”
- 5.
Exactly, Russell pointed the paradox in the expression of functions rather than sets, in the letter to Frege (van Heijenoort 1967).
- 6.
For example, in Haskell, \([x\uparrow 2\mid x\leftarrow [1..5]]\) produces [1, 4, 9, 16, 25]. (Hutton 2007).
- 7.
The set theory in NBG for individuals and sets can be regarded as a sort of first order logic, and then classes can be regarded as first order. However, RDFS can be regarded as much higher order logic as shown at Sect. 3.
- 8.
Predicates are functions that return truth value.
- 9.
In this paper ‘\(\uparrow \)’ is used to indicate the type of variable instead of colon that is usually used in type theory, as a colon is confusing with the notation for namespace in the syntax of Semantic Web.
- 10.
A question arises in the case of no statements of owl:sameAs and owl:differentFrom for atomic nodes in comparison of two different graphs. We proposed the algorithm named UNA for atomic objects in the non-UNA condition. See the motivation and the detail in Koide and Takeda 2011.
- 11.
See the simple interpretation 3 in RDF Semantics (Hayes 2004).
- 12.
- 13.
- 14.
References
Aczel, A.D.: The Mystery of the Aleph, Four Walls Eight Windows (2000)
Boolos, G.S.: The Iterative Conception of Set. J. Philosophy 68–8, 215–231 (1971)
Bourbaki, N.: Éléments de Mathématique, Chapitres 1 et 2. Hermann (1966)
Cantor, G.: Beiträge zur Begründung der transfiniten Mengenlehre. Mathematische Annalen, Bd.46, S.481-512 (1895). Contributions to the Founding of the Theory of Transfinite Numbers, Dover (1955)
Cantor, G.: Letter to Dedekind (1899) in “From Frege to Gödel A Source Book in Mathematical Logic, 1879–1931”. In: van Heijenoort, J. (ed.). Harvard (1967)
Doets, H.C.: Zermelo-Fraenkel Set Theory (2002). http://staff.science.uva.nl/vervoort/AST/ast.pdf
Hutton, G.: Programming in Haskell. Cambridge University Press, New York (2007)
Hayes, P.: RDF Semantics. W3C Recommendation (2004). http://www.w3.org/TR/2004/REC-rdf-mt-20040210/
Kamareddine, F., Laan, T., Nederpelt, R.: A Modern Perspective on Type Theory. Kluwer, New York (2004)
Genesereth, M.R., Fikes, R.E.: Knowledge Interchange Format version 3.0 Reference Manual (1994). http://logic.stanford.edu/kif/Hypertext/kif-manual.html
Koide, S., Takeda, H.: Common Languages for Web Semantics, Evaluation of Novel Approaches to Software Engineering. In: Communications in Computer and Information Science, vol. 230, pp. 148–162. Springer (2011)
Motik, B.: On the properties of metamodeling in OWL. J. Logic Comput. 17(4), 617–637 (2007). doi:10.1093/logcom/exm027
Patel-Schneider, P.F., Hayes, P., Horrocks, I.: OWL web ontology language semantics and abstract syntax Sect. 5. RDF-Compatible Model-Theoretic Semantics (2004a). http://www.w3.org/TR/owl-semantics/rdfs.html
Patel-Schneider, P.F., Horrocks, I.: OWL web ontology language semantics and abstract syntax Sect. 3. Direct Model-Theoretic Semantics (2004b). http://www.w3.org/TR/owl-semantics/direct.html
Schneider, M.: OWL 2 web ontology language RDF-based semantics, 2nd edn. (2014). http://www.w3.org/TR/owl-rdf-based-semantics/
van Heijenoort, J. (ed.): Russell, B.: Letter to Frege (1902) in “From Frege to Gödel A Source Book in Mathematical Logic, 1879–1931”. Harvard (1967)
Whitehead, A.N., Russell, B.: Principia Mathematica, vol. 1. Merchant Books (1910)
Graham, S.: Re-examining Russell’s paralysis: ramified type-theory and Wittgenstein’s objection to Russell’s theory of judgment. J. Bertrand Russell Stud. 23, 5–26 (2003)
Tarski, A.: Introduction to Logic. Dover (1946/1995). This book is an extended edition of the book title “On Mathematical Logic and Deductive Method,” appeared at 1936 in Polish and 1937 in German
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Koide, S., Takeda, H. (2016). Inquiry into RDF and OWL Semantics. In: Li, YF., et al. Semantic Technology. JIST 2016. Lecture Notes in Computer Science(), vol 10055. Springer, Cham. https://doi.org/10.1007/978-3-319-50112-3_2
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