Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-27T00:05:23.156Z Has data issue: false hasContentIssue false

First-order logic based on inclusion and abstraction

Published online by Cambridge University Press:  12 March 2014

John Bacon*
Affiliation:
The University of Sydney, New South Wales, Australia2006

Extract

Quine has shown that set theory may be based on inclusion and abstraction [1937], [1953]. He quantifies over (or abstracts upon) sets of all kinds, of course, including sets of sets. Here I confine Quine's approach to quantification over (abstraction upon) individuals alone, or at any rate their unit classes. Forsaking quantification over sets undercuts Quine's definition of negation, however. Smullyan sketches a first-order restriction of Quine's approach with no bound class variables for which inclusion and abstraction alone are adequate logical primitives [1957, n. 10]. However, the definition of negation requires more than one element in the universe of discourse. This requirement is met for Smullyan because he is doing arithmetic. Here, on the other hand, I presuppose only that the universe is nonempty. Accordingly, I assume a third primitive notion, the empty class. I will show that this threefold basis suffices both for classical first-order logic and for a version of “free” many-sorted logic. The monadic fragment, which I call Boolean logic with abstraction, is intermediate in strength between Boolean class logic and Lesniewski's Ontology. It affords a novel perspective on descriptions, particularly in their generic use.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1965] Bacon, John, An alternative contextual definition for descriptions, Philosophical Studies, vol. 16, pp. 7576.CrossRefGoogle Scholar
[1974] Bacon, John, The untenability of genera, Logique et Analyse, vol. 17, pp. 197208.Google Scholar
[1956] Church, Alonzo, Introduction to mathematical logic, vol. 1, Princeton University Press, Princeton, New Jersey.Google Scholar
[1952] Fitch, Frederic Brenton, Symbolic logic: an introduction, Ronald Press, New York.Google Scholar
[1892] Frege, Gottlob, Ueber Begriff und Gegenstand, Funktion, Begriff, Bedeutung: fünf logische Studien (Patzig, G., Editor), 2nd ed., Vandenhoeck & Ruprecht, Göttingen, 1966, pp. 6680 [original pages cited].Google Scholar
[1983] Frege, Gottlob, Grundgesetze der Arithmetik, vol. 1, Hermann Pohle, Jena.Google Scholar
[1977] Lejewski, Czesław, Systems of Leśnewski's Ontology with the functor of weak inclusion as the only primitive term, Studia Logica, vol. 36, pp. 323349.CrossRefGoogle Scholar
[1930] Leśniewski, Stanisław, Über die Grundlagen der Ontologie, Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe iii, vol. 23, pp. 111132.Google Scholar
[1937] Quine, Willard Van Orman, Logic based on inclusion and abstraction, this Journal, vol. 2, pp. 145152; reprinted with revisions in Selected logic papers. Random House, New York, 1966, pp. 100–109.Google Scholar
[1953] Quine, Willard Van Orman, Supplementary remarks to New foundations for mathematical logic, From a logical point of view: 9 logico-philosophlcal essays, Harvard University Press, Cambridge, pp. 94101.Google Scholar
[1963] Sellars, Wifrid, Abstract entities, Philosophical perspectives, Part 2, Charles C. Thomas, Springfield, 1967, pp. 229269.Google Scholar
[1957] Smullyan, Raymond M., Languages in which self reference is possible, this Journal, vol. 22, pp. 5567; The philosophy of mathematics (J. Hintikka, Editor), Oxford University Press, London, 1969, pp. 64–77.Google Scholar
[1970] Thomason, Richmond H., Symbolic logic: an introduction, Macmillan, New York.Google Scholar