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All or none; A novel choice of primitives for elementary logic1

Published online by Cambridge University Press:  12 March 2014

R. H. Thomason  and
H. Leblanc
R. H. Thomason
Affiliation:
Yale University Bryn Mawr College
H. Leblanc
Affiliation:
Bryn Mawr College
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Extract

In [1] Ludwik Borkowski takes a quantifier symbol ‘Q1’ (e.g., the familiar ‘∀’) to permit definition of another quantifier symbol ‘Q1’ if, where ‘f’ is a singulary predicate variable, there exists a formula A of QC1—a first-order quantificational calculus (without identity and individual constants) having ‘Q1’ as its one primitive quantifier symbol—such that: (1) under the intended interpretations of ‘Q1’ and ‘Q1’ the biconditional (Q1X)f(X) = A is valid, (2) no individual variable occurs free in A, and (3) A contains no propositional variable, and no predicate variable other than ‘f.’

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 32 , Issue 3 , 09 October 1967 , pp. 345 - 351
Copyright
Copyright © Association for Symbolic Logic 1967

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Footnotes

1

The research leading to this paper was sponsored by the National Science Foundation, Grant GS-190 (R. Thomason), Grant GS-973 (H. Leblanc), and by the John Simon Guggenheim Memorial Foundation (H. Leblanc). The authors wish to thank Professor A. Mostowski, who drew their attention to the problem discussed here, as well as the referee, who provided many useful comments.

References

[1]Borkowski, L., On proper quantifiers I, Studia logica, vol. 8 (1958), pp. 65–130.CrossRefGoogle Scholar
[2]Church, A., Introduction to mathematical logic, vol. 1, Princeton Univ. Press, Princeton, N.J., 1956.Google Scholar
[3]Kleene, S., Introduction to metamathematics, Van Nostrana, New York, 1952.Google Scholar
[4]Leblanc, H. and Thomason, R. H., Syntactically free, semantically bound, to appear.Google Scholar
[5]Mostowski, A., On a generalization of quantifiers, Fundamenta mathematicae, vol. 44 (1957), pp. 12–36.CrossRefGoogle Scholar