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Logic with truth values in A linearly ordered heyting algebra1

Published online by Cambridge University Press:  12 March 2014

Alfred Horn
Alfred Horn*
Affiliation:
University of California, Los Angeles
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Extract

It is known that the theorems of the intuitionist predicate calculus are exactly those formulas which are valid in every Heyting algebra (that is, pseudo-Boolean algebra). The simplest kind of Heyting algebra is a linearly ordered set. This paper concerns the question of determining all formulas which are valid in every linearly ordered Heyting algebra. The question is of interest because it is a particularly simple case intermediate between the intuitionist and classical logics. Also the interpretation of implication is such that in general there exists no nondiscrete Hausdorff topology for which this operation is continuous.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 34 , Issue 3 , 17 November 1969 , pp. 395 - 408
Copyright
Copyright © Association for Symbolic Logic 1969

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Footnotes

1

This research was supported in part by National Science Foundation Grant No. 5600.

References

[1]Dummett, M., A propositional calculus with denumerable matrix, this Journal, vol. 24 (1959), pp. 97–106.Google Scholar
[2]Horn, A.. The normal completion of a subset of a complete lattice and lattices of continuous functions, Pacific journal of mathematics, vol. 3 (1953), pp. 137–152.CrossRefGoogle Scholar
[3]Rasiowa, H. and Sikorski, R., The mathematics of metamathematics, Monografie Matematyczne, vol. 41, PWN, Warsaw, 1963.Google Scholar