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On an algebra of lattice-valued logic

Published online by Cambridge University Press:  12 March 2014

Lars Hansen
Lars Hansen*
Affiliation:
Laerkehegnet 155, 2670 Greve, Denmark, E-mail: larsh@mobilixnet.dk
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Abstract

The purpose of this paper is to present an algebraic generalization of the traditional two-valued logic. This involves introducing a theory of automorphism algebras, which is an algebraic theory of many-valued logic having a complete lattice as the set of truth values. Two generalizations of the two-valued case will be considered, viz., the finite chain and the Boolean lattice. In the case of the Boolean lattice, on choosing a designated lattice value, this algebra has binary retracts that have the usual axiomatic theory of the propositional calculus as suitable theory. This suitability applies to the Boolean algebra of formalized token models [2] where the truth values are, for example, vocabularies. Finally, as the actual motivation for this paper, we indicate how the theory of formalized token models [2] is an example of a many-valued predicate calculus.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 1 , March 2005 , pp. 282 - 318
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[1]Birkhoff, Garrett, Lattice theory, 3rd ed., American Mathematical Society, Providence, R.I., 1967.Google Scholar
[2]Hansen, Lars, Formalized token models and duality in semantics: an algebraic approach, this Journal, vol. 69 (2004), pp. 443–477.Google Scholar
[3]Mendelson, Elliott, Introduction to mathematical logic, D. Van Nostrand Co., 1963.Google Scholar
[4]Rosser, J. B. and Turquette, A. R., Many-valued logics, North-Holland. 1958.Google Scholar
[5]Xu, Yanget al., Lattice-valued logic, Springer-Verlag, Berlin-Heidelberg, 2003.CrossRefGoogle Scholar