Journal of Symbolic Logic

On an algebra of lattice-valued logic

Lars Hansen

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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.

Article information

Source
J. Symbolic Logic Volume 70, Issue 1 (2005), 282-318.

Dates
First available in Project Euclid: 1 February 2005

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1107298521

Digital Object Identifier
doi:10.2178/jsl/1107298521

Mathematical Reviews number (MathSciNet)
MR2119134

Zentralblatt MATH identifier
05004799

Citation

Hansen, Lars. On an algebra of lattice-valued logic. J. Symbolic Logic 70 (2005), no. 1, 282--318. doi:10.2178/jsl/1107298521. http://projecteuclid.org/euclid.jsl/1107298521.


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References

  • Garrett Birkhoff Lattice theory, 3rd ed., American Mathematical Society, Providence, R.I.,1967.
  • Lars Hansen Formalized token models and duality in semantics: an algebraic approach, Journal of Symbolic Logic, vol. 69 (2004), pp. 443--477.
  • Elliott Mendelson Introduction to mathematical logic, D. Van Nostrand Co.,1963.
  • J. B. Rosser and A. R. Turquette Many-valued logics, North-Holland,1958.
  • Yang Xu et al. Lattice-valued logic, Springer-Verlag, Berlin-Heidelberg,2003.