Germinal theories in Łukasiewicz logic

Abstract

Differently from boolean logic, in Łukasiewicz infinite-valued propositional logic Ł∞ the theory Θmax⁡,v consisting of all formulas satisfied by a model v∈[0,1]n is not the only one having v as its unique model: indeed, there is a smallest such theory Θmin⁡,v, the germinal theory at v, which in general is strictly contained in Θmax⁡,v. The Lindenbaum algebra of Θmax⁡,v is promptly seen to coincide with the subalgebra of the standard MV-algebra [0,1] generated by the coordinates of v. The description of the Lindenbaum algebras of germinal theories in two variables is our main aim in this paper. As a basic prerequisite of independent interest, we prove that for any models v and w the germinal theories Θmin⁡,v and Θmin⁡,w have isomorphic Lindenbaum algebras iff v and w have the same orbit under the action of the affine group over the integers.