# On an infinite-valued Lukasiewicz logic that preserves degrees of truth

Universitat Pompeu Fabra, Barcelona

Infinite-valued Lukasiewicz can be defined from a class of matrices
constituted by Wajsberg algebras (also called MV-algebras) where the set of
designated values is an arbitrary implicative filter. This logic is
protoalgebraic, algebraizable but not selfextensional. It does not satisfy the
Deduction-Detachment Theorem, nor the Graded Deduction Theorem, but it
satisfies the Local Deduction-Detachment Theorem.

In this paper we will study a new logic, determined also by Wajsberg algebras,
but focusing on the order relation instead of the implication. This new logic
is an example of a ``logic that preserves degrees of truth'', and has the
following properties: it can be defined from the class of matrices constituted
by Wajsberg algebras where the set of designated values is an arbitrary lattice
filter, it is not protoalgebraic, not algebraizable and selfextensional. It
does not satisfy the Deduction-Detachment Theorem, nor the Local
Deduction-Detachment Theorem, but it does satisfy the Graded Deduction
Theorem.

Since the new logic is selfextensional and has conjuntion, it has a Gentzen
system
that is both fully adequate for it and algebraizable, having the same algebraic
counterpart as the logic. Wegive a a sequent calculus of the Gentzen system
corresponding to this logic, we study its properties and models, and we
determine several relationships between the Gentzen system and the infinite
valued Lukasiewicz logic.

#### Footnotes

- ...Angel J. Gil
^{1}
- joint work with Josep Maria Font, Antoni Torrens and Ventura Verdú

2005-05-23