Forcing in Lukasiewicz Predicate Logic
In this paper we study the notion of forcing for Lukasiewicz predicate logic (L\forall, for short), along the lines of Robinson’s forcing in classical model theory.We deal with both finite and infinite forcing. As regard to the former we prove a Generic Model Theorem for L\forall, while for the latter, we study the generic and existentially complete standard models of L\forall.
Forcing in Lukasiewicz Predicate Logic
LPi Logic with Fixed Points
We study a system, $\mu$L$\Pi$, obtained by an expansion of L$\Pi$ logic with fixed points connectives. The first main result of the paper is that $\mu$L$\Pi$ is standard complete, i.e. complete with regard to the unit interval of real numbers endowed with a suitable structure.
We also prove that the class of algebras which forms algebraic semantics for this logic is generated, as a variety, by its linearly ordered members and that they are precisely the interval algebras of real closed fields. This correspondence is extended to a categorical equivalence between the whole category of those algebras and another category naturally arising from real closed fields.
Finally, we show that this logic enjoys implicative interpolation.
A short introduction to formal fuzzy logic via t-norms
A VERY short introduction to formal fuzzy logic via t-norms.
A short introduction to formal fuzzy logic via t-norms
muMV-algebras: an approach to fixed points in Lukasiewicz logic
We study an expansion of MV-algebras, called $\mu$MV-algebras, in which minimum and maximum fixed points are definable. The first result is that $\mu$MV-algebras are term-wise equivalent to divisible MV$_\Delta$ algebras, i.e. a combination of two known MV-algebras expansion: divisible MV-algebras and MV$_\Delta$ algebras. Using methods from the two known extensions we derive a number of results about $\mu$MV-algebras; among others: subdirect representation, standard completeness, amalgamation property and a description of the free algebra.
muMV-algebras: an approach to fixed points in Lukasiewicz logic
Continuous approximations of MV-algebras with product and product residuation: a category-theoretic equivalence
A new class of $MV$-algebras with product, called L$\Pi_q$-algebras, has been introduced. In these algebras, the discontinuous product residuation $\to_\pi$ is replaced by a continuous approximation of it. These algebras seem to be a good compromise between the need ofexpressiveness and the need of continuity of connectives. Following a good tradition in many-valued logic, in this paper we introduce a class of commutative $f$-rings with strong unit and with a sort of weak divisibility property, called $f$-quasifields, and we show that the categories of L$\Pi_q$-algebras and of $f$-quasifields are equivalent.
