Preservation theorems in {\L}ukasiewicz \\model theory

Document Type : Research Paper


1 Department of Pure Mathematics, Faculty of Mathemat- ical Sciences, Tarbiat Modares University, P.O. Box 14115-134, and Institute for Re- search in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran

2 Department of Mathematics, Shahid Beheshti University, G. C., Evin, Tehran, Iran


We present some model theoretic results for {\L}ukasiewicz
predicate logic by using the methods of continuous model theory
developed by Chang and Keisler.
We prove compactness theorem with respect to the class of all
structures taking values in the {\L}ukasiewicz $\texttt{BL}$-algebra.
We also prove some appropriate preservation theorems concerning universal and inductive theories.
Finally, Skolemization and Morleyization in this framework are discussed and
some natural examples of fuzzy theories are presented.


\bibitem{Barwise} J. Barwise (editor), {\it Handbook of mathematical logic}, North-Holland, 1977.

\bibitem{CK} Chang and Keisler, {\it Continuous Model Theory}, Annals of Mathematical Studies, Princeton University Press, {\bf58} (1966).

\bibitem{CH} P. Cintula and P. H\'{a}jek, {\it Triangular norm based predicate fuzzy logics},
Fuzzy sets and systems, {\bf161} (2010), 311-346.

\bibitem{DE} P. Dellunde and F. Esteva, {\it On elementary extensions for fuzzy predicate logics},
In Proceedings of IPMU, (2010), 747-756.

\bibitem{G} G. Gerla, {\it The category of the fuzzy models and Lowenheim-Skolem theorem},
Mathematics of Fuzzy Systems, (1986), 121-141.

\bibitem{Hajek1} P. H\'{a}jek, {\it Metamathematics of Fuzzy Logic, Trends in Logic},
 Kluwer Academic Publishers, Dordercht, {\bf4} (1998).

\bibitem{Hajek2} P. H\'{a}jek and P. Cintula, {\it On theories and models in fuzzy predicate logics},
Journal of Symbolic Logic, {\bf 71} (2006), 863-880.

\bibitem{Montagna} F. Montagna, {\it On the predicate logics of continuous t-norm BL-algebras},
Archive for Mathematical Logic, {\bf44} (2005), 97-114.

\bibitem{Yaacov} I. B. Yaacov, A. P. Pedersen, {\it A Proof of completeness for continnuous
first-order logic}, Journal of Symbolic Logic, {\bf75} (2010), 168-190.

\bibitem{YBHU} I. B. Yaacov, A. Berenstein, C. W. Henson and A. Usvyatsov,
{\it Model theory for metric structures}, In Model Theory with Applications to Algebra and Analysis,
Vol. II, eds. Z. Chatzidakis, D. Macpherson, A. Pillay, and A.Wilkie, Lecture Notes series
of the London Mathematical Society, No. 350, Cambridge University Press, (2008), 315-427.