THE CYLINDRIC ALGEBRAS OF 4-VALUED LOGIC
Rogier Jacobsz · 2011
In this thesis the syntax and semantics of four-valued first-order predicate logic are introduced. When we define the semantics, we use 4-cylindric set algebras. Then we define 4-cylindric algebras which are supposed to reflect the algebraic properties of this logic. We give a method for constructing 4-cylindric algebras out of cylindric algebras and prove that in fact every 4-cylindric algebra is isomorphic to a 4-cylindric algebra that is constructed in this way. It will turn out that every locally finite 4-cylindric algebra is a subdirect product of a family of 4-cylindric set algebras. This result will be used in order to prove a completeness theorem with respect to a proof system we introduce. At last, we compare 4-cylindric algebras to 3-cylindric algebras. It turns out that every 4cylindric algebra contains a 3-cylindric algebra as a subreduct. Moreover, every 3-cylindric algebra is isomorphic to a subreduct of some 4-cylindric algebra. Acknowledgements This thesis was written under supervision of Piet Rodenburg. I would like to thank him for putting a lot of time and effort into supervising me. He was prepared to help at any moment with any issue. I imagine there exist worse supervisors. Next, I am very grateful to Benedikt Lowe. Not only did he serve as chair of the Thesis Committee, he also supervised me during my first attempt to write a thesis. During this first attempt, he was always prepared to help me with anything. The reason I did not succeed the first time was certainly not him. I should also thank Yurii Khomskii for his help during my first attempt. He spent a considerable amount of time on helping me and I am grateful to him for that. Furthermore, I thank Inge Bethke, Alexandru Baltag and Vincenzo Ciancia for being willing to serve as members of the Thesis Committee. Finally, I would like to express my gratitude towards my parents, Fred and Marleen, for both their mental and their financial support, and towards my brother, Marvin, who pushed me to go study mathematics. He was right. Rogier Jacobsz Amsterdam, August 31st, 2011