ABSTRACT INVESTIGATION OF THE TOPOLOGICAL INTERPRETATION OF MODAL LOGICS This thesis studies interpretations of modal logics in topological spaces. We prove that adding a new axiom (P ! P ) to S4 gives us a sound and complete axiom system over all discrete topological spaces. We show non-completeness of S4 over three families of topological spaces: particular point, excluded point, and another natural generalization of the Sierpi´nski spaces. Namely, we find extensions of S4 that are sound over these families, although their completeness remains an open question. We show that given any set X and any interpretation of in X that satisfies S4, the image of this interpretation is a topology on X. We also study the influence of the modal axioms of S4 on topological properties of the image. Bing Xu May 2016 INVESTIGATION OF THE TOPOLOGICAL INTERPRETATION OF MODAL LOGICS by Bing Xu A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts in Mathematics in the College of Science and Mathematics California State University, Fresno May 2016 APPROVED For the Department of Mathematics: We, the undersigned, certify that the thesis of the following student meets the required standards of scholarship, format, and style of the university and the student's graduate degree program for the awarding of the master's degree. Bing Xu Thesis Author Maria Nogin (Chair) Mathematics Katherine Kelm Mathematics Marat Markin Mathematics For the University Graduate Committee: Dean, Division of Graduate Studies AUTHORIZATION FOR REPRODUCTION OF MASTER'S THESIS I grant permission for the reproduction of this thesis in part or in its entirety without further authorization from me, on the condition that the person or agency requesting reproduction absorbs the cost and provides proper acknowledgment of authorship. X Permission to reproduce this thesis in part or in its entirety must be obtained from me. Signature of thesis author: ACKNOWLEDGMENTS First, I would like to thank my advisor, Dr. Maria Nogin, for all the meetings we had throughout the past two semesters and all the hours you spent on reading and commenting on my drafts. I am very grateful for all the valuable advice and timely feedback. Thank you for being so patient listening to all my thoughts and answering all the questions I asked. Also, I really appreciated all the encouragement you gave me to pursue the thesis option every time I thought about writing a project. Thank you for pushing me through this great accomplishment. I could never have finished my thesis without your guidance and support. It was a great honor and pleasure for me to work with you. I would also like to thank Drs. Marat Markin and Katherine Kelm for being on my thesis committee. Thank you for reading all the drafts of my work and for your invaluable mathematics expertise. I also want to thank my mother and my husband for being so supportive to my education. It was my mother who watcedh my baby all the hours in which I was writing the thesis, and my husband who believes there is nothing greater to advance the family that to get advanced degrees. Last but not least, I would like to apologize to my two-year-old daughter Medleyana for not spending much time with you lately. Thank you for being so understanding and not being mad at me at all. You are the reason that everything I did was meaningful! TABLE OF CONTENTS Page LIST OF TABLES . vi LIST OF FIGURES . vii INTRODUCTION . 1 The Classical Propositional Logic . 1 Interpretations of Formulas in Subsets . 5 Modal Logics . 6 Interpretations of Modal Logics in Topological Spaces . 9 THE LOGIC OF DISCRETE TOPOLOGICAL SPACES . 12 NON-COMPLETENESS OF S4 OVER SOME FAMILIES OF TOPO- LOGICAL SPACES . 18 THE RELATIONSHIP BETWEEN THE TOPOLOGICAL PROPER- TIES AND COMMON MODAL LOGICS . 26 CONCLUSIONS . 39 REFERENCES . 40 LIST OF TABLES Page Table 1. The Truth Table of :F; F ^ G; F _ G, F ! G, and F $ G 1 Table 2. The Truth Table of the Formulas F ! G and :F _ G . 2 Table 3. Some Axioms in Modal Logics . 7 Table 4. Common Modal Logics and the Corresponding Added Axioms 8 Table 5. The Effect of Modal Axioms on Topological Properties . 27 LIST OF FIGURES Page Figure 1. The Topological Spaces Xi for i = 2; 3; and n . 20 Figure 2. The Topological Space X1 . 21 Figure 3. The Sublogic Relationship among Some Modal Logics . 28 INTRODUCTION The Classical Propositional Logic Classical propositional logic is based on propositional variables and logical connectives. A propositional variable is a variable that can be assigned a truth value: true or false. Symbols :; ^; _, ! and $ are called logical connectives. The logical connectives enable compound formulas to be built from simpler formulas. A formula is defined as follows: (i) Every propositional variable is a formula. (ii) Given a formula F , the negation :F (not F ) is a formula. (iii) Given two formulas F and G, the conjunction F ^ G (F and G), the disjunction F _ G (F or G), the implication F ! G (F implies G), and the biconditional F $ G (F if and only if G) are formulas. The truth value of the compound formula is defined as a certain function of the truth values of the simpler formulas. Namely, the truth values of :F; F ^ G; F _ G; F ! G, and F $ G are defined as shown in Table 1. Table 1. The Truth Table of :F; F ^ G; F _ G, F ! G, and F $ G F G :F F ^ G F _ G F ! G F $ G T T F T T T T T F F F T F F F T T F T T F F F T F F T T In classical propositional logic, an interpretation of a formula is an assignment of truth values to its component variables. It is possible that a formula has different truth values under different interpretations. A formula F 2 is valid (also called a tautology) if F is true under every interpretation. If G is false under every interpretation, then G is called a contradiction. For example, the formula S _:S is a tautology and the formula S ^ :S is a contradiction. A truth table of a formula is a table that lists all its interpretations. For example, Table 2 is the truth table of the formulas F ! G and :F _ G. Table 2. The Truth Table of the Formulas F ! G and :F _ G F G :F :F _ G F ! G T T F T T T F F F F F T T T T F F T T T Notice from the truth table above that (F ! G) and (:F _ G) have the same truth values for any combination of truth values of formulas F and G. We say that F ! G and :F _ G are logically equivalent, and denote this by (F ! G) ≡ (:F _ G). Notice also that for formulas A and B, A ≡ B if and only if A $ B is a tautology. An axiom system consists of a set of formulas (called axioms) and some rules (called rules of inference). We say that an axiom system is sound if every formula that is derivable from this axiom system is valid. An axiom system is complete if every valid formula can be derived from this axiom system. The following is one of the sound and complete axiom systems for classical propositional logic: Axioms (1) (X ^ Y ) ! X 3 (2) (Y ^ X) ! X (3) X ! (X _ Y ) (4) X ! (Y _ X) (5) ::X ! X (6) X ! (Y ! X) (7) X ! (Y ! (X ^ Y )) (8) ((X ! Y ) ^ (X !:Y )) !:X (9) ((X ! Z) ^ (Y ! Z)) ! ((X _ Y ) ! Z) (10) ((X ! Y ) ^ (X ! (Y ! Z))) ! (X ! Z) Rule of inference X; X ! Y Modus Ponens Y Remark 1. The rule of Modus Ponens means that if X and X ! Y are derivable from the axiom system, then so is Y . Example 2. It is easy to show using a truth table that (A ^ B) ! (B ^ A) is a tautology. Let us show how it can be derived from the above the axiom system. 1. B ! (A ! (B ^ A)) axiom (7) 2. (A ^ B) ! B axiom (2) 3. (((A ^ B) ! A) ^ ((A ^ B) ! (A ! (B ^ A))) ! ((A ^ B) ! (B ^ A)) axiom (10) by substituting X with (A ^ B), Y with A, and Z with (B ^ A) 4. (((A ^ B) ! B) ^ ((A ^ B) ! (B ! (A ! (B ^ A))))) ! ((A ^ B) ! (A ! (B ^ A))) axiom (10) by substituting X with (A ^ B), Y with B, and Z with [A ! (B ^ A)] 5. (B ! (A ! (B ^ A))) ! ((A ^ B) ! (B ! (A ! (B ^ A)))) 4 axiom (6) by substituting X with (B ! (A ! (B ^ A))) and Y with (A ^ B) 6. (A ^ B) ! (B ! (A ! (B ^ A))) Modus Ponens, 1, 5 7. ((A ^ B) ! B) ! (((A ^ B) ! (B ! (A ! (B ^ A)))) ! (((A ^ B) ! B) ^ ((A ^ B) ! (B ! (A ! (B ^ A))))) axiom (7) by substituting X with ((A ^ B) ! B) and Y with ((A ^ B) ! (B ! (A ! (B ^ A)))) 8. ((A ^ B) ! (B ! (A ! (B ^ A)))) ! (((A ^ B) ! B) ^ ((A ^ B) ! (B ! (A ! (B ^ A)))) Modus Ponens, 2, 7 9. ((A ^ B) ! B) ^ ((A ^ B) ! (B ! (A ! (B ^ A)))) Modus Ponens, 6, 8 10. (A ^ B) ! (A ! (B ^ A)) Modus Ponens, 4, 9 11. (A ^ B) ! A axiom (1) 12. ((A ^ B) ! A) ! (((A ^ B) ! (A ! (B ^ A))) ! (((A ^ B) ! A) ^ ((A ^ B) ! (A ! (B ^ A))))) axiom (7) by substituting X with ((A ^ B) ! A) and Y with ((A ^ B) ! (A ! (B ^ A))) 13.