Contents Preface Chapter 1 Introduction 1 1.1 Overview of the standard completeness for fuzzy logics 1 1.2 Standard completeness of IUL 2 1.3 My work on the standard completeness for IUL and reviewers'comments on it 3 1.3.1 Reviewers'comments in 2015 by APAL 3 1.3.2 Reviewers'comments in 2016 by FSS 5 1.3.3 Reviewers'comments in 2018 by FSS 7 1.3.4 Reviewers'comments in 2018 on my second paper by JSL 9 1.3.5 Communication with editors and colleagues 10 Chapter 2 The Logic HpsUL* and Its Chain Completeness 12 2.1 The logic HpsUL* 12 2.2 Proof by cases property and prelinearity property of HpsUL 22 2.3 The chain completeness for HpsUL 27 2.4 Extensions of HpsUL 31 Chapter 3 Jenei and Montagna's Algebraic Method 38 3.1 The logic MTL and MTL-algebras 38 3.2 Standard completeness for MTL 40 3.3 Standard completeness for IMTL 47 Chapter 4 Wang's Constructions on HpsUL*-Chains 52 4.1 New extensions of HpsUL* and new algebras 52 4.2 Structures X on HpsUL*-chains 55 4.3 Standard completeness for CnIUL and IULw 62 4.4 A proof of associativity of in X for IUL 71 Chapter 5 Metcalfe and Montagna's Proof-Theoretic Method 95 5.1 Standard completeness and the density rule 95 5.2 Hypersequent calculi 106 5.3 Cut-elimination 112 5.4 Density elimination 117 Chapter 6 Wang's Proof-Theoretic Method 132 6.1 Introduction 132 6.2 Proof of the Main theorem: A computational example 133 6.3 Preprocessing of proof tree 143 6.4 The generalized density rule (D) for GL 152 6.5 Extraction of elimination rules 158 Chapter 7 Standard Completeness for IUL 166 7.1 Separation of one branch 166 7.2 Separation algorithm of multiple branches 172 7.3 The proof of Main theorem 187 7.4 Final remarks and open problems 189 Chapter 8 An Extension of Wang's Proof-Theoretic Method 190 8.1 Introduction 190 8.2 GpsUL** and its cut-elimination 192 8.3 GpsUL and its generalized density rule (D) 202 8.4 Density elimination for GpsUL* 216 Chapter 9 Two Big Examples for Separation Algorithms 224 9.1 Proof of Theorem 7.2.2: A big example 224 9.1.1 Preprocessing of proof tree 225 9.1.2 Elimination rules 227 9.1.3 Separation of one branch 228 9.1.4 Separations of two branches 230 9.1.5 A concise algorithm 235 9.1.6 Separation of four branches 237 9.2 Proof of Lemma 8.4.7: A big example 238 9.2.1 Maximal nodes, complete sets and their heights 241 9.2.2 Some copies of can't be contracted in G1(J) 241 9.2.3 Illustration of Lemma 8.4.7: Construct 1;2 246 9.2.4 Remark on construction of 250 9.2.5 Constructing of 251 9.2.6 is undefined and can't be constructed by the new main algorithm 254 Bibliography 256
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