R-Calculus, II: Many-Valued Logics

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Author(s):	Li, Wei
ISBN No.:	9789811692963
Pages:	xiii, 271
Year:	202304
Format:	Trade Paper
Price:	$ 234.59
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1 Introduction 11 1.1 Belief revision . 11 1.2 R-calculus . 13 1.3 Contents in the first-volume . 14 1.4 Contents in this volume .

17 1.5 Notations . 21 2 R-Calculus For Propositional Logic 24 2.1 Basic definitions . 25 2.2 Monotonic tableau proof systems . 26 2.2.

1 Tableau proof system Tf . 26 2.2.2 Tableau proof system Tt . 29 2.3 Nonmonotonic tableau proof systems . 31 2.3.

1 Tableau proof system St . 32 2.3.2 Tableau proof system Sf . 34 2.4 R-calculi . 35 2.4.

1 R-calculus Rt . 36 2.4.2 R-calculus Rf . 40 2.5 Projecting R-calculi to tableau proof systems . 41 2.6 Notes .

43 3 R-Calculus For L3-Valued Propositional Logic 45 3.1 Basic definitions . 46 3.2 Monotonic tableau proof systems . 49 3.2.1 Tableau proof system Tt . 49 3.

2.2 Tableau proof system Tm . 50 3.2.3 Tableau proof system Tf . 51 3.3 Nonmonotonic tableau proof systems . 52 3.

3.1 Tableau proof system St . 54 3.3.2 Tableau proof system Sm . 55 3.3.3 Tableau proof system Sf .

55 3.4 R-calculi . 56 3.4.1 R-calculus Rt . 57 3.4.2 R-calculus Rm .

60 3.4.3 R-calculus Rf . 63 3.5 Satisfiability and unsatisfiability . 65 3.5.1 t-satisfiability and t-unsatisfiability .

65 3.5.2 m-satisfiability and m-unsatisfiability . 67 3.5.3 f-satisfiability and f-unsatisfiability . 68 3.6 Projecting R-calculi to tableau proof systems .

70 3.7 Notes . 71 4 R-Calculus For L3-Valued PL,II 75 4.1 Monotonic tableau proof systems . 75 4.1.1 Tableau proof system Tt . 76 4.

1.2 Tableau proof system Tm . 77 4.1.3 Tableau proof system Tf . 78 4.2 Nonmonotonic tableau proof systems . 79 4.

2.1 Tableau proof system St . 79 4.2.2 Tableau proof system Sm . 80 4.2.3 Tableau proof system Sf .

81 4.3 R-calculi . 82 4.3.1 R-calculus Rt . 82 4.3.2 R-calculus Rm .

85 4.3.3 R-calculus Rf . 87 4.4 Validity and invalidity . 90 4.4.1 t-invalidity and t-validity .

90 4.4.2 m-invalidity and m-validity . 92 4.4.3 f-invalidity and f-validity . 94 4.5 Projecting R-calculi to tableau proof systems .

96 5 R-Calculus For B22-Valued PL 98 5.1 Basic definitions . 100 5.2 Monotonic tableau proof systems . 101 5.2.1 Tableau proof system Tt . 103 5.

2.2 Tableau proof system Ttop . 104 5.2.3 Tableau proof system Tâ¥. 105 5.2.4 Tableau proof system Tf .

107 5.3 Nonmonotonic tableau proof systems . 108 5.3.1 Tableau proof system St . 110 5.3.2 Tableau proof system Stop .

112 5.3.3 Tableau proof system Sâ¥. 113 5.3.4 Tableau proof system Sf . 114 5.4 R-calculi .

116 5.4.1 R-calculus Rt . 117 5.4.2 R-calculus Rtop . 121 5.4.

3 R-calculus Râ¥ . 125 5.4.4 R-calculus Rf . 128 5.5 Projecting R-calculi to tableau proof systems . 132 5.6 Notes .

135 6 R-Calculus For B22-Valued PL,II 138 6.1 Monotonic tableau proof systems . 140 6.1.1 Tableau proof system Tt top . 142 6.1.2 Tableau proof system Ttâ¥ .

143 6.2 Tableau proof systems . 145 6.2.1 Tableau proof system Tt top . 146 6.2.2 Tableau proof system Ttâ¥ .

148 6.3 Nonmonotonic tableau proof systems . 149 6.3.1 Tableau proof system St top . 150 6.3.2 Tableau proof system Stâ¥.

151 6.3.3 Tableau proof system St top . 152 6.3.4 Tableau proof system Stâ¥. 153 6.4 R-calculi .

155 6.4.1 R-calculus Rt top . 155 6.4.2 R-calculus Rtâ¥. 157 6.4.

3 R-calculus Rf top . 159 6.4.4 R-calculus Rf.â¥. 161 6.5 Projecting R-calculi to tableau proof systems . 163 6.

6 Notes . 165 7 Complementary R-Calculus For PL 168 7.1 Co-R-calculi in propositional logic . 169 7.1.1 Co-R-calculus Ut . 169 7.1.

2 Co-R-calculus Uf . 171 7.2 Co-R-calculi in L3-valued PL . 173 7.2.1 Co-R-calculus Ut . 173 7.2.

2 Co-R-calculus Um . 176 7.2.3 Co-R-calculus Uf . 179 7.3 Co-R-calculi in B22-valued propositional logic . 181 7.3.

1 Co-R-calculus Ut . 182 7.4 Notes . 187 8 Multisequents and Hypersequents 188 8.1 Tableau proof systems . 189 8.1.1 Tableau-typed proof system Tt .

189 8.1.2 Tableau proof system Tt . 190 8.2 Sequents in L3-valued propositional logic . 192 8.2.1 Gentzen deduction system for Î=>â.

193 8.2.2 Gentzen deduction system for Î=> Î. 195 8.2.3 Gentzen deduction system for Î=> Î . 196 8.3 Multisequents in L3-valued PL .

197 8.3.1 Multisequents . 199 8.3.2 Co-multisequents . 202 8.4 Hypersequents in L3-valued PL .

206 8.5 Notes . 214 9 Product of Two R-Calculi 216 9.1 Tableau proof systems in modalized PL . 217 9.1.1 Monotonic tableau proof systems . 217 9.

1.2 Nonmonotonic tableau proof systems . 218 9.2 Product of B2-valued PLs . 219 9.2.1 Tableau pro.

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