| Nonstandard Inferences in Description Logics: The Story So Far | p. 1 |
| Introduction | p. 2 |
| Description Logics and Standard Inferences | p. 6 |
| Nonstandard Inferences-Motivation and Definitions | p. 11 |
| Motivation | p. 12 |
| Definitions | p. 15 |
| Techniques | p. 21 |
| A Structural Characterization of Subsumption | p. 23 |
| Getting started - The characterization for EL | p. 24 |
| Extending the characterization to ALE | p. 27 |
| Characterization of subsumption for other DLs | p. 31 |
| The Least Common Subsumer | p. 32 |
| The LCS for EL | p. 32 |
| The LCS for ALE | p. 34 |
| The LCS for other DLs | p. 36 |
| The Most Specific Concept | p. 36 |
| Existence and approximation of the MSC | p. 36 |
| The most specific concept in the presence of cyclic TBoxes | p. 44 |
| Rewriting | p. 44 |
| The minimal rewriting decision problem | p. 45 |
| The minimal rewriting computation problem | p. 46 |
| Approximation | p. 52 |
| Matching | p. 53 |
| Deciding matching problems | p. 54 |
| Solutions of matching problems | p. 55 |
| Computing matchers | p. 58 |
| Matching in other DLs and extensions of matching | p. 64 |
| Conclusion and Future Perspectives | p. 64 |
| References | p. 66 |
| Problems in the Logic of Provability | p. 77 |
| Introduction | p. 78 |
| Informal Concepts of Proof | p. 81 |
| Formal and informal provability and the problem of equivalence of proofs | p. 81 |
| Strengthening Hilbert's thesis | p. 85 |
| Coordinate-free proof theory | p. 87 |
| Basics of Provability Logic | p. 91 |
| Provability Logic for Intuitionistic Arithmetic | p. 93 |
| Propositional logics of arithmetical theories | p. 94 |
| Admissible rules | p. 97 |
| The provability logic of HA and related theories | p. 100 |
| Provability Logic and Bounded Arithmetic | p. 102 |
| Classification of Bimodal Provability Logics | p. 106 |
| Magari Algebras | p. 109 |
| Interpretability Logic | p. 114 |
| Graded Provability Algebras | p. 120 |
| List of Problems | p. 125 |
| References | p. 129 |
| Open Problems in Logical Dynamics | p. 137 |
| Logical Dynamics | p. 137 |
| Standard Epistemic Logic | p. 139 |
| Language | p. 140 |
| Semantics | p. 141 |
| Basic model theory | p. 142 |
| Axiomatics | p. 143 |
| Complexity | p. 143 |
| Open problems, even here | p. 144 |
| Public Announcement: Epistemic Logic Dynamified | p. 144 |
| World elimination: the system PAL | p. 145 |
| What are the real update laws? | p. 150 |
| Model theory of learning | p. 153 |
| Communication and planning | p. 155 |
| Group knowledge | p. 156 |
| Dynamic Epistemic Logic | p. 157 |
| Information from arbitrary events: product update | p. 158 |
| Update evolution | p. 160 |
| Questions of language design | p. 162 |
| Extensions of empirical coverage | p. 163 |
| Background in Standard Logics | p. 164 |
| Modal logic | p. 165 |
| First-order logic | p. 166 |
| Fixed-point logics | p. 168 |
| From Information Update to Belief Revision | p. 170 |
| From knowledge to belief | p. 170 |
| Dynamic doxastic logic | p. 171 |
| Better-known theories of belief revision | p. 175 |
| Probabilistic update | p. 177 |
| Temporal Epistemic logic | p. 178 |
| Broader temporal perspectives on update | p. 178 |
| Knowledge and ignorance over time | p. 179 |
| Representation of update logics | p. 181 |
| Connections with other parts of mathematics | p. 183 |
| Game Logics and Game Theory | p. 184 |
| Conclusion | p. 185 |
| References | p. 186 |
| Computability and Emergence | p. 193 |
| An Emergent World around Us | p. 194 |
| Descriptions, Algorithms, and the Breakdown of Inductive Structure | p. 195 |
| Ontology and Mathematical Structure | p. 202 |
| Where does It All Start? | p. 204 |
| Towards a Model Based on Algorithmic Content | p. 209 |
| Levels of Reality | p. 214 |
| Algorithmic Content Revisited | p. 221 |
| What Is to Be Done? | p. 224 |
| References | p. 228 |
| Samsara | p. 233 |
| Introduction | p. 234 |
| The structure of the paper | p. 234 |
| An Example of a Process | p. 235 |
| What Logics Do We Need? | p. 236 |
| Extracting constructions from proofs | p. 242 |
| The Lambda Calculus and the Curry-Howard correspondence | p. 243 |
| Proofs as types | p. 245 |
| Strong normalization and program extraction | p. 248 |
| Beyond traditional logic in program extraction | p. 250 |
| Proofs from programs | p. 256 |
| Programs then proofs | p. 257 |
| What are Logical Systems and What Should They Be? | p. 258 |
| Higher order logic | p. 259 |
| A note on set theory | p. 261 |
| Computation and proof | p. 262 |
| The Nature of Proof | p. 262 |
| The question of scale and the role of technology | p. 263 |
| Foundations | p. 268 |
| Final Remarks | p. 269 |
| References | p. 270 |
| Two Doors to Open | p. 277 |
| Logic and Cognitive Science | p. 279 |
| Spatial intuition | p. 281 |
| Kurt Gödel and the choice of representation | p. 285 |
| A sample cognitive description of reasoning | p. 293 |
| Frege versus Peirce: comparison of representations | p. 298 |
| Medieval Arabic Semantics | p. 300 |
| References | p. 313 |
| Applied Logic: A Manifesto | p. 317 |
| What is Applied Logic? | p. 317 |
| Mathematics and Logic, but Different from Mathematical Logic | p. 319 |
| Mathematical Logic and Mathematics | p. 320 |
| Where applied logic differs | p. 322 |
| Applied mathematics is good mathematics | p. 324 |
| Applied logic is applied mathematics | p. 325 |
| Applied Philosophical Logic | p. 326 |
| Applied philosophical logic = theoretical AI | p. 328 |
| What Does Computer Science Have to Do with It? | p. 328 |
| Logic is the calculus of computer science | p. 329 |
| Computer science motivates logic | p. 330 |
| Going beyond the traditional boundaries of logic | p. 331 |
| Other Case Studies | p. 332 |
| Neural networks and non-monotonic logic | p. 332 |
| Dynamic epistemic logic | p. 333 |
| Linguistics, logic, and mathematics | p. 335 |
| But is it dead? | p. 339 |
| Being as catholic as Possible | p. 341 |
| References | p. 343 |
| Index | p. 345 |
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