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This comprehensive book provides an adequate framework to establish various calculi of logical inference. Being an 'enriched' system of natural deduction, it helps to formulate logical calculi in an operational manner. By uncovering a certain harmony between a functional calculus on the labels and a logical calculus on the formulas, it allows mathematical foundations for systems of logic presentation designed to handle meta-level features at the object-level via a labelling mechanism, such as the D Gabbay's Labelled Deductive Systems. The book truly demonstrates that introducing of 'labels' is useful to understand the proof-calculus itself, and also to clarify its connections with model-theoretic interpretations.
(source: Nielsen Book Data)

• 1. Labelled natural deduction. 1.1. The role of the labels. 1.2. Canonical proofs and normalisation
• 2. The functional interpretation of implication. 2.1. Introduction. 2.2. Origins. 2.3. Types and propositions. 2.4. [symbol]-abstraction and implication. 2.5. Consistency proof. 2.6. Systems of implication and combinators. 2.7. Finale
• 3. The existential quantifier. 3.1. Motivation. 3.2. Quantifiers and normalisation. 3.3. Other approaches to existential quantification. 3.4. Model-theoretic semantics. 3.5. Finale. 3.6. Examples of deduction
• 4. Normalisation. 4.1. Introduction. 4.2. Proof transformations in labelled deduction. 4.3. Equivalences between proofs in LND. 4.4. The term rewriting system for LND. 4.5. Examples of transformations between proofs. 4.6. Final remarks
• 5. Natural deduction for equality. 5.1. Introduction. 5.2. Labelled deduction. 5.3. Finale
• 6. Normalisation for the equality fragment. 6.1. General rules. 6.2. The 'subterm substitution' rule. 6.3. The [symbol]- and [symbol]-rules. 6.4. Term rewriting systems. 6.5. The transformations between proofs in the equational fragment of the LND. 6.6. The rewriting system for the LND equational logic. 6.7. The normalization procedure: some examples. 6.8. Final remarks. 6.9. Appendix: The [symbol]- and [symbol]-reductions for the LND system. 6.10. Termination property for LND(EQ)-TRS
• 7. Modal logics. 7.1. The functional interpretation. 7.2. Modal logics and the functional interpretation. 7.3. Finale
• 8. Meaning and proofs: a reflection on proof-theoretic semantics. 8.1. Proof-theoretic semantics. 8.2. Meaning, use and consequences. 8.3. Meaning and purpose. 8.4. Meaning and use. 8.5. Meaning and the explanation of consequences. 8.6. Use and the explanation of consequences. 8.7. Early signs of 'meaning-use/usefulness-consequences'. 8.8. Normalisation of proofs: the explanation of the consequences. 8.9. Concluding remarks.

This comprehensive book provides an adequate framework to establish various calculi of logical inference. Being an 'enriched' system of natural deduction, it helps to formulate logical calculi in an operational manner. By uncovering a certain harmony between a functional calculus on the labels and a logical calculus on the formulas, it allows mathematical foundations for systems of logic presentation designed to handle meta-level features at the object-level via a labelling mechanism, such as the D Gabbay's Labelled Deductive Systems. The book truly demonstrates that introducing 'labels' is useful to understand the proof-calculus itself, and also to clarify its connections with model-theoretic interpretations.

• Scott Consequence Relations
• Biconsequence Relations
• Four-Valued Logics
• Nonmonotonic Semantics
• Default Consequence Relations
• Argumentation Theory
• Production and Causal Inference
• Epistemic Consequence Relations
• Modal Nonmonotonic Logics.
• (source: Nielsen Book Data)

Many approaches in the field of nonmonotonic and "commonsense" reasoning are actually different representations of the same basic ideas and constructions. This book gives a logical formalization of the original, explanatory approach to nonmonotonic reasoning. It uses the basic formalism of biconsequence relations, as well as derived systems of default, autoepistemic and causal inference, to cover in a single framework such diverse systems as default logic, autoepistemic and modal nonmonotonic logics, input/output and causal logics, argumentation theory, and semantics of general logic programs with negation as failure. This approach provides a clear separation between logical (monotonic) and nonmonotonic aspects of nonmonotonic reasoning. The separation allows, in particular, to single out the logics underlying modern logic programming and restore thereby the connection between logic programming and logic.
(source: Nielsen Book Data)

• Scott Consequence Relations
• Biconsequence Relations
• Four-Valued Logics
• Nonmonotonic Semantics
• Default Consequence Relations
• Argumentation Theory
• Production and Causal Inference
• Epistemic Consequence Relations
• Modal Nonmonotonic Logics.
• (source: Nielsen Book Data)

Many approaches in the field of nonmonotonic and "commonsense" reasoning are actually different representations of the same basic ideas and constructions. This book gives a logical formalization of the original, explanatory approach to nonmonotonic reasoning. It uses the basic formalism of biconsequence relations, as well as derived systems of default, autoepistemic and causal inference, to cover in a single framework such diverse systems as default logic, autoepistemic and modal nonmonotonic logics, input/output and causal logics, argumentation theory, and semantics of general logic programs with negation as failure. This approach provides a clear separation between logical (monotonic) and nonmonotonic aspects of nonmonotonic reasoning. The separation allows, in particular, to single out the logics underlying modern logic programming and restore thereby the connection between logic programming and logic.
(source: Nielsen Book Data)

• Definite Descriptions
• Virtual Objects
• Cardinal Arithmetic
• The Canonical Simulation
• Iterated Virtuality in Cardinal Arithmetic
• Ordinals.
• (source: Nielsen Book Data)

Reductionism is one of those philosophical myths that are either enthusiastically embraced or wholeheartedly rejected. And, like all other philosophical myths, it rarely gets serious consideration. Reasoning About Theoretical Entities strives to give reductionism its day in court, as it were, by explicitly developing several versions of the reductionist project and assessing their merits within the framework of modern symbolic logic. Not since the days of Carnap's Aufbau has reductionism received such close attention (albeit in a necessarily restricted and regimented setting such as that of modern mathematical logic). As such this book fills a void in the philosophical literature and presents a challenge to every would-be (anti-)reductionist. It should be required reading for every first-year graduate student in philosophy.
(source: Nielsen Book Data)

• Definite Descriptions
• Virtual Objects
• Cardinal Arithmetic
• The Canonical Simulation
• Iterated Virtuality in Cardinal Arithmetic
• Ordinals.
• (source: Nielsen Book Data)

Reductionism is one of those philosophical myths that are either enthusiastically embraced or wholeheartedly rejected. And, like all other philosophical myths, it rarely gets serious consideration. Reasoning About Theoretical Entities strives to give reductionism its day in court, as it were, by explicitly developing several versions of the reductionist project and assessing their merits within the framework of modern symbolic logic. Not since the days of Carnap's Aufbau has reductionism received such close attention (albeit in a necessarily restricted and regimented setting such as that of modern mathematical logic). As such this book fills a void in the philosophical literature and presents a challenge to every would-be (anti-)reductionist. It should be required reading for every first-year graduate student in philosophy.
(source: Nielsen Book Data)

• Ch. 1. Introduction and motivations. 1.1. Software specification, binary relations and fork
• ch. 2. Algebras of binary relations and relation algebras. 2.1. History and definitions. 2.2. Arithmetical properties
• ch. 3. Proper and abstract fork algebras. 3.1. On the origin of fork algebras. 3.2. Definition of the classes. 3.3. Arithmetical properties
• ch. 4. Representability and independence. 4.1. Representability of abstract fork algebras. 4.2. Independence of the axiomatization of fork
• ch. 5. Interpretability of classical first-order logic. 5.1. Basic definitions. 5.2. Interpreting FOLE
• ch. 6. Algebraization of non-classical logics. 6.1. Basic definitions and properties. 6.2. The fork logic FL. 6.3. Modal logics. 6.4. Representation of constraints in FL. 6.5. Interpretability of modal logics in FL. 6.6. A proof theoretical approach. 6.7. Interpretability of propositional dynamic logic in FL. 6.8. The fork logic FL'. 6.9. A Rasiowa-Sikorski calculus for FL'. 6.10. A relational proof system for intuitionistic logic. 6.11. A relational proof system for minimal intuitionistic logic. 6.12. Relational reasoning in intermediate logics
• ch. 7. A calculus for program construction. 7.1. Introduction. 7.2. Filters and sets. 7.3. The relational implication. 7.4. Representability and expressiveness in program construction. 7.5. A methodology for program construction. 7.6. Examples. 7.7. A D & C algorithm for MAXSTA. 7.8. Comparison with previous work.

Fork algebras are a formalism based on the relational calculus, with interesting algebraic and metalogical properties. Their representability is especially appealing in computer science, since it allows a closer relationship between their language and models. This book gives a careful account of the results and presents some applications of Fork algebras in computer science, particularly in system specification and program construction. Many applications of Fork algebras in formal methods can be applied in many ways, and the book covers all the essentials in order to provide the reader with a better understanding.

• Contents: Introduction and Motivations
• Algebras of Binary Relations and Relation Algebras
• Proper and Abstract Fork Algebras
• Representability and Independence
• Interpretability of Classical First-Order Logic
• Algebraization of Non-Classical Logics
• A Calculus for Program Construction.
• (source: Nielsen Book Data)

Fork algebras are a formalism based on the relational calculus, with interesting algebraic and metalogical properties. Their representability is especially appealing in computer science, since it allows a closer relationship between their language and models. This book gives a careful account of the results and presents some applications of Fork algebras in computer science, particularly in system specification and program construction. Many applications of Fork algebras in formal methods can be applied in many ways, and the book covers all the essentials in order to provide the reader with a better understanding.
(source: Nielsen Book Data)

• Fine-Grained Theories of Time (P Blackburn)
• Revision Sequences and Computers with an Infinite Amount of Time (B Lowe)
• On Frege's Nightmare: A Combination of Intuitionistic, Free and Paraconsistent Logics (S Rahman)
• Truthmakers, Entailment and Necessity (S Read)
• Global Definability in Basic Modal Logic (M de Rijke & H Sturm)
• Ackermann's Implication for Typefree Logic (K Robering)
• Why Dialogical Logic? (H Ruckert)
• Semantics for Constructive Negations (Y Shramko)
• Recent Trends in Paraconsistent Logic (M Urchs)
• Obligations, Authorities, and History Dependence (H Wansing).
• (source: Nielsen Book Data)

This book covers a broad range of up-to-date issues in non-classical logic that are of interest not only to philosophical and mathematical logicians but also to computer scientists and researchers in artificial intelligence. The problems addressed range from methodological issues in paraconsistent and deontic logic to the revision theory of truth and infinite Turing machines. The book identifies a number of important current trends in contemporary non-classical logic. Among them are dialogical and substructural logic, the classification of concepts of negation, truthmaker theory, and mathematical and foundational aspects of modal and temporal logic.
(source: Nielsen Book Data)

• Fine-Grained Theories of Time (P Blackburn)
• Revision Sequences and Computers with an Infinite Amount of Time (B Lowe)
• On Frege's Nightmare: A Combination of Intuitionistic, Free and Paraconsistent Logics (S Rahman)
• Truthmakers, Entailment and Necessity (S Read)
• Global Definability in Basic Modal Logic (M de Rijke & H Sturm)
• Ackermann's Implication for Typefree Logic (K Robering)
• Why Dialogical Logic? (H Ruckert)
• Semantics for Constructive Negations (Y Shramko)
• Recent Trends in Paraconsistent Logic (M Urchs)
• Obligations, Authorities, and History Dependence (H Wansing).
• (source: Nielsen Book Data)

This book covers a broad range of up-to-date issues in non-classical logic that are of interest not only to philosophical and mathematical logicians but also to computer scientists and researchers in artificial intelligence. The problems addressed range from methodological issues in paraconsistent and deontic logic to the revision theory of truth and infinite Turing machines. The book identifies a number of important current trends in contemporary non-classical logic. Among them are dialogical and substructural logic, the classification of concepts of negation, truthmaker theory, and mathematical and foundational aspects of modal and temporal logic.
(source: Nielsen Book Data)