Lambda Calculi: A Guide Interpolation and Definability Discourse Representation Theory

such questions for centuries (unrestricted by the capabilities of any ha- ware). Theprinciplesgoverningtheinteractionofseveralprocesses, forexample, are abstract an similar to principles governing the cooperation of two large organisation. A detailed rule based e?ective but rigid bureaucracy is very much similar to a complex computer program handling and manipulating data. My guess is that the principles underlying one are very much the same as those underlying the other. Ibelievethedayisnotfarawayinthefuturewhenthecomputerscientist will wake up one morning with the realisation that he is actually a kind of formal philosopher! The projected number of volumes for this Handbook is about 18. The subjecthasevolvedanditsareashavebecomeinterrelatedtosuchanextent that it no longer makes sense to dedicate volumes to topics. However, the volumes do follow some natural groupings of chapters. I would like to thank our authors and readers for their contributions and their commitment in making this Handbook a success. Thanksalso to our publication administrator Mrs J. Spurr for her usual dedication and excellence and to Kluwer Academic Publishers (now Springer) for their continuing support for the Handbook. Dov Gabbay King's College London x PREFACE TO THE SECOND EDITION Logic IT Natural Program Arti?cialin- Logic p- language control spec- telligence gramming processing i?cation, veri?cation, concurrency Temporal Expressive Expressive Planning. Extension of logic power of tense power for re- Time depen- Horn clause operators. currentevents. dent data. with time Temporal Speci?cation Eventcalculus. capability. indices. Sepa- of tempo- Persistence Eventcalculus. ration of past ral control. through time- Temporal logic from future Decision prob- the Frame programming.

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good! The first edition was the second handbook published for the logic com- nity. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook of Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organi- tion on the one hand and to provide the theoretical basis for the computer program constructs on the other.

such questions for centuries (unrestricted by the capabilities of any hard ware). The principles governing the interaction of several processes, for example, are abstract an similar to principles governing the cooperation of two large organisation. A detailed rule based effective but rigid bureaucracy is very much similar to a complex computer program handling and manipulating data. My guess is that the principles underlying one are very much the same as those underlying the other. I believe the day is not far away in the future when the computer scientist will wake up one morning with the realisation that he is actually a kind of formal philosopher The projected number of volumes for this Handbook is about 18. The subject has evolved and its areas have become interrelated to such an extent that it no longer makes sense to dedicate volumes to topics. However, the volumes do follow some natural groupings of chapters. I would like to thank our authors are readers for their contributions and their commitment in making this Handbook a success. Thanks also to our publication administrator Mrs J. Spurr for her usual dedication and excellence and to Kluwer Academic Publishers for their continuing support for the Handbook."

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook of Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other."

suchquestionsforcenturies(unrestrictedbythecapabilitiesofanyhard- ware). Theprinciplesgoverningtheinteractionofseveralprocesses,forexample, areabstractansimilartoprinciplesgoverningthecooperationoftwolarge organisation.Adetailedrulebasedeffectivebutrigidbureaucracyisvery muchsimilartoacomplexcomputerprogramhandlingandmanipulating data. Myguessisthattheprinciplesunderlyingoneareverymuchthe sameasthoseunderlyingtheother. Ibelievethedayisnotfarawayinthefuturewhenthecomputerscientist willwakeuponemorningwiththerealisationthatheisactuallyakindof formalphilosopher! TheprojectednumberofvolumesforthisHandbookisabout18.The subjecthasevolvedanditsareashavebecomeinterrelatedtosuchanextent thatitnolongermakessensetodedicatevolumestotopics.However,the volumesdofollowsomenaturalgroupingsofchapters. Iwouldliketothankourauthorsarereadersfortheircontributionsand theircommitmentinmakingthisHandbookasuccess. Thanksalsoto ourpublicationadministratorMrsJ.Spurrforherusualdedicationand excellenceandtoKluwerAcademicPublishersfortheircontinuingsupport fortheHandbook. DovGabbay King'sCollegeLondon x Logic II IT Natural Program Artificialin- Logic p- language controlspec- telligence gramming processing ification, verification, concurrency Temporal Expressive Expressive Planning. Extension of logic poweroftense power for re- Time depen- Horn clause operators. currentevents. dent data. with time Temporal Specification Eventcalculus. capability. indices. Sepa- of tempo- Persistence Eventcalculus. rationofpast ral control. throughtime- Temporallogic fromfuture Decisionprob- the Frame programming. Problem.Tem- lems. Model checking. poral query language. temporal transactions. Modal logic. generalised Actionlogic Beliefrevision. Negation by Multi-modal quantifiers Inferential failure and logics databases modality Algorithmic Discourse rep- New logics. Generaltheory Proceduralap- proof resentation. Generic theo- of reasoning. proachtologic Direct com- remprovers Non-monotonic putation on systems linguisticinput Non- Resolving Loopchecking. Intrinsiclogical Negation by monotonic ambigui- Non-monotonic discipline for failure.Deduc- reasoning ties. Machine decisionsabout AI. Evolving tivedatabases translation. loops. Faults and com- Document insystems. municating classification. databases Relevance theory Probabilistic logicalanalysis Realtimesys- Expert sys- Semantics for and fuzzy oflanguage tems tems.Machine logicprograms logic learning Intuitionistic Quantifiers in Constructive Intuitionistic Horn clause logic logic reasoning and logicisabetter logic is really proof theory logical basis intuitionistic.

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic artiele in the Encyelopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good. ! The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook 0/ Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook 0/ Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence cireles. These areas were under increasing commercial press ure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other.

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as weIl as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook 0/ Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook 0/ Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other."

The essays in this collection are the outgrowth of a workshop, held in June 1976, on formal approaches to the semantics and pragmatics of natural languages. They document in an astoundingly uniform way the develop ments in the formal analysis of natural languages since the late sixties. The avowed aim of the' workshop was in fact to assess the progress made in the application of formal methods to semantics, to confront different approaches to essentially the same problems on the one hand, and, on the other, to show the way in relating semantic and pragmatic explanations of linguistic phenomena. Several of these papers can in fact be regarded as attempts to close the 'semiotic circle' by bringing together the syntactic, semantic and pragmatic properties of certain constructions in an explanatory framework thereby making it more than obvious that these three components of an integrated linguistic theory cannot be as neatly separated as one would have liked to believe. In other words, not only can we not elaborate a syntactic description of (a fragment of) a language and then proceed to the semantics (as Montague pointed out already forcefully in 1968), we cannot hope to achieve an adequate integrated syntax and semantics without paying heed to the pragmatic aspects of the constructions involved. The behavior of polarity items, 'quantifiers' like any, conditionals or even logical particles like and and or in non-indicative sentences is clear-cut evidence for the need to let each component of the grammar inform the other."

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic artiele in the Encyelopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good. ! The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook 0/ Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook 0/ Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence cireles. These areas were under increasing commercial press ure to provide devices which help andjor replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other.

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good, The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook of Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other."

The fourteenth volume of the Second Edition covers central topics in philosophical logic that have been studied for thousands of years, since Aristotle: Inconsistency, Causality, Conditionals, and Quantifiers. These topics are central in many applications of logic in central disciplines and this book is indispensable to any advanced student or researcher using logic in these areas. The chapters are comprehensive and written by major figures in the field.

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as weIl as to consumers of logic in many applied areas. The main logic artiele in the Encyelopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good. ! The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook 0/ Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook 0/ Philosophical Logic, published 1983-1989 came at a fortunate at the evolution of logic. This was the time when logic temporal junction was gaining ground in computer science and artificial intelligence cireles. These areas were under increasing commercial pressure to provide devices which help andjor replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other.

such questions for centuries (unrestricted by the capabilities of any ha- ware). Theprinciplesgoverningtheinteractionofseveralprocesses, forexample, are abstract an similar to principles governing the cooperation of two large organisation. A detailed rule based e?ective but rigid bureaucracy is very much similar to a complex computer program handling and manipulating data. My guess is that the principles underlying one are very much the same as those underlying the other. Ibelievethedayisnotfarawayinthefuturewhenthecomputerscientist will wake up one morning with the realisation that he is actually a kind of formal philosopher! The projected number of volumes for this Handbook is about 18. The subjecthasevolvedanditsareashavebecomeinterrelatedtosuchanextent that it no longer makes sense to dedicate volumes to topics. However, the volumes do follow some natural groupingsof chapters. Iwould liketothank our authorsand readersfor their contributionsand their commitment in making this Handbook a success. Thanksalso to our publication administrator Mrs J. Spurr for her usual dedication and excellence and to Kluwer Academic Publishers (now Springer) for their continuing support for the Handbook. Dov Gabbay King's College London 10 Logic IT Natural Program Arti?cial in- Logicp- language control spec- telligence gramming processing i?cation, veri?cation, concurrency Temporal Expressive Expressive Planning. Extension of logic power of tense power for re- Time depen- Horn clause operators. current events. dent data. with time Temporal Speci?cation Eventcalculus. capability. indices. Sepa- of tempo- Persistence Event calculus. ration of past ral control. through time- Temporal logic from future Decision prob- theFrame programming. lems. Model Problem. T- checking. poral query language. temporal transactions.

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good. ! The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise, The four volume Handbook of Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other.

This second edition of the Handbook of Philosophical Logic reflects great changes in the landscape of philosophical logic since the first edition. It gives readers an idea of that landscape and its relation to computer science and formal language and artificial intelligence. It shows how the increased demand for philosophical logic from computer science and artificial intelligence and computational linguistics accelerated the development of the subject directly and indirectly. This development in turn, directly pushed research forward, stimulated by the needs of applications. New logic areas became established and old areas were enriched and expanded. At the same time, it socially provided employment for generations of logicians residing in computer science, linguistics and electrical engineering departments which of course helped keep the logic community to thrive. The many contributors to this Handbook are active in these application areas and are among the most famous leading figures of applied philosophical logic of our times. "

Belief Revision Refutation and systems in Propositional Logic. A Quantifier Scope in Formal Linguistics and Non-deterministic Semantics for Logical Systems.

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook of Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other."

This second edition of the Handbook of Philosophical Logic reflects great changes in the landscape of philosophical logic since the first edition. It gives readers an idea of that landscape and its relation to computer science and formal language and artificial intelligence. It shows how the increased demand for philosophical logic from computer science and artificial intelligence and computational linguistics accelerated the development of the subject directly and indirectly. This development in turn, directly pushed research forward, stimulated by the needs of applications. New logic areas became established and old areas were enriched and expanded. At the same time, it socially provided employment for generations of logicians residing in computer science, linguistics and electrical engineering departments which of course helped keep the logic community to thrive. The many contributors to this Handbook are active in these application areas and are among the most famous leading figures of applied philosophical logic of our times.

Lambda Calculi: A Guide Interpolation and Definability Discourse Representation Theory

Belief Revision Refutation and systems in Propositional Logic. A Quantifier Scope in Formal Linguistics and Non-deterministic Semantics for Logical Systems.

suchquestionsforcenturies(unrestrictedbythecapabilitiesofanyhard- ware). Theprinciplesgoverningtheinteractionofseveralprocesses,forexample, areabstractansimilartoprinciplesgoverningthecooperationoftwolarge organisation.Adetailedrulebasedeffectivebutrigidbureaucracyisvery muchsimilartoacomplexcomputerprogramhandlingandmanipulating data. Myguessisthattheprinciplesunderlyingoneareverymuchthe sameasthoseunderlyingtheother. Ibelievethedayisnotfarawayinthefuturewhenthecomputerscientist willwakeuponemorningwiththerealisationthatheisactuallyakindof formalphilosopher! TheprojectednumberofvolumesforthisHandbookisabout18.The subjecthasevolvedanditsareashavebecomeinterrelatedtosuchanextent thatitnolongermakessensetodedicatevolumestotopics.However,the volumesdofollowsomenaturalgroupingsofchapters. Iwouldliketothankourauthorsarereadersfortheircontributionsand theircommitmentinmakingthisHandbookasuccess. Thanksalsoto ourpublicationadministratorMrsJ.Spurrforherusualdedicationand excellenceandtoKluwerAcademicPublishersfortheircontinuingsupport fortheHandbook. DovGabbay King'sCollegeLondon x Logic II IT Natural Program Artificialin- Logic p- language controlspec- telligence gramming processing ification, verification, concurrency Temporal Expressive Expressive Planning. Extension of logic poweroftense power for re- Time depen- Horn clause operators. currentevents. dent data. with time Temporal Specification Eventcalculus. capability. indices. Sepa- of tempo- Persistence Eventcalculus. rationofpast ral control. throughtime- Temporallogic fromfuture Decisionprob- the Frame programming. Problem.Tem- lems. Model checking. poral query language. temporal transactions. Modal logic. generalised Actionlogic Beliefrevision. Negation by Multi-modal quantifiers Inferential failure and logics databases modality Algorithmic Discourse rep- New logics. Generaltheory Proceduralap- proof resentation. Generic theo- of reasoning. proachtologic Direct com- remprovers Non-monotonic putation on systems linguisticinput Non- Resolving Loopchecking. Intrinsiclogical Negation by monotonic ambigui- Non-monotonic discipline for failure.Deduc- reasoning ties. Machine decisionsabout AI. Evolving tivedatabases translation. loops. Faults and com- Document insystems. municating classification. databases Relevance theory Probabilistic logicalanalysis Realtimesys- Expert sys- Semantics for and fuzzy oflanguage tems tems.Machine logicprograms logic learning Intuitionistic Quantifiers in Constructive Intuitionistic Horn clause logic logic reasoning and logicisabetter logic is really proof theory logical basis intuitionistic.

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good! The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook of Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other.

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic artiele in the Encyelopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good. ! The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook 0/ Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook 0/ Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence cireles. These areas were under increasing commercial press ure to provide devices which help andjor replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other.

The aim of the first volume of the present Handbook of Philosophical Logic is essentially two-fold: First of all, the chapters in this volume should provide a concise overview of the main parts of classical logic. Second, these chapters are intended to present all the relevant background material necessary for the understanding of the contributions which are to follow in the next three volumes. We have thought it to be of importance that the connections between classical logic and its 'extensions' (covered in Volume 11) as well as its most important 'alternatives' (covered in Volume Ill) be brought out clearly from the start. The first chapter presents a clear and detailed picture of the range of what is generally taken to be the standard logical framework, namely, predicate (or first-order quantificational) logic. On the one hand, this chapter surveys both propositionai logic and first-order predicate logic and, on the other hand, presents the main metalogical results obtained for them. Chapter 1. 1 also contains a discussion of the limits of first-order logic, i. e. it presents an answer to the question: Why has predicate logic played such a formidable role in the formalization of mathematics and in the many areas of philo sophical and linguistic applications? Chapter 1. 1 is prerequisite for just about all the other chapters in the entire Handbook, while the other chapters in Volume I provide more detailed discussions of material developed or hinted at in the first chapter.