Algebraic Aspects of Entailment�

Approximation� Belief Revision

and Computability

Cyril Lee Flax �BSc� MSc� MBA�

A dissertation submitted to Macquarie University for the degree of

Doctor of Philosophy in the Department of Computing� Division of

Information and Communication Sciences�

January ����

ii

Contents

Thesis Statement v

Certi�cation ix

Acknowledgements xi

� Introduction �

� First�order Syntax and Semantics �

� The FOE Languages ��

� Restricted Semantics ��

� Restricted Entailments as a Lattice ��

Restricted entailments as a domain ��

� Modal Restricted First�order Semantics �

� Tableau Proofs ��

Soundness of the modal proof system ��

�� Completeness ��

�� The Preboolean Algebra PSENL

�� Belief Revision Using PSENL ��

�� Nomonotonic Entailment �

�� Computability ���

�� Conclusion ��

A Set theory ���

iii

iv CONTENTS

Bibliography ���

Thesis Statement

This work examines the entailment relation of �rst�order logic from two per�spectives� one involving the idea of restriction and the other algebraic�

First� I have shown that the entailment relation between sets of sentencescan be restricted in the following sense� One imagines an agent with limitedcapacity checking that a set of sentences X entails a set Y � The agent is limitedin that it can only check that a model of X also satis�es Y for models belongingto a subset of the family of all models ������ That is the agent is restricted in itsmodel checking� In de�ning restricted entailment I have been motivated by themental model theory of human reasoning in cognitive science due to Johnson�Laird �� � �� �� where it is claimed that humans reason by building mentalmodels but the model checking is not exhaustive at any stage because of limitedhuman capacities �see Introduction��

Secondly� I have shown that restricted entailment can be regarded as a lat�tice preorder which induces a preboolean algebra whose elements are sets ofsentences � � ���

Thesis

My thesis is that the logical properties of restricted entailment can be rigor�ously analysed and that the induced algebra can be used to give an account ofbelief contraction in the AGM theory of belief revision as well as nonmonotonicentailment in the context of a belief set�

Belief Revision

I have adopted the AGM approach to belief revision due to Alchourr�on� G�ardenforsand Makinson ��� �� ���� Belief revision is about changing or updating a be�lief set� A belief set is taken to be a set of �rst�order sentences closed underentailment� That is� if the belief set is K and K entails the sentence �� then �belongs to K� We imagine a situation where we make changes to a belief set inthe face of changing circumstances or facts we receive as time progresses�

AGM allows three types of changes to a belief set� Expansion� a new sentenceis added to the belief set� Revision� a new sentence is added to the belief set� but

v

vi THESIS STATEMENT

to retain consistency some existing sentences are removed from it� Contraction�a sentence � is removed from the belief set� but then other sentences whichentail � must also be removed� There is a way to calculate expansions� add thenew sentence and then take the closure of the resulting set under entailment�However AGM does not specify a unique method for calculating changes in thecase of revision or contraction� Rather a set of postulates is given for eachcase� one for revision and another for contraction� Any method of belief setchange that satis�es its set of postulates is deemed to be rational� It can beshown ��� ��� that a rational revision together with expansion determine aunique rational contraction and similarly a rational contraction together withexpansion determine a rational revision� For this reason� when characterisingbelief revision� it is su�cient to concentrate on contraction and I have done soin my algebraic treatment�

Subtheses

The above thesis is elaborated in the following subtheses�

� Restricted entailment is a generalisation of entailment ���� and remarkfollowing� and provides a mechanism for formalising the kind of reasoningperformed by an agent with �nite reasoning capacities ��� ���

�� Restricted entailment is conceptually respectable because it can be shownto approximate entailment in a well�de�ned and precise manner� certainof the restricted entailments comprise the compact elements in a Scottdomain of which entailment is a member ��� ��� In this domain entailmentis the least upper bound of these compact elements�

� Restricted entailment is logically respectable because it has a deductivecounterpart and soundness and completeness hold �chapters � to ���

�� Restricted entailment has enough power to induce preboolean operationsbetween pairs of sets of sentences � ���� The induced operations com�prise a preboolean algebra� PSEN � � ��� The members of the elementsof PSEN are sentences of an extension of �rst�order logic� These sentencesconsist of �rst�order sentences� arbitrary conjunctions of �rst�order sen�tences and arbitrary disjunctions of those arbitrary conjunctions�

�� The algebra PSEN is powerful enough to support a set of algebraic postu�lates couched in terms of the operations of PSEN which are equivalent tothe AGM postulates for contraction� These algebraic postulates generalisein a natural way to allow contraction by a set of sentences rather than justa single sentence �chapter ���

�� The algebra PSEN allows for a natural and simple treatment of belief bases� �����

vii

�� The algebra PSEN allows for a natural and simple treatment of nonmono�tonic entailment in the context of a belief set �chapter ��

�� Restricted entailment facilitates a simple and natural approach to com�putability of expressions in PSEN and hence of contraction operators andnonmonotonic entailment in the context of a belief set �chapter ���

Thesis Contribution

This thesis breaks new ground as follows�

� A notion of restricted entailment is semantically de�ned�

�� A modal proof theory for the modality approximately true is provided asa counterpart to restricted entailment and the pair of relations� restrictedentailment and provability of approximately true� is shown to be soundand complete�

� Compact restricted entailments have been characterised� and it is shownthat compact entailments approximate ordinary entailment in the senseof domain theory�

�� It is shown that a restricted entailment induces a preboolean algebra�PSEN� whose elements are sets of sentences�

�� It is shown that PSEN enables algebraic characterisations to be given ofAGM belief revision and nonmonotonic entailment in the context of abelief set�

viii THESIS STATEMENT

Certi�cation

I hereby declare that this work has not been submitted for a higher degree atthis or any other university or institution�

Cyril Lee Flax�

ix

x CERTIFICATION

Acknowledgements

My supervisors were Mike Johnson and Mike McGrath� I thank them for theirguidance and for the sense of balance and perspective they provided� especiallytowards the end� I would also like to thank Mike Johnson for taking me onas a student at the start and for providing patient encouragement whenever Ineeded it� It is a pleasure to acknowledge the colleagues and friends who havehelped me� Norman Foo� Raj Gor�e� Ruth Mawson� Abhaya Nayak� MehmetOrgun� Morri Pagnucco� Igor Shparlinsky and Dom Verity� I would also like tothank my employer� Macquarie University and its Department of Computing�for providing me with a semester free of teaching in ����

Years ago Guillaume Br�ummer took me on as a masters student and saw tomy preparation with dedication and care� I am the better for his teaching� andI thank him�

Most importantly� Deborah has supported my research for years with love�dedication and good humour� She has taken on extra work at home and seen toit that I was always well looked after� I thank her from the bottom of my heartand dedicate this thesis to her�

xi

xii ACKNOWLEDGEMENTS

Chapter �

Introduction

Johnson�Laird and his co�workers �� � �� �� claim that human reasoningand deduction is carried out by people forming mental models of the situationbeing reasoned about and then manipulating the mental models in certain ways�The precise nature of these mental models is not important to our argumentso we do not describe them� but the theory is regarded seriously by cognitivescientists� Eysenck and Keane have evaluated it �they call it model theory�along with two other theories of human reasoning and summarise the situationas follows � page �����

� � � each of the three main groups of theories can be evaluated� Atpresent� the model theory fares better than the abstract�rule theoryand both of these fare better than concrete�rule accounts�

However in � page �� Johnson�Laird and Byrne say of mental modeltheory�

The theory is compatible with the way in which logicians formu�late a semantics for a calculus � � � But� logical accounts dependon assigning an in�nite number of models to each proposition� andan in�nite set is far too big to �t inside anyone�s head � � � peopleconstruct a minimum of models� they try to work with just a singlerepresentative sample from the set of possible models� � � �

So Johnson�Laird claims that humans reason by building mental models butthe model checking is not exhaustive at any stage because of limited humancapacities� One can map this situation into the context of an agent doing se�mantics of �rst�order logic and say that in checking the validity of a �rst�ordersentence the agent is only able to check the truth of a subset of all models� Thatis the agent checks a restricted set of models�

We take this as our point of departure and ask what can be done with anapproach to the semantics of �rst�order logic limited in this way� We set out

�My emphasis�

� CHAPTER �� INTRODUCTION

our ideas on what restricted semantics should be and how satisfaction should bede�ned for it� We also give a corresponding syntax and proof theory� So �rst�restricted entailment is de�ned and its semantics examined� Then to provide asyntax to go with the semantics� it is shown how restricted entailment can bemodelled in a modal language using a notion of �approximately true� denotedby a modal operator �� A proof theory is then given for the modal languageincorporating �approximately true��

Having de�ned restricted entailment one can ask how it relates to ordinaryentailment� The results of chapters � to � show that a restricted entailmentapproximates ordinary entailment in the sense that ordinary entailment consistsof all the �nite restricted entailments �joined together�� Concepts from booleanalgebra and domain theory are used to make these notions precise�

Chapters � and � deal with the proof theory of modal operators correspond�ing to restricted entailments� A restricted entailment corresponds to a modaloperator � in the following way� �We use the word �model� to mean modalmodel� while structure means �rst�order model�� To every subset R of struc�tures there corresponds a restricted entailment �R and if R is �nite there is alsoa modal operator �R� The set R is said to be the restriction of both �R and�R� For every �nite R� �R and �R are related by the fact that for �rst�ordersentences �� satisfaction of � by �R is equivalent to modal satisfaction of �R ��Also given �R� a tableau�based proof theory is given for the modal languagebased on it and in chapters � and � soundness and completeness are proved forthe restricted system�

We now leave domain�theoretic and logical matters and turn to investigatealgebraic aspects of restricted entailment� when we regard it as a lattice preorder�We show that it induces a preboolean algebra called PSEN� This algebra isapplied to give characterisations of two constructs arising from the treatment ofbelief revision due to Alchourr�on� G�ardenfors and Makinson �AGM� ��� �� ����belief contraction and nonmonotonic entailment in the context of a belief set�We are also able to use the fact that these notions are expressed algebraicallyto examine their computability� The notions can be brie�y described as follows�

AGM belief revision is about changing or updating a belief set� We take abelief set to be a set of sentences closed under entailment� That is� if the belief setis K and K entails the sentence �� then � belongs to K� We imagine a situationwhere we make changes to a belief set in the face of changing circumstances orfacts we receive as time progresses�

AGM allows three types of changes to a belief set� Expansion� a new sentenceis added to the belief set� Revision� a new sentence is added to the belief set� butto retain consistency some existing sentences are removed from it� Contraction�a sentence � is removed from the belief set� but then other sentences whichentail � must also be removed� There is a way to calculate expansions� add thenew sentence and then take the closure of the resulting set under entailment�However AGM does not specify a unique method for calculating changes in thecase of revision or contraction� rather a set of postulates is given for each case�one for revision and one for contraction� Any method of belief set change thatsatis�es its set of postulates is said to be rational� It can be shown ��� ��� that a

rational revision together with expansion determine a rational contraction andsimilarly a rational contraction and expansion determine a rational revision�For this reason we have found it su�cient to concentrate on contraction in ouralgebraic treatment of belief revision�

Nonmonotonic entailment is a kind of entailment that allows one to �jumpto conclusions�� It is nonmonotonic because ��

� � � increasing the amount of information available as premises maysometimes lead to loss of some of the conclusions that can be drawn�

In ��� it is shown how nonmonotonic entailment in the context of a beliefset can be de�ned in terms of rational revision� Using an expression called theLevi identity this can be translated into a certain contraction followed by anexpansion� and this is the basis of our approach�

As we have said� the vehicle for our discussion and proofs is the prebooleanalgebra PSEN� We de�ne it by using the entailment relation to induce PSEN�slattice preorder� Algebraic methods are then used to show the following� beliefcontraction can be done for sets of sentences rather than just a single sentence�the AGM belief contraction postulates can be retrieved from a certain set ofalgebraic postulates and in turn� the algebraic postulates can be retrieved fromthe AGM ones� nonmonotonic entailment in the context of a belief set can beexpressed in terms of an algebraic expression involving ordinary �monotonic�entailment� Also as a byproduct of the algebraic formulation� belief bases canbe handled in a natural way� A belief base� H � is a set of sentences which ismeant to record the explicit beliefs of an agent� The set H is a base for the beliefset K if and only if K is the closure of H under consequence� In our algebra a setof sentences and its closure under consequence are indistinguishable in the sensethat they are equivalent� So in our algebra a belief base and its correspondingbelief set are equivalent�

We have pointed out that the main techniques used in our discussion arealgebraic ones� In order to discuss the computability of our constructions weuse restricted entailment� The related notion of a restricted algebraic operatoris also used� It turns out that subject to some assumptions� given in detail inchapter �� restricted entailment and the restricted algebraic operators are allcomputable and hence so are the restricted versions of belief contraction andnonmonotonic entailment in the context of a belief set�

The contents of chapters � to � are brie�y as follows� In chapter � the syntaxand semantics of �rst�order logic are reviewed� The approach is standard�

In chapter �rst�order syntax and semantics are extended to give the FOE

languages� which allow de�nition of separator sentences� These extensions arewell�known �� page �� and our use of them in de�ning separator sentences isbasic to our approach to tableau rules and completeness in chapters � and �and to de�ning the operations in the algebra PSEN in chapter �

Chapter � introduces the notion fundamental to this work� restricted en�tailment� It is de�ned semantically and is shown to satisfy the properties ofre�exivity� cut and monotony� The related notion of a restricted consequence op�erator is also de�ned and shown to satisfy inclusion� idempotence and monotony�

� CHAPTER �� INTRODUCTION

Many results in this chapter are lemmas which are used repeatedly in the restof this work� Well�known facts about consequence and entailment are collectedtogether in restricted form� while others deal with the behaviour of entailmentand consequence when moving up and down between a �rst�order language andits FOE extension�

In chapter � the family� ENT� of restricted entailments is regarded as a par�tially ordered set under inclusion� �� A notion of fullness of a set of structuresis de�ned and is used to show that �ENT��� is a complete lattice under suitablyde�ned operations� This is slightly more than is needed for chapter � where themain requirement is that �ENT��� have an in�nitary meet operator�

Chapter � discusses approximation of restricted entailments using elemen�tary de�nitions of domain theory� Some boolean properties of ENT are usedto de�ne �ENT��� as a complete partial order �CPO�� A compact element in aCPO is one which is �nite in some sense� A condition is given for a restrictedentailment to be compact� Also it is shown that any restricted entailment isapproximated by the compact restricted entailments greater than it�

Chapter � sets up the syntax of a modal language to model restricted entail�ment semantics� It turns out that any restricted entailment is approximated byentailments with �nite restrictions� Accordingly given a �nite set of structuresR� a modal language is de�ned with modal operator �R having the intuitivemeaning of �approximately true�� Loosely speaking the main results of thischapter show that for �rst�order sentences� restricted entailment and modalentailment involving �R amount to the same thing�

In chapter � a tableau proof method is presented for modal sentences� Themethod uses signed sentences and the general line of argument in this chapteras well as chapters � and � follows Fitting and Mendelsohn ��� The well�known tableau branch extension rules for �rst�order sentences �� ��� ��� areaugmented to include ones for tableau extensions of sentences issuing from ����Proofs from premises are also considered and a premise branch extension ruleis given�

Chapters� � and �� deal with soundness and completeness� which takentogether show that a modal sentence has a proof if and only if it is forced byevery model�

Chapter de�nes the preboolean algebra PSENL using separators �de�nedin chapter �� and gives some of its boolean properties� Proposition � � de�scribes the relationship between the boolean operations� meet and join� and settheoretical union and intersection� The lemmas � � and � � have to do witha �rst�order language F and its FOE extension L introduced in chapter � The�rst shows how the consequence operator restricts from L down to F � whilethe second shows how it lifts from F to L� The last proposition in the chapterexamines an isomorphism between a certain lattice de�ned on F and an equiva�lence of a sublattice of PSENL� This result rounds out the discussion of PSENL

regarded as a lattice� but is not used in this thesis�Chapter � treats the foundations of belief contraction from an algebraic

point of view� This is done using rejection functions� A set of algebraic pos�tulates is given from which the classical AGM belief contraction postulates can

�

be retrieved� The AGM postulates for belief contraction are generalised slightlyand these are shown to be equivalent to the algebraic ones� The algebraic postu�lates introduce two generalisations� First� contraction of a belief set can be donewith respect to a set of sentences instead of just a single sentence� Secondly�in e�ect the last two AGM postulates �K�� and K��� can be made to deal witharbitrary conjunctions of sentences instead of just �nite ones�

Chapter gives an algebraic treatment of the foundations of nonmono�tonic entailment in the context of a given belief set� An algebraic expressioncan be set up for nonmonotonic entailment because there is a standard way oftranslating a nonmonotonic entailment into a certain contraction of that beliefset� The method used in ��� gives a nonmonotonic entailment relation betweensentences� The algebraic approach used here allows this to be generalised to anonmonotonic entailment relation between sets of sentences� There is a corre�sponding nonmonotonic consequence operator� This is characterised and someof its properties are proved� It is interesting to note that idempotence andcumulativity of this consequence operator are not automatic� but are equiva�lent to some special properties of certain related rejection functions� �Rejectionfunctions are de�ned in chapter ���

In chapter � the computability of belief contraction and nonmonotonic en�tailment in the context of a belief set are examined� The discussion is limitedto elementary questions of computability and decidability� computational com�plexity is not examined� Basically a belief contraction is computable and non�monotonic entailment is decidable provided algebraic operations and relationsare restricted to �nite sets of computable structures with �nite domains andonly �nite sets of sentences appear in algebraic expressions�

Appendix A explains our approach to set theory� It gives the motivationfor adopting a set theory based on a universe of sets satisfying the Zermelo�Fraenkel axioms and summarises the underlying assumptions� This approach isnecessary� Without it� di�culties are introduced into some of our constructionsby the fact that the family of all structures is a class and not a set�

To recapitulate� Johnson�Laird postulates a reasoning process in humansbased on a limited amount of model checking� We propose the notion of re�stricted entailment as a formalisation of this� In the chapters which follow weset out to show that this proposal is not an unreasonable one� We lay out a chainof notions with results connecting one notion to another as follows� We showthat the semantics of restricted entailment is well�behaved and that the familyof entailments with �nite restrictions approximates any arbitrary restricted en�tailment in the sense of domain theory� Then we provide a link from entailmentswith �nite restrictions� �R� to a language with a modal operator� �R� havingthe restriction as a parameter� We claim that it is reasonable to say that �Rmodels �R because they are semantically equivalent on �rst�order formulas� Atableau proof theory is provided for the modal language incorporating �R andit is shown to be sound and complete� Then restricted entailment is used toinduce an algebra in which formulations are given for belief contraction andnonmonotonic entailment in the context of a belief set� The algebraic expres�sions for contraction and nonmonotonic entailment are shown to be computable

� CHAPTER �� INTRODUCTION

when some plausible assumptions are made�

Chapter �

First�order Syntax and

Semantics

In this chapter the syntax and semantics of �rst�order logic are reviewed� Thepurpose of the review is to provide orientation for the reader and to �x nota�tion� Later� �rst�order syntax is expanded so that further developments andarguments can proceed� so it is important to be clear about what we regard asconstituting a �rst�order language� The approach we use is standard�

The family of �rst�order languages is denoted FO� A �rst�order language�see �� � �� ��� ���� consists of formulas� and formulas are built from terms�De�nitions are given below� The constituents are as follows�

� A countable number of individual variables�

� A countable number of constant symbols�

� A countable number of function symbols of any �nite arity�

� A countable number of relation symbols of any �nite arity�

� The connectives �� �� �� ��

� The quanti�ers � �

Terms are built recursively from individual variables� constants and functionsymbols as follows�

� An individual variable is a term�

� A constant is a term�

� If f is a function symbol of arity n� and t�� � � � � tn are terms� then f�t�� � � � � tn�is a term�

�

� CHAPTER �� FIRST�ORDER SYNTAX AND SEMANTICS

Formulas are built recursively from terms� relation symbols� connectives andquanti�ers as follows�

� If p is a relation symbol of arity n and t�� � � � � tn are terms� then p�t�� � � � � tn�is a formula� Formulas of this kind are called atomic formulas�

� If � and � are formulas� then so are ��� � � �� � � � and �� ��

� If � is a formula and x an individual variable� then �x�� is a formula�

� If � is a formula and x an individual variable� then �x�� is a formula�

Note � �

� If � is a formula and x is an individual variable of � then x is free in � ifx does not fall within the scope of a quanti�er appearing in �� If x doesfall within the scope of a quanti�er of � then it is bound� It is possible fora variable to be both free and bound in the same formula�

�� A sentence is a formula with no free variables�

� The following notation is used in various places in what follows� Theexpression ��x� is taken to mean that if � has a free variable then it is x�and if c is a constant or parameter then ��c� is the result of replacing xby c in ��

�� We assume throughout that any language is nontrivial� that is it has atleast one relation symbol�

The semantics of this language is examined in the usual way by consideringthe behaviour of interpretations on formulas � ��� There are two equivalentways of regarding the behaviour of an interpretation on a formula� First� onecan de�ne what it means for an interpretation to evaluate a formula as eitherTRUE or FALSE� Secondly� one can de�ne what it means for an interpretationto satisfy a formula� As usual� one then �nds that a formula is TRUE underan interpretation if and only if the formula is satis�ed by the interpretation�We take the idea of satisfaction as primary and an interpretation�s action onformulas is de�ned in terms of satisfaction �see below��

An interpretation U consists of two parts� a structure� S say� and an assign�ment of variables� u say� So the interpretation U is an ordered pair� U � �S� u��

A structure S consists of the following four things�

� A set D associated with S called its domain� The set D is also denoteddom�S��

� A mapping of each constant� c� to an element cS of D�

� A mapping of each function symbol� f � of arity n to a function fS withdomain Dn and codomain D�

�

� A mapping of relation symbols of arity n into n�fold relations on D� If pis a relation symbol of arity n� then S maps p to an n�fold relation on Ddenoted pS � that is pS is a subset of Dn�

We remark that the set D � dom�S� should not be confused with the do�main of de�nition of the structure S when it is considered as a mapping� Whenregarding S as a mapping it has a domain of de�nition that contains the con�stants� the function symbols� the relation symbols as well as all the terms� Theset D is the �domain of interpretation� of S�

An assignment of variables u maps each individual variable to an element ofD�

The explanation of how U satis�es formulas is given in two steps� First theaction of U on terms is described� This is a combination of the actions of Sand u� Then the de�nition of U satisfying a formula is given in ���� This is arecursive de�nition�

The interpretation U maps a term �other than an individual variable or aconstant� recursively to an element of D as follows�

� If f is a function symbol of arity n and t�� � � � � tn are terms mappedby U to elements of D respectively denoted by tU� � � � � � t

Un � then U maps

f�t�� � � � � tn� to the element fS�tU� � � � � � tUn � of D�

Satisfaction is a relation between the set of all interpretations and the set ofall formulas of a �rst�order language� The fact that an interpretation U satis�esa formula � is written U � �� If U does not satisfy � it is written U � �� Thesatisfaction relation is de�ned as follows�

De�nition � � �First�Order Satisfaction� Let U � �S� u� be an interpreta�tion�

�� Let x be an individual variable� The interpretation V with assignment ofvariables v is a x�variant of U if V has the same structure� S� as U andif v is the same as u except that it possibly maps the individual variable xto a di�erent element of dom�S� than u does�

�� If p is a relation symbol of arity n and t�� � � � � tn are terms� then U �

p�t�� � � � � tn� if and only if �tU� � � � � � tUn � is an element of the relation pS �

�� The satisfaction relation between U and an arbitrary formula is de�nedrecursively as follows�

a U � �� i� U � ��

b U � � � � i� U � � or U � ��

c U � � � � i� U � � and U � ��

d U � �� � i� U � � and U � �� otherwise U � �� ��

e U � �x�� i� for every x�variant V of U � V � ��

f U � �x�� i� for some x�variant V of U � V � ��

� CHAPTER �� FIRST�ORDER SYNTAX AND SEMANTICS

�� If U � � we also write S �u � and say that S satis�es � under u�

�� If S �u � for every assignment of variables u then S is said to satisfy ��and this is denoted S � ��

� The formula � is valid� written ��� if and only if U � � for every inter�pretation U �

For ease of reference� notation for the set of all interpretations and struc�tures of a �rst�order language is recorded below� Note that our set�theoreticalconstructions are made within a universe of sets� So every collection of objectsis a set� there are no classes� �See appendix A�� As a result the statements inthe following de�nition make sense�

De�nition � � Let F be a �rst�order language�

�� Denote the set of all interpretations of F by INTF �

�� Denote the set of all structures of F by STRUC�

�� Denote the set of all subsets of STRUC by PSTRUC�

The description of the syntax and semantics of �rst�order logic given aboveforms the basis for all that follows� In the next chapter FO syntax and semanticswill be extended slightly to de�ne the FOE languages� These form the founda�tion for the de�nition of the separator sentences that are used later in tableaubranch extension rules and in de�ning the preboolean algebras which providethe machinery for the algebraic description of contraction and nonmonotonicentailment�

Chapter �

The FOE Languages

The syntax and semantics of the FOE languages are described here� �FOE standsfor ��rst�order extended��� These languages are well�known and are extensionsof the languages in FO �� page ��� They have formulas which are certain kindsof in�nite conjunctions and disjunctions� The advantage of working with FOE

languages is that they satisfy a certain separation property� each FOE languageL has the property that it can separate any elementarily disjoint pair of setsof stuctures of L� What this means is that if A and B are sets of structuresof L such that no member of A is elementarily equivalent to a member of B�two structures are elementarily equivalent if they satisfy the same sentences��then there is a sentence �A�B in L for which S � �A�B for each S in A andS � � �A�B for each S � in B� This sentence �A�B is called the separator of A andB� Separators are used in three important ways� in the tableau branch extensionrules of chapter �� in proving completeness in chapter �� and in chapter tode�ne algebraic operations for the preboolean algebra PSEN�

De�nition � � �FOE extension� Let F be an FO language�

�� The language L is said to be the FOE extension of F i� L has the sameindividual variables� constants� function symbols and relation symbols asF � and the following constitute the formulas of L�

a Any formula of F is a formula of L�

b �Conjunctive FOE formula� For any X � F �VX is a formula of

L�

c �Disjunctive FOE formula� For any set Y of conjunctive FOE for�mulas�

WY is a formula of L�

�� The language L is said to be in FOE if and only if L is the FOE extensionof some language F in FO�

The following convention is adopted�

� CHAPTER �� THE FOE LANGUAGES

Convention � � A sentence such as �Let L be the FOE of F �� will be takento mean that F is in FO and that L is the FOE of F �

The semantics of languages in FOE will now be examined brie�y� Let L bean FOE extension of F � If U is an interpretation of L then U �s assignment ofvariables and structure will also be an assignment of variables and a structurefor F � and conversely a structure and assignment of variables for F will alsoserve for L� So it follows that U satis�es the formulas of F according to therules given in ��� for satisfaction in F and the following de�nition makes sense�

De�nition � � �FOE semantics� Let L be the FOE of F � and let U be aninterpretation for L� U satis�es formulas of L as follows�

�� Formulas of F are satis�ed by U according to the rules for FO given in ����

�� If � is of the formVX where X � F � then U � � i� U satis�es each

member of X�

�� If � is of the formWX� where for each � � X� � is a conjunctive FOE

formula� then U � � i� U � � for some � � X�

Suppose L is the FOE of F then there is a one�to�one correspondence betweeninterpretations of F and of L� This is stated the next proposition�

Proposition � � Let L be the FOE of F � then any interpretation of F has aunique extension to an interpretation of L�

Proof� Clearly� any assignment of variables� structure and domain for F willserve for L� and the rules given in the above de�nition� �� uniquely determinesatisfaction for an FOE formula� �

Corollary � � Let L be an FOE of F � There is a one�to�one correspondencebetween interpretations of F and interpretations of L�

Two di�erent interpretations that satisfy the same formulas are said to beelementarily equivalent �see ��� This important notion is crystallised in thenext de�nition�

De�nition � �Elementary equivalence� Let L be a language in FO orFOE�

�� Two interpretations� U and V� of L are elementarily equivalent if and onlyif they satisfy the same formulas of L� This is denoted U � V�

�� Let A and B be sets of interpretations of L� A and B are elementarilydisjoint if and only if no member of A is elementarily equivalent to amember of B�

�� Two structures� S and S �� of L are elementarily equivalent if and only ifthey satisfy the same sentences of L� This is denoted S � S ��

�� Let A and B be sets of structures of L� A and B are elementarily disjointif and only if no member of A is elementarily equivalent to a member ofB�

Note that if A and B are elementarily disjoint sets of structures then theyare disjoint as sets because if S � A B then A and B are not elementarilydisjoint�

In what follows we shall deal with sentences rather than formulas and struc�tures rather than interpretations� We now show that the languages of FOE areable to separate elementarily disjoint sets of their structures in the sense ofde�nition �� below� We assume from now on that the �rst�order sentences ofany language in FO are enumerated in a �xed countable sequence� which maydepend on the language being considered� Let L be the FOE of F � As has beennoted above the set of structures for F is the same as for L and it is denotedSTRUC �see ����

Proposition � � Let L be the FOE of F �

�� There is a sentence� ���STRUC� in L satisfying S � ���STRUC for each S �STRUC� Denote ���STRUC by ��

�� There is a sentence� �STRUC��� in L satisfying S � �STRUC�� for each S �STRUC� Denote �STRUC�� by ��

�� Let A and B be non�empty sets of elementarily disjoint structures of L�There is a sentence� �A�B� in L satisfying S � �A�B for each S � A andS � � �A�B for each S � � B�

�� The construction of �A�B produces a unique sentence� so there is a functionfrom pairs of subsets of STRUC to L given by �A�B� �� �A�B�

Proof�

� Take ���STRUC to be the �rst sentence which is a tautology in the standardenumeration of sentences of F �

�� Take �STRUC�� to be the �rst sentence which is a fallacy in the standardenumeration of sentences of F �

� For each �S�S �� � A � B let �S�S� � F be the �rst sentence in the enu�meration of sentences of F which satis�es S � �S�S� and S � � �S�S� � Thisis possible because A and B are elementarily disjoint so there is a formulawith this property� if the formula has free variables then universal quan�ti�cation over each free variable will produce a sentence with the requiredproperty� Set �A�B �

WS��B�

VS�A �S�S��� This last expression de�ning

�A�B is well formed according to de�nition � �

Let S � � B and S �� � A� Now S �� � �S���S� so S �� �VS�A �S�S� � This

holds for each S � � B and so S �� � �A�B � This is true for each S �� � A� Asimilar argument shows that S �� � �A�B for each S �� � B�

� CHAPTER �� THE FOE LANGUAGES

�� Each �S�S� is unique because of the way it is chosen� It follows that foreach S � � B the set f�S�S� � S � Ag is unique� �

Note � � Note that � and � are sentences of F � and if both A and B are�nite then �A�B is a sentence of F �

We extract some notation from the proof of proposition �� and �x it togetherwith some terminology in the following de�nition�

De�nition � �Separator� Let L be the FOE of F � and let A and B be setsof elementarily disjoint structures of L�

�� Given �S�S �� � A�B� the �rst sentence �S�S� in the enumeration of sen�tences of F that satis�es S � �S�S� and S � � �S�S� � is called the separatorof S� and S ��

�� The separator of A and B is �A�B �WS��B�

VS�A �S�S���

The following easy result is needed for lemma �� in proving the complete�ness of restricted semantics�

Proposition � �� Let A and B be sets of stuctures and suppose A is elemen�tarily disjoint from B and contains every structure not elementarily equivalentto some member of B� Let �A�B be the separator of A and B� then the structureS � �A�B if and only if S is elementarily equivalent to some member of B�

Proof� Suppose S � �A�B then by ��� S cannot belong to A� It follows fromthe way that A and B are chosen that S is elementarily equivalent to somemember of B�

Conversely suppose S � S � � B� Now from de�nition ���

�A�B ��

Sy�B

��Sz�A

�Sy�Sz��

By �� S � � �A�B � so there is Sy � B with S � � �Sy�Sz for every Sz � A�Since each �Sy�Sz is a �rst�order sentence and S � S �� it therefore follows thatS � �A�B � �

Note that all the �S�S� are �rst�order formulas and there are at most count�ably many of them� For this reason� for �xed S �� the set f�S�S� � S � Ag isat most countable and

VS�A �S�S� can be written as

Vn�N � �n�S� � where N �

depends on S � and either stands for all integers or some subset of the integers�

Note � ��

� The subscripts in �A�B will be dropped and it will be written as � if noconfusion results�

�� Without loss of generality� the separator �A�B can be writtenWS��B �

Vn�N � �n�S�� where N � is some subset �possibly all� of the in�

tegers and depends on S ��

�

� Without loss of generality� if B is �nite then �A�B can be written �A�B �Wmi�� Ci� where Ci �

Vn�Ni

�n�i for Ni some subset �possibly all� of theintegers which depends on i� and each i corresponds uniquely to somemember of B�

The fundamental notion of a separator has been de�ned in this chapter� Sep�arators are used later in tableau branch extension rules� in proving completenessof our modal language� and in producing algebraic operations in PSEN�

Separators are sentences in FOE languages� which are extensions of �rst�order languages incorporating in�nite conjunctions of �rst�order formulas andin�nite disjunctions of those conjunctions� If L is the FOE of F � then bothF and L have the same structures and there is a one�to�one correspondencebetween their interpretations� The two main results of this chapter are propo�sitions �� and � � which describe the satisfaction behaviour of separators� Inthe next chapter �rst�order semantics is generalised by introducing the idea ofa restriction�

� CHAPTER �� THE FOE LANGUAGES

Chapter �

Restricted Semantics

In this chapter we de�ne restricted entailment which is a generalisation of en�tailment� The de�nition we give of entailment is semantic� If X and Y are setsof formulas of a �rst�order language F � the semantic view of entailment saysone can check that X entails Y by verifying that each interpretation of F whichmakes all of X true also makes all of Y true�

The de�nition of restricted entailment is motivated by the mental modeltheory of human reasoning in cognitive science due to Johnson�Laird �� � ����� He and his co�workers claim that humans reason by building mental modelsbut the model checking is not exhaustive at any stage because of limited humancapacities� A restricted entailment arises by reducing the amount of veri�cationone does in the set of all interpretations of F � By restricting veri�cation tothose interpretations belonging to a subset R of the set of all interpretations ofF � one obtains an entailment restricted to R�

In the approach adopted here instead of restricting to a set R of interpreta�tions� the set R is taken to consist of structures� The reason for this is that thesoundness and completeness results of chapters � and � are proved by consid�ering the semantics of sentences� When dealing with the semantics of sentencesit is more natural to work with structures because if an interpretation satis�es asentence then so will any other interpretation with the same structure �see �����So one can focus on the structure and ignore any particular assignment of vari�ables� An entailment restricted to R is just called a restricted entailment whenthere is no danger of ambiguity�

Aside from restricted entailment� the other fundamental notions de�ned hereare restricted consequence �or closure� and restricted logical equivalence betweensets of sentences� Making the de�nitions in restricted form allows a restrictedpreboolean algebra to be de�ned in chapter � and the fact that it is restrictedallows discussion of its computability� This in turn allows discussion of thecomputability of belief revision and nonmonotonic entailment�

Properties of the restricted notions mentioned above and relationships be�tween them are laid out in this chapter in a series of lemmas� These lemmasare used repeatedly in the arguments of later chapters�

�

� CHAPTER �� RESTRICTED SEMANTICS

We recall that a sentence is a formula having no free variables� If an interpre�tation satis�es a sentence it will do so no matter what assignment of variablesit has� To show this we need the following result which is well�known�

Proposition � � Let � be a formula and let S be a structure� If any pair ofassignments of variables� u and v say� agree on the free variables of �� thenS �u � i� S �v ��

Proof outline � For this proof we adopt the following convention on nota�tion for interpretations� If U is an interpretation with structure S� say� thenthe assignment of variables of U is denoted u� so U � �S� u�� We extend theconvention slightly and also write U � � �S� u���

The proof is by induction� First� by induction on the structure of termsit is straightforward to see that for any term t whose variables are free vari�ables of �� tU � tV � Secondly� for formulas the base case when � is atomic iseasily proved� let p be a predicate symbol of arity n� then �p�t�� � � � � tn��U �pS�tU� � � � � � t

Un � � pS�tV� � � � � � t

Vn � � �p�t�� � � � � tn��V � Next� the cases for which �

is ���� where � is a binary connective and both � and � satisfy the proposition�are straightforward�

Finally we deal with the cases involving quanti�ers� For the existential casesuppose � is �x��� where � satis�es the proposition� �By � satisfying theproposition we mean that if an arbitrary pair of assignments of variables agreeon the free variables of � then the corresponding interpretations with structureS both satisfy ��� Suppose U satis�es �x�� and let W be an x�variant of Usatisfying �� De�ne the assignment of variables v� as follows�

v��z� �

�v�z� if z �� xw�z� if z � x

The assignments w and v� agree on the free variables of �� so because �satis�es the proposition W and V � both satisfy �� But V � is an x�variant of V soit follows that V satis�es �x��� The argument is symmetrical so the existentialcase of the proposition is proved� The proof for the universal quanti�er case issimilar� �

Corollary � �

�� If � is a sentence and the interpretation U � �S� u� satis�es � then sowill V � �S� v� for any other assignment of variables v�

�� The truth of a sentence depends only on the stucture of an interpreta�tion and reference to the assignment of variables can be dropped withoutintroducing any ambiguity�

Proof� By proposition �� � since � has no free variables� �

In the case of the sentence � in the corollary above we can simply say thatS satis�es � and write S � �� This explains why de�nition ��� is made in termsof structures and not interpretations�

�

De�nition � � Let L be a language�

�� SENL is the set of sentences of L�

�� PSENL is the set of all subsets of SENL�

�� INTL is the set of all interpretations of L� This is consistent with the �rstpart of de�nition ���

When there is no danger of confusion the superscript� L� is dropped in theabove de�nition�

The following de�nition could be made in terms of a set of formulas Xinstead of a set of sentences� but for the reasons pointed out in the introductionit suits our purposes better to make it in terms of sentences� We now de�ne setof models� validity and entailment�

De�nition � � Let L be a language and let X and Y be sets of sentences of L�

�� Let S � STRUC� then S � X if and only if S � � for each � � X�

�� The set of models of X� denoted mod�X�� is mod�X� � fS � STRUC �S � Xg�

�� The sentence � is valid� denoted � �� if and only if STRUC � mod����

�� X entails Y � denoted X � Y � if and only if mod�X� � mod�Y ��

It is well�known that entailment satis�es three properties called re�exivity�cut and monotony which are de�ned as follows for sentences � and �q �q � Q�and sets of sentences X and Y �

Re�exivity If � � X � then X � ��

Cut If X � �q for all q � Q and X � f�q � q � Qg � �� then X � ��

Monotony If Y � � and Y � X � then X � ��

It is also well�known that there is a consequence operator� denoted Cn� whichcorresponds to entailment� It is de�ned as follows� Let X be a set of sentences�then Cn�X� � f� � mod�X� � mod���g� The consequence operator satis�es thefollowing three properties� where X and Y are sets of sentences�

Inclusion X � Cn�X��

Idempotence Cn�X� � Cn�Cn�X���

Monotony If X � Y � then Cn�X� � Cn�Y ��

Now the following fundamental restricted notions are de�ned� set of models�entailment� the consequence operator and logical equivalence� The de�nitionsare made by analogy with the unrestricted ones of ����

�� CHAPTER �� RESTRICTED SEMANTICS

De�nition � � Let L be a language� let X and Y be members of PSENL andlet R � STRUC�

�� The set of models of X restricted to R� denoted modR�X�� is modR�X� �fS � R � S � Xg�

�� X entails Y with restriction R i� modRX � modRY � this is written asX �R Y �

�� CnR�X� � f� � SENL � modRX � modR�g� The operator CnR is calleda restricted consequence or restricted closure operator�

�� X �R Y i� X �R Y and Y �R X� X is said to be logically equivalentwith restriction to Y if X �R Y �

The statement that restricted entailment generalises entailment is justi�edby noting that � equals �STRUC�

A straightforward argument shows that restricted entailment also satis�esre�exivity� cut and monotony� Restricted consequence satis�es inclusion� idem�potence and monotony� this is part of lemma ��� below�

Proposition � Any restricted entailment satis�es re�exivity� cut and monotony�

Proof� Straightforward from the de�nitions� �

The next three lemmas collect several results together� They are used re�peatedly in what follows� especially from chapter onwards�

Lemma � � Let L be a language� let X� Y and Zq �q � Q� be members of

PSENL and R � STRUCL�

�� The operator CnR satis�es the following�

Inclusion� X � CnRX�

Idempotence� CnRX � CnRCnRX�

Monotony� If X � Y � then CnRX � CnRY �

�� CnR�Tq�Q CnRZq� �

Tq�Q CnRZq�

�� modRX � modRCnRX�

�� The following are equivalent�

a CnRX � CnRY �

b modRY � modRX�

c Y �R X�

�� X �R Y i� CnRX � CnRY �

� X �R CnRX�

�

Proof�

� Let V � W � X � Y and Z be sets of sentences� It is easy to see from thede�nition of �R that it is transitive� That is if X �R Y and Y �R Z� thenX �R Z� Also it is easy to see from the de�nition of �R that

W � CnR�V � i� V �R W� ���

INCLUSION� X �R X and so by ���� X � CnR�X��

IDEMPOTENCE� First CnR�X� � CnR�CnR�X�� by inclusion� Next wemust show that CnR�CnR�X�� � CnR�X�� Now CnR�Z� � CnR�Z� and soby ����

Z �R CnR�Z�� ����

Replacing Z by CnR�X� in ���� we see that CnR�X� �R CnR�CnR�X���So by transitivity of �R it follows� on taking Z to be X in ����� that X �R

CnR�CnR�X��� Idempotence follows by using ��� to get CnR�CnR�X�� �CnR�X��

MONOTONY� Suppose X � Y � From inclusion Y � CnR�Y �� so X �CnR�Y � and from ���� Y �R X � Also CnR�X� � CnR�X� so by ����X �R CnR�X�� It follows from transitivity of �R that Y �R CnR�X� andso from ��� again� CnR�X� � CnR�Y ��

�� First CnR�Tq�Q CnRZq� �

Tq�Q CnRZq� This is seen as follows�T

q�Q CnRZq � CnRZn for each n � Q� so CnR�Tq�Q CnRZq� � CnRCnRZn

� CnRZn by monotony and idempotence� and so CnR�Tq�Q CnRZq� �T

q�Q CnRZq � The reverse inclusion follows from part � Inclusion� com�pleting the proof�

� By inclusion X � CnRX � so it is immediate that modRCnRX � modRX �On the other hand from the de�nition of CnR� modRX � modRCnRX �

�� �a� implies �b�� From CnRX � CnRY it follows that modRCnRY �modRCnRX � and because modRX � modRCnRX �this is part above�the result follows�

�b� implies �a�� Suppose modRY � modRX and let � � CnRX � We mustshow � � CnRY � But from the de�nition of CnRX � modRX � modR��and because modRY � modRX the result follows�

�b� i� �c�� Follows from the de�nition of �R�

�� From part � above� CnRX � CnRY i� Y �R X and X �R Y �

�� By idempotence CnRX � CnRCnRX and the result follows from part ��

�

�� CHAPTER �� RESTRICTED SEMANTICS

Remark � � Let L be the FOE of F � We recall from de�nition � that theconstants� function symbols and relation symbols of L and F are the same andso the set of structures� STRUC� of F and L are the same�

Lemma � Let L be the FOE of F and let X and Y be members of PSENF

and R � STRUC� The following are equivalent�

�� Y � CnFRX�

�� X �FR Y �

�� X �LR Y �

Proof� � � implies ���� Using monotony and idempotence of CnR we have thatCnFRY � CnFRX and so by ��� part � X �FR Y �

��� implies � �� This follows from ��� part � and the inclusion property ofCnR�

��� i� ��� Because X and Y are sentences of F � and using ��� it followsthat modFRX � modFRY i� modLRX � modLRY � �

The next lemma is used in the arguments of chapter � on belief revision�where one moves back and forth between closures in F and its extension L�

Lemma � �� Let L be the FOE of F � let X � PSENF and R � STRUC�

�� �CnLRX� SENF � CnFRX�

�� CnLRCnFRX � CnLRX�

Proof�

� By �� any interpretation of F has a unique extension to an interpretationof L� So for any sentence � � F � modLR�X� � modLR��� i� modFR�X� �modFR����

�� By ��� part � X �FR CnFRX and so the result follows by using ��� and ���

part �� �

The de�nitions of the fundamental concepts of restricted entailment andrestricted consequence were given semantically in this chapter� The restrictednotions can be regarded as generalisations of the ordinary ones by taking therestriction to be the set of all structures� Also� the restricted objects havethe same properties that their unrestricted counterparts do� re�exivity� cutand monotony for entailment� and inclusion� idempotence and monotony forconsequence�

The rest of this work is concerned with rami�cations arising from the analysisof restricted entailment� in chapters � to �� how restricted entailments can beapproximated by others and how they can be modelled in a modal languagewhich is sound and complete� and in chapters to �� how they can be used toconstruct an algebra within which belief revision and nonmonotonic entailmentcan be done in a computable way�

�

The next chapter� chapter �� analyses the lattice�theoretic properties of thefamily of restricted entailments as a preliminary to the domain�theoretic analysisof the approximation of restricted entailments in chapter ��

�� CHAPTER �� RESTRICTED SEMANTICS

Chapter �

Restricted Entailments as a

Lattice

We shall show in chapter � that the entailment relation is approximated bythe set of restricted entailments in the sense of domain theory� In this chapterwe take the �rst step in this direction by showing that any set of restrictedentailments has a restricted entailment which is its greatest lower bound under�� For the cost of little extra e�ort we show that restricted entailments underset inclusion form a complete lattice� The last result in this chapter rounds outthe theory� but it is not used� It shows that a certain sublattice is a completeboolean algebra isomorphic to a certain subalgebra of the set of all subsets ofSTRUC� A convenient reference for lattice theory is ���� another is ���� Foralgebra see ����

We have chosen an approach to set theory that operates within a universe ofsets �see appendix A�� This approach overcomes some di�culties introduced bythe fact that STRUC is not a set� but a class� The appendix gives an example�Working in a universe of sets overcomes these di�culties and allows results inthis chapter to go through in a satisfactory way� particularly �� � and the resulton joins in �� �

Some preliminary facts are needed� as well as the notion of fullness � a setof structures is full if it is maximal with respect to elementary equivalence�see ���� The next proposition shows that the restricted set of models operatoris monotonic in its restriction parameter�

Proposition � � If X is a set of formulas and I � J � STRUC then modI �X� �modJ �X��

Proof� modI�X� � I mod�X� � J mod�X� � modJ �X� �

Propositions ��� and ��� are needed for the important propositions ��� and �� �on lower and upper bounds of families of restricted entailments� Proposition ���says that if a set of structures is enlarged then the associated restricted entail�ment is reduced� while proposition ��� provides a converse to proposition ���

��

�� CHAPTER �� RESTRICTED ENTAILMENTS AS A LATTICE

under a fullness condition on the restriction of the entailment�

Proposition � � Let I � J � STRUC� then �J � �I �

Proof� Let X and Y be sets of sentences and suppose that X �J Y � that ismodJ�X� � modJ �Y �� We must show modI�X� � modI�Y ��

modI�X� � I mod�X�� �I J� mod�X� since I � J� I �J mod�X�� I modJ�X�� I modJ�Y � since modJ�X� � modJ�Y �� I �J mod�Y ��� �I J� mod�Y �� I mod�Y � since I � J� modI�Y �

�

There is a converse to proposition ��� provided J is full� This is de�nednext�

De�nition � � Let I � STRUC� then I is full if and only if given elements Sand S � of STRUC� if S � I and S is elementarily equivalent to S � then S � is inI�

Proposition � � Let I and J be subsets of STRUC and suppose J is full� If�J � �I � then I � J �

Proof� We suppose I �� J and prove that �J �� �I � There are two cases toconsider� either J is empty or not empty�

CASE J � �� Let S � I and let � be a sentence with S � �� Set X � f�gand Y � f��g� then S � X and S � Y � So S � modI�X� but S �� modI�Y ��Since J � �� X �J Y but as we have seen X �I Y �

CASE J �� �� Suppose S � I and S �� J � then S is not elementarilyequivalent to any S � � J � This is so because if S is elementarily equivalent tosome S � � J then since J is full S � J � which is contrary to our supposition�So for each S � � J there is a sentence �S� � say� with S � �S� and S � � �S� � SetX � f�S� � S � � Jg� then S � modI�X� but modJ �X� � �� Also pick S �� � Jand set Y � f��S��g� then S �� modI�Y ��

Now modJ�X� � � � modJ �Y �� so X �J Y � But S � modI�X� and S ��modI�Y � so X �I Y � �

Next we de�ne the greatest lower bound and the least upper bound of a setof restricted entailments and then show how to calculate them� We recall thata partial order is a re�exive� transitive� antisymmetric relation�

De�nition � �

�� ENT � f�I � I � STRUCg�

��

�� �ENT��� is the partially ordered structure where the elements of ENT arepartially ordered by set inclusion�

�� Let E � ENT� An element �H of ENT is a lower bound of E if �H � �I

for each �I � E�

�� An element �G of ENT is the greatest lower bound of E � ENT if �G isa lower bound of E and it is a superset of� or equal to any other lowerbound of E� The greatest lower bound of E� if it exists� is denoted

VE�

�� Let E � ENT� An element �H of ENT is an upper bound of E if �I � �H

for each �I� E�

� An element �L of ENT is the least upper bound of E � ENT if �L is anupper bound of E and it is a subset of� or equal to any other upper boundof E� The least upper bound of E� if it exists� is denoted

WE�

When E is a �nite set in�x notation is mostly used� soVf�I ��Jg is written

�I � �J � We also use the terms meet or inf for greatest lower bound and joinor sup for least upper bound�

We want to show that regarding � as a lattice order the meet of �I and �Jis �I�J � We do this with the help of an operator that takes any subset I ofSTRUC and turns it into a full subset containing I denoted I �� The operator iscalled the full expansion operator�

We recall that PSTRUC stands for the set of all subsets of STRUC �seede�nition ���� If E � PSTRUC then each element of E is a subset of STRUC�SE means the union and

TE means the intersection of all members of E�

De�nition �

�� Let I � STRUC� The full expansion of I� denoted I �� is fS � STRUC �S � � I � S � S �g�

�� Let E � PSTRUC� then E� � fI � � I � Eg�

Proposition � �

�� The full expansion of a set of structures is full�

�� The full expansion of a set of structures is unique�

�� The full expansion operator satis�es inclusion� idempotence and monotony�

a I � I ��

b I �� � I ��

c If I � J � then I � � J ��

Proof� Straightforward� �

Proposition � � Let I � STRUC� then �I � ��I��

�� CHAPTER �� RESTRICTED ENTAILMENTS AS A LATTICE

Proof� Full expansion satis�es inclusion so I � I � and by proposition �����I� � �I �

On the other hand suppose X �I Y and let S � � I � mod�X�� We mustshow that S � � I � mod�Y �� There is S � I such that S � � S� Also S � mod�X�because� as is easily seen� mod�X� is full and S � � mod�X�� So S � modI�X�and therefore by our supposition S � modI�Y �� By the inclusion property offull expansion I � I �� therefore S � I mod�Y � � I � mod�Y �� But since� asis easily seen� I � mod�Y � is full� we have that S � � I � mod�Y �� �

The following results allow one to calculate meets and joins� The meetof a family of restricted entailments is generated by taking the union of therestriction sets� The join� however� is generated by taking the intersection ofthe full expansions of the restriction sets�

Proposition � Let E � PSTRUC� The following are true�

�� ��E � �I � for each I � E�

�� Suppose there is G � STRUC with ��E � �G � �I for each I � E� then�G � ��E�

Proof�

� This follows from proposition ��� because I �SE for each I � E�

�� We have supposed that

��E � �G � �I so

���E� � ��G� � �I �by ���� and

I � G� � �E� �by ���� so

�E � G� � �E� and

�E� � G� � �E� �by ��� monotony and idempotence� giving

G� � �E� and so

�G � ��E �using ����

�

The next proposition depends on the evaluation of the expression E� whereE � PSTRUC� The discussion in appendix A shows that if a Bernays�vonNeumann�G�odel set theory �such as in Kelley ��� is adopted E� can be emptyeven when E is not� If this is the case then E� is not a set� it is the classof all sets� We prefer to work in an environment without such �explosions��To achieve this we could either dispense with de�ning the least upper boundaltogether or use an approach to set theory yielding more tractable results� Wehave decided to work within a universe of sets �see �� ���� where the results ofall constructions are sets� Basic properties are summarised in appendix A� Ina universe of sets E� is not empty if E is not and also E� is a set�

��

�ENT��� �PSTRUC���

Unit �� STRUC

Zero �STRUC �Complement �I � STRUC� IMeet �I � �J � �I�J I � J � I JJoin �I � �J � ��I���J� I � J � I � JIn�nite meet

VI�E �I � ��E

VI�E �

TE

In�nite joinWI�E �I � ���E�

WI�E �

SE

Figure �� � The lattice ENT and boolean algebra PSTRUC�

Proposition � �� Let E � PSTRUC� The following are true�

�� �I � ���E�� for each I � E�

�� Suppose there is L � STRUC with �I � �L � ���E� for each I � E� then�L � ���E��

Proof�

� For each I � E we have E� � I �� so by ��� ��I� � ���E� and by �����I� � �I �

�� We have supposed that

�I � �L ����E� so

��I� � ��L� � ���E� �by ���� and

E� � L� � I � �by ���� so

E� � L� � E� giving

L� � E� and so

�L� ���E� �using ����

�

Corollary � �� Let E � PSTRUC� then

��VI�E �I � ��E�

��WI�E �I � ���E��

It is easy to see that �ENT��� is a complete lattice and �PSTRUC��� isa complete boolean algebra� Their distinguished elements and operators areshown in �gure �� �

There is a homomorphism from �PSTRUC�����S

� to �ENT�����V

� whenboth structures are considered as abstract algebras� When� however� the re�strictions� I � of the entailments �I are taken to be full �ENT��� becomes

� CHAPTER �� RESTRICTED ENTAILMENTS AS A LATTICE

a complete boolean algebra ENTfull � �ENTfull���������V�W

�� The setENTfull � f�I � I � STRUC � I is fullg� The complement in ENTfull is given by��I � �STRUC�I � Denoting the restriction of PSTRUC to full subsets of STRUCby PSTRUCfull� there is an isomorphism of algebras from the boolean algebraPSTRUCfull � �PSTRUCfull������� �

S�T

� to the boolean algebra ENTfull� Themorphisms are not boolean morphisms because meet and join in PSTRUC cor�respond respectively to join and meet in ENT� We recall some basic de�nitions�

De�nition � ��

�� Let X be a set� An operation on X of arity m is a function o � Xm � X�

�� An algebra is a set with some operations de�ned on it� If the set is Xand for each i � I� oi is an operation of the algebra then the algebra isdenoted �X� foi � i � Ig�� In the �nite case where there are n operations�o�� o�� � � � � on� the algebra is denoted �X� o�� o�� � � � � on��

�� Two algebras �X� foi � i � Ig� and �Y� fpi � i � Jg� are similar if I � Jand for each i � I the arities of oi and pi are the same�

�� A function h � X � Y is a homomorphism from the algebra X ��X� foi � i � Ig� to the algebra Y � �Y� fpi � i � Jg� if X is sim�ilar to Y and h preserves all the operations� That is for each i � I�h�oi�x�� x�� � � � � xn�� � pi�h�x��� h�x��� � � � � h�xn��� where n is the arityof both oi and pi�

�� An isomorphism is an invertable homomorphism�

Proposition � ��

�� Regarding PSTRUC � �PSTRUC�����S

� and ENT � �ENT�����V

� asabstract algebras there is a homomorphism of algebras ���� � PSTRUC�ENT de�ned as follows� ��I� � �I �

�� The algebras PSTRUCfull � �PSTRUCfull������� �S�T

� and ENTfull ��ENTfull���������

V�W

� are complete boolean algebras�

�� Regarding the complete boolean algebras PSTRUCfull and ENTfull as abstractalgebras there is an isomorphism ���� � PSTRUCfull � ENTfull de�ned by��I� � �I �

Proof�

� We must show that the function ���� preserves corresponding operations�

�a� ��I � J� � �I�J � �I � �J �

�b� The in�nite operations are treated similarly�

�� First it is easy to see that PSTRUCfull is a boolean algebra� Next it is easyto check that ENTfull is a distributive lattice because for any �I � ENTfull

we have that I � � I � and so it can easily be seen that ENTfull is a booleanalgebra�

� First we must show that ���� preserves corresponding operations�

�a� ���I� � �STRUC�I � � �I �

�b� ��I � J� � �I�J � �I � �J �

�c� ��I J� � �I�J � �I � �J �

�d� The in�nite operations are treated similarly�

Next ���� is obviously surjective� Finally for any full I and J if �I � �J

then� using ���� we see that I � J and so ���� is one�to�one� �

This chapter examined the lattice properties of the family of restricted en�tailments� ENT � �ENT���� Set inclusion is a partial order for ENT� In�nitaryleast upper and greatest lower bounds exist in ENT and their existence dependson the order�behaviour of restricted entailment based on its restriction parame�ter� There are three results relevant to this� Two of them use the full expansionoperator�

� Restricted entailment is anti�monotonic in its restriction parameter� IfI � J � then �J � �I �

� The converse to the above is true if J is full�

� The restricted entailment �I� ��I��

The approach to set theory we have chosen was exploited in making thede�nition of in�nitary lattice join� all our set�theoretical constructions are madein a universe of sets� This ensures that the set�theoretical problems discussed inappendix A do not arise when de�ning the �set�theoretical� in�nitary greatestlower bound�

That ENT is a complete lattice enables the theory to be rounded out withsome discussion of ENT as a boolean algebra �see �� �� However the fact thatENT has an in�nitary meet operator is of fundamental importance for the nextchapter� where the approximation properties of ENT using domain theory arediscussed�

� CHAPTER �� RESTRICTED ENTAILMENTS AS A LATTICE

Chapter �

Restricted entailments as a

domain

Here we show that an arbitrary restricted entailment of ENT can be approxi�mated by compact ones� These are of the form �I where I � STRUC has only a�nite number of structures which are not elementarily equivalent to each other�That is� I has �nitely many equivalence classes under the relation of elementaryequivalence� As a byproduct of this approach it will be shown how ordinaryentailment� �� can be approximated in this way� It turns out that ENT is adomain �sometimes called a Scott domain�� which means roughly that it is apartially ordered set in which each element is approximated by the compactelements greater than it� To say that a set E of elements of ENT approximatesan element x �with respect to ENT�s partial order� means that x equals the leastupper bound of E�

When regarding ENT as a domain the domain partial order will be taken tobe � �see �gure �� for a summary of ENT as a complete lattice�� This is becausethe natural direction for approximation is downwards in the lattice� where theordinary entailment is the lattice zero� � � �STRUC� Also the lattice has anin�nitary meet operation which is taken to be the domain least upper boundoperator with respect to the partial order �� The least element for the domainis taken to be the lattice unit� ��� The domain de�nitions come from ���� Othersources for this area are �� ��� ��� ���

We recall that a partial order is a re�exive� anti�symmetric and transitiverelation�

De�nition � �Complete Partial Order� Let D � �D�v��� be a partiallyordered set with least element ��

�� �Directed Set� Let A � D� then A is directed if whenever w and x aremembers of A there is y � A satisfying w v y and x v y�

�� �Complete Partial Order� D is called a complete partial order �CPO�

� CHAPTER �� RESTRICTED ENTAILMENTS AS A DOMAIN

if whenever A � D and A is directed then the least upper bound of A�denoted

FA� exists in D�

Proposition ��� and corollary �� part justify the use ofV

asF

in thede�nition below�

De�nition � The set ENT � f�I � I � STRUCg can be regarded as a CPOENT � �ENT�v��� by taking

�� v to be ��

�� � to be ���

��F

to beV�

The compact elements in a CPO play a major role in approximation ofmembers of the CPO so compactness is de�ned next�

De�nition � �Compact element� Let D be a CPO�

�� An element d � D is compact if and only if the following condition issatis�ed�

For all A � D if A is directed and d vFA� then there is x � A

with d v x�

�� The set of compact members of D is denoted comp�D��

Next a precise meaning is given to the word �approximation� in a CPO� Thede�nition comes from ��

De�nition � Let y and z be elements of a CPO� D� The element y approx�imates z� denoted y � z� if for all directed subsets A of D� z v tA impliesy v x for some x � A�

It can be easily seen from the above de�nitions that in a CPO� D� d � Dis compact i� d � d� That is the compact elements of a CPO approximatethemselves�

In order to characterise the compact elements in proposition �� � some re�sults ���� to ���� are needed on the cardinality of the set of equivalence classesof expansions� and related matters� Elementary equivalence was de�ned in ���Let I be a subset of STRUC� the family of equivalence classes of I under ele�mentary equivalence is denoted I�� If S � I the equivalence class of S in I�is denoted kSkI �

Lemma � Suppose I � H � � STRUC�

�� There is a one�to�one function f � I� � H ���

�� If I � H� then f is onto�

�

Proof�

� De�ne f�kSkI� � kSk�H�� The function f is well�de�ned because if kSkI �kS �kI then S � S � and so f�kSkI� � kSk�H� � kS �k�H� � f�kS �kI��

ONE�TO�ONE� Suppose kSk�H� � kS �k�H�� then S � S � so kSkI � kS �kI �

�� Every member of H �� contains an element of H because I � H � �

For any set X let card�X� denote the cardinality of X �

Proposition Let H � STRUC�

�� Both H �� and H� are sets satisfying

a card�H ��� � card�H���

b card�H�� � card�H��

�� If I � H �� then card�I�� � card�H��

Proof�

� By lemma ��� there is a one�to�one function from H� onto H ��� Alsoby picking a unique element from each equivalence class in H� one seesthat there is a one�to�one function from H� onto a subset of H �

�� By lemma ��� I� maps one�to�one onto a subset of H �� so the resultfollows from what has been proved above� �

De�nition � �Reduction� Let I � STRUC�

�� J � STRUC is a reduction of I� denoted J � I� if and only if

a J � I�

b For each element x � I�� there is an element S � J satisfyingS � x�

c No two elements of J are elementarily equivalent�

�� I is said to be reduced if and only if I � I�

The proof of the following is straightforward�

Proposition � Let J and I be subsets of STRUC�

�� If I is reduced� then J � I i� J � I�

�� I has a reduction�

Proposition Suppose J � I� then the following are true�

�� J is reduced�

� CHAPTER �� RESTRICTED ENTAILMENTS AS A DOMAIN

�� J � � I ��

�� If I� is �nite� then so is J �

Proof�

� No two elements of J are elementarily equivalent so J� consists of sin�gletons and every member of J belongs to a unique element of J�� Theresult now follows easily from the de�nition�

�� From the de�nition of reduction J � I so by monotony of expansionJ � � I �� On the other hand� suppose S � I �� We must show S � J ��

There is SI � I with S � SI � But because J is a reduction of I there isSJ � J satisfying SJ � SI � It follows that SJ � S and so S � J ��

� Because J is reduced there is a one�to�one function from J onto J�� AlsoJ � I � I � so from lemma ��� there is a one�to�one function from J�into I ��� But from ��� card�I ��� � card�I��� �

The next result characterises compact restricted entailments�

Proposition �� The restricted entailment �I is compact if and only if I�is �nite�

Proof� Suppose �I is compact� Let E � fF � F � I � F is �niteg andconsider T � f�F � F � Eg� T is directed and

VT � ��E � �I because

I �SE� Since �I is compact there is �H� T with �H � �I � By de�nition of

T � H is �nite� By propositions ��� and ��� I � H � and so I� is �nite by ����This proves one half of the proposition�

To prove the other half suppose I� is �nite� We must show that �I iscompact� Let T � f�R� R � Eg be directed where E � PSTRUC is arbitrary�Suppose

VT � �I � We must show there is a member of T that is a subset of

�I � Now ���E� � ��E �VT � �I so I �

SE� by ����

Let J � I � then by proposition ��� J is �nite� Also J � I so J � SE��

It follows that for each S � J there is a member of E� denote it RS � suchthat S � RS �� But J is �nite and T is directed so there is �G� T with�G � �RS

for each S � J � Because ��G� � �G we have RS � G� by ��� andidempotence� and by monotony of expansion and idempotence RS � � G�� Sowe have J �

SfRS � � S � Jg � G� and it follows that �G � ��G� � �J � ��J��

Since ��J� � ��I� by ���� we have that �G � ��I� � �I � �

A CPO �D�v��� is algebraic if any element of D is equal to the the leastupper bound of the compact elements below it� Speaking loosely we say that�compact members approximate�� We recall that the compact elements of Dare denoted by comp�D��

De�nition �� �Algebraic CPO� Let D � �D�v��� be a CPO� For x � D�denote fc � comp�D� � c v xg by approx�x�� D is algebraic if and only if thefollowing condition holds�

�

If x � D� then approx�x� is directed and x �Fapprox�x��

The next proposition shows that any member of approx�x� approximates x�

Proposition �� Let D be a CPO� let z � D and suppose all the members ofF � D are compact� If tF � z� then any f � F satis�es f � z�

Proof� Let A � D be directed and suppose z v tA� We must show that forany f � F there is x � A satisfying f v x� Now tF � z v tA so for any f � F �f v tA� But f is compact so f � f and by de�nition ��� there is x � A withf v x� �

The next lemma is used to show that ENT is algebraic�

Lemma �� Let I and J be subsets of STRUC� If both I� and J� are �nite�then so is �I � J���

Proof� If H � STRUC and G � H � then it is easy to see that card�G� �card�H��� Let G � I � J � Because G � I � J every element of G belongs tosome member of I� � J�� so there is a function from G into I� � J� andtherefore G is �nite� But as mentioned above� card�G� � card��I � J���� �

Proposition �� The CPO ENT � �ENT�v��� is algebraic�

Proof� From de�nition ��� v is � so the set approx��R� � f�I � �R � �I

� I� is �niteg� It is directed because if �I and �J are elements of approx��R�� then �R � �I � �J � �I�J � which is an element of approx��R� since bylemma �� �I � J�� is �nite if both I� and J� are�

Finally we must show thatVapprox��R� � �R� Any �nite set F � STRUC

has �nite F�� Now consider E � fF � F � R � F is �niteg� For each F � E��F � approx��R� and

SE � R� So �R �

Vapprox��R� �

Vf�F � F �

E � F is �niteg � �SE � �R� �

Using proposition �� �� the last line in the proof of �� � shows that anyrestricted entailment can be approximated by a subset of the compact onesgreater than it� namely by the entailments with �nite restrictions which aregreater than it�

Proposition �� Let F � R � STRUC and let F be �nite� then �F � �R�

Proof� The last line of proposition �� � shows that

�R ��f�F � F � R � F is �niteg�

Also for �nite F � �F is compact and the result follows from �� �� �

A domain is an algebraic CPO satisfying a certain consistency condition�

De�nition � �Consistent Set� Let D � �D�v� be partially ordered andlet A � D� then A is consistent if and only if A has an upper bound in D�

� CHAPTER �� RESTRICTED ENTAILMENTS AS A DOMAIN

De�nition �� �Domain� Let D � �D�v��� be a CPO� then D is a domainif and only if the following two conditions are satis�ed�

�� D is an algebraic CPO�

�� If d and e are compact members of D and fd� eg is consistent� then d � eexists and lies in D�

We have seen in proposition �� � that ENT � �ENT�v��� is algebraic�By ��� any pair of elements of ENT has a least upper bound given by meet�The following proposition summarises the situation�

Proposition �� The CPO ENT � �ENT�v��� de�ned in ��� is a domain inwhich

�� The compact members of ENT are the restricted entailments �I � whereI� is �nite�

�� For �R� ENT� approx��R� � f�I � �R � �I � I� is �niteg�

This chapter starts by de�ning the CPO �ENT�v���� Then some results andde�nitions concerning the equivalence classes of full expansions and related mat�ters are given in items ��� to ���� These are used in the characterisation of thecompact restricted entailments� The important proposition �� � characterises arestricted entailment �I as compact if and only if the set of equivalence classesI� is �nite� It is shown that any restricted entailment can be approximatedby compact ones� In fact� �ENT�v��� is algebraic� that is any restricted entail�ment is the least upper bound of the compact restricted entailments less thanit in the CPO order� We abbreviate this by saying that �compact elementsapproximate�� Indeed� we have what is perhaps the simplest approximationresult� �entailments with �nite restrictions approximate�� That is� in the par�tially ordered set �ENT���� any restricted entailment �R is approximated bythe restricted entailments �F with �nite restrictions satisfying �R � �F �thisfollows from �� � and ����� This is signi�cant because it provides a link to alogic with a modal operator� �F � de�ned in terms of a �nite set F of structures�In chapter � the semantics of �F is shown to be equivalent to the semantics of�F for �rst�order formulas� Finally the notion of a domain is de�ned and thedomain properties of �ENT�v��� summarised�

The next chapter discusses the modelling of restricted semantics using amodal language�

Chapter

Modal Restricted

First�order Semantics

We now set out to provide a proof theory as a counterpart to the semanticsof restricted entailments introduced in chapter �� We do this by enriching thesyntax of �rst�order logic by adding a modal operator� This enlarged languagehas both a semantics and a proof theory� In this chapter the semantics isexamined� The proof theory is dealt with in chapter ��

Proposition �� � shows that any restricted entailment is approximated byentailments restricted to �nite sets of structures� so it makes sense to seek aproof method based on them� Accordingly given a �nite set of structures R� amodal language is de�ned with modal operator �R having the intuitive meaningof �approximately true�� The use of the phrase �approximately true� highlightsthe fact that there is a connection between �R and restricted entailment� infact for �rst order sentences restricted entailment semantics is shown to beequivalent to the semantics of �approximately true� �see proposition �� ���

We have modelled the notion �approximately true� in a modal languagebecause it allows the idea to be dealt with explicitly in the syntax and alsoit is possible to develop a proof theory which is a counterpart of the semanticsatisfaction of restricted entailment� We concentrate here on the modality of�approximately true� and do not treat the modalities of necessity and possibility�This simpli�es the semantics�

Approach

In dealing with modal semantics �and proof theory� we adopt the approach ofFitting and Mendelsohn ��� However there is a signi�cant di�erence in ourtreatment� we admit as modal formulas �rst�order formulas containing functionsymbols and constants� So for example if f is a one place function symbol�c a constant and p a two place predicate then the following is accepted as amodal sentence� �x�p�x� f�c��� Fitting and Mendelsohn do not allow constants

�

�� CHAPTER � MODAL RESTRICTED FIRST�ORDER SEMANTICS

to appear in formulas in this form� The reason has to do with the fact that themeaning of a constant can vary in Kripke style semantics� We need to explainwhy this happens in their treatment but not in ours� We do this by giving aslightly modi�ed example from �� page �� ���

Suppose the constant symbol c is to denote the tallest person in the world�Modally if the operator � is taken to mean the temporal operator at all futuretimes and the possible worlds are instants of time� then taking b to be thepredicate has blue eyes the sentence �b�c� can be read in two ways� One reading�takes the constant as primary focus to designate a �xed person now� Joy say�supposing she is the tallest person in the world now� Let us also suppose Joylives for ever� then this reading says Joy always has blue eyes� In this readingthe statement is true if c now has blue eyes� Another reading takes � as primaryand takes c to denote the person who is the tallest in a world designated by aparticular time� In this reading c�s identity may change from world to worldand so c�s eye colour may vary from world to world �that is� as time progresses��So the sentence will be false if c�s eye colour varies over time� The reason forthis ambiguity is that the �Kripke� interpretation of a modal formula involvingthe necessity operator� �� looks at how the formula behaves in several di�erentpossible worlds� The same holds for the possibility operator� ��

To remove this ambiguity Fitting and Mendelsohn use varying domain mod�els and also expand the syntax of �rst�order modal logic to include predicateabstracts �� �� �which we do not de�ne here� in formulas involving terms�The semantics is also expanded to deal with these formulas� However as canbe seen in de�nition ��� the semantics of �R only involves one world� So noambiguity of the type mentioned above will arise out of the use of �R as longas � and � are not also used in modal formulas� Our de�nition �� of modalformulas does not include � or � and so the terms of our modal language cansafely include constants and function symbols�

If � and � are to be included in the syntax as well as terms involving con�stants and function symbols then the syntax and semantics should be expandedas in ��� say� On the other hand if it is acceptable for terms to consist only ofindividual variables �leaving out constants and function symbols� then no ex�pansion is needed� Hughes and Cresswell � use this approach� Alternativelythe approach in Nerode and Shore ��� page ���� could be followed� which allowsconstants but not function symbols�

To summarise� in our approach we allow constants� function symbols andthe modal operator �R but not � or ��

Modal Syntax and Semantics

Modal formulas are de�ned in the way given below so as to allow the complete�ness proof of chapter � to go through smoothly� It turns out that care hasto be taken when putting � as the �rst symbol in an expression� This is thereason for �rst de�ning approximation sentences and then using them to de�nemodal formulas� We mention that although any �rst�order formula is a modal

�

formula� �rst�order formulas are as de�ned in chapter �� The phrase ��rst�orderformula� always means a formula as de�ned in chapter ��

De�nition � � �Modal Formula�

�� Let R � STRUC� The approximation sentences are de�ned inductively asfollows�

a Any �rst�order sentence is an approximation sentence�

b If � is an approximation sentence� then so is �R �� When there isno danger of confusion the subscript R in �R will be dropped�

c If � and � are approximation sentences� then so are ��� ���� ���and �� ��

�� The modal formulas are de�ned inductively as follows�

a Any approximation sentence is a modal formula�

b Any atomic �rst�order formula p�t�� � � � � tn� is a modal formula�

c If � and � are modal formulas� then so are ��� � � �� � � � and�� ��

d If � is a modal formula and x an individual variable� then �x�� isa modal formula�

e If � is a modal formula and x an individual variable� then �x�� isa modal formula�

�� The modal language with formulas de�ned as above is denoted by ML�

Remark � � Let � be a �rst�order formula whose only free variable is x� andlet � be a �rst�order sentence�

� The expression �x�����R�� is a modal formula�

� The expression �x��R� is not a modal formula because �R� is not anapproximation sentence�

For the soundness and completeness proofs of later chapters we need toextend the language ML� We add a countably in�nite set fd�� d�� � � � � dn� � � � gof constant symbols called parameters to ML� The resulting language is calledMLpar� The parameters are all assumed to be di�erent from the constants ofML�

De�nition � � �The Language MLpar� The symbolMLpar denotes the lan�guage obtained from the language ML by adding to it a countably in�nite setof constant symbols� fd�� d�� � � � � dn� � � � g� which are disjoint from the constantsymbols of ML�

�� CHAPTER � MODAL RESTRICTED FIRST�ORDER SEMANTICS

Note � �

� Approximation sentences never contain parameters�

� A �rst�order sentence never contains ��

� A �rst�order sentence of MLpar may� but need not� contain parameters�

We will de�ne forcing for modal formulas� The semantics is unusual in thatit uses a single possible world� The de�nintion is made in terms of a frame�which is de�ned below�

De�nition � � �Frame� A frame is an ordered triple �G�R�D� where�

� G is a non�empty set with one element� G say� G is called a world or apossible world�

� R is a binary relation de�ned on G called the accessability relation� Thereare only two possibilities for R in our system� it is either empty or itequals �G�G��

� D is a set called the domain of the frame�

De�nition � �Model�

�� A model is an ordered four�tuple M � �G�R�D�S�� where �G�R�D� is aframe and S is a �rst�order structure with domain dom�S� � D�

�� The model M is said to be based on the frame �G�R�D��

We recall that an assignment of variables u with domain dom�u� � D is amapping from the set of individual variables to D and an interpretation I is anordered pair I � �S� u� where S is a structure� u is an assignment of variablesand dom�S� � dom�u� �see chapter ��� Also if u and v are assignments ofvariables� then v is an x�variant of u if u and v agree on all variables exceptpossibly the variable x �see de�nition �����

The idea of forcing in modal logic is analogous to satisfaction in �rst�orderlogic� Forcing is a relation between models� worlds and assignments of variableson the one hand and modal formulas on the other� When G consists of morethan one possible world� the fact that the model M and assignment of variablesu force the formula � at world G is denoted M� G �u �� However according toour de�nition G has only one world so mention of G can be omitted from theforcing expression above thus� M �u ��

De�nition � � �Modal Forcing� Let M � �G�R�D�S� be a model� I ��S� u� be an interpretation� G � fGg and let � and � be modal formulas� Theforcing relation� �� is de�ned recursively as follows�

�� If � is an atomic formula� then M �u � i� I � ��

�� M �u �� i� M �u ��

�

�� M �u � � � i� M �u � or M �u ��

�� M �u � � � i� M �u � and M �u ��

�� M �u �� � i� M �u � and M �u �� otherwise M �u �� ��

� Let R � STRUC be the restriction of �� then M �u�� i� the followingholds� if GRG and S � R then M �u ��

�� M �u �x�� i� for every x�variant v of u� M �v ��

�� M �u �x�� i� for some x�variant v of u� M �v ��

In order to de�ne validity of modal sentences a result similar to proposi�tion �� is needed�

Proposition � � Let � be a modal formula� let M � �G�R�D�S� be a modeland let u and v be assignments of variables with domain D which agree on thefree variables of � then M �u � i� M �v ��

Proof outline� The proof is by induction on the structure of ��BASE CASE� � is an atomic formula� By proposition �� the interpretation

�S� u� � � i� �S� v� � �� The proposition follows for this case from de�nition ����CASE � is �� where the proposition holds for �� Easy�CASE � is �� where the proposition holds for and �� and � is one of ��

� or �� Easy�CASE � is �R� where the proposition holds for �� Suppose GRG and S � R�

By the inductive hypothesis M �u � i� M �v � and so by de�nition ��� theresult follows� If �G�G� � R or S � R then the proposition is true by default�

CASE � is �x�� where the proposition holds for �� Similar to the corre�sponding case of proposition �� �

CASE � is �x�� where the proposition holds for �� Similar to proof for�x��� �

Corollary � Let � be a modal sentence� then M �u � for some assignmentof variables u if and only if M �u � for every assignment of variables u�

Proof� A sentence has no free variables� �

De�nition � �� Let M � �G�R�D�S� be a model and � a modal sentence�

�� The model M forces the sentence � if and only if M �u � for someassignment of variables u� In view of corollary ��� this is independent ofthe assignment of variables and so is denoted M � ��

�� The sentence � is valid in the frame �G�R�D�� denoted �G�R�D� � �� ifit is forced by every model based on that frame� In this case �G�R�D� isalso said to force ��

�� CHAPTER � MODAL RESTRICTED FIRST�ORDER SEMANTICS

�� The sentence � is valid� denoted ��� if it is valid or forced in everyframe�

In corollary �� a connection is made between the validity of a �rst�ordersentence � in a subset of structures and the validity of ��� but �rst the followingresults are needed� We recall that the notion of the structure S satisfying the�rst�order formula � was de�ned in ���� It is written S � ��

Proposition � �� Suppose M � �G�R�D�S� is a model and let � be a �rst�order sentence� then S � � i� M � ��

Proof� A straightforward induction on the structure of �� �

Proposition � �� Suppose M � �G�R�D�S� is a model with a non�emptyaccesibility relation R and suppose R � STRUC is the restriction of �� Let � bea �rst�order sentence� then S �R � i� M � ���

Proof� Straightforward using proposition �� � �

Corollary � �� Let � be a �rst�order sentence and let R � STRUC be therestriction of �� then �R � i� � ���

In proposition �� � we show that restricted entailment is equivalent to theforcing of approximately true sentences under an appropriate extension of themeaning of forcing� which is de�ned next�

De�nition � �� �Modal Entailment� Let M � �G�R�D�S� be a model andlet X be a set of modal sentences and � a modal sentence�

�� M � X i� M � � for every � � X� If M � X� then M is said to forceX�

�� X � � if and only if for every model M� M � X implies M � ��

The next proposition and its corollary justify the claim that �R models thesemantics of �R for �rst�order sentences�

De�nition � �� If X is a set of sentences� then �X � f�� � � � Xg�

Proposition � � Let X be a set of �rst�order sentences� � a �rst�order sen�tence and let R be the restriction of �� then X �R � i� �X � ���

Proof� First suppose X �R � and let M � �G�R�D�S� be a model whichforces �X � We must show that M � ��� From de�nition ��� it follows that ifS �� R then M ��� by default� If S � R then using proposition �� � S � X soby supposition S � �� Using proposition �� � again we have M � ���

Conversely suppose �X � ��� let S � R and suppose S � X � We must showthat S � �� From our assumption as well as proposition �� � the model M ��Xso M � ��� It follows that S � �� �

��

Corollary � �� Let X and Y be sets of �rst�order sentences� then X �R Y i��X � �Y �

Note � �� The operator � has the rather odd property of simultaneously ex�hibiting features characteristic of the standard modal operators � and �� Thepossibility of a modal operator exhibiting a duality of this kind is not new� Fairt�lough and Mendler �� de�ne a modal operator which also exhibits features ofboth � and � to model the notion of a formula �holding under a constraint��Their resulting modal logic can be applied to hardware veri�cation� The sim�ilarities and di�erences between �approximately true� and �holding under aconstraint� are not investigated here�

Some observations on the behaviour of � follow� Naming conventions forthe following axiom schemas can be found in Gor�e ��� and Chellas ��� Theoperators � and � are regarded as being mutually interde�nable by taking �to be equivalent to ���� In the following schemas� the word in parenthesesindicates the condition that the axiom imposes on the accessibility relation�The axiom K imposes no restriction at all� The list below is not exhaustive�but it is su�cient to illustrate the dual nature of ��

K ���� �� � ���� ���T �Re�exive� ��� �D �Serial� ��� ��� �Transitive� ��� ���� �Euclidean� ��� ���B �Symmetric� �� ���

Each axiom schema has two versions� labelled either with � or �� whichare derived as follows� To obtain the box version� replace every occurrence of� by ��� and simplify the expression remembering that � and � can be anysentences whatever� To obtain the diamond version replace � by ���� So� forexample� T� is ��� �� while T� is �� ��� We will say that � satis�es T��for example� if � satis�es the expression obtained from T� by replacing everyocurrence of � by �� A similar procedure is used to see if � satis�es T��

One can show that � satis�es the axiom schemas listed below� The relevantaxiom schema� with � replacing � or �� is shown if � satis�es it� The proofs areeasily done semantically or by the tableaux of chapter ��

K� � �� � �� � ���� � ��Re�exive T� �� ��Serial D� � ���� ��Transitive �� �� �� � � �� � �� �� �Euclidean �� � � �� �� ��Symmetric B� �� �� ���

�

In this chapter a modal language was de�ned whose semantics can modelthe semantics of a restricted entailment� The chapter is mainly concerned with

�� CHAPTER � MODAL RESTRICTED FIRST�ORDER SEMANTICS

the mechanics of the language and with justifying the claim that the modallanguage models restricted �rst�order semantics� In our de�nition of modalformulas� �rst�order formulas incoporating constants and function symbols areallowed� This was justi�ed in the discussion on approach� at the beginning ofthe chapter� Also the modal operator �R with the meaning �approximatelytrue� is introduced� �It is noted that the operator � exhibits features of both �and ���

Discussion of modal semantics uses the notions of a frame and a model� Aframe �G�R�D� has a set of worlds G� an accessibility relation R and a set D�the domain of the frame� Notable features of our frame de�nition are that�there is only one world� G say� in the set of worlds� the accessibility relationis either empty or it equals �G�G� and the domain consists of a single set� Amodal model is a four�tuple consisting of a frame and a �rst�order structure�

Modal forcing �analogous to �rst�order satisfaction� is de�ned and the chap�ter culminates in a result that shows for �rst�order sentences� restricted en�tailment is equivalent to forcing of �approximately true�� let X and Y besets of �rst�order sentences and let R be a set of structures� then X �R Yi� �RX � �RY �

Now� having a modal language with a semantics� a proof method is required�this is given next in chapter ��

Chapter

Tableau Proofs

We have seen in the previous chapter �proposition �� �� that restricted entail�ment can be modelled in a modal logic having an operator� denoted �� with theintuitive meaning �approximately true�� In this chapter we give a proof theorybased on tableaux for this logic� The tableau method can be readily adapted todeal with di�erent kinds of logics� as the following works show �� ��� ��� ��� ����

We use signed expressions �explained below� in our tableaux� Tableau rulesfor �rst�order logic are augmented to include ones for sentences issuing from ��These are sentences of the following types� T��� F��� T� and TCi �see de�ni�tion �� and �gure �� �� We base our approach on Fitting and Mendelsohn ���although the neccessity and possibility operators �� and �� are not discussedhere because they are not used�

Two kinds of proof are considered� proofs without premises and proofs frompremises� A set of premises is just a set of sentences� A proof without premisesis a proof from the empty set of premises� that is a proof without assumptions�A proof from premises uses the premises as assumptions� The fact that � can beproved without premises is written � �� while the fact that � is provable frompremises X is written X � �� A brief explanation of the machinery of tableauproofs is given below starting with proofs without premises�

Tableau Nodes� Signed Expressions

A tableau is a tree� The nodes in the tree are signed expressions� A signedexpression begins with T or F� There are two kinds of expression used in buildingtableaux� One kind consists of signed modal sentences� These are of the formT� or F�� where � is a modal sentence� The other kind is used to deal withformulas containing �� This kind is a signed separator � T�� The separator� ��is de�ned as follows� Let R be the restriction of � and suppose R is a �niteset of structures� Take A to consist of all structures that are not elementarilyequivalent to some member of B� We can write an expression for � by referringto note � part �

��

�� CHAPTER � TABLEAU PROOFS

De�nition � � Let R be the restriction of � and suppose R is �nite with melements� Let A consist of those structures not elementarily equivalent to anymember of R� The separator � can be written � � �A�R �

Wmi�� Ci� where

Ci �Vn�Ni

�n�i for Ni some subset of the integers possibly all the integerswhich depends on i� and each �n�i is a �rst�order sentence�

Tableau Proof

A tableau proof without premises of a modal formula � proceeds by buildinga tree with root node F�� This a negated form of the formula � and acts asa starting assumption� One tries to derive a contradiction by applying certainrules for growing the tree by appending child nodes to existing leaves in thetree� A branch in a tree is a path starting at the root and ending at a leaf�Once the tree has been built� every branch is checked to see if there is a pairof contradictory nodes on that branch� Two nodes are contradictory if they arethe same except for their pre�xes� that is if one has the form Fw and the otherTw where w is some string of symbols� If every branch in the tree has a pairof contradictory nodes then the intuitive idea is that the root node� F�� leadsto contradiction and so the formula � must be true� A tree built as describedbelow is called a tableau� A tableau branch is said to be closed if it has a pairof contradictory nodes and a tableau is closed if all of its branches are closed�A closed tableau with root node F� is said to be a tableau proof of ��

Building the Tableau

A tableau is built in stages using a nondeterministic process� The �rst stageconsists of a tree having a single root node of the kind F�� Now suppose thetree has been built to a certain stage� It will have a number of leaves� Childrencan be appended to a leaf by picking a signed sentence� call it the parent� lyingon the branch terminating in that leaf and deciding which children to append tothe leaf� The children to append depend on the form of the parent� Figure �� covers all the possibilities� where the forms of the parent and children to beappended are given as generic cases called tableau branch extension rules�

Here are some explanatory comments on some of the branch extension rulesof �gure �� � For example�

� The parent is T�� where � is an atomic �rst�order sentence� then nochildren are appended�

� The parent is T� � �� then the leaf branches� A left node is appended tothe leaf of the form T� and a right node is appended to the leaf of theform T��

� The parent is F���� then two nodes are appended to the leaf in sequence�F� and F��

��

T� F� T�� F��any atomic � any atomic � F� T�

T� � � F� � � T� � � F� � �T� T� F� T� F� F�

F� T�

T�� � F�� �F� T� T�

F�

Tx��x� Fx��x� Tx��x� Fx��x�T��c� F��t� T��t� F��c�

some new c any closed term t any closed term t some new c

T �� F ��T� F�

If T� is on branch T�

T� TCiTC� � � � TCm T���i

���T�n�i

���

Figure �� � Tableaux Branch Extension Rules�

�� CHAPTER � TABLEAU PROOFS

� The parent is Tx��x�� a node is appended to the leaf of the form T��c��where c is a parameter of the language MLpar and new to the tableaubranch�

� The parent is Fx��x�� a node is appended to the leaf of the form F��t��where t is any closed term of the language MLpar�

� The parent is Tx��x�� a node is appended to the leaf of the form T��t��where t is any closed term of the language MLpar�

� The parent is Fx��x�� a node is appended to the leaf of the form F��c��where c is a parameter of the language MLpar and new to the tableaubranch�

� The parent is T ��� the node T� is appended to the leaf if T� alreadyappears on the branch� otherwise nothing is appended�

� The parent is F ��� then two nodes are appended in sequence� F� andT��

� The parent is T�� then m nodes are appended in parallel� TC�� � � � �TCm�

� The parent is TCi� then nodes T�n�i are appended in sequence� for eachn � Ni �see de�nition �� �� We recall that each of the �n�i is a �rst ordersentence�

We assume that development of a branch stops as soon as a contradictorypair of nodes is detected on that branch�

When proving � from a set of premises� X � the idea is to produce a tableauin which members of X are forced� We call this a tableau from X � A tableauto prove the sentence � from X is built in exactly the same way as describedabove except that an extra rule is used to allow nodes to be appended to a leaf�

Premise Extension Rule

When building a tableau from premises X � if � � X then T� can be appendedto any leaf of the tableau� �

The focus in this chapter was on the tableau proof method for the languagewith modal operator �R� where R is a �nite set of structures� As we saw inchapter � the operator �R models the restricted entailment �R� This chapterprovided a proof theory that links back to the semantic notion of restrictedentailment� We use signed sentences in our tableaux� that is sentences pre�xedby the letters T or F�

Tableau branch extension rules for �rst�order logic were augmented to in�clude rules for sentences issuing from �� they are sentences of the following kinds�T��� F��� T� and TCi �see de�nition �� and �gure �� �� The premise extensionrule for tableau proofs from premises was also given�

�

Having set up the �syntactic� machinery of proof in this chapter and the�semantic� machinery of forcing in chapter � it remains to show soundness inchapter � �every provable sentence is forced by every model� and its converse�completeness� in chapter ��

�� CHAPTER � TABLEAU PROOFS

Chapter �

Soundness of the modal

proof system

We now prove soundness of the modal proof system of chapter �� The form ofthe argument is based on Fitting and Mendelsohn ���

We must show that if a modal sentence is provable then it is valid� that is�forced by every model� The notions of signed sentence and signed separator werede�ned in chapter �� the nodes of a tableau consist of these� In what follows wetake the meaning of the phrase �set of signed sentences� to include the possibilitythat the set can contain a signed separator� The proof of soundness hinges on thenotion of satis�ability� This is de�ned next� Note that we distinguish betweenR in calligraphic font for an accessibility relation and R in non�calligraphic fontfor a restriction�

De�nition � �Satis�able�

�� Let T be a set of signed sentences� then T is said to be satis�able by themodel M � �G�R�D�S� if and only if the following hold�

a If T� is in T � then R � f�G�G�g and S � R� where G � fGg and Ris the restriction of �R�

b If T� is in T � then M � ��

c If F� is in T � then M � ��

�� A tableau branch is said to be satis�able if and only if its set of signedsentences is satis�able�

�� A tableau is satis�able if and only if some branch of it is satis�able�

To show soundness two results about satis�ability are needed�

�

�� CHAPTER �� SOUNDNESS OF THE MODAL PROOF SYSTEM

Proposition � A closed tableau is not satis�able�

Proof� Suppose a tableau is both closed and satis�able� Since it satis�able ithas a satis�able branch� Let T be the set of signed sentences on that branchand suppose the model M satis�es T � The branch is closed so there is a pairof contradictory signed sentences in T � Let them be T� and F�� Then we willhave both M � � and M � �� This is a contradiction� �

In �gure �� � tableau branch extension rules were given for the modal lan�guage considered here�

Proposition � If a tableau branch extension rule is applied to a leaf in asatis�able tableau� then the result is another satis�able tableau�

Proof outline� Suppose a branch extension rule is applied to a satis�ablebranch B of a tableau� Denote the set of signed sentences on B by T andsuppose the model M satis�es T � There are a number of cases to examine� onefor each branch extension rule in �gure �� �

CASE T��� Suppose T�� is on the branch� then it is satis�able by M andso M � �� It follows that T � fF�g is also satis�ed by M� The case F�� istreated similarly�

CASE T���� Suppose T��� is on the branch� then because it is satis�ableby M it follows that M � � or M � �� From the de�nition of satis�ability� Mthen satis�es T � fT�g or T � fT�g�

The other cases involving binary connectives have similar proofs�CASE Tx��x�� Suppose Tx��x� is on the branch� then is satis�able by

M� where M � �G�R�D�S�� Let the paramter c be new to the branch� Wemust show that ��c� is satis�able� We shall produce a model M� � �G�R�D�S ��which satis�es ��c��

Now for any interpretation of variables u� M �u x��x� and so there is an x�variant u� of u for which M �u� ��x�� De�ne the structure S � to be the same asS except that S ��c� � u��x�� then M� � ��c�� Also for any other interpretationof variables w let w� be the x�variant of w with w��x� � u��x�� Then w� agreeswith u� on the free variables of ��x� �namely x because x��x� is a sentence��and so M� � ��c� i� M �u� ��x� i� M �w� ��x� �using ���� i� M �w ��x��Because c does not occur in T it follows that M� satis�es T � f��c�g�

The other quanti�er cases are proved similarly�CASE T�R�� Suppose that T� is already on the branch then we must show

that the extended branch� T � fT�g� is satis�able by M� Now the set T issatis�able by M and contains T�R� so M ��R�� But also T� � T and so byde�nition �� � �G�G� � R and S � R� It follows from de�nition ��� of modalforcing that M � �� If T� is not on the branch then nothing is appended� andT remains satis�able�

CASE F �R�� The tableau branch extension rule appends F� and T� insequence� Now the set T is satis�able and contains F�R� so M ��R�� Fromde�nition ��� of forcing it follows that �G�G� � R� S � R and M � �� Thismeans that the signed sentences on the extended branch� T � fF�g � fT�g� are

��

satis�able by M� F� is satis�able because M � � and T� is satis�able becauseby proposition � � S � R means that S � �� and using proposition �� itfollows easily that M � � � �

Proposition � �Soundness� If a modal sentence � has a tableau proof� thenit is valid�

Proof� Suppose � has a tableau proof but it is not valid� Since � has a tableauproof there is a closed tableau T with root F�� Set the tableau T� to be thisroot node� The tableau T results from T� by applying branch extension rules�

Since � is not valid there is a modelMwithM � �� Then from de�nition �� fF�g � T� is satis�able� The tableau T results from T� by application ofbranch extension rules and so T is also satis�able by proposition ��� This is acontradiction because by proposition ��� no closed tableau is satis�able� �

We have shown the soundness of the modal proof system� if a modal sentenceis provable� then it is forced by every model� The proof hinges on the notion ofsatis�ability� The argument then proceeded as follows�

� A closed tableau is not satis�able�

� A tableau branch extension rule applied to a leaf of a satis�able tableaugives another satis�able tableau�

� Finally� using the above two points� soundness was proved by contradic�tion�

In the next chapter the converse of soundness� completeness� is proved�

�� CHAPTER �� SOUNDNESS OF THE MODAL PROOF SYSTEM

Chapter ��

Completeness

In this chapter our argument is again based on Fitting and Mendelsohn ���We will prove completeness of modal sentences not containing parameters�

Completeness is the converse of soundness� if a sentence is forced by everymodel then it is provable� Completeness is proved by contradiction� if a sentence� does not have a proof� then there is a model which does not force it� Thismodel is obtained from an open branch of a tableau with F� as root node� So webegin by giving a standard and exhaustive method of constructing a tableau�the �standard tableau construction method�� If a standard tableau is not aproof it will have a branch without a pair of contradictory entries on it� Such abranch is called an open branch� The argument then proceeds as follows�

� Given an open branch� B� a structure� SB� related to B is constructed� �Itis actually a Herbrand structure��

� A model� MB� based on SB is constructed taking into account whetherthe separator T� lies on B or not�

� The model MB is a construct that helps in the proof of completeness� Letthe sentence �B be the same as � except that �R is replaced by �fSBg�The main property of MB is stated in proposition ���� if T� is on B thenMB � �B� and if F� is on B then MB � �B�

� Next� in ��� it is shown that if MB � �B� then M � �� where � is asentence without parameters� and M is a model with a structure S � Rthat is elementarily equivalent to SB� The structure S is shown to existin ���

� Completeness is proved in ���� if � does not have a proof then there is atableau having F� as root with an open branch B� According to the aboveMB � �B� and so M � �� This shows that M does not force �� whichproves completeness�

��

�� CHAPTER ��� COMPLETENESS

Finally soundness and completeness of proofs from premises are proved�see �� � and �� ��

We make some comments on the role of parameters in our arguments ofthis chapter� Parameters are used as auxiliary items which appear in sentencesin our proofs� They arise from applying tableau expansion rules to quanti�edsentences� The structure SB is built from terms appearing in sentences lyingon B� and so the domain of SB turns out� in general� to contain parameters�Our goal� however� is to prove completeness of the language ML� which doesnot have parameters� We use terms and sentences of the language MLpar inintermediate steps of our completeness argument for ML�

The standard tableau construction method will need to refer to each ofthe closed terms of the language MLpar in an unambiguous way� There arecountably many of them so we assume that they have been enumerated once andfor all and are in one�to�one correspondence with the positive integers� Denotethe �nite set of the �rst n of these closed terms by TERMn� The de�nition ofTERMn is recorded below�

De�nition �� � Suppose the countably in�nite set of closed terms of MLparhas been given a �xed enumeration� The symbol TERMn denotes the �rst n ofthese terms in the �xed enumeration�

Standard Tableau Construction Method

The method proceeds in steps and inductively describes the construction of onestep from another starting at step � There may be an in�nite number of stepsin the construction and the tableau may turn out to be in�nite� Suppose thesentence to be proved is �� For step form the single�node tableau having F�as its root node� Now suppose step n has been constructed and that the tableauis not closed� The following is the construction for step n � �

Start with the left�most open branch of the tableau and treat each signedsentence� �� on the branch� going from top to bottom� in the following way�

� If � is T��� then add F� to the end of the branch as long as F� does notalready occur on it�

� If � is F��� then add T� to the end of the branch as long as T� does notalready occur on it�

� If � is T�� �� then as long as neither T� nor T� lies on the branch� splitthe end of the branch and add T� on the left and T� on the right� Treatthe cases F� � � and T�� � similarly splitting the end of the branch toF�� F� and F�� T� respectively�

� If � is F� � �� then add F� and F� to the end of the branch as long asthey are not already on it� Treat the cases T� � � and F� � � similarlyadding T�� T� and T�� F� to the end of the branch respectively�

��

� If � is Tx��x�� then if T��d� is not on the branch for any parameter d�choose the �rst parameter di of the list d�� d�� � � � of parameters which isnot used on that branch and append T��di� to the leaf� Treat the caseFx��x� similarly appending F��di��

� If � is Tx��x� then for each t � TERMn� de�ned above� add the sentenceT��t� to the end of the branch as long as T��t� does not already occur onit� Treat the case Fx��x� similarly adding the sentence F��t� for eacht � TERMn�

� If � is T��� then add T� to the end of the branch as long as T� alreadyoccurs on it and T� does not�

� If � is F��� then add T� to the end if it is not already on it� and add F�to the end if it is not already on it�

� If � is T� then split the end of the branch and add TC�� � � � �TCm inparallel� omitting any of the TCi which are already on the branch�

� If � is TCi then add T���i� � � � �T�n�i� � � � in sequence� omitting any of theT�n�i which are already on the branch�

This completes the treatment of the left�most open branch� Now do thesame to the next open branch to the right and so on� This completes step n� �The same procedure is repeated for steps n� �� n� � � � � � �

The standard tableau constructed from � may have open branches� Anyopen branch B of a tableau constructed by the standard method has the follow�ing properties�

Open Branch Properties

� Not both T� and F� are on B because it is not closed�

�� If T�� is on B then so is F��

� If F�� is on B then so is T��

�� If T� � � is on B then so is one of T� or T��

�� If F� � � is on B then so are F� and F��

�� If T� � � is on B then so are T� and T��

�� If F� � � is on B then so is one of F� or F��

�� If T�� � is on B then so is one of F� or T��

�� If F�� � is on B then so are T� and F��

�� If Tx��x� is on B then so is T��d� for some parameter d�

�� CHAPTER ��� COMPLETENESS

� If Fx��x� is on B then so is F��t� for every closed term t of MLpar�

�� If Tx��x� is on B then so is T��t� for every closed term t of MLpar�

� If Fx��x� is on B then so is F��d� for some parameter d�

�� If T �� is on B then so is T� if T� is already on B�

�� If F �� is on B then so are T� and F��

�� If T� is on B then so is TCi for some i � f � � � � �mg�

�� If TCi is on B then so are T�n�i for all n � Ni �see de�nition �� �� �

If the modal sentence � does not have a tableau proof� then the standardtableau built from F� has open branches� Let B be such an open branch� Theclosed terms and relation symbols ocurring on B can be used to construct anassociated structure SB with a domain DB� de�ned below�

Construction of SB

The domain DB is de�ned inductively as follows�

� Every constant of ML appearing in a signed sentence on B is in DB�

� Every parameter of MLpar appearing in a signed sentence on B is in DB�

� If f is a function symbol of arity n appearing in a signed sentence on Band t�� t�� � � � � tn are in DB� then f�t�� t�� � � � � tn� is in DB�

Note that according to the above de�nition� every element of DB is a termof MLpar�

The structure SB is de�ned as follows�

� Any constant or parameter of ML or MLpar which also belongs to DB ismapped onto itself by SB�

� Other constants and parameters can be arbitrarily mapped into DB bySB�

� If f is a function symbol of arity n appearing in a signed sentence onB and t�� t�� � � � � tn are each in DB� then fSB maps �t�� t�� � � � � tn� tof�t�� t�� � � � � tn� � DB�

� For any other function symbol f � fSB can be any mapping into DB�

� If p is a relation symbol of arity n and t�� t�� � � � � tn are each in DB� then�t�� t�� � � � � tn� � pSB if and only if Tp�t�� t�� � � � � tn� is on B�

�

This completes the construction of the structure SB� �

In view of de�nition ��� and corollary ��� the following lemma� used to provelemma ��� is true�

Lemma �� � Let SB be the structure de�ned above�

�� The following hold for �rst�order sentences � and �� which may containparameters�

a SB � �� i� SB � ��

b SB � � � � i� SB � � or SB � ��

c SB � � � � i� SB � � and SB � ��

d SB � �� � i� SB � � and SB � �� otherwise SB � �� ��

�� Let ��x� be a �rst�order formula in which x is the only free variable� thenthe following hold�

a SB � x��x� i� SB � ��t� for some term t � DB�

b SB � x��x� i� SB � ��t� for every term t � DB�

The next lemma gives some important properties of SB which are used inthe proof of the crucial proposition ���� The property that is most importantis the one mentioned in part of the lemma� It shows there is a structure� S�which belongs to the restriction R of �R and is elementarily equivalent to SB�provided the signed separator� T�A�R� lies on the open branch B�

We recall that L denotes a �rst�order language without parameters� and �denotes elementary equivalence in L�

Lemma �� � Suppose B is an open branch of a standard tableau� suppose Ris the restriction of � and let SB� as constructed above� be the structure corre�sponding to B�

�� Let � be a �rst�order sentence of MLpar�

a If T� lies on B then SB � ��

b If F� lies on B then SB � ��

�� Let �A�R be the separator of A and R where A contains every structurenot elementarily equivalent in L to some member of R� If T�A�R lies onB then SB � �A�R�

�� If T�A�R lies on B� then there is a structure S � R such that SB � S�

�� CHAPTER ��� COMPLETENESS

Proof�

� This is proved by induction on the complexity of �� as follows�

BASE CASE � is an atomic sentence� Suppose � is p�t�� t�� � � � � tn�� wherethe ti are closed terms of MLpar� Let Tp�t�� t�� � � � � tn� be on B� It followsfrom the de�nition of SB that SB � p�t�� t�� � � � � tn��

Now let Fp�t�� t�� � � � � tn� be on B� By open branch property � Tp�t�� t�� � � �� tn� cannot be on B so by the de�nition of SB� SB � p�t�� t�� � � � � tn��

CASE � is �� where the proposition holds with � replacing �� SupposeT�� is on B� then by open branch property �� F� is on B and so byhypothesis SB � �� But by ��� this means that SB � ��� A similarargument proves the proposition when F�� is on B�

CASE � is � �� where the proposition holds for and � replacing ��Suppose T�� is on B� then by open branch property � both T and T�are on B� By hypothesis SB � and SB � �� so SB � � � � A similarargument using open branch property � proves the case when F � � ison B�

The rest of the propositional cases have similar proofs�

CASE � is x��x�� Suppose Tx��x� is on B� then by open branchproperty � T��d� is on B for some parameter d� By inductive hypothesisSB � ��d�� It follows from lemma ��� that SB � x��x��

Now suppose Fx��x� is on B� By open branch property F��t� ison B for every closed term t of MLpar and so by inductive hypothesis�SB � ��t� for every closed term t of MLpar� It follows from lemma ���that SB � x��x��

CASE � is x��x�� Suppose Tx��x� is on B� By open branch prop�erty � T��t� is on B for every closed term of MLpar and so by inductivehypothesis SB � ��t� for every closed term t of MLpar� It follows fromlemma ��� that SB � x��x�� The proof of the proposition for Fx��x�is similar�

�� This is proved by induction on the complexity of �A�R�

BASE CASE T�n�i lies on B� �Note that �n�i can only occur on B in theform T�n�i�� The sentence �n�i is �rst�order and so according to part ofthe proposition SB � �n�i�

CASE TCi lies on B� By open branch property �� T�n�i lies on B foreach n � Ni and by inductive hypothesis SB � �n�i for each n � Ni� Itfollows from the de�nition of satisfaction of �A�R in � that SB � Ci�

CASE T�A�R lies on B� Using open branch property �� suppose TCilies on B� By inductive hypothesis SB � Ci and from the de�nition ofsatisfaction of �A�R in � it follows that SB � �A�R�

�

� Suppose T�A�R lies on B� then according to part � SB � �A�R and theresult follows from proposition � �� �

Suppose the modal sentence � does not have a proof� then the standardtableau built from F� will have an open branch B� A model must be de�nedfor use in propositions ��� and ���� Depending on whether or not T� lies onB we de�ne the model MB as follows�

De�nition �� � Let B be an open branch of a standard tableau and let thestructure SB be as constructed above� De�ne the model MB � �G�R�DB�SB�as follows�

�� The set of worlds G for the model MB is de�ned to be the singleton f g�G � f g�

�� If T� � B� then set MB � �f g� f� � �g�DB�SB��

�� If T� � B� then set MB � �f g� ��DB�SB��

The model MB has the properties mentioned in proposition ��� below�which are used in the proof of ���� completeness�

Proposition �� � Let the open branch B and the model MB � �G�R�DB�SB�be as in ��� above� and let � be a modal sentence� possibly with parameters�Set RB � fSBg� and de�ne �B to be the same as � except that everywhere therestriction R is replaced by RB�

�� If T� is on B� then MB � �B�

�� If F� is on B� then MB � �B�

Proof� By induction on the complexity of ��BASE CASE � is an atomic sentence� Suppose � is p�t�� t�� � � � � tn�� where

the ti are closed terms of MLpar� Let Tp�t�� t�� � � � � tn� be on B� It is clearthat the atomic sentence �p�t�� t�� � � � � tn��B � p�t�� t�� � � � � tn�� It follows fromthe de�nition of SB that SB � p�t�� t�� � � � � tn�� and by de�nition ��� that MB �

p�t�� t�� � � � � tn��Now let Fp�t�� t�� � � � � tn� be on B� By open branch property �

Tp�t�� t�� � � � � tn� cannot be on B so by the de�nition of SB� SB � p�t�� t�� � � � � tn��Once more using de�nition ���� MB � p�t�� t�� � � � � tn��

CASE � is �� where the proposition holds with � replacing �� SupposeT�� is on B� then by open branch property �� F� is on B and so by hypothesisMB � �B� But by de�nition ��� this means that MB � ��B� A similarargument proves the proposition when F�� is on B�

Proofs of the rest of the propositional cases follow a similar pattern�CASE � is Ci �

Vn�Ni

�n�i �see �� � where the proposition holds for each�n�i in place of �� Note that each �n�i is �rst�order� so �Ci�B � Ci� Now Ci onlyoccurs on the branch in the form TCi� so suppose TCi is on B� By open branchproperty � for each n � Ni� T�n�i is on B and so by hypothesis� and using �� �

�� CHAPTER ��� COMPLETENESS

MB � �n�i� It follows� again using �� and the de�nition of satisfaction in ��that MB � Ci�

CASE � is � �Wmi�� Ci �see �� � where the proposition holds for each Ci in

place of �� Note that �B � �� Again� � can only occur on the branch B in theform T�� so suppose T� is on B� By open branch property � TCi is on B forsome i � f � � � � �mg� so by hypothesis MB � Ci� It follows from �� and thede�nition of satisfaction in � that MB � ��

CASE � is x��x�� Suppose Tx��x� is on B� then by open branch prop�erty � T��d� is on B for some parameter d� By inductive hypothesis MB �

���d��B � It follows from de�nition ��� that MB � �x��x��B �

Now suppose Fx��x� is on B� By open branch property F��t� is on Bfor every closed term t of MLpar and so by inductive hypothesis� MB � ���t��Bfor every closed term t of MLpar� It follows that MB � �x��x��B �

CASE � is x��x�� Suppose Tx��x� is on B� By open branch prop�erty � T��t� is on B for every closed term of MLpar and so by inductivehypothesis MB � ���t��B for every closed term t of MLpar� It follows thatMB � �x��x��B � The proof of the proposition for Fx��x� is similar�

CASE � is �R�� where the proposition holds with � replacing �� SupposeT �R� is on B� We must show MB ��RB

�� There are two sub�cases to consider�either T� is on B or T� is not on B�

If T� is on B� then by de�nition ��� R � f� � �g� Also the structure SBof MB satis�es SB � RB� Now by open branch property � T� is also on Band so by inductive hypothesis MB � �B� It follows from de�nition ��� thatMB � ��R��B�

If T� is not on B� then R � � and MB � ��R��B holds by default�

Now suppose F�R� is on B� Using open branch property �� F� and T�are also on B� By inductive hypothesis MB � �B and because T� is on B� therelation R � f� � �g� We have SB � RB and so by de�nition ��� it follows thatMB � ��R��B� �

As we have said� supposing the modal sentence � does not have a proof� thestandard tableau built from F� will have an open branch B� A model must bede�ned for use in propositions ��� and ���� This model must have a structurethat lies in R� the restriction of �R� so that forcing assertions can be provedabout sentences of the form �R�� Depending on whether or not T� lies on Bwe de�ne the structure S of the model M � �G�R�D�S� as follows� If T� is onB� then by lemma �� part there is a structure SR � R that satis�es SB � SR�In this case the structure of the model M is taken to be SR� If T� is not on Bthen the structure of M is taken to be SB� The de�nition is as follows�

De�nition �� Let B be an open branch of a standard tableau� let the structureSB be as constructed above and suppose R is the restriction of �R� De�ne themodel M � �G�R�D�S� as follows�

�� The set of worlds G for the model M is de�ned to be the singleton f g�G � f g�

��

�� If T� � B� then set M � �f g� f� � �g� dom�SR��SR�� where usinglemma �� SR is chosen to satisfy SB � SR and SR � R�

�� If T� � B� then set M � �f g� ��DB�SB��

Note that in both cases above� S � SB�The next proposition proves something about a modal sentence without pa�

rameters� Proofs of the cases involving quanti�ers examine subcases involvingapproximation sentences �see de�nition �� �� This is di�erent from the approachto the proof of ��� where the inductive hypothesis allowed consideration of sen�tences with parameters� The structure SR which is used in the next propositionis not de�ned for sentences containing parameters� Hence a sentence startingwith a quanti�er has to be broken down into subcases involving some formulaswith a single free variable and others which are approximation sentences� Infact we were led to the de�nition of modal formulas involving approximationsentences so that the proofs of the cases involving quanti�ers would go smoothly�

Proposition �� � Suppose � is a modal sentence without parameters and letR be the restriction of �R� De�ne �B to be the same as � except that everywhereR is replaced by RB � fSBg� Let MB and M be as de�ned in ��� and ����The following is true�

If MB � �B� then M � ��

Proof sketch� By induction on the complexity of �� The following cases aresu�cient to induce proofs for all modal formulas�

BASE CASE � is �rst�order� then �B � �� By �� SB � �� But S � SB soS � �� and M � ��

CASE � is the approximation sentence �R�� where the result holds for ��Suppose that MB � �B� and that T� lies on B� We must show that M � ��By de�nition of forcing for �RB

�B� MB � �B� So using our hypothesis M � ��If T� does not lie on B� then R � � and M ��R� by default�CASE � is x����� where is a �rst�order formula with x as its only free

variable and � is an approximation sentence� We assume that the propositionholds for both �x� and � in place of �� Let MB � �x� � ���B� Thesentence �x� � ���B � x� � �B� so there is an assignment of variables� u�such that MB �u and MB �u �B� Therefore MB � �x� and MB � �B�By our assumption� M � �x� and M � �� So again there is an assignmentof variables� v� with M �v and M �v �� Thus M �v � � and it followsthat M � x� � ��� since M � x� � �� is a sentence�

The cases x���� and x�� �� are treated similarly as well as the casesx� � �� etc�

CASE � is ��� where � is a modal sentence and the proposition holdsfor � in place of �� We must show that MB � �B implies M � �� Toreach a contradiction suppose MB � �B implies M � �� then M � � impliesMB � �B� By hypothesis it follows that M � �� which is a contradiction�

The cases where and � are modal sentences and the proposition holds forthem� and � is � �� � � or � �� are straightforward�

�� CHAPTER ��� COMPLETENESS

CASE � is x� � ��� where and � are modal formulas whose only freevariable is x� and the proposition holds for x and x�� This case is handledsimilarly to the existential case discussed above� The cases for x����� x���� and the universally quanti�ed cases are handled similarly� As is the casex��� �

Proposition �� � �Completeness� Let � be a modal sentence without pa�rameters� If � is valid� then it has a tableau proof�

Proof� By contradiction� Suppose � does not have a tableau proof then thestandard tableau starting with F� as root node will have an open branch B� Letthe models MB and M be as de�ned in ��� and ��� resectively� Also let �Bbe obtained from � by replacing �R with �fSBg�

The signed formula F� is on B and according to proposition ���� MB � �B�So MB � ��B� and by proposition ��� M � ��� It follows that M � �� andthis is a contradiction because � is valid� �

Soundness and Completeness of Proofs from

Premises

In chapter � the idea of a proof of a sentence from premises was introduced anda tableau expansion rule was given to handle the introduction of premises intoa proof� The fact that the sentence � is provable from the set of premises X isdenoted X � �� This de�nes � as a binary relation between sets of sentencesand individual sentences� There is another forcing relation� �� between sets ofsentences and individual sentences de�ned in �� �� We show here that these tworelations are the same� that is �� �� This will be proved in two steps� First wewill show soundness� �� � and then completeness� �� �� To prove soundnesswe need a result similar to proposition ���

Proposition �� Let X be a set of modal sentences and let T be a tableau�Suppose that the model M forces X and that T is satis�able in M� If� takingX as premises� a premise extension rule is applied to a leaf of T then the resultis another tableau satis�able in M�

Proof� Let � � X and suppose T� is appended to the leaf of a branch B ofT satis�able in M� The model M � � because by hypothesis M forces thepremise �� It follows that the set of signed sentences B � fT�g satis�es all therequirements of de�nition �� and so is also satis�able in M� �

Proposition �� �� �Soundness� Let X be a set of modal sentences and � amodal sentence� If X � �� then X � ��

Proof� Let M be a model and suppose M � X � To get a contradictionsuppose M � ��

��

Now let T� be the trivial tableau consisting just of F�� The tableau T� issatis�able by M because M � �� By hypothesis there is a proof of � frompremises X � In other words T� can be expanded to a closed tableau T usingtableau and premise expansion rules� According to propositions �� and ���T is also satis�able� but this is not possible because by ��� a closed tableau isnot satis�able� �

Now we prove completeness� �� ��

Proposition �� �� �Completeness� Let X be a set of modal sentences and� a modal sentence� none of which contain parameters� If X � �� then X � ��

Proof� The steps in the proof are similar to those leading to the proof of com�pleteness without premises� proposition ���� A contradiction will be derivedby supposing there is no tableau proof of � from premises X �

The standard method of tableau construction is modi�ed as follows to incor�porate the sentences in X � The building of the tableau is started in the sameway� This completes stage � Even stages consist of appending the next unusedsentence in X � � say� to the leaf of every open branch of the tableau in the formT�� �There are countably many sentences in X � use the nth sentence of X atstage �n�� The remaining odd stages are handled in the way described abovefor building a standard tableau�

Because we have assumed there is no tableau proof of � from premises X thistableau will have an open branch B� This branch will have the same open branchproperties as listed above and again the model MB of ��� can be de�ned� andproposition ��� will hold� Every sentence � of X appears on B in the form T�and so by ���MB � �B for every � � X � LetM be as de�ned in ���� It followsfrom ��� that M � � for every � � X � Therefore M � X � By hypothesisthen� M � �� But the root node F� is on B giving� by ��� and ���� M � �which is a contradiction� �

The introduction to this chapter gave a summary of the completeness argu�ment in point form� we shall not repeat it here� The essence of the argument isthat assuming a sentence � does not have a proof� a contradiction is obtainedby producing a model� M� which both forces and does not force �� The modelhas a structure� S� which is related to a structure SB built from parts of signedsentences lying on an open branch B of a tableau with F� as root node�

A fundamental feature of the completeness argument which is special to thiswork is lemma �� which shows that if the signed separator T� lies on B thenthere is a structure S � R �where R is the restriction of �R� such that SB � S�This captures the intuitive idea that if a structure forces � then it must beelementarily equivalent to one lying in R� This allows the cases involving �R tobe handled� In summary� if T� is on B then M � �� and if F� is on B thenM � �� Another special feature is that the de�nition of the model M takesinto account whether or not T� lies on B�

In retrospect we see that any restricted entailment is approximated by oneswith �nite restrictions� and with the proof of completeness a chain of equivalent

�� CHAPTER ��� COMPLETENESS

ideas has been set up� satisfaction of �R to forcing of �R to proof involving �Rand back again�

We now leave soundness and completeness� In the next chapter we againuse separators� this time to de�ne operations in a preboolean algebra� Thiswill be used to give an algebraic treatment of belief revision and nonmonotonicentailment�

Chapter ��

The Preboolean Algebra

PSENL

This chapter de�nes a preboolean algebra associated with an FOE language�The elements of the algebra are sets of sentences and the lattice preorder of thealgebra is the entailment relation� This prelattice is shown to be a completepreboolean algebra� So it turns out that each FOE language has a completepreboolean algebra associated with it� For the FOE language L the prebooleanalgebra is denoted PSENL� In chapters � and PSEN is used to give anaccount of belief contraction and nonmonotonic entailment�

We mention that our idea of using entailment to de�ne a preorder is anal�ogous to �although not the same as� using �rst�order implication to de�ne apreorder� Rasiowa and Sikorski ��� page ���� do this to de�ne their algebra ofa formalised theory� The operations and propeties of their algebra are developedsyntactically using methods due to Lindenbaum whereas we work semantically�Another detailed study of consequence operations whose methods we have notpursued is W�ojcicki ����

In what follows de�nitions of operators and relations are given in restrictedform� All results of this chapter� except � �� are used in the following chap�ters� As we have said� PSENL is shown to be a complete preboolean algebrawith preorder �L� A series of preliminary results and de�nitions lead up to this�First it is shown that �L is a preorder antisymmetric up to equivalence then�using separators� lattice operations as well as an inverse are de�ned� Havingconstructed PSENL� some results about the inverse and some standard resultsabout the boolean operators are proved� Then proposition � � shows how torepresent in�nite �and hence �nite� meets and joins in terms of set theoreticunions and intersections� Lemma � � provides a kind of distributive law forconsequence operators� it shows that intersection respects the consequence op�erator� Finally in � � a lattice de�ned on PSENF is shown to be isomorphicto a certain lattice induced by PSENL� While interesting� this result is not usedin what follows�

��

�� CHAPTER ��� THE PREBOOLEAN ALGEBRA PSENL

In the following chapters my arguments have been based on an approach viathe preboolean algebra de�ned on PSENL rather than on the lattice PSENF �where L is the FOE of F � This is because� to use an approach based on PSENF �I would have to de�ne an inverse operation in it in order to obtain a prebooleanalgebra and I do not know how to do this�

Restricted entailment is a preorder� it is a re�exive and transitive relation� Italso induces an equivalence relation� This is the subject of the next two results�

Note that from now on languages are assumed to be non�modal� that is theydo not have the � operator�

Lemma �� � Let L be a language� R � STRUC and let X� Y and Z be membersof PSENL� then

�� X �R X�

�� If X �R Y and Y �R Z� then X �R Z�

Proof� Straightforward from the de�nition of �R� �

Proposition �� � Let L be a language and R � STRUC� The relation �Rsee ��� between members of PSENL is an equivalence relation�

Proof� Straightforward using � � �

A restricted entailment is almost a partial order� instead of being antisym�metric it is antisymmetric up to equivalence�

Proposition �� � Let L be a language and R � STRUC� then for X� Y andZ members of PSENL� �R satis�es

�� X �R X�

�� If X �R Y and Y �R Z then X �R Z�

�� If X �R Z and Z �R X then X �R Z� The entailment �R is said to beantisymmetric up to equivalence�

Proof� Straightforward� �

The relation �R will be a lattice preorder on PSEN if every pair of elementsof PSEN has a greatest lower bound �or meet� and a least upper bound �orjoin� with respect to �R� These bounds can be constructed using separators�see ��� associated with pairs of elements of PSEN� The important point about aseparator is that it is de�ned with respect to pairs of sets of structures which areassumed to be elementarily disjoint� So appropriate sets of structures are chosento be elementarily disjoint in proposition �� where the algebraic operations onPSEN are de�ned� A property called relative fullness is used which guaranteeselementary disjointness of the sets of structures used� Relative fullness is ageneralisation of fullness� which was de�ned in ��� In the de�nition below��relative fullness� reverts to being �fullness� if R is taken to be STRUC�

�

De�nition �� � �Relative Fullness� Let L be a language� Let I � R �STRUC� Then I is full relative to R if and only if given members S and S � ofR� if S � I and S is elementarily equivalent to S � then S � � I�

The following gives a simple criterion for elementary disjointness�

Proposition �� � Suppose A and B are full relative to R� and A and B aredisjoint� then A and B are elementarily disjoint�

Proof� Straightforward� �

Relative fullness is preserved under arbitrary intersections� arbitrary unionsand the taking of complements� if X is a set of sentences then a restricted setof models of X is relatively full�

Proposition �� Let L be a language and let I and J be subsets of R �STRUC and let Hq for q � Q be subsets of R�

�� If both I and J are full relative to R then so are I J � I � J and R� I�

�� If each Hq is full relative to R� then so areTq Hq and

Sq Hq�

�� Let X be a set of formulas of L� then modR�X� is full relative to R�

Proof� Straightforward� Three of the cases will be proved�CASE R � I where I is full relative to R� Let S � R � I � S � � R and S

be elementarily equivalent to S �� If S � � I then from the de�nition of relativefullness so is S� which is not the case�

CASETqHq where each Hq is full relative R� Let S �

TqHq � S

� � R andS be elementarily equivalent to S �� Using the de�nition of relative fullness S � isa member of each Hq�

CASE modR�X�� Let S � modR�X�� S � � R and S be elementarily equiva�lent to S �� It follows that S � � modR�X�� �

The next proposition de�nes some algebraic operations on sets of sentences ofan FOE L� The results of these operations are speci�c sentences in L� The notionof a separator �see ��� is used to make the de�nitions in a uniform way� Recallthat the separator of A and B is �A�B � where A and B are elementarily disjointsubsets of STRUC� The results in the proposition are based on a restrictedentailment �R which is regarded as a preorder� where R � STRUC� For thesake of precision� even though it leads to cluttered notation� all appropriateoperators and relations are subscripted by R�

Proposition �� � Let L be an FOE and R � STRUC� let Q be an in�nite indexset and let X� Y and Wq �q � Q� be members of PSENL�

�� Let B � modR�X� modR�Y � and A � R�B the set�theoretic di�erence�then �A�B is a lower bound of X and Y with respect to �R greater thanany other lower bound of X and Y � Denote �A�B by X ��R Y �

�� CHAPTER ��� THE PREBOOLEAN ALGEBRA PSENL

�� Let B � modR�X� � modR�Y � and A � R � B� then �A�B is an upperbound of X and Y with respect to �R less than any other upper bound ofX and Y � Denote �A�B by X ��R Y �

�� Let B �Tq�QmodR�Wq� and A � R�B� then �A�B is a lower bound of

Wq �q � Q� with respect to �R greater than any other lower bound of Wq

�q � Q�� Denote �A�B byV�RfWq � q � Qg�

�� Let B �Sq�QmodR�Wq� and A � R � B� then �A�B is an upper bound

of Wq �q � Q� with respect to �R less than any other upper bound of Wq

�q � Q�� Denote �A�B byW�RfWq � q � Qg�

�� X �R � and � �R X�

� Let A � modR�X� and B � R � A� then �A�B is denoted �RX and thefollowing hold�

a The sentence �RX satis�es �RX ��R X �R � and is greater withrespect to �R than any set of sentences Z satisfying Z ��R X �R ��

b The sentence �RX satis�es �RX��RX �R � and is less with respect

to �R than any set of sentences Z satisfying Z ��R X �R ��

Proof�

� According to �� B � modR�X� modR�Y � is full and so is A � R ��modR�X� modR�Y ��� Both A and B are therefore elementarily disjointsets of interpretations and so �A�B is the separator of A and B by ���Now �A�B is a lower bound of X and Y because modR��A�B� � B �modR�X� modR�Y �� Let Z be another lower bound of X and Y � thenmodR�Z� � modR�X� modR�Y � � B � modR��A�B��

�� Straightforward� dual to the above�

� Similar to the argument for �� in �

�� Straightforward� dual to the above�

�� By ��� the formula ���STRUC � � is a tautology so modR� � R� By ���the formula �STRUC�� � � is a fallacy so modR� � ��

�� �a� First� modR��RX� modR�X� � �R�R modR�X�� modR�X� � ��and so �RX �� X � �� Now suppose Z satis�es Z �� X � �� Wemust show that Z �R �RX � It follows from our assumption on Zthat modR�Z� modR�X� � � and so modR�Z� � R�RmodR�X� �modR��RX��

�b� We have modR��RX��modR�X� � �R�RmodR�X���modR�X� �R� Now Z satis�es modRZ � modRX � R so modR��RX� � R �RmodR�X� � modRZ� �

�

The next proposition establishes that PSENL is a preboolean algebra� Firstthe de�nition of a boolean algebra is given �see �����

De�nition �� � A boolean algebra B is a distributive lattice� under binary op�erations �� and ��� which also has universal bounds � and � and a unaryoperation � satisfying� for each b � B� b �� �b � � and b �� �b � ��

Because PSENL is a preboolean algebra rather than a boolean algebra� twoantisymmetric elements are equivalent �under �R� rather than being equal� Forthis reason it is necessary to be able to talk about elements of the algebra whichare in some sense the largest or smallest of their kind and so the ideas of maximallower bound and minimal upper bound are de�ned� In the following a maximallower bound� Z say� with respect to �R of Xq �q � Q�� where each Xq � PSENL�is a lower bound of Xq �q � Q� which satis�es W �R Z for any other lowerbound W of Xq �q � Q�� The notion minimal upper bound is de�ned dually�

Proposition �� Let R � STRUC and L be an FOE with entailment �R�

�� The relation �R between members of PSENL is re�exive� transitive andantisymmetric up to equivalence�

�� �PSENL��R��R���R���R� is a prelattice with order �R and operators �R� ��R

and ��R which are unique up to equivalence� That is� the following are sat�

is�ed for members X� Y and Z of PSENL�

a If Z satis�es Z ��R X �R � and Z ��R X �R �� then Z �R �RX�

b If Z is a maximal lower bound of X and Y � then Z �R X ��R Y �

c If Z is a minimal upper bound of X and Y � then Z �R X ��R Y �

�� Let Q be an in�nite index set� There are in�nitary operatorsV�R and

W�R

de�ned on �PSENL��R� which are unique up to equivalence� That is� thefollowing are satis�ed for members Xq �q � Q� of PSENL�

a If Z is a maximal lower bound of Xq �q � Q�� then Z �RV�RfXq �

q � Qg�

b If Z is a minimal upper bound of Xq �q � Q�� then Z �RW�RfXq �

q � Qg�

�� The following distributive laws are satis�ed� where X� Y � Z and Yq �q � Q�

are members of PSENL and Q is in�nite�

a X ��R �Y ��R Z� �R �X ��R Y � ��R �X ��R Z��

b X ��R �Y ��R Z� �R �X ��R Y � ��R �X ��R Z��

c X ��R �W�RfYq � q � Qg� �R

W�RfX �� Yq � q � Qg�

d X ��R �V�RfYq � q � Qg� �R

V�RfX ��R Yq � q � Qg�

�� CHAPTER ��� THE PREBOOLEAN ALGEBRA PSENL

Proof�

� See ��

�� By proposition �� �R� ��R and ��R can be regarded as operators de�ned

on PSENL�

�a� By �� as well as the assumptions on Z� Z �R �RX and �RX �R Zand so Z �R �RX �

�b� By �� as well as the assumptions on Z both Z and X ��R Y aremaximal with respect to �R�

�c� Similar to the above�

� Similar to the above�

�� The set operators and � distribute in the set STRUC� �

Corollary �� �� The algebra �PSENL��R��R���R���R�V�R�W�R� is a complete

preboolean algebra with preorder �R and distinguished elements and operatorswhich are unique up to equivalence� �R� as follows�

Zero� � �see ����

Unit� � �see ����

Inverse� �R�

Binary meet� ��R�

Binary join� ��R�

In�nitary meet�V�R�

In�nitary join�W�R�

The following lemma relates the boolean operator �R and logical negationfor individual sentences� also the law of the excluded middle holds for sets ofsentences�

Lemma �� �� Let L be the FOE of F and R � STRUC�

�� For any � � SENF � �� �R �R��

�� For any X � PSENL� �R �R X �R X�

Proof�

� We must show modR���� � modR��R���

��

modR����

� R mod����

� R �mod�� �because S � �� i� S � ��

� R �modR�

� modR��R���

�� We have

modR��R �R X�

� R �modR��RX�

� R � �R�modRX�

� modRX � �

Lemma �� �� �De Morgan�s laws� Let L be an FOE� R � STRUC and X�Y and Zq �q � Q� be members of PSENL� where Q is an index set�

�� �R�X ��R Y � �R �RX ��R �RY �

�� �R�X ��R Y � �R �RX ��R �RY �

�� �RV�RfZq � q � Qg �R

W�Rf�RZq � q � Qg�

�� �RW�RfZq � q � Qg �R

V�Rf�RZq � q � Qg�

Proof� Apply De Morgan�s laws to the appropriate sets in STRUC� �

The following are restricted adaptations of well�known results and are usedin chapter � on belief revision�

Lemma �� �� Let L be an FOE� R � STRUC and let X and Y be members ofPSENL� The following are true�

�� X �R Y i� X ��R Y �R Y �

�� X �R Y i� X ��R Y �R X�

�� X �R Y i� X ��R �RY �R ��

�� X �R Y i� �Y �R �X�

Proof�

� Suppose X �R Y � On the one hand X ��R Y �R Y by the minimalityproperty of X ��R Y because Y is an upper bound of both X and Y � Onthe other hand by de�nition Y �R X ��R Y � So it follows that X ��R Y �RY � The reverse implication is immediate because if X ��R Y �R Y thenX �R X ��R Y �R Y �

�� By duality�

�� CHAPTER ��� THE PREBOOLEAN ALGEBRA PSENL

� Suppose X ��R �RY �R �� then �X ��R �RY � ��R Y �R � ��R Y �R Y �But �X ��R �RY ���R Y �R �X ��R Y ���R ��RY ��R Y � �R X ��R Y � So wehave X �R X ��R Y �R �X ��R �RY � ��R Y �R Y �

On the other hand if X �R Y � then X ��R �RY �R Y ��R �RY �R ��

�� X �R Y

i� X ��R �RY �R �� by part

i� �R��RX� ��R �RY �R �

i� �RY ��R �R��RX� �R �

i� �RY �R �RX � by part �

�

The next proposition establishes some relationships between the boolean op�erators of PSEN and set theoretical operations� They are used in the treatmentof belief revision in chapter ��

Proposition �� �� Let L be an FOE� let R � STRUC� let Q be an index setand for each q � Q let Xq � PSENL� The following are true�

��V�RfXq � q � Qg �R

Sq�QXq�

��W�RfXq � q � Qg �R

Tq�QXq�

�� If for each q � Q� CnRXq � Xq� thenW�RfXq � q � Qg �R

Tq�QXq�

Proof�

� We shall show that modR�Sq�QXq� �

Tq�QmodRXq from which �see

��� �� and � �� the result will follow�

On the one hand for each n � Q� Xn �Sq�QXq so modR�

Sq�QXq� �

modRXn from which it follows that modR�Sq�QXq� �

Tq�QmodRXq�

On the other hand if S �Tq�QmodRXq then S � modRXq for each q � Q�

and so S � modR�Sq�QXq�� From this it follows that

Tq�QmodRXq �

modR�Sq�QXq��

�� For each n � Q�Tq�QXq � Xn so Xn �R

Tq�QXq� It follows thatW�

RfXq � q � Qg �RTq�QXq�

� We shall show thatTq�QXq �R

W�RfXq � q � Qg then the result will

follow from what has already been proved in � above�

For each n � Q� Xn �RW�RfXq � q � Qg� Also we have shown in �

thatW�RfXq � q � Qg �R

Tq�QXq� So it follows from ��� part �

that CnRTq�QXq � CnR

W�RfXq � q � Qg � CnRXn� By inclusionT

q�QXq � CnRTq�QXq and by assumption CnRXn � Xn so

Tq�QXq �

CnRW�RfXq � q � Qg �

Tq�QXq� Thus CnR

W�RfXq � q � Qg �

Tq�QXq

and so CnRTq�QXq � CnRCnR

W�RfXq � q � Qg � CnR

W�RfXq � q � Qg�

The proof is completed by using ��� part �� �

��

The next result is similar to part of lemma �� �� except it deals with therestriction of a join�

Lemma �� �� Let L be the FOE of F � let R � STRUC and let X and Y bemembers of PSENF � then CnLR�X ��R Y � SENF � CnFRX CnFRY �

Proof� �X ��R Y � �R �CnLRX ��R CnLRY � �R �CnLRX CnLRY � by ��� part �

and � �� So CnLR�X ��R Y � � CnLR�CnLRX CnLRY � � CnLRX CnLRY by ���

parts � and �� It now follows that CnLR�X ��R Y � SENF � CnLRX CnLRY

SENF � CnFX CnFY by �� �� �

The next example shows that the entailment ��� of � � part � cannotbe replaced by equivalence ���� and that an intersection of closures is not� ingeneral� a closure of an intersection�

Example �� � A literal is a formula which is of the form or �� Let �and � be literals which are sentences of the �rst�order language F � let L bethe FOE of F � in �R take the restriction R to be STRUC and suppose � � �is neither a fallacy nor a tautology� Subscripts will be dropped for all operatorsand relations� so for example �R will be written �� Set X � f��� �� ���g andY � f��� �� � � �g� Now in L

� mod�X Y � � mod�� � �� �� ��

� mod�X Y � �� STRUC�

� Cn�X Y � �� SEN�

� mod�X� � � � mod�Y ��

� CnX � CnY � SEN�

� mod�X �� Y � � mod�X� �mod�Y � � ��

So we see that in L

� X Y � X �� Y �

� CnX CnY �� Cn�X Y ��

All the relations not involving �� also hold in F where mod means modF andCn means CnF � �

The following lemma is crucial to some of the arguments in chapter � onbelief revision� Contrary to the case examined in � �� the lemma gives anexample of when �intersection respects closure��

Lemma �� �� Let L be the FOE of F � let X and Y be members of PSENF andlet R � STRUC� then CnLR�CnFRX CnFRY � � CnLRX CnLRY �

�� CHAPTER ��� THE PREBOOLEAN ALGEBRA PSENL

Proof� First de�ne X dY � f��� � � � X� � � Y g� The proof of the lemmais then broken down as follows�

� CnFRX CnFRY � CnFR�X d Y ��To see this it is su�cient to show that CnFRX CnFRY �R �X d Y �� in view

of ��� part �� Accordingly let S be a structure�First suppose S �R CnFRX CnFRY and let � � X� � � Y be arbitrary� We

must show S �R � � �� But � � � is a member of both CnFRX and CnFRY andso belongs to CnFRX CnFRY � From this it follows that S �R � � ��

Conversely suppose S �R X d Y � We must show S �R CnFRX CnFRY � Nowif S �R � for some � � X � then S �R � for all � � Y �because� �xing ��f� � � � � � Y g � X d Y �� Likewise if S �R � for some � � Y � then S �R �for all � � X � So there are three cases to consider� S �R Y � S �R X � S �R Xand S �R Y �

CASE S �R Y � Because S �R Y � S �R CnFRY � But CnFRX CnFRY � CnFRYand the result follows in this case�

The second case is treated similarly and the third is a special case of eitherof the �rst two�

� CnLR�CnFRX CnFRY � � CnLRCnFR�X d Y � � CnLR�X d Y ��

This follows from the bullet point above and �� ��

� CnLR�X d Y � � CnLRX CnLRY �This is seen as follows� In L� modLRCn

LR�XdY � � modLRXdY � by ��� part �

Now

modLRX d Y

� modLRX � modLRY � by similar reasoning to that in the converseargument of the �rst bullet point

� modLR�X �� Y �� by ��

� modLR�CnLRX CnLRY �� by ��� part � and � � part

So X d Y �R �CnLRX CnLRY ��The result now follows from the last two bullet points� �

Consider a �rst�order language F � The entailment �F can also be regardedas a lattice preorder for PSENF and meet and join can be de�ned� up to equiva�lence� for members of PSENF � This can be done in F without using the machin�ery of separators� �A lattice can also be set up in L without using separatorsbut separators are used to de�ne complements which are needed for a booleanprealgebra�� The way to do this is suggested by proposition � � parts and �To compare this lattice with PSENL it is easier to form equivalence classes under� and then to compare the lattices obtained by using the equivalence relation toinduce lattice operations on the equivalence classes� So when one forms equiv�alence classes under �F one obtains a lattice� This lattice is isomorphic to thelattice obtained from a certain sublattice of PSENL by taking equivalence classesunder �L� This is the topic of the next proposition� It is not used in any ofthe arguments that follow but is given to round o� the discussion of this topic�

��

The proposition is stated for the unrestricted case where the restriction R of �Ris taken to be STRUC� and so the subscript R is dropped from operators andrelations�

Proposition �� �� Let L be the FOE of F �

�� �PSENF ��F ������� is a prelattice where X �� Y is de�ned by X � Y andX �� Y by CnFX CnFY �

�� Let PF denote the set of equivalence classes of PSENF under �F andif X � PSENF let kXk � PF denote the equivalence class of X� then�PF ��F ������� is a lattice under the following de�nitions the operationson the right are the ones de�ned on PSENF�

� kXk �F kY k i� X �F Y �

� kXk �� kY k � kX �� Y k�

� kXk �� kY k � kX �� Y k�

The relation �F in PF is a partial order� that is it is re�exive� antisym�metric and transitive�

�� Let kXk denote the equivalence class of X under �L� Set PL � fkXk �X � PSENL � Z with X �L CnLZ � Z � PSENFg� then �PL��L

������� is a lattice under the following de�nitions lattice operations onthe right are operations of PSENL�

� kXk �L kY k i� X �L Y �

� kXk �� kY k � kX �� Y k�

� kXk �� kY k � kX �� Y k�

The relation �L in PL is a partial order�

�� The lattice �PF ��F ������� is isomorphic to �PL��L������� under thelattice isomorphism CnL � PF � PL with CnL � kXk �� kCnLXk�

The inverse is � SENF � PL � PF with � SENF � kXk �� k�CnLZ� SENFk where for some Z � PSENF � X �L CnLZ�

Proof� Straightforward but tedious� The proof of the last part needs to showthat the function CnL preserves meets and joins� but this follows using � �and � � parts and � Also for the last part it is straightforward to show� thecomposition of functions �� SENF� � �CnL� equals the identity on PF �thisuses �� ��� and �CnL� � �� SENF� equals the identity on PL� �

This chapter has set up the basic boolean machinery for the algebraic treat�ment of belief contraction and nonmonotonic entailment� Again de�nitions ofoperators and relations have been given in restricted form so as to allow exam�ination of a computable fragment of the algebraic formulation in chapter ��

�� CHAPTER ��� THE PREBOOLEAN ALGEBRA PSENL

The fundamental result is proposition ��� which uses separators to providethe operations for the algebra PSENL� The notion of relative fullness is neededto ensure that relevant sets of structures are elementarily disjoint in the proofof ��� In�nitary meet and join operators are de�ned and several results typicalof boolean algebras are proved and used repeatedly in the following chapters�

Two other fundamental results are � � and � �� The �rst characterisesin�nitary meet and join in PSEN in terms of the set�theoretic operators unionand intersection� The second shows that when L is the FOE of F � the closureoperator of L respects intersection in the sense that it carries an intersectionof closures in F up to an intersection of closures in L� These results are usedrepeatedly in what follows� The chapter ends with an isomorphism result in�volving PSENF �

The next chapter deals with belief revision�

Chapter ��

Belief Revision Using PSENL

We will not give a detailed summary of belief revision here� Instead we highlightsome basic aspects of belief revision and describe our algebraic treatment ofit� We have adopted the AGM approach to belief revision as expounded byG�ardenfors ��� ��� G�ardenfors and Rott ��� and Hansson ���� The �eld is byno means static and it sees continual development and re�nement as evidencedby ��� �� � � ��� � �� for example�

We suppose a reasoning agent possesses a set of beliefs which it is capableof modifying as time progesses� Following the lead of AGM� we identify theagent�s set of beliefs with a belief set which is a set of sentences closed underconsequence� According to AGM there are three kinds of operation which canbe applied to the belief set to modify it in a rational way� expansion� revisionand contraction� These were brie�y described in the Introduction� chapter �

It turns out that revision can be recovered from contraction via the Leviidentity �see ��� and chapter � and so we work only with contraction here�Contraction can be described as follows� Given a belief set K and a sentence ��we contract K by � when we remove from K� in a rational way� anything thatentails �� A method of contraction is rational if it satis�es the AGM postulatesfor contraction �see below��

In this work we refer to AGM contraction as expounded in ��� �� ���as classical AGM contraction� all operations in classical AGM contraction areperformed in the context of a �rst�order language and involve �rst�order struc�tures� The postulates for classical AGM contraction are given in de�nition �� ��part � In contrast� �non�classical� AGM contraction is de�ned in �� �� part ��It generalises classical AGM contraction in that it contracts by a set of sentences�rather than by a single sentence�

The approach we use to contraction� in contrast to the classical AGM ap�proach� involves an FOE L and the preboolean algebra PSENL� Our contractionconstruction is based upon an algebraic expression involving a rejector� Thebasic property of a rejector for a belief set K and a sentence � is that it en�tails ��� The de�nition is actually made in terms of a rejection function� Thename� rejection function� comes from Britz �� Some discussion comparing our

�

�� CHAPTER ��� BELIEF REVISION USING PSENL

approach to Britz� appears near the end of this chapter�For ease of reference we call our algebraic de�nitions of contraction algebraic

contraction� We show that algebraic contraction and AGM contraction are in�terde�nable and that classical AGM contraction can be retrieved from algebraiccontraction�

From now on the convention will be adopted that a phrase such as � � � � letoperations and relations be restricted to R � � � � will mean that any operation orrelation of the algebra PSENL will be understood to be restricted toR � STRUC�So for example �� without a subscript� should be read as �R� the subscript isunderstood to be there by convention�

Convention �� � A phrase such as

� � � let operations and relations be restricted to R � � �

will mean that R � STRUC and that the operations and relations of the pre�boolean algebra PSENL see � � are understood to have R as a subscript�even though the subscript is dropped� So �� �� ��� ���

V��W� and � are

understood to mean �R� �R� ��R� ��R�V�R�W�R and �R� Also Cn and mod

mean CnR and modR�

The argument in this chapter is made easier by couching the algebraic de��nitions of rejection and contraction in terms of partial functions� First we recallthe de�nition of a partial function�

De�nition �� � A partial function f � A� B is a function� f � which may notbe de�ned for some possibly all elements of A�

De�nition �� � Let L be the FOE of F � let operations and relations be re�stricted to R see convention �� and let X and Y be members of PSENL�

�� A partial function M � PSENL � PSENL � PSENL� where the value of Mat �K�X� is denoted MK�X � is a rejection function if it satis�es BA�� toBA� below�

BA�� If X � � or K � X� then MK�X � �� provided MK�X is de�ned�

BA�� If MK�X � �� then X � � or K � X� provided MK�X is de�ned�

BA�� MK�X � �X� provided MK�X is de�ned�

BA�� If X � Y � then MK�X � MK�Y � provided that if one of MK�X

or MK�Y is de�ned then so is the other�

BA�� MK�V�fXq �q�Qg �

W�fMK�Xq� q � Qg� where Q is an index

set and for each q � Q� Xq � PSENL� provided MK�V�fXq �q�Qg and

MK�Xqfor q � Q are all de�ned�

BA� If MK�V�fXq �q�Qg � Xn� then MK�Xn

� MK�V�fXq �q�Qg� where

Q is an index set� n � Q and for each q � Q� Xq � PSENL� providedMK�

V�fXq �q�Qg and MK�Xq

for q � Q are all de�ned�

�

�� Let M be a rejection function� K � PSENL and X � PSENL� Thepartial function con�L���M��� � PSENL � PSENL � PSENL� is de��ned by con�L���M��� � �K�X� �� CnL�K �� MK�X�� The value ofcon�L���M��� at �K�X� is denoted con�L�K�M�X� and is called the al�gebraic contraction ofK byX underM � It is assumed that con�L�K�M�X�is de�ned if and only if MK�X is de�ned� If there is no danger of confu�sion� for example when L� K and M have been declared and are kept �xed�then con�L�K�M�X� will be written as con�X��

�� If K� X and MK�X are all members of PSENF � then con�L�K�M�X�

SENF is denoted con�F �K�M�X��

The above de�nition of a rejection function in �� is not vacuous� In factfor any X � PSEN there is a canonical de�nition of MK�X � take it to be �X �

Proposition �� � There is a rejection function M � PSENL � PSENL �PSENL�

Proof� Let K be any set of sentences� If X � � or K � X � then set MK�X � ��otherwise set MK�X � �X � This de�nition is easily seen to satisfy BA� to BA��� To show that BA�� and BA�� are satis�ed� note that by � � �

V�fXq � q �Qg �

W�f�Xq � q � Qg� Hence for BA��� MK�V�fXq �q�Qg �

W�fMK�Xq� q �

Qg� Also for BA�� MK�Xn�MK�

V�fXq �q�Qg� �

While the above proposition shows that there is at least one rejection func�tion� it is not hard to see that each di�erent classical AGM contraction functiongives rise to a di�erent rejection function �see �� ���

Contraction can be characterised set theoretically using rejection functions�

Proposition �� � Let L be the FOE of F and let operations and relations berestricted to R�

�� Let K � PSENL� then con�L�K�M�X� � CnLK CnLMK�X �

�� Let K� X and MK�X be members of PSENF � then con�F �K�M�X� �

CnFK CnFMK�X �

Proof�

� con�L�K�M�X� � CnL�K �� MK�X� � CnLK CnLMK�X using ���part �� � � part and ��� part ��

�� K and MK�� consist of sentences of F � Also

con�F �K�M���

� con�L�K�M��� SENF

� �CnLK CnLMK�X� SENF � by part �

Applying � � completes the proof� �

�� CHAPTER ��� BELIEF REVISION USING PSENL

The process of adding a sentence to a set of sentences is called expansion�see ����� Instead of adding a single sentence� the approach taken here is toadd a set of sentences� Expansion is de�ned algebraically below� but the ideathat an expansion adds sentences is borne out explicitly by the set theoreticalcharacterisation in proposition ����

De�nition �� Let L be the FOE of F and let operations and relations berestricted to R�

�� Let K and X be members of PSENL� The expansion of K by X� denotedexpn�L�K�X�� is CnL�K �� X��

�� If K and X are members of PSENF � then expn�L�K�X� SENF is denotedexpn�F �K�X��

Expansion can be characterised set theoretically�

Proposition �� � Let L be the FOE of F and let operations and relations berestricted to R�

�� Let K � PSENL� then expn�L�K�X� � CnL�K �X��

�� Let K and X be members of PSENF � then expn�F �K�X� � CnF �K �X��

Proof�

� We have that expn�L�K�X� � CnL�K �� X�� � CnL�K �X� using � �and ��� part ��

�� We have that expn�F �K�X� � expn�L�K�X� SENF � �CnL�K �X�� SENF by part � The proof is completed by applying �� �� �

As we have said� a belief set is a set of sentences which is closed underconsequence�

De�nition �� � Let L be a language and let operations and relations be re�stricted to R� A set of sentences K � PSENL is a belief set if K � CnLK�

In the introduction we said a belief base� H � is a set of sentences which ismeant to record the explicit beliefs of an agent� The set H is a base for thebelief set K if and only if K is the closure of H under consequence�

Remark �� If K is a belief set� that is CnK � K� and if CnH � K� thenin PSEN� H � K� So in PSEN a belief base and its corresponding belief set areequivalent�

G�ardenfors ��� de�nes contraction in terms of an unrestricted belief set anda sentence� the sentence �and anything entailing it� is removed from the beliefset� �Recall that an unrestricted belief set X is one which satis�es CnX � X � therestrictionR � STRUC�� He gives eight postulates that a belief set should satisfy

��

to be a contraction� The postulates are given below for contraction of a belief setby a set of sentences X satisfying a restricted closure� CnRX � X � When X isa singleton set� that is it consists of a single sentence� and there is no restriction�R � STRUC� then the postulates revert to those given by G�ardenfors� Thenotation used in �� � parts and � is based on G�ardenfors ����

De�nition �� ��

�� Let K be a belief set in a language L and let operations and relationsbe restricted to R� In the style of notation used in G�ardenfors ���� K�

X

denotes the expansion of K by X�

�� Let K be a belief set in a language L and let operations and relations berestricted to R� A partial function K� � PSENL � PSENL� where thevalue of K� at X is denoted K�

X � is called an AGM contraction functionif it satis�es the following�

K�� K�X is a belief set� provided K�

X is de�ned�

K�� K�X � K� provided K�

X is de�ned�

K�� If X � K� then K�X � K� provided K�

X is de�ned�

K�� If X � �� then X � K�X � provided K

�X is de�ned�

K�� If X � K� then K � �K�X��X � provided K

�X is de�ned�

K� If X � Y � then K�X � K�

Y � provided that if one of K�X or K�

Y isde�ned then so is the other�

K��Tq�QK

�Xq

� K�V�fXq �q�Qg

� where Q is an index set and for each

q � Q� Xq � PSENL� provided K�Xq

for q � Q and K�V�fXq �q�Qg

are

all de�ned�

K�� If Xn � K�V�fXq �q�Qg

� then K�V�fXq �q�Qg

� K�Xn

� where Q is an

index set� n � Q and for each q � Q� Xq � PSENL� provided K�Xq

for q � Q and K�V�fXq �q�Qg

are all de�ned�

�� An AGM contraction function K� is said to be classical if the languageis �rst�order� there is no restriction on operations and relations R �STRUC� the function K� is a total function on its domain� the sets Xand Y are singletons� f�g and f�g say� and K�� and K�� are replaced byK�� � and K�� ��

K��� K�� K

�� � K�

���

K��� If � �� K���� then K�

�� � K�� �

Note that notation is simpli�ed by writing K�� � for example� instead of

K�f�g� The postulates just given for a classical AGM contraction are the

same as those in ����

�� CHAPTER ��� BELIEF REVISION USING PSENL

Now we show that any rejection function gives rise to an AGM contractionfunction� a contraction in an FOE language L de�ned in terms of a rejectionfunction as in de�nition �� induces an AGM contraction function in L� Also�the original rejection function can be recovered� up to equivalence� from theinduced AGM contraction function�

Proposition �� �� Let L be an FOE� let K be a belief set in L� let operationsand relations be restricted to R� let M be a rejection function and let X and Ybe members of PSENL� The following are true�

�� The function con � con�L�K�M��� is an AGM contraction function inL� that is it satis�es the following�

K�� con�X� is a belief set�

K�� con�X� � K�

K�� If X � K� then con�X� � K�

K�� If X � �� then X � con�X��

K�� If X � K� then K � expn�L� con�X�� X��

K� If X � Y � then con�X� � con�Y ��

K��Tq�Q con�Xq� � con�

V�fXq � q � Qg�� where Q is an index set

and for each q � Q� Xq � PSENL�

K�� If Xn � con�V�fXq � q � Qg�� then con�

V�fXq � q � Qg� �con�Xn�� where Q is an index set� n � Q and for each q � Q� Xq �

PSENL�

�� The rejection function MK�X �L �con�L�K�M�X� �� �K��

Proof� K�� Immediate from the de�nition of con in ���K�� K � K �� MK�X so Cn�K �� MK�X� � CnK by ��� part ��K�� Using ��� parts � and � we have

X � K i� K � X �because K is a belief set� so

X � K implies

K � X which implies

MK�X � � �by BA� � which implies

K �� MK�X � K which implies

K � con�X� �by ��� part ���

K�� Suppose X � �� We consider separately two mutually exclusive cases�either K � X or K � X �

CASE K � X � In this case MK�X � � by BA� � So from the de�nition ofcon� con�X� � K � X � It follows from ��� part � that X � con�X��

��

CASE K � X � In this case MK�X � � by BA��� since we have supposedX � �� Now suppose� contrary to what we want to prove� that

X � con�X�� ���

Then X � Cn�K ��MK�X� and so using ��� part � K ��MK�X � X which givesby � part

�K �� MK�X� �� �X � �� ����

But �K �� MK�X� �� �X simpli�es as follows�

�K �� MK�X� �� �X

� �K �� �X� �� �MK�X �� �X� �by distributivity�

� ��� �MK�X �� �X� �by � part � because K � X�

� ��� MK�X �by � part �� because of BA��

�MK�X �

We have just shown that MK�X � �K ��MK�X� �� �X � � by ����� whichis a contradiction because we observed at the start of this case that MK�X � ��Therefore we cannot suppose ���� X � con�X�� It follows that X � con�X��

K�� Suppose that X � K� We shall prove that expn�L� con�X�� X� � K�

expn�L� con�X�� X�

� con�X� �� X

� �K �� MK�X� �� X

� �K �� X� �� �MK�X �� X�

� K �� �MK�X �� X�� since X � K giving K � X so K �� X � Kby � part �

� K �� �� since MK�X � �X by BA� giving MK�X �� X � �using � part

� K�

The result follows from ��� part ��K� If X � Y � then by BA�� MK�X �MK�Y and the result follows�K�� BA�� is MK�

V�fXq �q�Qg �

W�fMK�Xq� q � Qg� From this it follows

that K �� MK�V�fXq �q�Qg �

W�fK �� MK�Xq� q � Qg� The result follows on

using � � part � and ��� part ��K�� First let W and Z be members of PSENL�

W � con�Z�

i� K �� MK�Z �W �using ��� part ��

i� K �W and MK�Z �W

�� CHAPTER ��� BELIEF REVISION USING PSENL

and so

W � con�Z� i� K �W or MK�Z �W� ���

We now suppose Xn � con�V�fXq � q � Qg� and conclude from ��� that

K � Xn or MK�V�fXq �q�Qg � Xn� ����

The proof now proceeds by considering the two cases in �����CASE K � Xn� By BA� MK�Xn

� �� so K �� MK�Xn� K� But K �

K �� MK�V�fXq �q�Qg so K �� MK�Xn

� K ��MK�V�fXq �q�Qg�

CASE MK�V�fXq �q�Qg � Xn� Then from BA�� MK�Xn

� MK�V�fXq �q�Qg

and so K �� MK�Xn� K �� MK�

V�fXq �q�Qg� The result for K�� now follows

from ��� part ��It remains to show that MK�X �L �con�L�K�M�X� �� �K�� First

con�X� �� �K

� �K �� MK�X� �� �K

� �K �� �K� �� �MK�X �� �K�

�MK�X �� �K�

Next there are two cases to consider� K � X and K � X �CASE K � X � By BA� MK�X � � so MK�X �� �K � � and the result

follows in this case�CASE K � X � By BA� MK�X � �X � �K by � part � since K � X

for this case� So by � part � MK�X � MK�X �� �K � MK�X � givingMK�X �� �K �MK�X and the result follows� �

We now show the converse of �� � Any AGM contraction function in anFOE L induces a rejection function which can be used as in �� to de�ne acontraction function which equals the original AGM contraction function�

Proposition �� �� Let L be an FOE and let operations and relations be re�stricted to R� Any AGM contraction function K� in L gives rise to a rejectionfunction M in L de�ned by MK�X � K�

X �� �K satisfying con�L�K�M�X� �K�X �

Proof� First we shall show that M satis�es BA� to BA�� and then thatcon�L�K�M�X� � K�

X � By hypothesis K� satis�es K� to K�� of de�ni�tion �� ��

BA�� CASE X � �� K � � � X so K � X and by K�� K�X �� X � K�

So K�X � K�

X ��� � K�

X ��X � K� But also by K�� K � K�

X and so it followsthat K�

X � K and from the de�nition of MK�X � MK�X � ��CASE K � X � Using ��� part � the result follows from K��BA�� We shall prove the contrapositive� that is assuming X � � and that

K � X we shall show MK�X � ��First since X � � it follows from K��� using ��� part �� that

K�X � X� ���

��

Now to reach a contradiction suppose MK�X � �� then by de�nition of MK�X �K�X �

� �K � MK�X � � and so K�X � K by � � But by assumption K � X

so K�X � X which contradicts ���� It follows that MK�X � ��BA�� There are two cases to consider� K � X or K � X �CASE K � X � By BA� MK�X � � � �X �CASE K � X � By lemma � part MK�X � �X i� MK�X�����X� � ��

which we now show is true�

MK�X �� ���X�

� K�X �

� �K �� X �by de�nition of MK�X�

� �K�X �

� X� �� �K

� K �� �K �by the de�nition of expansion and K���

� ��

BA�� Suppose X � Y � then by K�� K�X � K�

Y � So MK�X � K�X �

� �K �K�Y �

� �K �MK�Y �BA�� First because an AGM contraction is closed under consequence it

follows from K�� and � � part that

K�V�fXq �q�Qg

� ��fK�Xq

� q � Qg� ���

Now

MK�V�fXq �q�Qg

� K�V�fXq �q�Qg

�� �K� by de�nition of MK�V�fXq �q�Qg

� �W�fK�

Xq� q � Qg� �� �K� using ���

�W�fK�

Xq�� �K � q � Qg� using �� part �

�W�fMK�Xq

� q � Qg� by de�nition of M �

BA� By the de�nition of M � MK�V�fXq �q�Qg � K�V

�fXq �q�Qg�� �K �

K�V�fXq �q�Qg

so if K�V�fXq �q�Qg

� Xn then MK�V�fXq �q�Qg � Xn� Now sup�

pose MK�V�fXq �q�Qg � Xn then K�V

�fXq �q�Qg� Xn� It then follows from ���

part � and K�� that K�Xn

� K�V�fXq �q�Qg

� So applying �� � K to both sides

of the entailment K�Xn

� K�V�fXq �q�Qg

� we have that MK�Xn� K�

Xn�� �K �

K�V�fXq �q�Qg

�� �K �MK�V�fXq �q�Qg� This completes the proof of BA���

Finally to show con�L�K�M�X� � K�X it is su�cient to show that con�L�K�M�X� �

K�X because the expressions on either side of the � sign are closed under CnL�

Now

con�L�K�M�X�

� K �� MK�X � using the de�nition of con

� K �� �K�X �

� �K�� using the de�nition of M

�� CHAPTER ��� BELIEF REVISION USING PSENL

� �K �� K�X� �� �K �� �K�

� �K �� K�X� �� �

� K �� K�X

� K�X � since by K��� K � K�

X �

�

If a rejection function takes values in PSENF � where F is a �rst�order lan�guage� then it induces a classical AGM contraction function in F when con�tracting by a singleton set� We recall that according to de�nition � if L is theFOE of F � then any formula of F is a formula of L�

Proposition �� �� Let L be the FOE of F and let operations and relations berestricted to R� Suppose M is a rejection function with the property that if � isa sentence in F and K is a belief set in F � then MK�� � PSENF � It then follows

that for K a belief set in F � con�F �K�M��� � SENF � PSENF is a classicalAGM contraction function in F and con�F �K�M��� � K CnFMK���

Proof outline� First we show that for � � SENF � con�L�CnLK�M��� SENF � con�F �K�M��� � K CnFMK��� We have by ��� that

con�L�CnLK�M��� � CnLCnLK CnLMK�� � CnLK CnLMK��� On in�

tersecting throughout with SENF and applying �� � and ��� we have what werequire�

Secondly we show that con�L�CnLK�M��� SENF � SENF � PSENF is aclassical AGM contraction function in F � Now� by �� con�L�CnLK�M���satis�es K� to K�� in L� Next it can be seen that con�L�CnLK�M��� SENF �SENF � PSENF satis�es K� to K�� in F � The results for K� and K�� areeasy and for K��� K�� K��� K�� and K�� the result follows on intersecting bySENF both on the left and the right of appropriate � and � signs�

The proof for K�� is as follows�

Suppose � � K� then � � CnLK and

CnLK

� expn�L� con���� �� �by the proof of K�� in �� �

� CnL��K �� MK��� �� ��

� CnL��CnLK CnLMK��� � �� �by � ��

� CnL��K CnFMK������ �using �� � and � �� since Cn�W�Z� �Cn�CnW � Z��

So K � expn�F � con�F �K�M���� ��� on intersecting with SENF and using �� �and what was shown in the �rst paragraph of the proof� �

Conversely to �� � any classical AGM contraction function� K�� inducesa rejection function from which K� can be recovered�

�

Proposition �� �� Let L be the FOE of F and let operations and relationsbe restricted to R� Suppose K � PSENF and K� � SENF � PSENF is aclassical AGM contraction function in F � then M � fKg � SENF � PSENL

de�ned by MK�� � K�� �

� �K is a rejection function in L and for � � SENF �con�F �K�M��� � K�

� �

Proof outline � First we recall that non�classical AGM contraction functionsare allowed to be partial �see �� � part �� and we show�

� CnLK� � SENF � PSENL de�ned by CnLK���� � CnLK�� is a contraction

of CnLK in L�This is seen as follows�

K�� Obvious�

K�� By monotonicity since K�� � K�

K�� If � �� CnLK then because K � CnLK� � �� K and so K�� � K in F � This

implies CnLK�� � CnLK�

K�� If � is not a tautology in L then it is not one in F � so � �� K�� and

so � �� CnLK�� �because� � � CnLK�

� implies � � CnLK�� SENF �

CnFK�� � K�

� ��

K�� If � � CnLK then � � K �because� � � F and CnLK SENF � K��So K � CnF �K�

� � f�g�� which implies CnLK � CnL�K�� � f�g� �

CnL�CnLK�� � f�g� � �CnLK�

� ��� �

K� Easy�

K�� CnLK�� CnLK�

� � CnL�K�� K

�� � � CnL�K�

����

K�� K��� � CnLK�

�� so � �� CnLK��� implies � �� K�

�� which implies

K��� � K�

� which implies CnLK��� � CnLK�

� �

� Next it follows from the �rst bullet point and �� � that M � fCnLKg �SENF � PSENL de�ned by MCnLK�� � CnLK�

� �� ��CnLK� is a rejection

fuction in L and also that con�L�CnLK�M��� � CnLK�� �

� CnLK�� �

� ��CnLK� � K�� �

� �K�

Because� mod���CnLK�� � STRUCL� mod CnLK � STRUCL� mod K �mod ��K�� and so �CnLK � �K�

� Finally using the last two bullet points� MCnLK�� � MK�� and con�L�K�M���

SENF � con�L�CnLK�M��� SENF � CnLK�� SENF � K�

� � �

There are as many rejection functions as there are di�erent contraction func�tions�

�� CHAPTER ��� BELIEF REVISION USING PSENL

Corollary �� �� Each unique contraction function induces a unique rejectionfunction up to equivalence�

Proof sketch � Two contraction functions are equal if and only if they areequivalent because contraction is closed under consequence� So equal contrac�tion functions induce equvalent rejection functions according to the formulade�ning rejection functions in the statement of proposition �� �� �

In � Britz de�nes an algebra based on a language equipped with proposi�tional connectives and a deducibility relation� �� The elements of the algebraare sets of elements of the language� Britz also de�nes two binary algebraicoperators �� and ��� Then Britz de�nes a notion of rejection function andshows that contraction can be de�ned in terms of a certain algebraic expressioninvolving the operator �� and rejection� It is then shown that contraction andrejection are interde�nable� �I have also shown this in �� to �� ���

There are two di�erences between my approach and Britz�� First Britz doesnot use boolean methods� Secondly our rejection functions are di�erent� theyare generally incompatible with each other �see �� ���

We suppose that Britz� de�nitions are made in a �rst�order language� Britz �de�nes a rejection function by giving six postulates it must satisfy� Four of themcorrespond to our BA� to BA�� �taking the set X in my postulates to be single�ton formulas and taking the �nitary versions of BA�� and BA���� The remainingtwo are given below� We have used our notation� K is a belief set� � is a sen�tence and M stands for the rejection function and K� � and MK�� are all takento be �rst�order�

Britz�� MK�� is inconsistent i� � � Cn����

Britz�� If � �� K then K �MK���

Britz� can be recast as follows�

Britz��� MK�� � � i� � is a tautology�

The following crystallises a di�erence between BA and Britz�

Proposition �� � If K � �� then the rejection function postulates BA�� andBA�� are incompatible with Britz� ��

Proof� First suppose BA�� holds and K � �� We show that Britz� � cannothold� Let MK�� � �� then by BA�� � � � or K � �� Since K � �� � � ��

On the other hand suppose Britz� � holds and K � �� We show that BA� cannot hold� Since K � �� � � � and therefore by Britz� � MK�� � �� SoBA� does not hold� �

Here we have used the algebra PSEN to give a treatment of the AGM ap�proach to belief contraction� The presentation is in terms of restricted oper�ations and relations so that computability of contraction �and nonmonotonicentailment� can be examined later� Another advantage of our approach is that

�

we are able to de�ne contraction by a set of sentences rather than by just asingle sentence� as is the case in classical AGM contraction� This is exploitedin our treatment of nonmomotonic entailment where we de�ne nonmonotonicentailment �based on contraction� between sets of sentences rather than justbetween single sentences�

The algebraic treatment of contraction is presented in terms of rejectionfunctions� The de�nition of rejection function is not vacuous� there is a �canon�ical� rejection function corresponding to any set of sentences� In fact there is adi�erent rejection function corresponding to each di�erent contraction function�

Algebraic contraction and AGM contraction are interde�nable in L� whereF is a �rst�order language and L is the FOE extension of F � The same is truein the �rst�order case� when all sentences are taken to be �rst�order� algebraiccontraction restricted to F and classical AGM contraction are interde�nable�Finally the algebraic approach due to Britz � is brie�y examined and it ispointed out how our approach di�ers from it�

The next chapter gives another application of the algebra PSEN� It is usedto characterise nonmonotonic entailment based upon belief revision�

�� CHAPTER ��� BELIEF REVISION USING PSENL

Chapter ��

Nomonotonic Entailment

Makinson in �� page �� describes frameworks for nonmonotonic reasoning asmethods for

� � � jumping to conclusions�� that is� for tentatively inferring fromgiven information rather more than is deductively implied� They arenonmonotonic in the sense that increasing the amount of informa�tion available as premises may sometimes lead to loss of some of theconclusions that can be drawn� This is in contrast with the situationfor purely deductive reasoning where the addition of further infor�mation can never lead to loss of a correctly established conclusion�

Our objective in this chapter is not to provide a detailed summary of thepurpose and methods of nonmonotonic reasoning� Rather� we �rst describe acertain well�known connection between belief revision and nonmonotonic entail�ment� which we call classical nonmonotonic entailment� Then we show that ouralgebraic approach can be used to describe this classical nonmonotonic entail�ment�

The �eld of classical nonmonotonic entailment is large� Some examples ofwork in the area having a distinctly logical �avour are �� �� �� � � �� �� ����A broader work covering methods using partial and modal logic is ���

The connection mentioned above is made by identifying a classical non�monotonic entailment depending on a �xed background belief set� K� with anexpression involving a certain contraction of K� This allows nonmonotonic en�tailment to be given an algebraic expression involving ordinary entailment� Infact� a feature of the arguments in this chapter is that they are all algebraic�and the algebraic expression of nonmonotonic entailment is an application ofthe algebra PSEN�

Classical nonmonotonic entailment via belief revision is a relation between�rst�order sentences� We show how the algebraic approach can generalise this tobe a relation between sets of sentences in an FOE language� L say� That is� non�monotonic entailment via belief revision becomes a relation between membersof PSENL�

��

�� CHAPTER ��� NOMONOTONIC ENTAILMENT

There is a nonmonotonic consequence operator� C� corresponding to non�monotonic entailment� We characterise it and give necessary and su�cientconditions for the equality CX � CnX to hold� where X is a set of sentences�We also show that C satis�es some properties mentioned in ��� inclusion�supraclassicality and distribution� Idempotence and cumulativity are not auto�matically satis�ed by C but are shown to be equivalent to certain conditions onrejection functions�

If � and � are sentences and K is a �xed background belief set� then thenonmonotonic entailment of � by � depending on K is denoted � j�K �� Arevision of K by � is denoted K�

�� According to the approach used here �see ����

the nonmonotonic entailment � j�K � is identi�ed with the expression � �K��� However the revision K�

� can be translated into an expression involvinga contraction followed by an expansion by using the Levi identity �see �����K�� � �K�

���� � So � j�K � is identi�ed with � � �K����� �

Now� because �K����� is a belief set� � � �K�

���� if and only if �K����� � ��

This can be converted into an expression in the preboolean algebra PSENL togive the following result which translates a nonmonotonic entailment into anordinary entailment�

Proposition �� � Suppose the classical AGM contractionK� � SENF � PSENF

is given and� according to �� �� let MK�� � SENF � PSENL be the induced re�jection function� then the following statements are equivalent�

�� � j�K ��

�� K�� �

� � � ��

�� �K �� �� �� MK��� � ��

Proof� � � i� ���� SENF � SENL� PSENF � PSENL and according to �� �and ���� �K�

���� � K�� �

� ����� i� ��� According to � �� � ��� Therefore� using BA��� MK�� �

MK��� and so K�� �

� � � �K �� MK���� �� � � �K �� �� �� �MK��� �� ���But by BA� MK��� � � so using � part � MK��� �

� � � MK��� and theresult follows� �

Corollary �� �

�� If K � ��� then � j�K � i� K �� � � ��

�� If K � ��� then � j�K � i� MK��� � ��

Proof� According to � �� � �� for � � SENF �

� By BA� MK��� � ��

�� Since K � ��� we have by � part K �� � � �� �

��

We now extend the de�nition of j�K to hold between sets of sentences inan FOE� L� The role played above by a sentence� �� is now taken by a set ofsentences� X � and also because �� � �� in PSENF �by � �� the analogy isextended and �X is substituted for ���

De�nition �� � Let L be an FOE� let X and Y be elements of PSENL and letK� � PSENL � PSENL be any AGM contraction� then X j�K Y i� �K�

�X ��

X� � Y � If there is no danger of confusion the superscript K in j�K will bedropped thus� X j� Y �

The following results and their proofs are analogous to � and ��� �Theequivalence of the �rst two statements in �� is really a restatement of de�ni�tion ���

Proposition �� � Let L be an FOE and let operations and relations be re�stricted to R� Suppose the AGM contraction K� � PSENL � PSENL is givenand� according to �� �� let MK�� � PSENL � PSENL be the induced rejectionfunction� then the following statements are equivalent�

�� X j�K Y �

�� K��X �

� X � Y �

�� �K �� X� �� MK��X � Y �

Corollary �� �

�� If K � �X� then X j�K Y i� K �� X � Y �

�� If K � �X� then X j�K Y i� MK��X � Y �

In view of the equivalence of parts and of proposition ��� restrictednonmonotonic entailment is de�ned as follows�

De�nition �� Let L be an FOE� let R � STRUC� let X and Y be elements ofPSENL� let K � PSENL satisfy CnRK � K and let MK�� � PSENL � PSENL bea rejection function which is total on its domain� The nonmonotonic entailmentrestricted to R is de�ned by X j�KR Y i� �K ��R X� ��RMK��RX �R Y �

There is no loss of generality in making the de�nition in terms of a rejec�tion function because if an AGM contraction function K� is given then settingMK�� � K� �� �K de�nes a rejection function for which K �� MK�� � K�

�see �� ���For the rest of this chapter convention �� is adopted so the restriction for

operations and relations will be understood implicitly�To justify calling X j�K Y a nonmonotonic entailment it should be shown to

satisfy some properties of nonmonotonic entailments� Propositions � � � and � � examine this� The de�nitions come from Makinson ��� where theyare made in terms of the consequence operator� denoted C� corresponding to thenonmonotonic entailment j�K � The operator C is de�ned next�

�� CHAPTER ��� NOMONOTONIC ENTAILMENT

De�nition �� � Let L be an FOE� let operations and relations be restrictedto R� let � � SENL� X � PSENL and MK�� � PSENL � PSENL a rejection

function which is a total function� then CX � f� � SENL � X j�K �g�

Using the de�nition of j�K the following observations can be made aboutthe structure of the nonmonotonic consequence operator C� It is interesting tonote that the �rst two parts of the proposition below express C in terms of Cn�

Proposition �� � Let L be an FOE� let operations and relations be restrictedto R and let X and Y be members of PSENL� The following are true�

�� CX � Cn�K��X �

� X��

�� CX � Cn��K �� X� �� MK��X��

�� X j�K Y i� Y � CX�

Proof�

� � � CX

i� X j�K �� �by ���

i� K��X �

� X � �� �by ��

i� Cn� � Cn�K��X �

� X�� �by ��� part ��

i� � � Cn�K��X �

� X�� �by ��� part ��

�� K��X �

� X

� �K �� MK��X� �� X

� �K �� X� �� �MK��X �� X�

� �K �� X� �� MK��X � �by � part �� since MK��X � X byBA���

� X j�K Y

i� K��X �

� X � Y � �by ��

i� CnY � Cn�K��X �

� X�� �by ��� part ��

i� Y � CX � �by ��� part and the �rst part above�� �

The following simpli�es the de�nition of C and is used in proofs of the resultsbelow�

Corollary �� Under the same assumptions as above� the following are true�

�� If K � �X� then CX � K �� X�

�� If K � �X� then CX �MK��X �

Proof�

� By BA� MK��X � �� then apply ��� part ��

��

�� By � part K �� X � �� then apply ��� part �� �

We can now ask the question� �When does CX � CnX!�� A partial answeris� �When MK��X � X�� That is� when MK��X is the canonical rejectionfunction of proposition ���� Necessary and su�cient conditions for CX � CnXto hold are given in the next proposition�

Proposition �� �� Let L be an FOE� let operations and relations be restrictedto R� let X be a member of PSENL and let C be induced by j�K � The followingare true�

�� If MK��X � X or X � K� then CX � CnX�

�� If CX � CnX� then

�MK��X � X if K � �X�X � K if K � �X�

Proof�

� If MK��X � X � then the result follows immediately on substituting X forMK��X in proposition �� part �� If X � K� then again the result followseasily from proposition �� part � on using BA��

�� If K � �X � then the result is immediate from corollary ��� If K ��X then using corollary �� K �� X � X � and the result follows fromlemma � � �

The next few propositions have to do with some important properties of thenonmonotonic consequence operator C� They are described in Makinson ���Proposition � shows that C satis�es inclusion� supraclassicality and distri�bution� Proposition � gives a necessary and su�cient condition for C tosatisfy idempotence and � � does the same for cumulativity�

Proposition �� �� Let operations and relations be restricted to R� let the non�monotonic consequence operator C be induced by j�K and let X and Y be mem�bers of PSENL� where L is an FOE� then C satis�es the following�

Inclusion� X � CX�

Supraclassicality� CnX � CX�

Distribution� CX CY � C�CnX CnY ��

Proof�

Inclusion� According to �� part and de�nition �� X � CX i� K��X �

�

X � X � But clearly K��X �

� X � X � so the result follows�

Supraclassicality� By �� part and de�nition �� CnX � CX i� X j�K

CnX i� K��X �

� X � CnX i� K��X �

� X � X � But clearly this is true�

�� CHAPTER ��� NOMONOTONIC ENTAILMENT

Distribution� We shall show C�CnX CnY � � CX CY from which it willfollow using ��� part � that CX CY � C�CnX CnY � because CnC � C

�to see this apply Cn to both sides of �� part �� Now

C�CnX CnY �

� �K �� �CnX CnY �� �� MK��CnX�CnY � �by �� part ��

� �K �� �X �� Y �� �� MK��X��Y � �by � � part and ���part ��

� �K �� �X �� Y �� �� MK��X��Y � �by � ��

� �K �� �X �� Y �� �� �MK��X �� MK��Y �� �by BA���

� �K ��X��� �K �� Y ���MK��X ��MK��Y � �distributing ��

over ���

� CX��CY � �grouping terms one and three together and termstwo and four together and using �� part ��

� CX CY � �by � � part � since CnC � C� �

The next lemma is used in the proof of � �

Lemma �� �� Let operations and relations be restricted to R� let C be thenonmonotonic consequence induced by j�K and let X � PSENL where L is anFOE� If K � �X� then the following holds�

CX � CCX i� MK��X �MK��CX �

Proof� Suppose K � �X � Because C satis�es inclusion� X � CX and soby ��� part �� CX � X � Thus by � part �� �X � �CX and so K � �CX �Therefore� by �� CX �MK��X and CCX �MK��CX �

Next let CX � CCX � In view of what has just been shown it is clear thatMK��X �MK��CX �

Conversely if MK��X � MK��CX � then CX � MK��X � MK��CX � CCX �The reverse entailment follows using BA�� CCX �MK��CX � CX � �

The next proposition examines idempotence of C�

Proposition �� �� Let L be an FOE� let operations and relations be restrictedto R� let C be the nonmonotonic consequence induced by j�K � and let X �PSENL� The following are equivalent�

�� C is idempotent� CX � CCX�

�� If K � �X� then MK��X �MK��CX �

Proof� � � IMPLIES ���� Let C be idempotent and let K � �X � We mustshow that MK��X �MK��CX �

By inclusion CX � X so� as we saw in the proof of � �� �X � �CX � givingK � �X � �CX � From �� and idempotence of C� MK��X � CX � CCX �MK��CX �

�

��� IMPLIES � �� �Note that ��� is only used in the last case of this proof��It will be shown that CX � CCX then because CnCX � CX by �� part �� itwill follow that CX � CCX �

Since C satis�es inclusion it follows that �X � �CX � The proof now pro�ceeds by examining three cases arising out of the relationship between K� �Xand �CX �

CASE K � �CX � Then K � �X � By �� CCX � K �� CX and CX �K �� X � From the last equivalence it follows that CCX � K �� K �� X �K �� X � CX �

CASE K � �CX and K � �X � By ��K��X � CX and CCX �MK��CX �But K � �CX so CX � �K� thus K �� X � CX � �K� It follows thatK �� X � �K �� X� �� K � �K �� K � �� This means that CX � ��Also from this� �CX � � and so by BA� MK��CX � �� We have shownCX � � � CCX �

CASE K � �X � Since we have that K � �X it follows from lemma � �that CX � CCX � �

The next lemma is used in the proof of � ��

Lemma �� �� Let operations and relations be restricted to R� let C be thenonmonotonic consequence induced by j�K and let X and Y be members ofPSENL where L is an FOE� If K � �X � �Y � �CX and MK��X � �Y � thenthe following holds�

CX � CY i� MK��X �MK��Y �

Proof� It may be assumed that K � �X � �Y � �CX �First� if CX � CY � then by �� CX � MK��X and CY � MK��Y so

MK��X �MK��Y �On the other hand� let MK��X � �Y and suppose MK��X � MK��Y � It

must be shown that CX � CY � Now �X � �Y so �X �� �Y � �Xby � part �� Therefore using the fact that MK��X � �Y and apply�ing BA��� MK��Y � MK��X � But we have supposed MK��X � MK��Y soMK��X �MK��Y � Using ��� CX �MK��X �MK��Y � CY � �

The next proposition examines the cumulativity of C�

Proposition �� �� Let L be an FOE� let operations and relations be restrictedto R� and let C be the nonmonotonic consequence induced by j�K � Let X andY be members of PSENL� The following are equivalent�

�� C is cumulative� X � Y � CX implies CX � CY �

�� If X � Y � CX and K � �X� then MK��X �MK��Y �

Proof� � � IMPLIES ���� Suppose X � Y � CX and K � �X � Also let C

be cumulative� It must be shown that MK��X �MK��Y � From our suppositionthat X � Y � CX � it follows that CX � Y � X and that �X � �Y � �CX �and so we have that K � �X � �Y � �CX � From �� we have CX �MK��X

�� CHAPTER ��� NOMONOTONIC ENTAILMENT

and CY � MK��Y � From the cumulativity of C it follows that CX � CY andso MK��X �MK��Y �

��� IMPLIES � �� �Note that ��� is only used in the last subcase of thisproof�� Suppose X � Y � CX � It will be shown that CX � CY � from which itwill follow� by �� part � that CX � CY �

Again� it follows from X � Y � CX that CX � Y � X and that �X �

�Y � �CX � The proof now proceeds by examining four cases arising out of therelationship between K� �X � �Y and �CX �

CASE K � �CX � Then K � �Y and K � �X � By �� CX � K �� Xand CY � K �� Y � Therefore K �� X � CX � Y � X and so operating oneach term with K�� one obtains K ��X � K �� Y � K ��X � This means thatK �� X � K �� Y and CX � CY �

CASE K � �CX and K � �Y � Then K � �X and by the same argumentas in the previous case CX � CY �

CASE K � �Y and K � �X � Then K � �CX � The fact that this casecannot arise can be seen as follows� K � �X so by �� CX � K �� X � ButK � �CX so CX � �K� and so K �� X � CX � �K� It follows from � part that K �� X � � and again from � part that K � �X � Thiscontradicts the assumption that K � �X �

CASE K � �X � Then K � �Y and K � �CX � By �� CX � MK��X �So� using the fact that X � Y � CX and hence CX � Y � X � one hasMK��X � CX � Y � There are now two further subcases to consider�

SUBCASE MK��X � �Y � Then MK��X � Y ���Y � �� By BA�� �X � �or K � �X � But by assumption for this case K � �X � so�X � � and thereforeX � �� From Y � X it follows that Y � �� and so MK��Y � � by BA� � Forthis subcase then� CX �MK��X � � �MK��Y � CY �

SUBCASE MK��X � �Y � By lemma � � CX � CY � �

This chapter exploited the algebraic formulation of nonmonotonic entailmentto express it in terms of ordinary entailment� to simplify its expression on a caseby case basis � �� and ���� to generalise it to be a relation between sets offormulas� and to expess it in restricted form� A nonmonotonic consequenceoperator C corresponding to nonmonotonic entailment is de�ned� characterisedand its characterisation simpli�ed on a case by case basis � ���� Necessary andsu�cient conditions are given for C � Cn to hold� and it is shown that C satis�esinclusion� supraclassicality and distribution� Idempotence and cumulativity ofC� however� are equivalent to certain conditions on rejection functions� Theseconditions are not examined further here�

The next chapter examines the computability of restricted algebraic op�erations and relations and hence of restricted contraction and nonmonotonicentailment�

Chapter ��

Computability

In this chapter some aspects of the computability of contraction and nonmono�tonic entailment are discussed by examining the computability of their con�stituents� the algebraic expressions and entailment relation of the prebooleanalgebra PSEN� We use the approach to decidability� computability and enumer�ability described by Ebbinghaus� Flum and Thomas � chapter X�� Computabil�ity questions will be considered to be on a par with decidability questions� If weare trying to see if something can be computed we can turn it into a decidabilityquestion and try to answer it and vice versa� given a decidability question wecan turn it into computability question and answer that� The notion of decid�ability is de�ned in terms of �nite words over a �nite alphabet� Enumerabilityand computability are also de�ned� and proposition ��� from � page ���shows that the three concepts are closely linked�

In order to begin discussion of computability in PSEN one has to start withsomething that is computable and so assumptions about basic computabilityproperties of structures and assignments of variables are made explicit in ����An important assumption we have made for this chapter is that all structureshave �nite domains� �The study of model theory incorporating this assumption�ts in to the subject�matter of �nite model theory �� �� ���� With these basicassumptions we show that it is decidable whether a given structure entails agiven �rst�order sentence�

The fact that operations and relations are capable of being restricted to a�nite set of structures with �nite domains is exploited to allow computabilityresults to be proved� The expressions X��RY � X��RY and �RX are computable�where R is a �nite set of structures with �nite domains and X and Y are �nitesets of sentences� the restriction of the relation �R to pairs of �nite sets ofsentences is decidable� and �nally contraction is computable and nonmonotonicentailment is decidable� subject to the computability assumptions made aboutstructures�

A decidability problem has two inputs� a countably in�nite list of wordsover a �nite alphabet and a subset� W � of that list� The set W is said to bedecidable if there is a �nite� algorithmic� procedure which is able to produce for

�

�� CHAPTER ��� COMPUTABILITY

each word of the in�nite list a �yes� or �no� answer to the question whetherthat word belongs to W �

De�nition �� � Let A be a �nite alphabet� let A� be the set of all �nite wordsover A� let W � A� and let P be a procedure� Suppose the symbol � � A��

�� P is a decision procedure for W if for every w � A� which is input to P�the procedure stops after outputting � if w � W or else some non�emptyword from A� if w �W �

�� W is decidable if there is a decision procedure for W �

The concept of enumerability is closely linked to decidability �as well ascomputability " see �����

De�nition �� � Let A be a �nite alphabet� let A� be the set of all �nite wordsover A� let W � A� and let P be a procedure�

�� P is an enumeration procedure for W if after starting� it eventually out�puts each member of W possibly with repetitions�

�� W is ennumerable if it has an ennumeration procedure�

Informally� to say that a function is computable means that its domain is atmost countable and the value of the function at each element in its domain canbe evaluated using a �nite algorithm�

De�nition �� � Given two alphabets A and B� a function f � A� � B� iscomputable if there is a procedure P which� given an input w � A�� yields anoutput in B� equalling f�w��

According to � page ��� computability� enumerability and decidabilitycan be linked in the following way�

Proposition �� � Let A and B be alphabets� suppose the symbol � � A � Band f is a function f � A� � B�� then the following are equivalent�

�� The function f is computable�

�� The set fw�f�w� � w � A�g is enumerable�

�� The set fw�f�w� � w � A�g is decidable�

To get started in examining the computability of the satisfaction relationfor �rst�order sentences we assume that some basic �raw materials� are com�putable� structures are assumed to be computable and only structures with�nite domains are considered� In this chapter the following properties relatingto the computability of structures are assumed�

��

Properties �� �

� All structures have �nite domains�

� All structures are computable� This means that all the parts making upthe structure are computable� its mapping of

� constants to domain members�

� function symbols to functions de�ned on its domain�

� relation symbols to relations over its domain�

� Any assignment of variables is computable�

� If S is a structure and f a function symbol of arity n� then the domain offS is �domS�n and the values of fS are computable over its domain�

� If S is a structure and p is a relation symbol of arity n� then pS is arelation on �domS�n and the relation pS is decidable over �domS�n�

With these assumptions the next proposition shows that the value of anyterm under a given interpretation is computable and the satisfaction of any �rst�order formula by a given interpretation is decidable� Therefore it is decidablewhether a structure satis�es a �rst�order sentence�

Proposition �� Let S be a structure with �nite domain dom�S� and let u bean assignment of variables�

�� The value of any term t under the interpretation �S� u� is computable�

�� Let � be a �rst�order formula� It is decidable whether or not �S� u� � ��

Proof outline� By induction on the complexity of terms and formulas� Eachterm and formula is made up of only �nitely many subterms or subformulas� soeach induction only has �nitely many steps�

� BASE CASE t is a constant� c say� By ��� the value cS is assumed to becomputable�

BASE CASE t is an individual variable� x say� By ��� the value xu isassumed to be computable�

CASE t is f�t�� � � � � tn� where f is a function symbol of arity n and

tS�u� � � � � � t

S�un are all computable �each t is built up of �nitely many

computable subterms� and are members of dom�S�� By ��� the value

fS�tS�u� � � � � � t

S�un � is assumed to be computable� This completes the

argument for terms�

�� BASE CASE � is p�t�� � � � � tn� where p is a relation symbol of arity n and

each tS�u� � � � � � t

S�un is computable� The value of �p�t�� � � � � tn��S�u is

pS�tS�u� � � � � � t

S�un � and by ��� this is decidable�

�� CHAPTER ��� COMPUTABILITY

CASE � is �� where �S� u� � � is decidable� Because �S� u� � � isdecidable so is �S� u� � � in one extra step�

Treatment of the propositional cases is also straightforward� where � is� � � � is one of �� � or � and both �S� u� � � and �S� u� � aredecidable�

CASE � is �x�� where �S� v� � � is decidable for each x�variant v ofu� Because dom�S� is �nite there are at most �nitely many x�variants ofu� so it possible to decide in a �nite number of steps by checking eachx�variant v of u whether �S� v� � � for one of them� and to decide whether�S� u� � � in one extra step�

The case where � is �x�� is similar� �

As a corollary� satisfaction of a speci�c sentence by a speci�c structure isdecidable�

Corollary �� � Let S be a structure with �nite domain and let � be a �rst�order sentence� It is decidable whether or not S � ��

Proof� Let u be an assignment of variables� Because � is a sentence and hasno free variables� �S� u� � � if and only if �S� v� � � for any other assignmentof variables v� But �S� u� � � is decidable by ���� �

The next de�nition is convenient because it allows the following results tobe stated more succinctly than without it�

De�nition �� �

�� FPSTRUCFIN is the family of all �nite sets of structures with �nite do�mains�

�� FPSENFO is the family of all �nite sets of �rst�order sentences�

�� If R � FPSTRUCFIN� the restriction of the relation �R to FPSENFO �FPSENFO is denoted �RFO�

Subject to the standard assumptions of this chapter� that the properties ���hold� the restricted set of models of a �nite set of sentences is computable�

Proposition �� Let R � FPSTRUCFIN and let X � FPSENFO� then modRXis �nite and computable�

Proof� First because R is �nite� modRX is �nite� Next� for each S � R and� � X � S � � is decidable by ���� so modRX can be computed in a �nitenumber of steps� �

The next result is a crucial step in the argument showing that the booleanoperators are computable�

Proposition �� �� Let A and B be elementarily disjoint members of FPSTRUCFIN�then the separator of A and B� �A�B� is computable�

��

Proof� The set A � B is �nite� For each �S�S �� � A � B the separator �S�S�

can be found in a �nite number of steps because the set of sentences is assumedto have been enumerated once and for all� Therefore the �rst�order sentence�A�B �see note ��� can be constructed in a �nite number of steps� �

The next two results show that the boolean operators are computable� sub�ject to the usual assumptions for this chapter�

Proposition �� �� Let R � FPSTRUCFIN and let modRX and modRY be com�putable� then X ��R Y � X ��R Y and �RX are computable�

Proof� Using ��� each of A� � modRX modRY � A� � modRX � modRYand A� � R�modRX is computable� as is each of R�A�� R�A� and R�A��By �� each Ai �i � � �� � is full and so each R � Ai �i � � �� � is full� Itfollows that each pair �R � Ai� Ai� �i � � �� � is elementarily disjoint and soby �� � each �R�Ai�Ai �i � � �� � is computable� The result follows usingproposition ��� �

Corollary �� �� For R � FPSTRUCFIN and X�Y � FPSENFO� X��RY � X��RY

and �RX are computable�

The next proposition outlines the proof that restricted entailment between�nite sets of sentences is decidable� This result together with the earlier onesallow one to conclude in �� � that contractraction is computable and nonmono�tonic entailment is decidable�

Proposition �� �� For R � FPSTRUCFIN� the relation �RFO is decidable�

Proof outline � First a computable encoding is sketched of a �nite set ofsentences to a word over the alphabet A � f � �� � �� �� �� �� �� �g� Next anoutline is given of how to decide whether or not X �RFO Y where X and Y aremembers of FPSENFO�

Every �rst�order sentence � corresponds to an integer in the enumerationsequence of the �rst�order sentences� the integer corresponding to the �rst oc�curence of �� Call this integer the correspondent of �� Let X � FPSENFO� Theset of correspondents of members of X is a �nite set of integers and is com�putable� Suppose it has n members� Let the set of correspondents of membersof X sorted into increasing sequence be fq�� � � � � qng� Encode X as the integerequal to the product of the following powers of primes� prime

q�� � � � � � prime

qnn �

where primen is the n�th prime number� Encode this integer in turn as a wordinA� whereA � f � �� � �� �� �� �� �� �g� Call this encoding of X � corr�X�� So corrcan be regarded as a function corr � FPSENFO � A� with corr � X �� corr�X��The function corr is computable�

Now expand the alphabet A to include the symbol �RFO and call it B�B � f � �� � �� �� �� �� �� ���RFOg� The set W � fcorr�X��RFOcorr�Y � � B� �X�Y � FPSENFO � X �RFO Y g is decidable� given w � B� it is decidable if itis in the form corr�X��RFOcorr�Y �� If it is then decode corr�X� and corr�Y � asX and Y � This is computable� The word w �W if modR�X� � modR�Y �� andthis is decidable� �

�� CHAPTER ��� COMPUTABILITY

The next proposition shows that when structures and sentences belong toFPSTRUCFIN and FPSENFO respectively� contraction and nonmonotonic entail�ment are computable� We recall that the canonical construction of ��� ensuresthat MK�X is computable for X � FPSENFO or for computable modRX � when�X is taken to be MK�X � Alternatively if MK�X � FPSENFO then it is alsocomputable� So the assumptions concerning M in the next proposition arefeasible�

Proposition �� �� For R � FPSTRUCFIN� W�X� Y � FPSENFO� andmodR�MCnL

RW�X � and modR�MCnL

RW��RX

� computable� we have

�� W ��RMCnLRW�X is computable�

�� �W ��R X� ��R MCnLRW��RX

is computable�

�� �W ��R X� ��R MCnLRW��RX

�RFO Y is decidable�

Proof� By �� and �� � �

Corollary �� �� For R � FPSTRUCFIN� W�X� Y � FPSENFO� andmodR�MCnL

RW�X � and modR�MCnL

RW��RX

� computable� we have

�� con�L�CnLRW�M�X� is computable�

�� X j�CnLRW Y is decidable�

Proof� Using �� and ��� since for any set of sentences Z� CnLRZ �LR Z

by ��� part �� �

This chapter has shown that the algebra PSEN is computable� subject tothe limitations mentioned at the start of the chapter� It follows that� subject tothe same limitations� belief contraction and nonmonotonic entailment are alsocomputable�

This completes an analysis of restricted entailment and its rami�cationsthat was motivated by Johnson�Laird�s approach to deduction and reasoning inhumans�

Chapter ��

Conclusion

This thesis started with the following quote from Johnson�Laird and Byrne �about mental models�

The theory is compatible with the way in which logicians formu�late a semantics for a calculus � � � But� logical accounts dependon assigning an in�nite number of models to each proposition� andan in�nite set is far too big to �t inside anyone�s head � � � peopleconstruct a minimum of models� they try to work with just a singlerepresentative sample from the set of possible models � � �

In other words in Johnson�Laird�s mental model theory� the number of se�mantic models that can be checked to verify logical truth is �nite� Abstractingfrom this idea led to the concept of restricted entailment� which generalisesordinary entailment�

The arguments in the preceeding chapters have established that restrictedentailment has a well�de�ned semantics� it has a corresponding modal operator�approximately true� semantic truth of a restricted entailment is equivalent tosemantic truth of a corresponding modal formula involving approximately true�the logic of approximately true is sound and complete� restricted entailmentinduces a preboolean algebra that can be used to formulate belief contractionand nonmonotonic entailment� and computability can be ensured by imposingsuitable limitations� These are� the properties ��� hold� restrictions are �nite�and sets of sentences appearing in algebraic expressions are �nite�

I conclude that my thesis has been substantiated� the logical properties ofrestricted entailment can be rigorously analysed and that the induced algebracan be used to give an account of belief contraction in the AGM theory of beliefrevision as well as nonmonotonic entailment in the context of a belief set � andits subtheses have also been substantiated�

��

� CHAPTER ��� CONCLUSION

Prospectus of future work

The syntactic counterpart of restricted entailment is a modal language havinga single modal operator �approximately true�� �R where R � STRUC is therestriction of �� This modal language results from adding the modal operator toan ordinary �rst�order logic including constants and function symbols� It wouldbe interesting to study this modal language when it is augmented in di�erentways� One way is to have many restrictions Rq � q � Q� and to have all the �Rqin the language simultaneously� One could then try to see whether there is asyntactic way of stating that ordinary entailment is approximated by a sequenceof approximately true operators�

Another augmentation would be to include � and �� To keep constant andfunction symbols in the language one would then include predicate abstracts ��in the language as well� This would handle the variation of constants andfunctions as the worlds varied�

Another direction for exploration is motivated by the following observation�As we know X � Y i� modX � modY � So perhaps it is possible to examinethe approximation of entailment by studying approximation in a space de�nedon STRUC� In fact� by studying topological neighbourhood structures de�neddirectly on STRUC� After all� our intuition about �neighbourhood� is that allpoints in a neighbourhood are closely related in some important sense� Maybesome points in the neighbourhood structure would approximate others� Howeverthe material in ��� pages ��� ��� suggests we proceed with caution� It is notencouraging as far as topologies are concerned� In any topology with the T�separation property �which is a rather weak property� the only point whichwill approximate a point x is x itself� �That is� the specialisation order on thetopology is the equality relation� See below for the de�nition of specialisationorder�� So instead of using a topology to approximate we propose using acontinuity structure called a quasiuniformity� In the case of a quasiuniformityon STRUC a neighbourhood is a relation between pairs of subsets of STRUC�

Before proceeding further� note that as mentioned above X � Y i� modX �modY � But this is so if and only if modX � � modY � because modX is full� Thissuggests exploring the following relation between subsets of STRUC� I �� J i�I � � J � It turns out that the relation �� on STRUC is a quasiuniformity� It isthe quasiuniformity induced by the re�exive relation of elementary equivalence��� between members of STRUC� These notions are de�ned and explored indetail by Cs�asz�ar ��� In fact� Smyth ��� ��� ��� has already done work in thisbroad area reconciling domains and metric spaces� and examining completenessof quasiuniform spaces�

The quasiuniform neighbourhood relation mentioned above can be used tode�ne some approximation notions on PSTRUC� We give some de�nitionsfrom ��� pages ��� ��� and sketch how this approach to approximation mightwork�

De�nition �� � Let A and P be sets�

�� A relation � from P to A is an approximation for A if it satis�es thefollowing�

a For each x � A there is p � P with p � x�

b For x� y � A the following holds� x � y i� fp � P � p � xg � fp �P � p � yg�

�� A preorder �P�v� is an approximation for A relative to � if

a �P��� is an approximation for A�

b p v q i� for each x � A� q � x implies p � x�

�� The specialisation order� �� from A to A is de�ned by x � y i� for eachp � P � p � x implies p � y�

It is easily seen from the de�nition of � that � is a partial order on A andfrom the �rst part of the de�nition� �A��� is an approximation for A�

In the case of PSTRUC de�ne the relation � from PSTRUC to the full mem�bers of PSTRUC as follows� J � I i� I � � J � It is easy to check that the pair�PSTRUC��� is an approximation for the full members of PSTRUC�

Next for I and J members of PSTRUC� de�ne I v J i� J� � I�� where I�

is the union of all full subsets of I � It is easy to see that for any I � PSTRUC�I� is full� It is also easy to see that �PSTRUC�v� is an approximation for thefull members of PSTRUC relative to ��

Finally� it is easy to see that by taking � to be the superset relation� �� �is the specialisation order on the full members of PSTRUC� This specialisationorder is di�erent from equality� So there is some hope that an approach toapproximation via types of continuity structures such as quasiuniformities willyield some results� One would also need to investigate what restrictions to placeon the approximations to make them computable� I conjecture that �nite setsof stuctures with �nite domains would ensure computability�

As a �nal direction for exploration we mention epistemic entrenchment�G�ardenfors and Rott ��� �� ��� and Nayak ��� ��� discuss it and use it asa method for constructing belief contractions� The way epistemic entrenchmentis de�ned in ��� shows that it is a re�exive relation� � say� de�ned on the setof sentences� As mentioned above it therefore induces a quasiuniformity on theset of sentences� But there is a connection to entailment because for sentences� and �� if � � � then � � � ���� The rami�cations of this are not yet clear�

When the connection mentioned above between entailment and neighbour�hood structures is properly articulated then for entailment� approximation willbe connected to convergence� This� in turn� will allow a kind of �numericalanalysis� of logic to be set up and programmed into computing agents� And sowe will see a symmetry between Johnson�Laird�s view of human problem solv�ing and the way agents mimic it� Thus we will have revisited our starting pointwith Johnson�Laird� but with deeper understanding and better capability�

� CHAPTER ��� CONCLUSION

Appendix A

Set theory

For most of the constructions in this work there is no need to worry about thespeci�cs of the set�theoretical foundations used� Unfortunately some construc�tions require careful consideration of the facilities provided by the set theorybeing used� these constructions are di�cult to de�ne even when there is re�course to classes which are �larger� than sets� We shall illustrate this pointwith two examples�

The �rst has to do with restricted entailment when the restriction is thefamily of all structures� STRUC� Some di�culties are introduced by the factthat STRUC is a class and not a set� However in some situations it is convenientto be able to treat STRUC as if it were a set� For example sometimes one maywish to regard a restricted entailment as ordinary entailment by taking therestriction to be STRUC�

For the second example we outline a simple de�nition of a notion requiredin chapter � and show how even a set theory that allows classes as well assets cannot necessarily provide a good de�nition� We shall then outline a well�known approach to foundations that removes problems such as these� the useof universes of sets�

Problem

Let us adopt for the moment the set theory described in the appendix of Kel�ley ��� According to Kelley this � � � � is a variant of systems of Skolem andof A� P� Morse and owes much to the Hilbert�Bernays�von Neumann system asformulated by G�odel�� In what follows we repeat just enough of Kelley�s ax�ioms� de�nitions and theorems to be able to explain our argument� A referencesuch as ��Kelley ���� refers to item �� of Kelley�s appendix in ��� Kelley�sset theory distinguishes between classes and sets� A set is a class which is amember of some class�

De�nition A � �Kelley �� x is a set i� for some y� x � y�

� APPENDIX A� SET THEORY

Classes can be formed from sets by using a classi�cation axiom scheme�

Axiom A � �Classi�cation� Kelley II� Suppose H�z� is a logical formulawith free variable z� For each x� x � fz � H�z�g i� x is a set and H�x��

In chapter � we wish to de�ne a least upper bound for in�nitely many re�stricted entailments�

WI�E �I where E � PSTRUC� the class of all subsets of

�rst�order structures� First we make the following de�nition� �The symbol �stands for elementary equivalence of structures� two structures are elementarilyequivalent if they satisfy the same sentences��

De�nition A � �See � �

�� Let I � STRUC� The full expansion of I� denoted I �� is fS � STRUC �S � � I � S � S �g�

�� Let E � PSTRUC� then E� � fI � � I � Eg�

Next de�ne the join as follows�

De�nition A � �See � ��� Let E � PSTRUC� SetWI�E �I � ���E��

A problem arises with E� in A�� because E� could be empty even if E isnot�

To see how E� could be empty �rst note that STRUC is not a set� To seethis some de�nitions and the axiom of substitution are needed� Suppose f is afunction�

De�nition A � �Kelley �� domain f � fx � for some y � �x� y� � fg�

De�nition A �Kelley � range f � fy � for some x � �x� y� � fg�

Axiom A � �Substitution� Kelley V� If f is a function and domain f is aset� then range f is a set�

The class of all sets is the universe� U� de�ned as follows�

De�nition A � �Universe� Kelley ��� U � fx � x � xg�

To show that STRUC is not a set� we suppose on the contrary that STRUCis a set and derive a contradiction� Consider the function f � f�S� dom�S�� �S � STRUCg� We have domain f � STRUC� Also range f � U � � � U

in view of A��� A� � and A� because there is a structure de�ned on every setexcept the empty set�

De�nition A �Kelley ��� x � y � x �� y��

Theorem A �� �Kelley ��� � � � U�

Theorem A �� �Kelley ��� x U � x�

�

By A�� and our supposition that STRUC is a set it follows that U is a set�But this is a contradiction in view of the next theorem�

Theorem A �� �Kelley �� U is not a set�

Now consider the use of the classi�cation axiom�scheme A�� as follows� putE � fIg where I � STRUC is such that the full expansion I � is not a set��We cannot be sure that this is never the case�� Using the classi�cation axiomscheme A�� E� � �� the empty set� because the only candidate for membershipof E� is I � which is not a set�

Next� the intersection of the empty set is the universe�

Theorem A �� �Kelley ��� � � U�

Thus if E� is empty then E� � U� the class of all sets� The class U �STRUC so join is not well de�ned in A���

It may be thought that perhaps the de�nition could be salvaged by de�ningWI�E �I � ���E��STRUC� because then the expression on the right evaluates

to �STRUC� But as we shall see this leads to a di�erent problem� First de�nein�nite meet and join as follows�

De�nition A �� �See � ��� Let E � PSTRUC�

��VI�E �I � ��E�

��WI�E �I � ���E��STRUC�

Now the problem is that in the lattice �ENT��� �see �gure �� �� when E� isempty we have

WI�E �I � �STRUC � ��E �

VI�E which is the wrong order

for meet and join in the lattice�

Remedy

To remedy the situation we work within a universe of sets �see �� ����� Thisis a set which supports the usual set theoretical constructions but the resultsof these constructions on subsets of the universe are always sets lying in theuniverse� Moschovakis ��� describes a universe� the least Zermelo universe Z �and says of it� � � � � we can develop classical mathematics and all the set theoryneeded for it as if all mathematical objects were members of Z��

A universe is a set V with the property that the restriction of the membershiprelation� �� to V satis�es the Zermelo�Fraenkel axioms for set theory� So V is aset satisfying the following �see ����

Properties A �� �Universe�

� If y � x and x � V then y � V�

� V contains the set of natural numbers�

� APPENDIX A� SET THEORY

� If x � V and y � V then fx� yg � V�

� The power set i�e� the set of all subsets and union of sets in V are in V�

� The image of a member of V under a function de�ned by a �rst�orderformula is in V�

We also assume the following�

Axiom A � �Mac Lane� There exists a universe�

We now sketch how to proceed to de�nitions A� and A�� as follows� Picka universe V and de�ne �rst�order interpretations and structures in it� Theset of all structures also lies in V � as does the set of all subsets of STRUC�STRUC � V � and PSTRUC � V �

Next� any subset of a member of V is a member of V � This can be seen asfollows� Let Y � V and X � Y � We must show X � V � By A� � the power setof Y � P�Y �� is a member of V � But X � P�Y � so by A� � X � V � Now it canbe seen that the following sets mentioned in A� are all members of V � I � I ��E and E�� Also the set E� is a subset of any member of E� and so is in V �De�nition A�� can now be made�

Approach� universe of sets

In summary� we assume the existence of a universe satisfying A� � and make allour constructions in it�

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