# UvA-DARE (Digital Academic Repository)

## Hele tekst

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UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)

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## Introduction n

### 1 1

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2 2 ChapterChapter 1. Introduction

### 1.11 W h a t to expect from this dissertation?

Inn this thesis we study definability and interpolation. These are properties of logics suchh as compactness or decidability that have been established as yardsticks by whichh to measure the behavior of logics. What do they look like? In a slogan, thee Beth (definability) property states that implicit definability equals explicit definability.. These notions will be explained in full detail in the thesis. The gist iss that implicit definability is a semantic concept whereas explicit definability is aa syntactic phenomenon. To say that the two forms of definability coincide (as thee Beth property does) may therefore be regarded as an indication that there is aa good balance between syntax and semantics of a logic.

Provingg that a given logic S has the Beth property usually proceeds by way of provingg the interpolation property for S. This property requires that any validity

^^ —> t' has an interpolant. That is. there exists a formula i) in the common languagee of ^ , i'. such that ^ —> 0 and i) —* v are again validities. Apart from itss connection with definability, interpolation is also an interesting notion in itself whichh points to a well-behaved deductive system.

T h ee objectives of this dissertation are fourfold. We successively

1.. Provide "everything you always wanted to know about definability and in- terpolationn but were afraid to ask."

2.. Relate definability to the algebraic property of surjectiveness of epimor- phisms. .

3.. Offer tools for proving and disproving definability theorems and interpolation theorems. .

4.. Present plenty of examples that show that the interpolation property is much strongerr than the definability property. To this end. we do two detailed case studies,, viz., of guarded fragments of first order logic and of interpretability logics. .

AA g e n e r a l p i c t u r e of definability and i n t e r p o l a t i o n The literature on de- finabilityy and interpolation is of a fragmentary nature. Investigations tend to con- centratee on particular logics, and results are scattered throughout the literature.

W h a tt is missing is an easily accessible introduction to the subject that sketches thee general picture. Chapter 2 has been written especially for this purpose.

A l g e b r a i cc c h a r a c t e r i z a t i o n of definability It is well-known that there is a closee correspondence between logic and algebra, which can and has been exploited t oo transfer methods and results between these two fields. A prime and well-known examplee is formed by the variety of Boolean algebras which forms the algebraic counterpartt of classical propositional logic. Actually, these algebras are named

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afterr G. Boole who was the first to study prepositional logic from this algebraic- perspective. .

Inn general, a logic S can often be investigated by means of studying an appropriate classs of algebras Mod*S. How much information does this yield? For one thing, itt may help to decide whether S has the interpolation property. For it turns out t h a tt for many logics S the question whether or not S has interpolation is answered ass soon as you know whether or not Mod*S has a certain algebraic property, viz.

thee amalgamation property. Since amalgamation has been well-investigated in thee field of universal algebra, this last question may well have been answered.

Otherr information on S that can be obtained by algebraic means concerns the compactnesss property, the deduction property, etc.

Inn this dissertation we give such algebraic characterizations of definability proper- ties.. Our main result states that, under mild conditions on the logic 5 , S has t h e Bethh property iff Mod*S has the property of surjectiveness of epimorphisms. As thee latter property is well-known from the algebraic literature, our characterization iss indeed useful. We supply plenty of applications to support this view, including applicationss to many-valued logics and relevance logics. Moreover, we show that thee proof of our characterization of the Beth property is generally applicable in provingg equivalences, mutatis mutandis, between all kinds of definability proper- tiess and surjectiveness of various kinds of epimorphisms. This gives us for example equallyy general characterizations of the weak Beth property and projective Beth property. .

T o o l ss for ( d i s ) p r o v i n g definability and i n t e r p o l a t i o n How to prove and disprovee definability theorems and interpolation theorems? In the literature many approachess can be found. In this dissertation wre discuss several of them.1 Besides thee algebraic method described above, these include the following.

1.. We extend a method for proving interpolation in modal logics (which resembles aa Henkin-style completeness proof) to modal logics with an extra non-standard operator.. We do this for the special case of interpretability logics, but the basic ideass are of a general nature.

2.. Wre explore another technique known from modal logic in order to obtain results onn interpolation for guarded fragments of first order logic. This technique involves bisimulations.. Our findings include both positive and negative results.

Thee above methods yield results on interpolation. W h a t about definability theo- rems?? Usually, definability theorems are derived from interpolation theorems via aa standard argument. But, as we will see, from the interpolation theorem for a Counting,, for the moment, the algebraic method as a semantic approach, we confine ourselves too semantic methods. This is the obvious approach if one wishes to establish a negative result.

Onn the other hand, positive results can be (and have been) obtained in abundance by syntactic meanss and so our adherence to the semantic side of the matter should be completely attributed too our personal liking.

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4 4 ChapterChapter 1. Introduction

givenn logic S we can infer much more than just the Beth definability theorem for S.S. This is illustrated in the case of interpretability logics where we derive the Bethh property for all interpretability logics from the interpolation property for thee (one) basic system IL. Putting it differently, we do not need the full strength off the interpolation theorem to derive the definability theorem. For example, our prooff of the Beth theorem for the guarded fragment (which does not have the interpolationn property) uses a limited form of interpolation. These results are nott just simple applications of the aforementioned standard argument but require somee extra effort.

C o n c l u s i o nn There is a widespread belief which lumps together the interpola- tionn property and the Beth definability property. This common fallacy might be rootedd in the fact that most (if not all) of the well-known logics have either both orr neither of these two properties. That is, there is a lack of examples t h a t in- dicatee the difference between Beth definability and interpolation. In this thesis wee present plenty of such examples. For example, as we already mentioned, all extensionss of the basic interpretability logic IL have the Beth property. On the otherr hand, besides t h e systems IL and ILP. so far no extensions have been found withh interpolation. Another example is presented by the guarded fragment and thee packed fragment of first order logic. We will prove the Beth theorem for these twoo fragments and also for all of their finite variable fragments. However, it will bee shown that none of these logics has interpolation, apart from the one- and two variablee case. T h e conclusion that we draw from all this is that the interpolation propertyy is much stronger than the definability property.

### 1.22 Organization of the dissertation

Below,, we give a brief survey of the thesis. A more detailed outline of the individual chapterss can be found on their respective title pages.

C h a p t e rr 2 The aim of the next chapter is to make the reader familiar with the mainn themes of this dissertation: definability and interpolation. The chapter is writtenn in a rather informal manner with an emphasis on giving simple examples.

Wee discuss the precise relationship between the relevant properties, summarize thee state of the art, and provide ample references to the literature. This chapter alsoo takes a look at the m a t t e r from an algebraic perspective.

C h a p t e rr 3 This chapter is of an abstract algebraic nature in which algebraic equivalentss of several Beth definability properties are given. We also supply many applicationss of these characterizations. The chapter contains an introduction to thee abstract algebraic framework(s) wre are working in.

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