By Yde Venema

It is a doctoral dissertation of Yde Venema below the supervision of prof. Johan van Benthem.

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By assumption we have v / / ( 0 ,77), and as / is a potential zigzag morphism we get a situation as showed in the right picture. By # |= N V 7 ( f ,(rj,r])), $ has a vv £ D with Hvvv and Vvvf'(rj,rj). We define /'O hO = V and set /'(£>£) as the unique diagonal if-successor of any/all of the f'(rjy£). It is straightforward to verify that with this definition the part of / ' defined up till now satisfies both conditions (1) and (2). 2. TWO-DIMENSIONAL CYLINDRIC MODAL LOGIC. D / \ tin 1 1 \v v,/ 1 1 Ki j / H V i, f'oo fi10 y / For the definition of /'(£, rj) (rj < £), we use the same trick as above to ensure /'(£, rj) £ D : as /'(£, £) is in D and f r(rj, £) is not, they cannot be identical.

In chapter 5 we will give a detailed account of how temporal logics of intervals fit in the two-dimensional picture. Many of the systems described above may be perceived as following a general trend in modal logic, namely of bridging the gap between the old-fashioned intensional framework, which is simple, elegant and has nice computational properties, and classical first order logic which is expressive and perhaps still more familiar. 1. INTRODUCTION. of the classical modal formalism are: Blackburn [19] and Gargov and Goranko [39] add ‘nominals’ or ‘names’ to the language, these being atomic propositions holding at unique possible worlds.

Section the standard correspondence map as defined for the similarity type. 3 for details) is not the same as 46 CHAPTER 3. THE SQUARE UNIVERSE. worlds consisting of a Carthesian square2 and the interpretation map having a fixed and uniform definition for this kind of universes. 2. (*>»)) I {((«>»). (»,»)) 1 {((u ,v ),(w ,x ),(y ,z)) | u = w ,v s* II & w II II 53 53 53 I(* ) 1(0 ) I(O ’) 1(0') 1 (0 ) 1(0) m m i(e ) I( o) = A two-dimensional or square model is a model based on a two-dimensional frame.

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