A new formal tool: Functorial variables representing assertions and presuppositions

A new formal toolt Functorlal variables representing assertions and presuppositions lngolf MAX Department of PhllosophylLoglc Martln-Luther-Unlverslty Halle-Wittenberg GroSs SteinetraBe 73~ 4010 HallelSaale, GDR Abstract The language of classical propositional iog- Ic is extended by functorial variables as a new syntactical category. Functorial vari- ables render to be a possible integrating representation of both assertion and presup- position in one and the same logical formula different from such using classical conjunc- tion. O~ Introduction From a computational point of vlew there is an important difficulty of an adequate formalization of both assertion and presup- position, It rests on dividing utterances into an explicit part (assertion) and an im- plicit one (presupposition). Frege 118921 refused any explication of this implicit part, because of hls ideallza= t/on that logic deals only with correct ~tatemerlts. At least in regard of definite descriptions Russell /1905/ pleaded for such an explication. But classical conjunction was one of its essential formal tools. To a certain extent assertion and presuppoeltion were represented at one and the same level. Strawson /1950/ claimed that presuppositions have to be explicated, but not at the same level like the implicit part. Many linguists = e, g. Klefer 11973/ - believe that logical means can be used to represent both parts of utterances aeparately~ but these representa- tions cannot be put together in one and the same logical expreeslon~ because Russell's solution Is unsatisfactory. My intention is to show that functorlal variables render to be possible tools for integrating representation of both assertion and presupposition in one and the same logi- cal formula, Moreover - unlike Bergmann i1981/, OunglKOstner /19861 - this represen- tation Is a svntactlcal one and different from that given by means of classical con- Junction (cp, Max l1986/i /forthcoming/). 1. The logical apparatus 1,1~ Functors as classical functions n Let ~t be the form of n-placed propositional 2n functors; 1 ~ t ~ 2 ~ I will use 1- and 2- placed functors only, These functors are in- terpreted as classical (i. e, 2-valued end extensional) functions, The value-tables are= 111 1 o--7- 1 1 1 1 0 I 1 0 1 0 110 I 1 1 1 1 0,11 1 1 0 0 0,0 I I 0 1 0 lll I~0 Oel OmO 1 0 0 0 0 0 0 0 0 1 I 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 1 .1. 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1.2, Functorlal variables I introduce functorial variables as e ne~ pyntactlcal category, and take the classical functors as values of these variables. Intro- ducing such variables we get a whole class of syntactical extensions of the classical pro- positional logic. Only functorlal varlablee of the following form are considered= G~=gi_ I ~ f,g ~ 16. The f's and g's are called components. The f-component (g-component) is called first (second) component. are values of 2 Semantically, ~ and …

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