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THE " DIALECTICA " INTERPRETATION AND CATEGORIES by PH~LIP J. SCOTT in Montreal (Canada) D e f i n i t i o n 2. A category@? is l o g i d if, for each object B ~$ 7 , the set of subobjects of B, denoted by '$(B), forms a lattice with smallest element OB . We also demand that if ) is preserved under pullbacks (cf. REYES [14, p. 231). Again in the category Sets, this reduces to saying f-I( ## The regular category associated with an intuitionistic theory Let T be a multi-sorted first-order intuitionistic theory. To each such theory we can associate a logical category @?(T) in a canonical way; we sketch the construction below. Our treatment follows the presentation in LAWVERE [9, p. 1391. D e f i n i t i o n 3. %(T)-the regular category accociated with T-is defined as follows: (i) 0 b j e c t s : formulae of L(T). (ii) Morphisms: equivalence classes of "provably functional relations"; that is, a morphism [F@, g)]: A(Z) -+ B@), where Z, g are vectors of variables, is an equivalence class of formulae (under the equivalence relation p N y if T k ~1 t+ y ) satisfying (I), ( 2 ) j C'WcWls, I , Zcys) A G'(P,YS, 9, y, ~c y s ) l , ( 3 ) ~ Vy. Z , z'[G(y, Z ) A G(y, z') -+ Z = Z ' ] , (3): 3Qc

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Author

Philip J. Scott

Publisher

John Wiley and Sons; Wiley (John Wiley & Sons); John Wiley & Sons Ltd.; Wiley (ISSN 0044-3050)

Published

1978

Language

EN

Field

Computer Science (Physical Sciences)