@. Transitivity, Let us consider ?, ? 1 , ? 2 , ? 3 ? ?, such that ? 1 ? ? 2 and ? 2 ? ? 3 , there are four cases

*. If, Due to E5 and U9 again, ))] = ? then using E5 and U9, ?(1, 2) = ?(1) = ?(1, 2, 3)? [(?(? 1 ) ? ?(? 3 ))], hence ? 1 ? ?

@. Reflexivity, (? 1 ) |= ?(? 1 ), hence either ? 1

. @bullet-?-respects, ? 1 ) ? ?(? 2 ))]. Due to E3, since (?, ? 1 ), (? 2 ))] = ? or ? 2 ? [?(?) Due to U1, ? 1 ? [?(?)? T ?(? 1 )]. Hence using E5 and U9, [?(?)? T (?(? 1 ) ? ?(? 2 )) ? ?(? 1 )] = [?(?)? T (?(? 1 )] = {?1}. Contradiction, hence ? 1 ? ? 2

@. If, (. A. , and R. , then t 1 indirectly defends the unique element of t 2 in (A, R) (which is denoted by (A, R) |= t 1 ~~ t 2 ) if and only if, ) |= (t 1 t 2 ) ? (?y ((t 1 y) ? (y ~~ t 2 )))

T. 3. Given and T. ?. , U a set of authorized transitions, there exists an operator ? T satisfying E3, U4, E5, E8, U9 if and only if there is an assignment respecting T such that ?G ? ? U , ??, ? ? YALLA U , References [1] C. Alchourrón, P. Gärdenfors, and D. Makinson. On the logic of theory change : partial meet contraction and revision functions, Journal of Symbolic Logic, vol.50, pp.510-530, 1985.

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