References to programming languages based on conditional logics
Check these references:

Programming languages CondLP and CondLP+: 

Gabbay, Giordano,  Martelli, Olivetti, Sapino, Conditional reasoning in logic programming,  Journal of Logic Programming, Volume 44, Issues 1-3, 1 July 2000, Pages 37-74 

Claudia, Oliveira, The implementation of CondLP, Lecture Notes in Computer Science, 1996, Volume 1085/1996, 713-715

Gabbay, Giordano, Martelli, Olivetti, Conditional logic programming, Proc. 11th Int. Conf. on Logic Programming, Santa Margherita Ligure, pages 272–289, 1994.

References to proof theory:

Olivetti, Pozzato, Schwind, A sequent calculus and a theorem prover for standard conditional logics, Journal ACM Transactions on Computational Logic (TOCL), Volume 8 Issue 4, August 2007