{Baker, Cooper, Fletcher, Miller, Smith};
(Unequal @@ %) && (And @@ (0 < # < 6 & /@ %)) &&
  Baker < 5 &&
  Cooper > 1 &&
  1 < Fletcher < 5 &&
  Miller > Cooper &&
  Abs[Smith - Fletcher] > 1 &&
  Abs[Cooper - Fletcher] > 1 //
 Reduce[#, %, Integers] &