  By considering type B analogs of permutations and tableaux, we extend
abstract dual equivalence to type B in two directions. In one direction, we
define involutions on signed permutations and shifted tableaux that give a weak
dual equivalence, thereby giving another proof of the Schur positivity of Schur
Q- and P-functions. In another direction, we define an abstract shifted dual
equivalence parallel to dual equivalence and prove that it can be used to
establish Schur P-positivity of a function expressed as a sum of shifted
fundamental quasisymmetric functions.
