  We introduce a new paradigm for proving the Schur $P$-positivity.
Generalizing dual equivalence, we give an axiomatic definition for a family of
involutions on a set of objects to be a queer dual equivalence, and we prove
whenever such a family exists, the fundamental quasisymmetric generating
function is Schur $P$-positive. In contrast with shifted dual equivalence, the
queer dual equivalence involutions restrict to a dual equivalence when the
queer involution is omitted. We highlight the difference between these two
generalization with a new application to the product of Schur $P$-functions.
