  It has been suggested that relational logic, a form of logic developed by C.
S. Peirce, is the common inner syntax of quantum mechanics and string theory. A
relation may be represented by a spinor and the Cartan-Penrose connection of
spinor to geometry, allows to abstract geometry from a calculus of
relations-spinors, reviving Wheeler\textquoteright{}s pregeometry. With a
single spinor related to the null cone of Minkowski space-time, we search for
the geometry emerging when we entangle a left-handed spinor and a right-handed
spinor. We find that the quantum entanglement generates an extra dimension and
the distance in the extra dimension is measured by the amount of entanglement.
The emerging geometry corresponds to a Milne space-time, with two branes
coexisting in the extra dimension. One brane hosts left-handed particles (our
brane), while the other brane hosts right-handed particles. A distinct
phenomenology accompanies our proposal. During the brane collision all points
are causally connected, making less pressing the inflationary scenario. The
left-right symmetry is achieved with having two "mirror" branes and the
neutrino appears as the ideal mediator between the branes. We may revisit also
the dark matter, dark energy issues, with everything on the other brane and in
the bulk appearing "dark" to us. If biohomochirality is due to parity
violation, we anticipate that the life-forms in the other brane demonstrate
opposite chiralities. Our scheme brings closer logic - quantum theory - string
theory -- geometry -- cosmology, while space-time, rather than an abstract and
a apriori construction, appears as the outcome of a quantum logical act.
