  In high-Lundquist-number plasmas, reconnection proceeds via onset of tearing,
followed by a nonlinear phase during which plasmoids continuously form, merge,
and are ejected from the current sheet (CS). This process is understood in
fully ionized, magnetohydrodynamic plasmas. However, many plasma environments,
such as star-forming molecular clouds and the solar chromosphere, are poorly
ionized. We use theory and computation to study tearing-mediated reconnection
in such poorly ionized systems. In this paper, we focus on the onset and linear
evolution of this process. In poorly ionized plasmas, magnetic nulls on scales
below $v_{\rm A,n0}/\nu_{\rm ni0}$, with $v_{\rm A,n0}$ the neutral Alfv\'{e}n
speed and $\nu_{\rm ni0}$ the neutral-ion collision frequency, will
self-sharpen via ambipolar diffusion. This sharpening occurs at an increasing
rate, inhibiting the onset of reconnection. Once the CS becomes thin enough,
however, ions decouple from neutrals and thinning of the CS slows, allowing
tearing to onset in a time of order $\nu_{\rm ni0}^{-1}$. We find that the
wavelength and growth rate of the mode that first disrupts the forming sheet
can be predicted from a poorly ionized tearing dispersion relation; as the
plasma recombination rate increases and ionization fraction decreases, the
growth rate becomes an increasing multiple of $\nu_{ni0}$ and the wavelength
becomes a decreasing fraction of $v_{\rm A,n0}/\nu_{\rm ni0}$.
