The robotic 2m Liverpool Telescope, based on the Canary island of La Palma, has a diverse instrument suite and a strong track record in time domain science, with highlights including early time photometry and spectra of supernovae, measurements of the polarization of gamma-ray burst afterglows, and high cadence light curves of transiting extrasolar planets. In the next decade the time domain will become an increasingly prominent part of the astronomical agenda with new facilities such as LSST, SKA, CTA and Gaia, and promised detections of astrophysical gravitational wave and neutrino sources opening new windows on the transient universe. To capitalise on this exciting new era we intend to build Liverpool Telescope 2: a new robotic facility on La Palma dedicated to time domain science. The next generation of survey facilities will discover large numbers of new transient sources, but there will be a pressing need for follow-up observations for scientific exploitation, in particular spectroscopic follow-up. Liverpool Telescope 2 will have a 4-metre aperture, enabling optical/infrared spectroscopy of faint objects. Robotic telescopes are capable of rapid reaction to unpredictable phenomena, and for fast-fading transients like gamma-ray burst afterglows. This rapid reaction enables observations which would be impossible on less agile telescopes of much larger aperture. We intend Liverpool Telescope 2 to have a world-leading response time, with the aim that we will be taking data with a few tens of seconds of receipt of a trigger from a ground- or space-based transient detection facility. We outline here our scientific goals and present the results of our preliminary optical design studies.