CL&C'12 was the fourth of a conference series on "Classical Logic and Computation", held as satellite to ICALP'12 on Sunday July 8, 2012 in Warwick, England. CL&C intends to cover all work aiming to explore computational aspects of classical logic and mathematics, and is focused on the exploration of the computational content of mathematical and logical principles. The scientific aim of this workshop is to bring together researchers from both fields and exchange ideas. The intention of the organisers is for CL&C to be an informal workshop. Participants are encouraged to present work in progress, overviews of more extensive work, and programmatic/position papers, as well as completed projects. Submission of both short abstracts and of longer papers were invited. Four submissions were accepted as full papers and are included in these proceedings. Three more were accepted only as communications for the conference. Paulo Oliva gave an invited talk on "Some connections between Game Theory and Proof Theory".