I regularly teach the required logic course in the PhD program at The Graduate Center. Here is a description of the course:
PHIL 72000 – Logic
This course is a philosophical introduction to classical symbolic logic. No prior background in logic is assumed. We will study sentential logic and first-order predicate logic with identity. Topics to be covered include semantics, syntax, and proof procedures. We will also cover the metalogical concepts of soundness and completeness. Along the way, we will take a look at some central problems in the philosophy of logic, motivations for non-classical logics, and the relevance of logic to other areas of philosophy. The goal is to achieve both a practical mastery and philosophical understanding of elementary logic.
One of two textbooks for the course is Greg Restall’s Logic, 2006. The book was published simultaneously by McGill-Queen’s University Press (978-0-773524231) and Routledge (978-0-415400688). It is available online from the publishers: McGill-Queen’s: <http://www.mqup.ca>; Routledge: <www.routledge.com>