The informal notion of "logically correct argument".
Informal strategies for producing logically correct arguments and counterexamples to fallacious arguments.
The propositional calculus as an example of a formal language, formal proofs, and the formalization of natural language arguments.
A discussion of the relationship of proof, truth, and counterexamples, including a discussion of the Soundness Theorem.
The predicate calculus extension of propositional logic.
At least an informal discussion of the Completeness Theorem, if time permits.
Here are some tips to help make these courses successful.
Most important, make sure the instructor is interested in and well grounded in logic.
Spend some time on amusing logic problems and puzzles of the sort Raymond Smullyan has made famous. Consider discussing some topics from this history of logic.
Spend some time on real-life examples of sound and unsound reasoning. (In the U.S. at least, this could include the kind of logic problems commonly found on GRE and LSAT exams.)
If formal rules are taught, present them as a mathematical model of informal reasoning methods.
Treat some applications that have been made of ideas in logic, say in computer science, in detail.