The course studied logical systems which are relevant to natural language semantics and also logical systems which try to use surface forms directly. That is, it presented logical systems for natural language inference which are closer to actual language than to standard logical languages like first-order logic. I have taught this material to audiences closer to linguistics, and also to beginning logic students, and I was excited to teach it at the NLS.
Even if you are not interested in the overall topic of the course, it introduced several topics that you might want to see: syllogistic logic, especially completeness theorems for various fragments; algebraic logic; decidable fragments of first-order logic; categorial grammar; the typed lambda calculus.
More specifically, the course is divided into a number of units. Some of these are independent after the first day.
Introduction: a list of test problems for natural logic, a summary of the results that we'll see in the course, and general history of the area. I also presented some background on decidable fragments of first-order logic.
Syllogistic proof systems: I'll summarize what is known about complete logical systems which can be called 'syllogistic' in the sense that they do not use variables or other devices besides the surface forms. It might be surprising that one can do any sort of linguistic reasoning this way. I presented a small number of the completeness proofs themselves in this part of the course.
Logics with relations: Moving on to logics with verbs and relative clauses brings a set of extra problems and opportunities.
Logic beyond the Aristotle boundary: I did not teach much of this material. It presents natural deduction-style systems which can handle interesting linguistic phenomena and at the same time remain decidable.
The third day was devoted to reasoning about the sizes of sets. This work concerns constructions like there are more books than magazines on the table. They are not expressible in first-order logic, yet their logic is decidable even when added to the other phenomena in this class.
Monotonicity and Polarity: The best-known work in the area of natural logic is based on the monotonicity calculus first identified and studied by Johan van Benthem. This part of the course presented much of what has been done in the area I stared with the needed background on categorial grammars and polarity phenomena in language.