Welcome to the 2017 Nordic Logic Summer School course on Natural Logic

The course will study logical systems which are relevant to semantics and also logical systems which try to use surface forms directly. It also will cover in detail how various standard logical systems work, including natural deduction systems, and others; the typed lambda calculus; and first-order logic and its decidable fragments. It will also present detailed studies of logical systems which are closer to natural language, such as extended syllogistic logics. These topics will be taught from the point of view of representing natural language inference.

More specifically, the course is divided into a general introduction and four more content-full parts.

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 will present 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 will try to present a fair 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: This part of the course presents natural deduction-style systems which can handle interesting linguistic phenomena and at the same time remain decidable.

The third day will focus on 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. The last two days are devoted to

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 will present much of what has been done in the area As part of the course, I'll present the needed background on categorial grammars and polarity phenomena in language.