The 2018 NASSLLI course on Logic for Natural Language, Logic in Natural Language

The course studies logical systems which are relevant to natural language semantics and also logical systems which try to use surface forms directly. That is, it presents 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 am excited to teach it at NASSLLI this coming June. But this will be changing.

Even if you are not interested in the overall topic of the course, the course will introduce 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 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 present 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 probably will 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 will be 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 presents much of what has been done in the area I include with the needed background on categorial grammars and polarity phenomena in language, and also on the related mathematical topics.

Computer Resources

I am preparing some Jupyter notebooks to be hosted on CoCalc. These would illustrate inference in action in several logical systems. In addition, some of the course homework will involve using these notebooks. They are written in Python and in Haskell, but you do not need to know any programming language to use them. If you want to use the notebooks, you will need to register with CoCalc. If you want to try them out before NASSLLI, please shoot me an email.

Lecture slides

Here are drafts of my slides for the course. During May and early June 2018, I expect to change them. I know that there is too much here.

My first lecture is an introduction to the topic as a whole. Also on Monday, I'll discuss the simplest logic in the world, the logic of sentences All x are y.

On Tuesday, the topic will be extended syllogistic logics, including logics with verbs, relative clauses, negation, and existential assertions.

In the middle of the week, I want to talk on logics for reasoning about the sizes of sets . This is a topic that goes beyond first-order logic in some ways.

The last two lectures are on the monotonicity calculus. I hope to also cover some very recent work on this topic, and to end the course with what people in the area are doing now.

Homework

Here is the homework set for Monday after the first class.

Here are the answers to that first homework set.

Here is the homework set for Tuesday.

Here is the homework set for Wednesday.

The homework set for Thursday is a good preliminary set on preorders, containing facts used in Friday's lecture.

If anyone wants to ask me about our homework at any point, please feel free: lsm@cs.indiana.edu.

More homework is coming.

Related Papers

Johan van Benthem, A Brief History of Natural Logic.

For the first lecture, here is some text.

LM,Completeness Theorems for Syllogistic Fragments.

Many of the results in the second lecture may be found in Logics for the Relational Syllogistic by Ian Pratt-Hartmann and LM. I also have some text material that I can send out if you are interested.

For our last unit (monotonicity), here is Thomas Icard and LM, Recent Progress on Monotonicity.