1st Semester 2023/24: Natural Logic

Larry Moss, Sonja Smets (Responsible)

Natural Logic means doing logic in natural language or something close, as opposed to doing it in a more standard kind of logical system.   It is concerned with inferential patterns in language and in designing and studying logical systems that reflect those patterns.

This course will present natural logic as a way to raise the level of the MoL students with proofs about logical systems, and it also will be of interest to MoL students interested in language-related topics.   The course will be roughly based on my logic course at Indiana University for 3rd and 4th year undergraduates.  The people who take this class are typically not well-versed in proofs, and so that course itself is a kind of 'bridge to proofs' mathematics course.   But also I have taught the same material in week-long classes at ESSLLI and NASSLLI, where only a fraction of the details are presented, resulting in a high-level view.   This MoL course will be somewhere in between.  It will be slow enough to go through examples and proofs in detail.  The goal for the one-month experience is that some students would build important skills in mathematics and get (further) attracted to logic.   

The plan for four long lectures would be
1. Basic syllogistic logics along with completeness theorems for them, leading to more and more expressive logical systems.
2. Logics for reasoning about the sizes of finite sets, making some connection to easy combinatorial ideas, and starting in on the next topic.
3. Monotonicity calculi, polarity marking, and connections to natural language syntax (categorial grammar and its interface with semantics) and semantics
 (Montague grammar, typed lambda-calculus) and to boolean algebras.
4. Computer implementation of the preceding material (or perhaps a time to complete one of the previous lectures).


The class will run during February 2024.   One reasonable plan would be to have lectures the first two weeks,
with lectures twice each week for 90 minutes each time.  Then the third week would be for students to work on a project, and for the instructor to be available for short meetings.
The final week should be reserved for presentations of those projects.   If not many students take the class, or if a great many do, then this plan might need to be revised.


Students should be comfortable with mathematical notation and with basic proofs, with the difference between syntax and semantics, and with structural induction on terms in a logical language.  Of course it would be better if students had seen and absorbed at least one completeness theorem, say for propositional logic.   But this is not strictly speaking a prerequisite.


Students will complete a project to be determined by them and the instructor.  There are a variety of possible projects.   Some students might want to work more deeply on the mathematical side, reading a paper and presenting (and expanding on) some key results.  Some will prefer to explore connections to linguistics or to computational semantics, reading state-of-the-art papers.   Others might work on implementing some of the proof-search and model-building algorithms.  


_Logic From Language_, draft textbook by the instructor

Johan van Benthem, 
A Brief History of Natural Logic,
in _Logic, Navya-Nyaya and Applications, Homage to Bimal Krishna Matilal_,
M. Chakraborty, B. Loewe, M. Nath Mitra and S. Sarukkai (eds), College Publications, 2028 

Tri Lai, Jorg Endrullis, and LM,   Majority Digraphs, 
Proceedings of the American Mathematical Society,  144 (2016), 3701-3715.

LM, Syllogistic Logic with Cardinality Comparisons,
 in Katalin Bimbo (ed.), J. Michael Dunn on Information Based Logics,
Springer Outstanding Contributions in Logic, 2016, 391--415.

Yifeng Ding, Matthew Harrison-Trainor, and Wesley Holliday,  
The Logic of Comparative Cardinality, J. Symbolic Logic 85 (2020), no. 3, 972–1005.

Thomas Icard and LM, Recent Progress on Monotonicity, Linguistic Issues in Language Technology, 9:7 (2014), 167-194.

Hai Hu and LM, 
Polarity Computations in Flexible Categorial Grammar,
Proceedings of the Seventh Joint Conference on Lexical and Computational Semantics, 2018, 124--129

Z Chen, Q Gao, LS Moss
Neurallog: Natural language inference with joint neural and logical reasoning.
"Proceedings of *SEM 2021: The Tenth Joint Conference on Lexical and Computational Semantics", 78-88, 2021.