SEMESTER 2 (30 ECTS)

Metaphysics claims to theorize “all things in general,” and formal ontology claims to be the science of “something in general,” which presupposes the availability of the notions of “things in general” and “something in general.” However, these notions are not self-evident; they are neither primitive nor obvious: this is the hypothesis that this course seeks to explore, particularly that philosophical generality is not separable from the forms that mathematics gives it.

The course will consist of three major parts. After distinguishing the two dimensions of generality—integrality (the aim of all things) and genericity (the aim of any thing)—we will begin by examining the first (absolute generality, i.e., the consideration of all things without exception), showing that, as much as its rejection, it gives rise to paradoxes. This will introduce the solidarity of the major registers of generality: philosophy, logic, and mathematics.

We will then focus on the notion of genericity, that is, the concept of any object, and its formal counterpart: the notion of variable.
Metaphysicians presuppose the possibility of referring to things in general, without becoming aware that the form of “something in general” that seems to deliver this possibility is an instrument borrowed from formal logic, and actually developed by logic in connection with mathematics. The second part of the course will examine the plural forms of the generic found in mathematics and their connection to philosophical figures of the general. It will defend the idea that the former partially under-determine the latter, and will support the priority of genericity over integrality.

The third and final part of the course will focus on the notions of variable and variation. While these have been separated by modern logic to avoid any confusion of generality with a real process, more recent developments, re-associating logic and geometry, allow these two notions to be conjoined in new ways. We will provide some illustrations by describing how generality can be conceived in terms of deformation, in modal logic and logical semantics.

Assessment Methods:
A personal research project on a theme related to the course.

This course aims to present, to students without particular competence in mathematics, a comprehensive set of mathematical tools that are notably useful for representing language and knowledge. The broader goal of the course is to provide access to the abstract methods of modern mathematics and to give an overview of both the variety of mathematical branches and their common conceptuality. The unifying thread for this will be the concept of space.

The course will begin with a foundation course on mathematical symbols and mathematical demonstrative discourse (1 session).

It will then cover four important approaches to the concept of space:
• Topology: space in the topological sense (5 sessions)
• Linear algebra: space as vector space (2 sessions)
• Differential geometry: space as manifold (2 sessions)
• Category theory: space as topos (2 sessions)

Assessment Methods:
A personal application project of a mathematical theory on a topic of choice.

LCHDY010: Syntax, Semantics, Discourse 2 (M2S4)
(Course taught in English) • 2 hours • 6 ECTS

The course is divided into three parts: it begins by examining Richard Montague’s approach to quantification, an approach that revolutionized formal semantics of natural languages and remains influential; we then discuss the theory of generalized quantification, in which a notion invented by logician Mostowski became the basis of the dominant approach for describing the meaning of noun phrases. Finally, we examine recent work that attempts to capture the insights of Montague and generalized quantifier theory, but in a way that is cognitively realistic and conceptually simpler, and therefore appropriate for describing dialogue.

Coordinator: Jonathan GINZBURG (yonatan.ginzburg@u-paris.fr)

Bibliography:
• Barwise, J. & Cooper, R. (1981). Generalized Quantifiers. Linguistics and Philosophy, 4, 159-219.
• Montague, R. (1974). The Proper Treatment of Quantification in Ordinary English. In R. Thomason (Ed.), Formal Philosophy. Yale University Press.
• Lücking, A. & Ginzburg, J. (2022). Referential Transparency as the proper treatment for quantification. Semantics and Pragmatics.
• Vendler, Z. (1962). Each and Every, Any and All. Mind, 71(282), 145–160.
• Westerståhl, D. (1985). Determiners and Context Sets. In A. ter Meulen & J. van Benthem (Eds.), Generalized Quantifiers in Natural Language (pp. 45–72). De Gruyter Mouton.

In this course, we will study computational semantics by focusing primarily on its symbolic aspects. We will therefore study the foundations of logic and formal semantics. After introducing lambda-calculus, we will address the syntax-semantics interface through Montague grammar and combinatory categorial grammars (CCG). Finally, we will shift perspective by studying, in particular, the task of coreference resolution and conducting a rapid overview of the different types of semantic representations used in computational linguistics. Lectures will be complemented by paper-based tutorial sessions, punctuated by practical computer-based work.

(Course taught in English)