Generalizations lie at the heart of all scientific thinking. When we say, “Every cell has a membrane,” or “Every electron carries a negative charge,” we move from talking about one particular case to making a statement that applies to a whole class of things. This ability to generalize — to see the pattern behind the particular — is essential for reasoning, building theories, and making predictions in every field of science.
The simplest kind of generalization is about objects: cells, electrons, numbers, and so on. In logic, this kind of generalization is captured by what are called quantifiers — words like “every” or “some” that allow us to speak about groups rather than individuals. But sometimes scientists, mathematicians and philosophers need to generalize at a higher level: not just over objects, but over statements or properties themselves. For example, mathematical induction and the laws of logic are both kinds of general claims about other statements.
There are two main ways to handle this kind of higher-level generalization, a direct and an indirect way. The direct method uses what is known as type theory or higher-order logic; the indirect one relies on self-applicable theories of truth, properties, and sets. Each approach comes with its own philosophical assumptions about what kinds of things exist and how they relate to one another. The choice between them has deep consequences for how we formulate theories in mathematics, science, and philosophy.
This research project will offer the first comprehensive study that compares these two approaches within a single framework. It aims to develop new logical tools that make it possible to construct stronger and more unified theories. Because of its foundational nature, this work will have implications across many areas — including mathematics, logic, linguistics, metaphysics, philosophy of language, and theoretical computer science.
