Generalisation into sentence and predicate positions

Introduction

Generalisations are fundamental to every scientific discipline: ‘Every cell has a plasma membrane’, ‘Every electron has a negative charge’, ‘Every natural number has a unique successor’. By means of generalisation, we turn a statement about a particular individual into a statement about a class of entities. Generalisations are essential to valid deductive reasoning. They are the building blocks of virtually every scientific theory, and therefore essential to understanding, explaining, and making predictions.

Basic Forms of Generalisation

The most basic and best understood form of generalisation is generalisation over objects (e.g. cells, electrons, numbers). In formal logic, this form of generalisation is achieved via first-order quantifiers, i.e. operators that bind variables in argument position.

Higher-Level Generalisation

However, many theoretical contexts require generalisation into sentence and predicate positions, a high-level form of generalisation where we make a general statement about a class of statements (e.g. mathematical induction, laws of logic).

Competing Methods

There are two competing methods for achieving this form of generality:

1. Higher-order logic
2. Self-applicable theories of truth, properties, and sets

As both methods come with their own ideological and ontological commitments, it makes a substantial difference which one is chosen as the framework for formulating our mathematical, scientific, and philosophical theories.

Current Research Status

Some research has been done in this direction, but it is still very much in its early stages.

Project Goals

This research project will significantly advance this foundational project. It will:

• Provide the first sustained systematic investigation of the two methods from a unified perspective.
• Develop novel formal tools to articulate deductively strong theories.

Impact

Due to its foundational character, it will have an impact on many disciplines, especially the foundations of mathematics, logic, formal semantics, metaphysics, philosophy of language, and theoretical computer science.