Generalisation into sentence and predicate positions
This project aims to systematically investigate and develop formal tools for higher-order generalisation methods, impacting foundational theories across mathematics, logic, and philosophy.
Projectdetails
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:
- Higher-order logic
- 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.
Research Significance
Some research has been done in this direction, but it is still very much in its early stages. This research project will significantly advance this foundational project.
Objectives of the Project
It will provide the first sustained systematic investigation of the two methods from a unified perspective and 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
- Theoretical computer science
Financiële details & Tijdlijn
Financiële details
Subsidiebedrag | € 1.493.715 |
Totale projectbegroting | € 1.493.715 |
Tijdlijn
Startdatum | 1-9-2023 |
Einddatum | 31-8-2028 |
Subsidiejaar | 2023 |
Partners & Locaties
Projectpartners
- UNIVERSITEIT VAN AMSTERDAMpenvoerder
Land(en)
Vergelijkbare projecten binnen European Research Council
Project | Regeling | Bedrag | Jaar | Actie |
---|---|---|---|---|
Foundations of GeneralizationThis project aims to explore generalization in overparameterized learning models through stochastic convex optimization and synthetic data generation, enhancing understanding of modern algorithms. | ERC Starting... | € 1.419.375 | 2024 | Details |
Coming to Terms: Proof Theory Extended to Definite Descriptions and other TermsExtenDD integrates proof theory and complex terms by developing formal theories of definite descriptions and enhancing sequent calculus, impacting automated deduction and philosophy of language. | ERC Advanced... | € 1.629.775 | 2022 | Details |
Logics and Algorithms for a Unified Theory of HyperpropertiesThis project aims to develop a unified theory and formal tools for hyperproperties in software, focusing on societal values like privacy and fairness, to enhance program verification and synthesis. | ERC Advanced... | € 2.227.500 | 2022 | Details |
The Formal Turn - The Emergence of Formalism in Twentieth-Century ThoughtThis project aims to provide an interdisciplinary study of the emergence and implications of formalism in early 20th-century science and logic through historical and comparative analyses. | ERC Consolid... | € 1.987.840 | 2022 | Details |
Unpacking Paradigmatic GapsUNPAG aims to explore and explain the rich landscape of universal paradigmatic gaps in language, focusing on the absence of negated quantifiers, to enhance understanding of cognition and communication. | ERC Advanced... | € 2.492.200 | 2024 | Details |
Foundations of Generalization
This project aims to explore generalization in overparameterized learning models through stochastic convex optimization and synthetic data generation, enhancing understanding of modern algorithms.
Coming to Terms: Proof Theory Extended to Definite Descriptions and other Terms
ExtenDD integrates proof theory and complex terms by developing formal theories of definite descriptions and enhancing sequent calculus, impacting automated deduction and philosophy of language.
Logics and Algorithms for a Unified Theory of Hyperproperties
This project aims to develop a unified theory and formal tools for hyperproperties in software, focusing on societal values like privacy and fairness, to enhance program verification and synthesis.
The Formal Turn - The Emergence of Formalism in Twentieth-Century Thought
This project aims to provide an interdisciplinary study of the emergence and implications of formalism in early 20th-century science and logic through historical and comparative analyses.
Unpacking Paradigmatic Gaps
UNPAG aims to explore and explain the rich landscape of universal paradigmatic gaps in language, focusing on the absence of negated quantifiers, to enhance understanding of cognition and communication.