Generalisation into sentence and predicate positions

Descripción del proyecto

Avanzar en la ciencia de la generalización

La generalización nos ayuda a convertir hechos concretos en principios generales, como «Todas las células tienen una membrana». Si bien entendemos la generalización para los objetos, como células o números, la generalización de alto nivel sobre los propios enunciados es más compleja. Se trata de elegir entre dos métodos: la lógica de orden superior o las teorías autoaplicables. El equipo del proyecto GENERALISE, financiado por el CEI, aborda estos retos investigando estos dos métodos clave. Con esta investigación pionera, promete perfeccionar la elaboración de teorías deductivamente sólidas, con repercusiones en campos que van desde las matemáticas y la lógica hasta la informática y la filosofía. Mediante la exploración de estos métodos fundamentales, en GENERALISE se intenta avanzar en nuestra comprensión de la generalización en la ciencia y más allá.

Objetivo

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.

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. But 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).

There are two competing methods for achieving this form of generality (i.e. higher-order logic and self-applicable theories of truth, properties, and sets respectively). 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.

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. 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. 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.