Overview

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 can turn a statement about some 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.

Generalisations are formed using quantifiers, i.e., expressions such as ‘every’, ‘for all’, ‘some’, ‘there is’, and ‘a’. Quantifiers raise many important questions. For instance, in logic and formal semantics we study what rules of inference involving a given quantifier are sound, what logical consequences a sentence involving a given quantifier has, and what the truth conditions of sentences involving a given quantifier are. In philosophy, we investigate, for example, the relation between quantification and existence. Many philosophers maintain that quantification and existence are intimately related: we should believe in the existence of Xs if quantification over Xs is required in our best scientific theories.

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 the syntactic position of singular terms (i.e., expressions that denote individual objects, such as ‘Socrates’ or ‘1’). However, many theoretical contexts require generalisation into sentence and predicate positions. Very roughly, generalisation into sentence and predicate positions is a high-level form of generalisation in which we make a general statement about a class of statements, For instance, it is required to articulate the principle of mathematical induction, the laws of logic, and the Ramsey sentence of a scientific theory.

We can distinguish two competing methods for achieving generalisation into sentence and predicate positions:

1. The direct method: By adding variables that can stand in the syntactic position of sentences and predicates, and quantifiers for them. This method is exemplified in the use of second- and higher-order logic (type theory).
2. The indirect method: By adding singular terms that are obtained from sentences and predicates by nominalising transformations, or by ascending to a metalanguage and attributing semantic properties to linguistic expressions or their contents. This method is exemplified in the use of formal theories of reified properties, sets, and classes, and formal theories of truth and satisfaction.

Since both methods come with their own distinctive 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.