Landini (2012) Frege’s Notations What They Are and How They Mean (1)

Landini, Gregory (2012). Frege's Notations: What They Are and How They Mean. London and Basingstoke: Palgrave-Macmillan. [1-14]

 

※ Author’s Note on the Use of Modern Logical Notations

The book endeavors to explain Frege’s notations in terms of modern nota tions. It is therefore worth making a few comments on the modern nota tions used in the book that might not be familiar to readers. In addition to brackets, dots are used for punctuation to help avoid a proliferation of brackets. I always use dots symmetrically for ease of reading. Thus, for example,

I also allow subscripting a variable to the logical particles → and ↔ as convenient notation for universal quantification. For example,

Letters such as F and F ψ G, ϕ, ψ are predicate variables of the object language, while letters A, B are schematic for well-formed formulas. The sign ∀ is our universal quantifier and the sign ∃ is our existential quanti fier. The book does not follow Frege in adopting Roman letters in addi tion to Gothic letters. As we shall see, the book maintains that this is a distinction without a difference. The book does follow Frege in using letters ξ, Φ, Φ Ψ as parametric letters which are not part of Frege’s intended t formal languages. This is a distinction with a very important difference.

 

Introduction

 

1. Main Goal: Rehabilitating Frege's Formalism

A revolution was on its way in logic whose implications for philosophy would be every bit as momentous as the Copernican revolution in physics. 

Boole’s pioneering Laws of Thought (1854) offered a new algebraic approach to logic, and together with Peirce’s work and Schröder’s three volume Vorlesungen über die Algebra der Logik (1890–1905), this algebraic tradition marks a remarkable advance over Aristotelian and medieval methods. But when it comes to the revolution in logic, it is the mathematician Gottlob Frege who played the role of Copernicus.

Frege’s work can be separated into three basic phases:

Begriffsscrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (1879),
Die Grundlagen der Arithmetik: eine logische-mathematische Untersuchung über den Begriff der Zahl (1884)
Grundgesetze der Arithmetic (vol. I 1893, vol. II 1903).

Main thesis: We endeavor to turn this on its head – Frege’s popular writings must be understood in terms of the formal deductive systems embedded in his notations.

His philosophy, and indeed, central derivations of theorems of his systems, cannot be recovered by investigating what might have been thought to be straightforward translations of his notations into those of a modern predicate calculi.

Why it is that differences are so important between a modern predicate logic of first or higher order and a formal system of logic that is a function-calculus (a phrase which we shall use synonymously with “function-script”).

 

2. Function

Frege adopts functions as primitive indefinable entities.

2.1 Defining a Function through a Relational Sign

The notion of a “function” used by philosophers today, however, has changed from its original mathematical meaning. Many today identify functions with relations R that have the special feature that

A function relates each element of a set with exactly one element (not 2 or more) of another set (possibly the same set).

Some relations do not have this feature. But those that do are said to be “functional.” Here we use the arrow → sign for “if ... then.” We shall subscript the arrow to conveniently abbreviate universal quantification. Thus, the above can be written as

This says that for all x, y, and z, if x bears relation R to y and x bears relation R to z, then y = z. Now consider the formula

This says that the unique entity to which x bears R is y.

Here the expression “R ‘x” is a term and not a formula. Only a term can flank the identity sign.

There are different ways to introduce function signs such as “R ‘x” to facilitate derivations in a formal system. One method is to employ Russell's theory of definite descriptions. A more artificial yet convenient method is to adopt an axiom such as the following:

This axiom supports the elimination of the sign “R ‘x” from all expressions, and it is non-creative (i.e., any theorem proved with the sign “R ‘x” can be proved without it).

This approach is called the “chosen object view,” for in all cases where the relation R is not functional, the axiom assigns a chosen object Δ to be the referent of “R ‘x ‘ .”

Thus, with the axioms for identity, we readily arrive at the familiar theorem

2.2 Frege's notion of a function

It is absolutely essential to understand that this is not Frege’s approach. Frege’s work does not identify functions with special relations of this sort. Frege’s “fx” must not, therefore, be conflated with “R ‘x.” For Frege, the ontology of functions, not relations, is the foundation of logic (and arithmetic and analysis).

This difference in orientation is of utmost importance. Yet it has been widely neglected. It is easy to become wholly blind to it. The very expression “function calculus” has been corrupted by some followers of Alonzo Church, who, slighting all important differences between a term “fx” and a formula “Fx,” make no distinction between a function calculus and a predicate calculus! But more charitably, perhaps this neglect is due to the belief that the difference is ultimately insignificant for understanding Frege. On this view, the achievements of Frege’s work can adequately be represented without respecting the functions-as primitives orientation of his work. This book will show that this belief is mistaken. It took years for me to see this myself. There seemed no point in learning the details of Frege’s strange notations and proofs. It appeared to be enough to be able to translate his theorems, ignoring the odd eccentricities of his systems, into the normal language of a predicate calculus.

Consider, for example, Frege’s definition of u ^ v. Frege has

Why, then, go to the trouble of working in Frege’s archaic system?

2.3 The extension of a relation

Readers of Frege will first discover a reason for going to the trouble when they encounter Frege’s treatment of the mathematical notion of the extension of a relation.

There was a serious problem at the turn of the last century concerning what is to be the extension of a relation, R, say of two terms. Relations order their terms. For example, if R is the relation “loves”, xRy is not always equivalent to yRx. In 1912/1914, Wiener and Kuratowski found ways of capturing order in the extension of a relation. Define as follows: