Evolution of Relations

Question

In Frege, one finds relations treated as predicates in complex terms. However, modern set theory appears to treat them as two-place relation. Is this correct? If so, when did this shift occur and to whom is it attributed? And, what is the difference?

Answer 1

In Gottlob Frege's mature thougth, the basic analysis of language is based on functions and objects; see Gottlob Frege, Philosophical Writings (Geach & Black edition - 1952) : Function and Concept (Über Funktion und Begriff - 1891), page 21-on.

He start his analysis from the "typical" mathematical fuctions, like :

$x + 1$

which get a number as "input" and produce as "output" a number, but then he generalize them to concepts, i.e. functions from objects to truth-value, like :

$x > 1$.

In this setting, a binary relation is a function from objects to truth-value with two argument places, like :

$x > y$.

See page 39 :

We have here a function whose value is always a truth-value. We called such functions of one argument concepts; we call such functions of two arguments relations.

In modern mathematical logic, the first-order language is based on individual variables and predicate letters.

An (atomic) expression is : $R(x,y)$, where $R$ is a binary predicate letter. In the language of arithmetic, we can use instead of the predicate letter $R$ the symbol ">" standing for the (binary) relation "greater than"; thus, we have the expression : $x > y$.

The basic difference between modern semantic for a first-order language an that of Frege is that today an expression like $x > y$ it is not interpreted as a function from couples of objects to truth-value but as the set of all couples satisying the relation.

If $\mathbb N$ is set of natural numbers which is the domain of our interpretation, we have that, writing $R^N$ for the interpretation of $R$, in the standard mathematical logic semantics :

$>^N \subseteq \mathbb N \times \mathbb N$

while for Frege :

$>^N : \mathbb N \times \mathbb N \rightarrow \{ TRUE, FALSE \}$.

The "shift" occurred progressively; already in Alfred North Whitehead and Bertrand Russell, Principia Mathematica (1910-1927) relations was a primitive notion (and functions only "special" relations : the "functional" ones).

In 1914 Norbert Wiener (A Simplification of the logic of relations) and in 1921 Kazimierz Kuratowski (Sur la notion de l'ordre dans la Théorie des Ensembles) find a way to define in the language of set theory the concept of ordered pair thus reducing the notion of relation to that of set.

Thus, simplifying a lot, while for Frege a relation was a kind of function, in current mathematical logic a function is a type of relation which, in turn, is a special kind of set (which, for Frege, was the extension of a concept).