# 4.1273 explained — Tractatus Logico-Philosophicus

> Chapter companion for Tractatus Logico-Philosophicus by Ludwig Wittgenstein.

## What happens

Wittgenstein argues that expressing the general proposition 'b is a successor of a' in logical symbolism requires a variable for the general term of a formal series, because the concept 'term of this formal series' is a formal concept. He criticizes Frege and Russell for overlooking this, claiming their method involves a vicious circle.

## Themes in this chapter

### Logical Atomism

The discussion of formal series and the need for a variable to express the general term reflects Wittgenstein's logical atomist approach to analyzing propositions.

### Critique of Traditional Philosophy

Wittgenstein directly criticizes Frege and Russell for their handling of general propositions, pointing out a vicious circle in their symbolism.

## Character check-ins

### Mr. Wittgenstein

Author of the proposition, presenting his critique of Frege and Russell's logical notation.

### Frege

Mentioned as one who overlooked the formal concept of the general term of a formal series.

### Bertrand Russell

Mentioned alongside Frege as having overlooked the formal concept, leading to a vicious circle in their expression of general propositions.

## Key lines

> If we want to express in logical symbolism the general proposition “b is a successor of a” we need for this an expression for the general term of the formal series: a R b , ( ∃ x ) : a R x . x R b , ( ∃ x , y ) : a R x . x R y . y R b .⁠ ⁠…

To symbolize 'b is a successor of a' in logic, we must have a way to represent the general term of the formal series that lists all possible chains of relations from a to b.

_Wittgenstein illustrates the need for a variable to capture the infinite series of relational steps._

> The general term of a formal series can only be expressed by a variable, for the concept symbolized by “term of this formal series” is a formal concept. (This Frege and Russell overlooked; the way in which they express general propositions like the above is, therefore, false; it contains a vicious circle.)

Because 'term of this formal series' is a formal concept, it cannot be captured by a function or predicate but only by a variable. Frege and Russell missed this, so their expressions for such propositions are circular and incorrect.

_Wittgenstein's critique centers on the distinction between formal and proper concepts, accusing Frege and Russell of a logical error._

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