Tractatus Logico-Philosophicus · Chapter
4.1273 explained
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.
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.
Characters to notice
- 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 passages
“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.