Tractatus Logico-Philosophicus · Chapter
5.5321 explained
Wittgenstein introduces a new notation for expressing that only one object satisfies a given function, replacing the identity sign with existential quantifiers and negation to avoid the use of '='.
What happens
Wittgenstein introduces a new notation for expressing that only one object satisfies a given function, replacing the identity sign with existential quantifiers and negation to avoid the use of '='.
Themes in this chapter
Logical Atomism
The notation reflects the analysis of propositions into elementary components and the elimination of unnecessary logical constants.
Nonsense and Meaning
By replacing the identity sign with quantifiers, Wittgenstein aims to avoid the nonsensical use of '=' in logical notation.
Characters to notice
- Mr. Wittgenstein
Author of the Tractatus, presenting a logical notation that eliminates the identity sign.
Key passages
“Instead of “ ( x ) : f x ⊃ x = a ” we therefore write e.g. “ ( ∃ x ) . f x . ⊃ . f a : ~ ( ∃ x , y ) . f x . f y ”.”
We replace the formula 'for all x, if f(x) then x equals a' with 'there exists an x such that f(x) implies f(a), and there do not exist two distinct x and y such that f(x) and f(y)'.
Wittgenstein demonstrates how to express uniqueness without using the identity sign.
“And if the proposition “ only one x satisfies f ( ) ” reads: “ ( ∃ x ) . f x : ~ ( ∃ x , y ) . f x . f y ”.”
The statement 'only one x satisfies f' is written as 'there exists an x such that f(x), and there do not exist two distinct x and y such that f(x) and f(y)'.
This notation captures the idea of exactly one object satisfying a property without using identity.