# 5.5321 explained — Tractatus Logico-Philosophicus

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

## 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.

## Character check-ins

### Mr. Wittgenstein

Author of the Tractatus, presenting a logical notation that eliminates the identity sign.

## Key lines

> 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._

## Links

- HTML: https://www.betterreads.online/discover/tractatus_logico_philosophicus_se/chapters/5-5321
- Book: https://www.betterreads.online/discover/tractatus_logico_philosophicus_se
- All chapters: https://www.betterreads.online/discover/tractatus_logico_philosophicus_se/chapters