# 5.513 explained — Tractatus Logico-Philosophicus

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

## What happens

Wittgenstein discusses the commonality among symbols that assert conjunctions and disjunctions, and argues that each proposition has only one negation because only one proposition lies entirely outside it. He also notes that in Russell's notation, 'q : p ∨ ~p' says the same as 'q', and 'p ∨ ~p' says nothing.

## Themes in this chapter

### Logic and Tautology

The discussion of 'p ∨ ~p' as saying nothing illustrates the nature of tautologies.

### Nonsense and Meaning

The proposition 'p ∨ ~p' says nothing, highlighting the boundary between sense and nonsense.

## Character check-ins

### Mr. Wittgenstein

Author of the proposition

### Bertrand Russell

Referenced for his notation

## Key lines

> What is common to all symbols, which assert both p and q , is the proposition “ p . q ”.

The common feature of all symbols that assert both p and q is the conjunction 'p and q'.

_Clarifies the shared logical form of conjunctions._

> Thus in Russell’s notation also it appears evident that “ q : p ∨ ~ p ” says the same thing as “ q ”; that “ p ∨ ~ p ” says nothing.

In Russell's notation, 'q and (p or not p)' is equivalent to 'q' alone, because 'p or not p' is a tautology that adds no information.

_Illustrates the vacuity of tautologies in logical notation._

## Links

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