# 5.152 explained — Tractatus Logico-Philosophicus

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

## What happens

Wittgenstein defines independent propositions as those sharing no truth-arguments, assigning them a probability of 1/2. He explains that logical consequence yields probability 1, making logical certainty a limiting case of probability, with application to tautology and contradiction.

## Themes in this chapter

### Logic and Tautology

The limiting case of probability is applied to tautology and contradiction, showing logical certainty as a boundary of probabilistic reasoning.

## Character check-ins

### Mr. Wittgenstein

Author of the Tractatus, presenting the logical relationship between independence, probability, and logical consequence.

## Key lines

> Propositions which have no truth-arguments in common with one another we call independent.

Propositions that share no truth-arguments are termed independent.

_Defines independence in terms of disjoint truth-arguments._

> Independent propositions ( e.g. any two elementary propositions) give to one another the probability ½.

Any two independent propositions, such as elementary propositions, each confer a probability of 1/2 on the other.

_Establishes the baseline probability for independent propositions._

> If p follows from q , the proposition q gives to the proposition p the probability 1.

When q entails p, q assigns probability 1 to p.

_Logical consequence yields certainty._

> The certainty of logical conclusion is a limiting case of probability.

Logical certainty represents the extreme case within the spectrum of probability.

_Probability theory subsumes logical deduction as a boundary case._

## Links

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