Tractatus Logico-Philosophicus · Chapter
5.152 explained
Wittgenstein defines independent propositions as those sharing no truth-arguments, assigning them a probability of 1/2.
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.
Characters to notice
- Mr. Wittgenstein
Author of the Tractatus, presenting the logical relationship between independence, probability, and logical consequence.
Key passages
“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.