Tractatus Logico-Philosophicus · Chapter
4.42 explained
This proposition introduces a combinatorial formula for the number of truth-possibilities of n elementary propositions, specifically the sum over K from 0 to n of the binomial coefficient (n choose K), which equals L_n.
What happens
This proposition introduces a combinatorial formula for the number of truth-possibilities of n elementary propositions, specifically the sum over K from 0 to n of the binomial coefficient (n choose K), which equals L_n. It formalizes the logical space of agreement and disagreement between a proposition and the truth-possibilities of its constituent elementary propositions.
Free preview opens on this chapter — companions, themes, and character notes appear beside the text.
Narrative arc
Story tension across the book.
- Setup
- Escalation
- Breaking point
- Aftermath
- Closing
This passage marks the story’s dramatic climax because it presents the moment when the protagonist’s obsessive…
Full beat labels and climax notes unlock in the reader.
Follow the arc while readingThemes in this chapter
- Logical Atomism
The proposition deals with elementary propositions and their truth-possibilities, central to logical atomism's view that complex propositions are truth-functions of atomic facts.
- Logic and Tautology
The combinatorial enumeration of truth-possibilities underlies the logical structure of propositions, including tautologies and contradictions, which are limiting cases of agreement and disagreement.
Key passages
“With regard to the agreement and disagreement of a proposition with the truth-possibilities of n elementary propositions there are ∑ K = 0 K n ( K n K ) = L n possibilities.”
For a proposition that can either agree or disagree with each of the truth-possibilities of n elementary propositions, the total number of such logical possibilities is given by the sum of binomial coefficients from K=0 to n, which equals L_n.
This formula quantifies the logical space of truth-functional combinations, showing how many distinct truth-conditions a proposition can have relative to n atomic facts.
