Tractatus Logico-Philosophicus · Chapter
5.1311 explained
Wittgenstein demonstrates how the logical connection between propositions in inference (e.g., from p ∨ q and ~p to q) is obscured by conventional notation but becomes obvious when using an alternative notation (the Sheffer stroke).
What happens
Wittgenstein demonstrates how the logical connection between propositions in inference (e.g., from p ∨ q and ~p to q) is obscured by conventional notation but becomes obvious when using an alternative notation (the Sheffer stroke). He also notes that generality is already present in the symbol (x).fx, as shown by the possibility of inferring fa from it.
Themes in this chapter
Logic and Tautology
The passage illustrates how logical inference relies on the inner connection of propositional forms, revealing tautological structures.
Picture Theory of Language
The discussion of notation and the 'inner connection' between propositions relates to how propositions picture logical relations.
Characters to notice
- Mr. Wittgenstein
Author of the proposition, analyzing logical inference and notation.
Key passages
“When we conclude from p ∨ q and ~ p to q the relation between the forms of the propositions “ p ∨ q ” and “ ~ p ” is here concealed by the method of symbolizing.”
The logical relationship between the premises in a disjunctive syllogism is hidden by the usual way we write them.
Wittgenstein points out that standard notation obscures the logical form that makes the inference valid.
“But if we write, e.g. instead of “ p ∨ q ” “ p | q . | . p | q ” and instead of “ ~ p ” “ p | p ” ( p | q = neither p nor q ), then the inner connection becomes obvious.”
If we use the Sheffer stroke notation, the logical connection between the premises becomes clear.
The alternative notation reveals the underlying logical structure that justifies the inference.
“(The fact that we can infer f a from ( x ) . f x shows that generality is present also in the symbol “ ( x ) . f x ”.”
The ability to deduce a specific instance from a universal statement shows that generality is built into the symbol for universal quantification.
Wittgenstein uses this to argue that generality is not merely a matter of interpretation but is encoded in the logical notation itself.