Tractatus Logico-Philosophicus · Chapter
5.43 explained
Wittgenstein argues that from a single fact p, an infinite number of logical propositions (like ~~p, ~~~~p, etc.) follow, which seems unbelievable.
What happens
Wittgenstein argues that from a single fact p, an infinite number of logical propositions (like ~~p, ~~~~p, etc.) follow, which seems unbelievable. He notes that it is equally remarkable that all logical and mathematical propositions follow from a few primitive propositions, but explains that all such logical propositions say the same thing—namely, nothing.
Themes in this chapter
Logic and Tautology
Wittgenstein asserts that all logical propositions say the same thing—nothing—highlighting their tautological nature.
Nonsense and Meaning
The claim that logical propositions say nothing touches on the boundary between sense and nonsense.
Characters to notice
- Mr. Wittgenstein
Author of the proposition; presents the argument about the infinite consequences of a fact and the nature of logical propositions.
Key passages
“That from a fact p an infinite number of others should follow, namely, ~ ~ p , ~ ~ ~ ~ p , etc. , is indeed hardly to be believed, and it is no less wonderful that the infinite number of propositions of logic (of mathematics) should follow from half a dozen 'primitive propositions.'”
It is hard to accept that from a single fact p, an endless series of negated propositions (like not-not-p, not-not-not-not-p, etc.) follows, and it is equally astonishing that all logical and mathematical truths can be derived from just a few basic axioms.
Wittgenstein highlights the seemingly paradoxical fecundity of logical inference from simple facts and axioms.
“But the propositions of logic say the same thing. That is, nothing.”
However, all logical propositions are equivalent in meaning—they express nothing at all.
This underscores the tautological and contentless nature of logical truths.