TLP 1 in Symbolic Logic — "Die Welt ist alles, was der Fall ist"

The Proposition

Tractatus Logico-Philosophicus, Proposition 1

"Die Welt ist alles, was der Fall ist."
"The world is everything that is the case."

A more precise reading: "The world is the totality of facts, not of things."

Wittgenstein's key move: Tatsachen (facts) vs Sachen (things/objects). The world is not a collection of objects — it is a collection of facts. A fact is an atomic state of affairs that obtains.

Formalisation

We define:

Then the proposition is a biconditional — both directions must hold:

(1)  (∀x) [ F(x) → W(x) ]      Every fact is in the world
(2)  (∀x) [ W(x) → F(x) ]      Everything in the world is a fact

Combined:

(3)  (∀x) [ W(x) ↔ F(x) ]      The world = exactly the facts that are the case

Or equivalently, with explicit predicates:

(4)  (∀x) [ W(x) ↔ C(x) ]      x is in the world IFF x is the case

The "All Facts" Completeness Condition (TLP 1.11)

The "all" in "everything that is the case" means the world is a closed totality:

(5)  W ≡ { x | C(x) }           The world IS the set of all things that are the case

The world is not about the facts — it is the facts. This is what makes it complete and closed.

The Three Key Moves

Wittgenstein's Point

This is not a synthetic claim (about what exists in the world) — it is an analytic clarification of what the word "world" means when used properly.

It is also, as Russell noted, surprisingly austere: if the world is only facts, then things (objects, substances) have no independent ontological status. Only their arrangements in facts matter.

TLP 1 in Context — The Propositions That Follow

That last one is doing something important: the world determines both positive and negative facts. What isn't the case is as much a feature of the world as what is.

Prepared by FAYE-9000 | 12 July 2026