Tractatus Logico-philosophicus

Wittgenstein defines a tautology as a proposition that is true regardless of what is and what is not the case. Such a proposition, he says, has no sense and says nothing. In claiming that the propositions of logic are tautologies, he is claiming that they are empty propositions that tell us nothing about the world, but only show us something about the logical form of the world. This view of logic contradicts Frege and Russell, both of whom saw logic as a set of propositions deduced from fundamental axioms and laws of inference. According to these two philosophers, the propositions of logic describe the laws of thought: they are the most general laws there are that prescribe the form that more specific laws and thoughts must take. If logic consists of tautologies, Frege and Russell are wrong on two main points. First, logical propositions could not be laws, because laws have content and tautologies are empty. Second, the propositions of logic cannot be derived from axioms that are more fundamental than they are, since all propositions of logic (axioms included) say the same thing (i.e. nothing) and so they are all of equal value.

Wittgenstein states his reasoning most clearly at 2.0211: "If the world had no substance [if there were no objects], then whether a proposition had sense would depend on whether another proposition was true." Objects define the logical form of the world: we can use "purple" in sentences dealing with color and "two" in sentences dealing with numbers because of the logical form of these words. If it were simply a contingent fact that "purple" is a color-word, then the sentence "purple is a color" would have to be established as true before we could even know if "my car is purple" makes sense. Wittgenstein insists that the formal properties of objects cannot be contingent, since it would then be impossible to know if what we were saying makes any sense.

Wittgenstein observes that there is one fundamental logical connective from which all propositions can be derived. This connective is called the "Sheffer stroke" and is defined as follows: "p|q" means "not p and not q." By successively applying the Sheffer stroke to propositions, we can derive any combination of truth-values we like. Wittgenstein interprets the Sheffer stroke as an operation ("N(p,q)"), and argues that the general form of the proposition is a repeated application of this operation to elementary propositions.