What exactly does tautology mean?

Question

I'm struggling to understand the actual meaning of tautology. I know how to determine tautology with a truth table. But I don't understand what it actually means, so consider the following reasoning:

"If I know I'm dead, then I'm dead" and "If I know I'm dead, then I'm not dead". Therefore "I don't know I'm dead.

This can be formalised to $\left( \left( P\Rightarrow Q\right) \wedge \left( P\Rightarrow \lnot Q\right) \right) \Rightarrow \lnot P$ where $P:$"I know I'm dead" and $Q:$ "I'm dead".

Using a truth table I can determine that this statement is indeed a tautology. But what does that actually mean? Does it mean that I can't ever know if I'm dead or are the statements simply worded in a way that results in a tautology? Is the conclusion true in the real world?

Answer 1

The statement is true in the real world; it just doesn’t actually say anything very interesting, and in particular it doesn’t say that its conclusion $\neg P$ is true. The statement says that

• if knowing that you’re dead implies that you really are dead,

• and knowing that you’re dead implies that you really aren’t dead,

• then you don’t know that you’re dead.

This is undeniably the case, since you cannot be both dead and not dead at the same time. However, it’s completely uninteresting unless you can imagine a situation in which knowing that you’re dead really does imply both that you are and that you are not dead. Until such a situation arises, the statement is vacuously true: it’s true because it’s an implication whose premise is false. As long as the condition specified in the premise is not met, the statement really says nothing much at all.

Answer 2

To simplify, a tautology in plain English is stating the same thing twice but in a different manner. So for example, the statement "this meaningless statement is non-meaningful" is a tautology, because it is essentially restating the same thing. This definition is analogous to the mathematical definition.

Mathematically, a statement $ S $ involving propositions $ P, Q $ is called a tautology iff for any truth-values of $ P, Q $, the statement $ S $ is true. For example, "proof by contradiction" is in fact a tautology.

Consider what we are actually doing when we write a proof by contradiction. Formally, it is given by $$ ( P \wedge (\neg Q \implies \neg P) ) \implies Q $$ What we are essentially doing is "stating" the same thing in two manners:

1. The statement $ S $ is true
2. The opposite of statement $ S $ cannot be true (because of the contradiction)

If you understand this then you'll become very comfortable understanding tautologies. Hope this helps!

Answer 3

In mathematical logic, tautology means a propositional valid formula :

a formula $\alpha$ is a tautoloy iff every truth assignment (for the sentence symbols in $\alpha$) satisfies it.

The terms in natural language means : "saying the same thing twice".

Thus, it stay for something redundant, not informative.

The term was used firstly by Ludwing Wittgenstein into his Tractatus Logico-Philosophicus (1921) :

4.4.6 Among the possible groups of truth-conditions there are two extreme cases. In one of these cases the proposition is true for all the truth-possibilities of the elementary propositions. We say that the truth-conditions are tautological.

We can understand W's use of the traditional term with a new "technical" meaning if we think at his view of the langugae : a tautology "says nothing" about the world because it is always true, i.e. it does not rule out any (truth-)possibility.