Are all tautologies logical truths

Tautology - Serlo "Math for non-freaks"

Tautology [edit]

The statement “If the rooster crows on the dung, then the weather changes or it stays as it is” is a tautology.

There are statements that are always true. The classic example of this is the farmer's rule: "When the rooster crows on the dung, the weather changes or it stays as it is." stand for the statement "The cock crows on the dung" and for "The weather is changing", we can formalize this pawn rule as follows:

Since statements are either "true" or "false", it is easy to see that the pawn rule is always true. It doesn't matter at all whether the rooster crows or not. Because or - one of these two statements is true. It also doesn't matter what exactly you are using what is meant is: it just has to be a statement! could also stand for the claim: "There are little green men on Mars."

Why is it that this statement is always true? It is because of how the statement is assembled from partial statements with joiners. We know the negation reverses the truth value: off becomes and vice versa becomes . The OR connection becomes true if one of the two partial statements is true. So is always true. The implication is only wrong if the premise is true and the conclusion wrong is. In our example, however, the conclusion is always true. Hence is also always true. Statements made up of connective elements that are always true become Tautologies or general statements called:

definition (Tautology)

A statement made up of junctions is called tautological or generalif it is true for every possible interpretation of its partial statements with truth values.

Particularly important tautologies are equivalences. Two statements and namely are equivalent if and only if the compound statement is a tautology. This is often used in evidence. Instead of the direct statement to prove becomes an equivalent statement shown.

example (Equivalent statements)

The following three statements are equivalent:

  • (Counterposition)
  • (Proof of contradiction)

So there are tautological:

An alternative formulation of the evidence of contradiction is incidentally .

Review of a tautology

We are now going to introduce three ways you can check whether a proposition is a tautology or not. All of these possibilities should be exemplified by the counterposition be demonstrated.

Create truth table [edit]

One method is to set up a truth table for the statement to be examined, see chapter "Truth table". If only “true” appears as the resulting truth value in the last column of the truth table, the statement being examined is a tautology. As soon as a resulting truth value is "false", the statement is not a tautology.

Task: Establish the truth table for on.

Result: The statement is a tautology.

Task: Establish the truth table for on.