Definition:Tautology

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Definition

A tautology is a statement form which is always true, no matter what the truth value of its component substatements.

A tautology is symbolised by $\top$, and referred to as a top.

This is also known as a logical truth.

It is possible to express a theorem $\vdash \phi$ as $\top \dashv \vdash \phi$, which can be broken down conceptually into:

$\top \vdash \phi$: this emphasises that logical truth logically implies $\phi$.
$\phi \vdash \top$: this emphasises that $\phi$ logically implies logical truth.

In the context of logical formulas the term valid is usually used:

A logical formula $P$ is valid if its value is True in all boolean interpretations.

If $P$ is valid it can be denoted $\models P$.

The notation here is that of model: the implication here is that $P$ is satisfied by any model.

Let $\mathbf A$ be a propositional WFF.

Then $\mathbf A$ is a tautology iff it is true in every model:

$\forall \mathcal M: \mathcal M \models \mathbf A$

There is only one boolean interpretation for $\top$:

$v \left({\top}\right) = T$