Definition:Tautology

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Definition

A tautology is a statement which is always true, independently of any relevant circumstances that could theoretically influence its truth value.

It is epitomised by the statement form:

$p \implies p$

that is:

if $p$ is true then $p$ is true.

An example of a "relevant circumstance" here is the truth value of $p$.


The archetypal tautology is symbolised by $\top$, and referred to as Top.


Tautologies in Formal Semantics

Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.


A logical formula $\phi$ of $\LL$ is a tautology for $\mathscr M$ if and only if:

$\phi$ is valid in every structure $\MM$ of $\mathscr M$


That $\phi$ is a tautology for $\mathscr M$ can be denoted as:

$\models_{\mathscr M} \phi$


Also known as

A tautology is also known as a logical truth.


Also defined as

Some sources define a tautology as a statement form which can be epitomised by:

$p \lor \lnot p$

which, while intuitively obvious, it not a universal definition as it does not apply in contexts in which Law of Excluded Middle does not necessarily hold.


Examples

Excluded Middle

The Law of Excluded Middle:

$p \lor \lnot p$

is an example of a tautology.


Arbitrary Example $1$

The WFF of propositional logic:

$\paren {\paren {\paren {\lnot p} \implies q} \implies \paren {\paren {\paren {\lnot p} \implies \paren {\lnot q} } \implies p} }$

is a tautology.


Arbitrary Example $2$

The WFF of propositional logic:

$\paren {\paren {\lnot p} \implies \paren {q \lor r} } \iff \paren {\paren {\lnot q} \implies \paren {\paren {\lnot r} \implies p} }$

is a tautology.


Also see

  • Results about tautology can be found here.


Sources

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