Law of Identity

Definition

Every proposition entails itself:

$p \vdash p$

From the modus ponendo ponens and the rule of implication, this is equivalent to:

$\top \dashv \vdash p \implies p$

or simply:

$\vdash p \implies p$

This is also known as the rule of repetition.

A seemingly trivial rule, but can be surprisingly useful to get a particular formula into the right place in a proof.

The context can be expanded slightly:

$p \dashv \vdash p$

from which we immediately obtain:

$\top \dashv \vdash p \iff p$

or simply:

$\vdash p \iff p$

This is also demonstrated in Equivalence Properties.

Proof

Proof by Natural deduction

By the tableau method:

$p \vdash p$

Line		Pool	Formula	Rule	Depends upon
1		1	$p$	P	(None)

$\blacksquare$

This is the shortest tableau proof possible.

$\vdash p \implies p$

Line		Pool	Formula	Rule	Depends upon
1		1	$p$	P	(None)
2			$p \implies p$	$\implies \mathcal I$	1, 1

$\blacksquare$

This is the second shortest tableau proof possible.

Proof by Truth Table

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, in both cases the truth value under the main connective is $T$ throughout for both models.

$\begin{array}{|ccc||ccc|} \hline p & \implies & p & p & \iff & p \\ \hline F & T & F & F & T & F \\ T & T & T & T & T & T \\ \hline \end{array}$

$\blacksquare$

Interpretation by Models

Clearly, every model of $P$ is a model of $P$.

Thus by definition of logical consequence:

$P \models P$

Comment

Some sources, for example D.J. O'Connor and Betty Powell: Elementary Logic (1980), use the statement:

$\vdash p \implies p$

to be the defining property of a tautology.

Sources

• Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{I}: \S 3$
• Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{I}: \S 5$: Theorem $\text{T1}$
• Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{II}: \S 5$: Theorem $\text{T91}$
• E.J. Lemmon: Beginning Logic (1965): $\S 1.5$: Theorem $29$
• Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.3$
• D.J. O'Connor and Betty Powell: Elementary Logic (1980): $\S 1.3$
• Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems (2000): $\S 1.2.1$