# Law of Identity

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## Theorem

Every proposition entails itself:

### Formulation 1

- $p \vdash p$

### Formulation 2

Every proposition entails itself:

- $\vdash p \implies p$

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

### Interpretation by Models

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

Thus by definition of semantic consequence:

- $P \models P$

## Also known as

This is also known as the **rule of repetition**.

## Also see

- Biconditional is Reflexive, where $\vdash p \iff p$ is shown.

## Technical Note

When invoking the **Law of Identity** in a tableau proof, use the `{{IdentityLaw}}`

template:

`{{IdentityLaw|line|pool|depends|statement}}`

or:

`{{IdentityLaw|line|pool|depends|statement|comment}}`

where:

`line`

is the number of the line on the tableau proof where Law of Identity is to be invoked`pool`

is the pool of assumptions (comma-separated list)`statement`

is the statement of logic that is to be displayed in the**Formula**column,**without**the`$ ... $`

delimiters`depends`

is the line of the tableau proof upon which this line directly depends`comment`

is the (optional) comment that is to be displayed in the**Notes**column.

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S \text{II}.12$: Laws of sentential calculus