Law of Identity

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

• Rule of Assumption

• 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