Law of Identity

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Theorem

Every proposition entails itself:

Formulation 1

$p \vdash p$

Formulation 2

$\vdash p \implies p$


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


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