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 invokedpool
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$ ... $
delimitersdepends
is the line of the tableau proof upon which this line directly dependscomment
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