Logical equality

Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.

A logical equality of propositions $p$ and $q$ may be written in various ways.  Among the most common are these:

• $p\iff q$
• $p\equiv q$
• $p=q$

A truth table for $p=q$ appears below:

$\text{Logical Equality}$

$p$	$q$	$p=q$
$\mathrm{F}$	$\mathrm{F}$	$\mathrm{T}$
$\mathrm{F}$	$\mathrm{T}$	$\mathrm{F}$
$\mathrm{T}$	$\mathrm{F}$	$\mathrm{F}$
$\mathrm{T}$	$\mathrm{T}$	$\mathrm{T}$

A logical graph for $p=q$ is shown below:

The traversal string of this graph is $\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mo stretchy="false">(</mo></mrow>}p\text{<mo>,</mo>}q\text{<mrow data-mjx-texclass="ORD"><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow>}.$  The proposition $p=q$ may be taken as a Boolean function $f(p,q)$ having the abstract type $f:\mathbb{𝔹}\times \mathbb{𝔹}\to \mathbb{𝔹},$ where $\mathbb{𝔹}=\{0,1\}$ is interpreted in such a way that $0$ means $\mathrm{f}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}$ and $1$ means $\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}.$

A Venn diagram for $p=q$ indicates the region where $p=q$ is true by means of a distinctive color or shading.  In this case the region consists of two single cells, as shown below:

Resources

• Logic Syllabus

Document history

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

• Logical Equality, InterSciWiki
• Logical Equality, Wikiversity
• Logical Equality, Wikipedia