Definition:Material Equivalence

Definition

Material Equivalence is a binary connective written symbolically as $p \iff q$ whose behaviour is as follows:

$p \iff q$ is defined as $\left({p \implies q}\right) \land \left({q \implies p}\right)$

Thus, $p \iff q$ means:

• $p$ is true if and only if $q$ is true

• $p$ is (logically) equivalent to $q$

• $p$ is true iff $q$ is true

$p \iff q$ can be voiced:

$p$ if and only if $q$.

Other names for this operator include:

• Biconditional

• Logical Equivalence

• Logical Equality

It can be written:

$\displaystyle {\left({p \implies q}\right) \quad \left({q \implies p}\right) \over p \iff q} \qquad \qquad {p \iff q \over p \implies q} \qquad \qquad {p \iff q \over q \implies p}$

Boolean Interpretation

From the above, we see that the boolean interpretations for $\mathbf A \iff \mathbf B$ under the model $\mathcal M$ are:

$\left({\mathbf A \iff \mathbf B}\right)_{\mathcal M} = \begin{cases} T & : \mathbf A_{\mathcal M} = \mathbf B_{\mathcal M} \\ F & : \text {otherwise} \end{cases}$

Complement

The complement of $\iff$ is the exclusive or operator.

Truth Function

The biconditional connective defines the truth function $f^\leftrightarrow$ as follows:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle f^\leftrightarrow \left({F, F}\right)\)	\(=\)	\(\displaystyle T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle f^\leftrightarrow \left({F, T}\right)\)	\(=\)	\(\displaystyle F\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle f^\leftrightarrow \left({T, F}\right)\)	\(=\)	\(\displaystyle F\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle f^\leftrightarrow \left({T, T}\right)\)	\(=\)	\(\displaystyle T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Truth Table

The truth table of $p \iff q$ and its complement is as follows:

$\begin{array}{|cc||c|c|} \hline p & q & p \iff q & p \oplus q \\ \hline F & F & T & F \\ F & T & F & T \\ T & F & F & T \\ T & T & T & F \\ \hline \end{array}$

Semantics of Equivalence

The concept of material equivalence has been defined as:

$p \iff q$ means $\left({p \implies q}\right) \land \left({q \implies p}\right)$

So $p \iff q$ means:

If $p$ is true then $q$ is true, and if $q$ is true then $p$ is true.

$p \iff q$ can be considered as a shorthand to replace the use of the longer and more unwieldy expression involving two conditionals and a conjunction.

If we refer to ways of expressing the conditional, we see that:

• $q \implies p$ can be interpreted as $p$ is true if $q$ is true, and

• $p \implies q$ can be interpreted as $p$ is true only if $q$ is true.

Thus we arrive at the usual way of reading $p \iff q$ which is: $p$ is true if and only if $q$ is true.

This can also be said as:

• The truth value of $p$ is equivalent to the truth value of $q$.

• $p$ is equivalent to $q$.

• $p$ and $q$ are equivalent.

• $p$ and $q$ are coimplicant.

• $p$ and $q$ are materially equivalent.

• $p$ is true exactly when $q$ is true.

• $p$ is true iff $q$ is true. This is another convenient and useful (if informal) shorthand which is catching on in the mathematical community.

Necessary and Sufficient

If $p \iff q$, we can say that $p$ is necessary and sufficient for $q$.

This is a consequence of the definitions of necessary and sufficient conditions.

Notational Variants

Various symbols are encountered that denote the concept of material equivalence:

Symbol	Origin
$p \iff q$
$p\ \mathsf{EQ} \ q$
$p \equiv q$	Bertrand Russell and Alfred North Whitehead: Principia Mathematica (1910)
$p = q$
$p \leftrightarrow q$
$\operatorname E p q$	Łukasiewicz's Polish notation

It is usual in mathematics to use $\iff$, as there are other uses for the other symbols.

See also

• Therefore
• Because
• Interderivable (Logical Equivalence)
• Conditional

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 2$: The Axiom of Specification
• Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{II}: \S 1, \ \S 3$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.2$
• E.J. Lemmon: Beginning Logic (1965): $\S 1.4$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Introduction
• Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.5$
• Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.2$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 4$
• D.J. O'Connor and Betty Powell: Elementary Logic (1980): $\S 1.12, \ \S 1.13$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.1$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): $\S 1.1$