Definition:Truth Function

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Definition

Let $\mathbb B$ be the Boolean domain $\set {\T, \F}$.

Let $k$ be a natural number.

A mapping $f: \mathbb B^k \to \mathbb B$ is called a truth function.


Truth Functions of Connectives

The logical connectives are assumed to be truth-functional.

Hence, they are represented by certain truth functions.


Logical Negation

The logical not connective defines the truth function $f^\neg$ as follows:

\(\ds \map {f^\neg} \F\) \(=\) \(\ds \T\)
\(\ds \map {f^\neg} \T\) \(=\) \(\ds \F\)


Logical Conjunction

The conjunction connective defines the truth function $f^\land$ as follows:

\(\ds \map {f^\land} {\F, \F}\) \(=\) \(\ds \F\)
\(\ds \map {f^\land} {\F, \T}\) \(=\) \(\ds \F\)
\(\ds \map {f^\land} {\T, \F}\) \(=\) \(\ds \F\)
\(\ds \map {f^\land} {\T, \T}\) \(=\) \(\ds \T\)


Logical Disjunction

The disjunction connective defines the truth function $f^\lor$ as follows:

\(\ds \map {f^\lor} {\F, \F}\) \(=\) \(\ds \F\)
\(\ds \map {f^\lor} {\F, \T}\) \(=\) \(\ds \T\)
\(\ds \map {f^\lor} {\T, \F}\) \(=\) \(\ds \T\)
\(\ds \map {f^\lor} {\T, \T}\) \(=\) \(\ds \T\)


Conditional

The conditional connective defines the truth function $f^\to$ as follows:

\(\ds f^\to \left({F, F}\right)\) \(=\) \(\ds T\)
\(\ds f^\to \left({F, T}\right)\) \(=\) \(\ds T\)
\(\ds f^\to \left({T, F}\right)\) \(=\) \(\ds F\)
\(\ds f^\to \left({T, T}\right)\) \(=\) \(\ds T\)


Biconditional

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

\(\ds \map {f^\leftrightarrow} {\F, \F}\) \(=\) \(\ds \T\)
\(\ds \map {f^\leftrightarrow} {\F, \T}\) \(=\) \(\ds \F\)
\(\ds \map {f^\leftrightarrow} {\T, \F}\) \(=\) \(\ds \F\)
\(\ds \map {f^\leftrightarrow} {\T, \T}\) \(=\) \(\ds \T\)


Exclusive Disjunction

The exclusive or connective defines the truth function $f^\oplus$ as follows:

\(\ds f^\oplus \left({F, F}\right)\) \(=\) \(\ds F\)
\(\ds f^\oplus \left({F, T}\right)\) \(=\) \(\ds T\)
\(\ds f^\oplus \left({T, F}\right)\) \(=\) \(\ds T\)
\(\ds f^\oplus \left({T, T}\right)\) \(=\) \(\ds F\)


Logical NAND

The NAND connective defines the truth function $f^\uparrow$ as follows:

\(\ds \map {f^\uparrow} {\F, \F}\) \(=\) \(\ds \T\)
\(\ds \map {f^\uparrow} {\F, \T}\) \(=\) \(\ds \T\)
\(\ds \map {f^\uparrow} {\T, \F}\) \(=\) \(\ds \T\)
\(\ds \map {f^\uparrow} {\T, \T}\) \(=\) \(\ds \F\)


Logical NOR

The NOR connective defines the truth function $f^\downarrow$ as follows:

\(\ds \map {f^\downarrow} {\F, \F}\) \(=\) \(\ds \T\)
\(\ds \map {f^\downarrow} {\F, \T}\) \(=\) \(\ds \F\)
\(\ds \map {f^\downarrow} {\T, \F}\) \(=\) \(\ds \F\)
\(\ds \map {f^\downarrow} {\T, \T}\) \(=\) \(\ds \F\)


Also known as

Some sources hyphenate: truth-function.

Others speak of a boolean function or a boolean operator, alluding to the fact that there are two possible outputs.

The name Boolean here is for George Boole, the pioneer of what is often referred to as Boolean algebra.


Also see

  • Results about truth functions can be found here.


Sources