Definition:Truth Function

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Definition

Let $\mathbb B$ be the set of truth values, and 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:

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


Logical Conjunction

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

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


Logical Disjunction

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

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


Conditional

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

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


Biconditional

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

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


Exclusive Disjunction

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

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


Logical NAND

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

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


Logical NOR

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

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


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.


Also see

  • Results about truth functions can be found here.


Sources