Definition:Truth Function/Connective

From ProofWiki
Jump to navigation Jump to search

Definition

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:

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