Definition:Truth Function/Connective
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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:
| \(\ds \map {f^\to} {\F, \F}\) | \(=\) | \(\ds \T\) | ||||||||||||
| \(\ds \map {f^\to} {\F, \T}\) | \(=\) | \(\ds \T\) | ||||||||||||
| \(\ds \map {f^\to} {\T, \F}\) | \(=\) | \(\ds \F\) | ||||||||||||
| \(\ds \map {f^\to} {\T, \T}\) | \(=\) | \(\ds \T\) |
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:
| \(\ds \map {f^\oplus} {\F, \F}\) | \(=\) | \(\ds \F\) | ||||||||||||
| \(\ds \map {f^\oplus} {\F, \T}\) | \(=\) | \(\ds \T\) | ||||||||||||
| \(\ds \map {f^\oplus} {\T, \F}\) | \(=\) | \(\ds \T\) | ||||||||||||
| \(\ds \map {f^\oplus} {\T, \T}\) | \(=\) | \(\ds \F\) |
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:
| \(\ds \map {f^\downarrow} {\F, \F}\) | \(=\) | \(\ds \T\) | ||||||||||||
| \(\ds \map {f^\downarrow} {\F, \T}\) | \(=\) | \(\ds \F\) | ||||||||||||
| \(\ds \map {f^\downarrow} {\T, \F}\) | \(=\) | \(\ds \F\) | ||||||||||||
| \(\ds \map {f^\downarrow} {\T, \T}\) | \(=\) | \(\ds \F\) |