Definition:Truth Function
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:
| \(\ds \map {f^\neg} \F\) | \(=\) | \(\ds \T\) | ||||||||||||
| \(\ds \map {f^\neg} \T\) | \(=\) | \(\ds \F\) |
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:
| \(\ds f^\lor \left({F, F}\right)\) | \(=\) | \(\ds F\) | ||||||||||||
| \(\ds f^\lor \left({F, T}\right)\) | \(=\) | \(\ds T\) | ||||||||||||
| \(\ds f^\lor \left({T, F}\right)\) | \(=\) | \(\ds T\) | ||||||||||||
| \(\ds f^\lor \left({T, T}\right)\) | \(=\) | \(\ds T\) |
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:
| \(\ds f^\leftrightarrow \left({F, F}\right)\) | \(=\) | \(\ds T\) | ||||||||||||
| \(\ds f^\leftrightarrow \left({F, T}\right)\) | \(=\) | \(\ds F\) | ||||||||||||
| \(\ds f^\leftrightarrow \left({T, F}\right)\) | \(=\) | \(\ds F\) | ||||||||||||
| \(\ds f^\leftrightarrow \left({T, T}\right)\) | \(=\) | \(\ds T\) |
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:
| \(\ds f^\uparrow \left({F, F}\right)\) | \(=\) | \(\ds T\) | ||||||||||||
| \(\ds f^\uparrow \left({F, T}\right)\) | \(=\) | \(\ds T\) | ||||||||||||
| \(\ds f^\uparrow \left({T, F}\right)\) | \(=\) | \(\ds T\) | ||||||||||||
| \(\ds f^\uparrow \left({T, T}\right)\) | \(=\) | \(\ds F\) |
Logical NOR
The NOR connective defines the truth function $f^\downarrow$ as follows:
| \(\ds f^\downarrow \left({F, F}\right)\) | \(=\) | \(\ds T\) | ||||||||||||
| \(\ds f^\downarrow \left({F, T}\right)\) | \(=\) | \(\ds F\) | ||||||||||||
| \(\ds f^\downarrow \left({T, F}\right)\) | \(=\) | \(\ds F\) | ||||||||||||
| \(\ds f^\downarrow \left({T, T}\right)\) | \(=\) | \(\ds F\) |
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.
Also see
- Definition:Truth Table, a common method for tabulating the definition of a truth function.
- Results about truth functions can be found here.
Sources
- 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica: Volume $\text { 1 }$ ... (previous) ... (next): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.13$: Symbolism of sentential calculus
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.2$: Truth-Functions
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): $\S 2.3$: Truth-Tables
- 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Negation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: truth function
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Boolean