Truth Functions
Truth functions are functions from sequences of truth values to truth values. A unary truth function, for example, takes a single truth value and maps it onto another truth value. Similarly, a binary truth function maps ordered pairs of truth values onto truth values, while a ternary truth function maps ordered triples of truth values onto truth values, and so on.
In the unary case, there are two possible inputs, viz. T and F, and thus four possible unary truth functions: one mapping T to T and F to F, one mapping T to F and F to F, one mapping T to T and F to T, and finally one mapping T to F and F to T, this last one corresponding to the familiar operation of logical negation. In the form of a table, the four unary truth functions may be represented as follows.
| p | p | F | T | ~p |
|---|---|---|---|---|
| T | T | F | T | F |
| F | F | F | T | T |
In the binary case, there are four possible inputs, viz. (T,T), (T,F), (F,T), and (F,F), thus yielding sixteen possible binary truth functions. Quite generally, for any number n, there are possible n-ary truth functions. The sixteen possible binary truth functions are listed in the table below.
| p | q | T | NAND | → | ~p | ← | ~q | ↔ | NOR | ∨ | XOR | q | N← | p | N→ | & | F |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| T | T | T | F | T | F | T | F | T | F | T | F | T | F | T | F | T | F |
| T | F | T | T | F | F | T | T | F | F | T | T | F | F | T | T | F | F |
| F | T | T | T | T | T | F | F | F | F | T | T | T | T | F | F | F | F |
| F | F | T | T | T | T | T | T | T | T | F | F | F | F | F | F | F | F |
Read more about this topic: Logic Alphabet
Famous quotes containing the words truth and/or functions:
“Mind and spirit together make up that which separates us from the rest of the animal world, that which enables a man to know the truth and that which enables him to die for the truth.”
—Edith Hamilton (1867–1963)
“Let us stop being afraid. Of our own thoughts, our own minds. Of madness, our own or others’. Stop being afraid of the mind itself, its astonishing functions and fandangos, its complications and simplifications, the wonderful operation of its machinery—more wonderful because it is not machinery at all or predictable.”
—Kate Millett (b. 1934)