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:
“During the late war [the American Revolution] I had an infallible rule for deciding what [Great Britain] would do on every occasion. It was, to consider what they ought to do, and to take the reverse of that as what they would assuredly do, and I can say with truth that I was never deceived.”
—Thomas Jefferson (1743–1826)
“When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.”
—Edward T. Hall (b. 1914)