Logic Alphabet - Truth Functions

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.

Unary truth functions
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.

Binary truth functions
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

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