Truth Function

A truth function is one that maps truth values (either true - $\top$ or false - $\perp$) to other truth values.

Definition §

Truth Function ^definition

A truth function $F$ is defined as:

$$F:\{\top ,\perp {\}}^n\to \{\top ,\perp \}$$

$$F:\{1,0{\}}^n\to \{1,0\}$$

• $\{{\}}^n$ refers to taking the cartesian product of a set $n$ times

$F$ can have $n$ inputs (making the function $n$-ary), provided they are all truth values.

Examples §

• Logical connectives are binary truth functions
  • For example, conjunction can be viewed as a truth function $\wedge =\{(1,0)\mapsto 0,(0,1)\mapsto 0,(0,0)\mapsto 0,(1,1)\mapsto 1,\}$
  • An exemption is negation, which is unary:

$$\neg =\{0\mapsto 1,1\mapsto 0\}$$