Extended Boolean Functions in OpenNARS

In NARS, many measurements take values from the [0, 1] interval, including all the components of truth-value, desire-value, budget-value, as well as the amount of evidence in single testing. These values are considered as "extended Boolean values", in the sense that the Boolean values in {0, 1} can be taken as special cases of [0, 1]. The Boolean operators “not”, “and” and “or” are extended into the real-number functions defined on [0, 1], too.

Semantics

Traditionally, the extended “and” and “or” are called Triangular norm (T-norm) and Triangular conorm (T-conorm), respectively. Each of them is commutative, associative, and monotonic in each variable and thus they can be extended to take arbitrary number of arguments in the following way:
and(x1, . . . , xn) = and(and(x1, . . . , xn−1), xn)
or(x1, . . . , xn) = or(or(x1, . . . , xn−1), xn)

In NARS, T-norm function y = and(x1, . . . , xn) is used when a quantity y is conjunctively determined by two or more other quantities x1, . . . , xn. For example y = 1 if and only if x1 = ... = xn = 1, and y = 0 if and only if x1 = 0 or ... or xn = 0.

Similarly, T-conorm function y = or(x1, . . . , xn) is used when a quantity y is disjunctively determined by two or more other quantities x1, ... , xn. For example, y = 1 if and only if x1 = 1 or . . . or xn = 1, and y = 0 if and only if x0 = · · · = xn = 0.

Intuitively, a variable y is conjunctively determined by variables x1,..., xn when all the x’s are its necessary factors, or numerically, if y is never bigger than any of them. Similarly, y is disjunctively determined by x1, . . . , xn when all the x’s are its sufficient factors, or numerically, it is never smaller than any of them.

The negation function is naturally taken to be the complement function.

Definition

In NARS the extended Boolean functions for "and", "or" and "not" operators are defined in the following way:

not(x) = 1 − x
and(x1, ..., xn) = x1 * ... * xn
or(x1, ..., xn) = 1 − [(1 − x1) * ... * (1 − xn)]

The and and or functions are used under the condition that x1, ..., xn are independent variables in the sense that the value of each of them cannot be determined from the values of the others.

It should be mentioned that though the extended Boolean functions of NARS share intuition and mathematical forms with probabilistic formula, they should not be understood as and(x, y) = P(x and y) and or(x, y) = P(x or y), simply because x and y are usually not random variables with probability distribution function P.