The discovery of the following underlying implication relation is due to D. T. Langendoen (private communication).

BASIC implication

In the BASIC programming language available on IBM personal computers (BASIC Compiler 2.00, Fundamentals, Boca Raton, Florida, 1985) there are many numeric functions called “logical operators.” These functions are defined over a finite set of integers that are encoded in binary notation as strings of sixteen bits of 0's and 1's. The first bit is 0 if the integer is positive, and 1 if negative. That leaves fifteen bits for encoding, so that the range of these functions consists of all the integers from −32768 to +32767 (−215 to +215). The representing string for any integer in this range, therefore, consists of an initial 0 or 1 according as it is positive or negative, and a tail end that depends on the binary representation of the integer. If the integer is positive, then the tail end consists of its binary representation; if the integer is negative, then the tail end is determined as follows: If the negative integer is −m, and the binary encoding of m has n digits, then calculate what integer together with −2(n−1) would sum to −m, and let its binary representation be at the tail end on the sixteen-bit string. Thus, 3, being positive, would be represented as 0000000000000011, the initial bit representing the positive character, and the tail end (11) being the binary representation of 3.