Question

It is standard in some programming languages for the number zero to represent one of the truth values (either true or false) and for positive numbers to represent the other truth value. If assigned correctly, the operations of addition and multiplication will then correspond to conjunction and disjunction, in some order. Figure out how to make this all work out neatly.

Answer 1

Without loss of generality, suppose that 0 represents the truth value FALSE and 1 represents the truth value TRUE.

Claim 1 : + corresponds to Disjunction (OR)

Proof : Observe that, 0 + 0 = 0. In case of disjunction, it is known that if a and b are both false, then so is a OR b.

Observe that, 0 + 1 = 1 + 0 = 1 + 1 = 1. In case of disjunction, it is known that if atleast one of a or b is true, then a OR b is true.

Thus, + corresponds to conjunction.

Claim 2 : . corresponds to Conjunction ( AND )

Observe that, 0.1 = 1.0 = 0.0 = 0. In case of conjunction, it is known that if atleast one of a or b is False, then a AND b is false.

Observe that, 1.1 = 1. In case of conjunction, it is known that P AND Q is true if both P and Q are true.

This proves the required claim.