We can give a simple numerical example of Boolean algebra and how it can calculate logical truth. Suppose that we take the truth value of a statement as 1 if it is true and 0 if it is false. Now we have numbers 1 and 0 instead of letters T and F. Since they are numbers, we can add them, subtract them, and multiply them. We can also make up simple numerical formulas that will let us calculate logical truth. If P and Q are statements, and if p and q are their truth values, respectively, we have [Table 7].
Table 7
| Statement | Truth Value |
|---|---|
| not-P | 1 - p |
| P and Q | pq |
| P or Q | p + q - pq |
| if P, then Q | 1 - p + pq |
| P if and only if Q | 1 - p - q + 2pq |
| P or else Q | p + q - 2pq |
For example, suppose that we have two statements P and Q:
- P: John Doe is eligible for insurance.
- Q: John Doe requires a medical examination.
To test that the truth value of “P or Q” is p + q-pq, let us put down the four cases, and calculate the result ([see Table 8]).
Table 8
| p | q | p + q - pq |
| 1 | 1 | 1 + 1 - 1 = 1 |
| 0 | 1 | 0 + 1 - 0 = 1 |
| 1 | 0 | 1 + 0 - 0 = 1 |
| 0 | 0 | 0 + 0 - 0 = 0 |
Now we know that P or Q is true if and only if either one or both of P and Q are true, and thus we see that the calculation is correct.
The algebra of logic ([see also Supplement 2]) is a more efficient way of calculating logical truth. But it is still a good deal of work to use the algebra. For example, if we have 10 conditions, we shall have 10 letters like p, q to handle in calculations. Thus we need a still more efficient way.