9.2 Numerical Truth Tables
What is essential about truth tables for Boolean logic is that we have two values, one for true and one for false.
It is not essential that we use “T” to mean true and “F” to mean false. We could use any symbol or code for true and false.
1 = True
0 = False
For example, it is common to use 1 for true and 0 for false.
Then the truth table for disjunction would look like this:
Notice this connection between numerical truth tables and binary numbers: the bottom row of the truth table is always 00, or zero. The next row up is 01, or one. The next row is 10, or two, and the last row is 11, or three.
That is, the truth table counts up from the bottom in binary from 0 to 3. In other words, the truth table counts down from the top in binary from 3-0.
The same thing happens with truth tables of any number of atomics. For example, here is a table with three atomics:
It’s important to remember that even though three atomic sentences can represent 8 different binary numbers, the largest number will always be one less than the total, since the bottom row represents 0.
The number of rows in a truth table is fixed by the number of atomic sentences.
It is just 2 to the power of the number of atomics. So 4 atomic sentences have 16 rows, and 5 atomics have 32 rows.
Atomic sentences can be true or false, 1 or 0. That means we can represent them with anything that has two states, like “on” and “off.”
For example, a light bulb.