T3.2: 4 of 7
Okay, a pattern emerges:
The point is this:
Each time an atomic component is added, the number of rows doubles.
So, the sentence '(A&B)v[C&(D>E)]' has a truth table with 32 rows. And if we needed to add an 'F' for a longer sentence, there would be 64 rows. And so forth.
If you continue to think about Agnes, Bob, and Carola, then add David and Edward to the mix (e.g., 'D' stands for "David will attend law school), then you will see that the same reasoning of the last page proves this "doubling" rule.
The doubling rule can be put in precise terms. Let 'n' be the number of atomic components in a sentence.
The number of rows in a table = 2n (2 to the nth power)
One last thing to notice. We have been putting the rows of a table in a standard form: You may have noticed, for instance, that the first row is always all "T"s for the atomic components. The standard form is not essential but will make our lives a little easier. Consider the simple table:
This table says that there are four possibilities (the rows) for the sentence '~B>C' and that this sentence is false only when both 'B' and 'C' are false. (Note the yellow mark, , under the main connective marking the one row in which the sentence is false.)
But there is no meaning to the order of the rows. We will just standardly put the 'F, F' row last.
The rule is this to produce a standard table with 2n rows: For the first atomic component (in alphabetical order), make the first half of the rows (2n /2) assign "T" and the rest "F". Thus, for '~B>C', we assign 'B' a "T" for the first two rows.
For the second atomic component (if there is one), assign the first fourth of the rows "T", the second fourth "F", the third fourth "T", and the last fourth "F". So, 'C' is assigned "T" for the first fourth of the four rows!
And so on: keep dividing by two. Notice how this applied to the tables we've been using for three atomic components:
And that it applies as well for four atomic components:
And so on. But let's spare ourselves the burden of looking at a table with 32 or 64 rows! Instead, let's apply our thinking.