T3.2: 4 of 7 Okay, a pattern emerges:
The point is this: Each time an atomic component is added, the number of rows
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 = 2 One last thing to notice. We have been putting the rows of a table in
a
This table says that there are four possibilities (the rows) for the
sentence '~B>C' and that this sentence is
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 2 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. |