Gappy Truth Tables
All the truth tables we've examined so far have no gaps. We compute the truth or falsity of every wff on every row of the truth table, and that's a good thing. Our truth tables give us complete information about the conditions under which the wffs concerned are truth or false. So why would we want anything else?
We use truth tables to determine whether an argument is valid or invalid. Remember test an argument for validity, we create a truth table of the premises and the conclusion. We examine the truth table and check to see if there is a row where the premises all come out true and the conclusion comes out false. If there is such a row, the argument is invalid. If there is no such row, the argument is valid. In Chapter 4 we looked at some very simple arguments. Let's consider one slightly more complex:
|
(A ⊃ (B v ~C)) |
This argument consists of two premises and a conclusion. There
are three
distinct simple propositions in the argument, namely A, B and
C. Thus we
must consider all the possible combinations of truth and falsity
for A, B and C
when setting up the truth table. When there are n
distinct simple
propositions, there will be 2n rows of the truth
table.
So for the argument we're considering, the truth table will have 8
rows.
| A | B | C | (A ⊃ (B v ~C)) | (C ≡ (A & B)) | ~(A & C) |
| T | T | T | |||
| T | T | F | |||
| T | F | T | |||
| T | F | F | |||
| F | T | T | |||
| F | T | F | |||
| F | F | T | |||
| F | F | F |
We set up the left columns, called the "base columns" as follows. We know that we'll have eight rows. So we begin with the left-most column, and we make the first half of the entries "T" and the second half "F". In the next column, we alternate by half of our pattern from the first column. Since the first four were "T" in the first column, we now alternate by two. Then we do half yet again, and alternate by one.
Now we're getting to the point: In arguments with lots of distinct atomic propositions, our truth-tables will be very long. (An argument with 6 distinct atomic propositions has 26 = 2x2x2x2x2x2 = 64 rows!) Cleary we'll want to minimize the number of rows we have to fill out!
Remember that an argument is invalid when there is at least one row of the truth table where the premises are true and the conclusion false. The Gappy Truth Table method exploits this fact. It tells us first to examine only those rows where the conclusion is false, since those are the only rows which can demonstrate invalidity. From among the rows where the conclusion turns out false, we need only look at those rows where all the premises are true. As soon as we find a row which has a false premise or a true conclusion, we can ignore the rest of that row, since it won't establish the invalidity of the argument. If we find an invalidating row, the argument is invalid. If we don't, the argument is valid.
Let's apply this strategy to the argument above, filling in the table only as needed:
| A | B | C | (A ⊃ (B v ~C)) | (C ≡ (A & B)) | ~(A & C) |
| T | T | T | T | T | F |
| T | T | F | |||
| T | F | T | F | F | |
| T | F | F | |||
| F | T | T | |||
| F | T | F | |||
| F | F | T | |||
| F | F | F |
We start with the conclusion. It is false on just the first and the third rows. So those are the only rows with which we need to be concerned. On the third row the first premise is false. So we can stop looking at that row. But on the first row, both premises are true. So the first row, with all true premises and a false conlusion, demonstrates the invalidity of the argument. That's all there is to the gappy truth-table method!
We can use the gappy truth table method to test our other semantic properties as well. The next section covers the application of this method to tautologies, contradictions, contingencies, consistency, and logical equivalence.
Here's a gappy truth table exercise!
