Section 4: Partial truth tables
In order to show that a sentence is a tautology, we need to show that it is 1 on every row. So we need a complete truth table. To show that a sentence is not a tautology, however, we only need one line: a line on which the sentence is 0. Therefore, in order to show that something is not a tautology, it is enough to provide a one-line partial truth table — regardless of how many sentence letters the sentence might have in it.
Consider, for example, the sentence ( \(U\) & \(T\) ) → ( \(S\) & \(W\) ). We want to show that it is not a tautology by providing a partial truth table. We fill in 0 for the entire sentence. The main connective of the sentence is a conditional. In order for the conditional to be false, the antecedent must be true (1) and the consequent must be false (0). So we fill these in on the table:
| \(S\) | \(T\) | \(U\) | \(W\) | (\(U\) & \(T\))→(\(S\) & \(W\)) |
| 1 0 0 |
In order for the (\(U\) &\(T\)) to be true, both \(U\) and \(T\) must be true.
| \(S\) | \(T\) | \(U\) | \(W\) | (\(U\) & \(T\))→(\(S\) & \(W\)) |
| 1 | 1 | 1 1 1 0 0 |
Now we just need to make (\(S\)&\(W\)) false. To do this, we need to make at least one of \(S\) and \(W\) false. We can make both \(S\) and \(W\) false if we want. All that matters is that the whole sentence turns out false on this line. Making an arbitrary decision, we finish the table in this way:
| \(S\) | \(T\) | \(U\) | \(W\) | (\(U\) & \(T\))→(\(S\) & \(W\)) |
| 1 | 1 | 0 | 1 1 1 0 0 0 0 |
Showing that something is a contradiction requires a complete truth table. Showing that something is not a contradiction requires only a one-line partial truth table, where the sentence is true on that one line.
A sentence is contingent if it is neither a tautology nor a contradiction. So showing that a sentence is contingent requires a two-line partial truth table: The sentence must be true on one line and false on the other. For example, we can show that the sentence above is contingent with this truth table:
| \(S\) | \(T\) | \(U\) | \(W\) | (\(U\) & \(T\))→(\(S\) & \(W\)) |
|
0 0 |
1 1 |
1 0 |
0 0 |
1 1 1 0 0 0 0 0 0 1 1 0 0 0 |
Note that there are many combinations of truth values that would have made the sentence true, so there are many ways we could have written the second line.
Showing that a sentence is not contingent requires providing a complete truth table, because it requires showing that the sentence is a tautology or that it is a contradiction. If you do not know whether a particular sentence is contingent, then you do not know whether you will need a complete or partial truth table. You can always start working on a complete truth table. If you complete rows that show the sentence is contingent, then you can stop. If not, then complete the truth table. Even though two carefully selected rows will show that a contingent sentence is contingent, there is nothing wrong with filling in more rows.
Showing that two sentences are logically equivalent requires providing a complete truth table. Showing that two sentences are not logically equivalent requires only a one-line partial truth table: Make the table so that one sentence is true and the other false.
Showing that a set of sentences is consistent requires providing one row of a truth table on which all of the sentences are true. The rest of the table is irrelevant, so a one-line partial truth table will do. Showing that a set of sentences is inconsistent, on the other hand, requires a complete truth table: You must show that on every row of the table at least one of the sentences is false.
Showing that an argument is valid requires a complete truth table. Showing that an argument is invalid only requires providing a one-line truth table: If you can produce a line on which the premises are all true and the conclusion is false, then the argument is invalid.
Here is a table that summarizes when a complete truth table is required and when a partial truth table will do.
tautology?
contradiction?
contingent?
equivalent?
consistent?
valid?
| YES | NO |
|
complete truth table complete truth table two-line partial truth table complete truth table one-line partial truth table complete truth table |
one-line partial truth table one-line partial truth table complete truth table one-line partial truth table complete truth table one-line partial truth table |
Table 3.2: Do you need a complete truth table or a partial truth table? It depends on what you are trying to show.