We can also construct a truth table for an entire sequent. The procedure is exactly like what we've just done, except that we start with several columns to the right: one for each premise and one for the conclusion. Here is a simple example:
Sequent:(P v Q) → R, ~R |- ~Q
Begin with columns for premises and conclusion:
| (P v Q) → R | ~R | |- ~Q |
|---|---|---|
Decompose the wffs:
| Q | (P v Q) → R | ~R | |- ~Q |
|---|---|---|---|
| Q | R | (P v Q) → R | ~R | |- ~Q |
|---|---|---|---|---|
| Q | R | (P v Q) | (P v Q) → R | ~R | |- ~Q |
|---|---|---|---|---|---|
| P | Q | R | (P v Q) | (P v Q) → R | ~R | |- ~Q |
|---|---|---|---|---|---|---|
Figure out the combinations (there are three sentence letters, so we will need 2 × 2 × 2 = 8 rows)
| P | Q | R | (P v Q) | (P v Q) → R | ~R | |- ~Q |
|---|---|---|---|---|---|---|
| T | T | T | ||||
| T | T | F | ||||
| T | F | T | ||||
| T | F | F | ||||
| F | T | T | ||||
| F | T | F | ||||
| F | F | T | ||||
| F | F | F |
Do the calculations:
| P | Q | R | (P v Q) | (P v Q) → R | ~R | |- ~Q |
|---|---|---|---|---|---|---|
| T | T | T | T | T | F | F |
| T | T | F | T | F | T | F |
| T | F | T | T | T | F | T |
| T | F | F | T | F | T | T |
| F | T | T | T | T | F | F |
| F | T | F | T | F | T | F |
| F | F | T | F | T | F | T |
| F | F | F | F | T | T | T |
Now, examine the truth values for the two premises and the conclusion in each row. Is there any row in which the conclusion is false and the premises both true?
| Premise 1 | Premise 2 | Conclusion | ||||
|---|---|---|---|---|---|---|
| P | Q | R | (P v Q) | (P v Q) → R | ~R | |- ~Q |
| T | T | T | T | T | F | F |
| T | T | F | T | F | T | F |
| T | F | T | T | T | F | T |
| T | F | F | T | F | T | T |
| F | T | T | T | T | F | F |
| F | T | F | T | F | T | F |
| F | F | T | F | T | F | T |
| F | F | F | F | T | T | T |
In fact, there isn't. There are four rows in which the conclusion is false (marked in red), but in each case at least one premise is also false. There is one case in which both premises are both true (marked in yellow), but in that case the conclusion is also true. So, we can say that in this argument, there is no way for the premises to be true and the conclusion also false at the same time. Therefore, this argument is valid.
Now, let's consider an argument that is not valid:
P v Q, Q → S, T |- S & T
| P | Q | S | T | P v Q | Q → S | T | |- S & T |
|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T | T |
| T | T | T | F | T | T | F | F |
| T | T | F | T | T | F | T | F |
| T | T | F | F | T | F | F | F |
| T | F | T | T | T | T | T | T |
| T | F | T | F | T | T | F | F |
| T | F | F | T | T | T | T | F |
| T | F | F | F | T | T | F | F |
| F | T | T | T | T | T | T | T |
| F | T | T | F | T | T | F | F |
| F | T | F | T | T | F | T | F |
| F | T | F | F | T | F | F | F |
| F | F | T | T | F | T | T | T |
| F | F | T | F | F | T | F | F |
| F | F | F | T | F | T | T | F |
| F | F | F | F | F | T | F | F |
There is one rows in which the premises are all true and the conclusion false. Therefore, this argument is not valid. It does not matter how many such rows there are as long as there is at least one: if there is at least one row in which the premises are true and the conclusion false, then the argument is invalid.
The truth values assigned to the atomic wffs (sentence letters) in a row of a truth table for a sequent in which the premises are true and the conclusion is false are called an invalidating assignment:
An INVALIDATING ASSIGNMENT for a sequent is an assignment of truth and falsity to its sentence letters that makes the premises true and the conclusion false.
If a sequent has an invalidating assignment, then it is invalid (do you see why?). Therefore, a valid sequent has no invalidating assignments. You could in fact define a valid sequent as one for which no assignment is an invalidating assignment.
As soon as we have found a row in which the premises are true and the conclusion false, we can stop: we know at that point that the argument is invalid, and filling in further rows will not add anything to this.
| P | Q | S | T | P v Q | Q → S | T | |- S & T |
|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T | T |
| T | T | T | F | T | T | F | F |
| T | T | F | T | T | F | T | F |
| T | T | F | F | T | F | F | F |
| T | F | T | T | T | T | T | T |
| T | F | T | F | T | T | F | F |
| T | F | F | T | T | T | T | F |
| T | F | F | F | ||||
| F | T | T | T | ||||
| F | T | T | F | ||||
| F | T | F | T | ||||
| F | T | F | F | ||||
| F | F | T | T | ||||
| F | F | T | F | ||||
| F | F | F | T | ||||
| F | F | F | F |
We can use this to develop an abbreviated truth-table test by trying to work backwards from the assumption that an argument is invalid. Taking the same example, suppose that it did have true premises and a false conclusion. We can represent this by starting out a "truth table" with the right side filled in first:
| P | Q | S | T | P v Q | Q → S | T | |- S & T |
|---|---|---|---|---|---|---|---|
| T | T | T | T | F |
What can we add to this? First, if S & T is false and T is true, then S must be false:
| P | Q | S | T | P v Q | Q → S | T | |- S & T |
|---|---|---|---|---|---|---|---|
| F | T | T | T | T | F |
Next, if Q → S is true and S is false, then Q must be false:
| P | Q | S | T | P v Q | Q → S | T | |- S & T |
|---|---|---|---|---|---|---|---|
| F | F | T | T | T | T | F |
But if Q is false and P v Q is true, then P must be true:
| P | Q | S | T | P v Q | Q → S | T | |- S & T |
|---|---|---|---|---|---|---|---|
| T | F | F | T | T | T | T | F |
In this case, we have figured out the only possible combination of truth values for the sentence letters in these wffs that makes the conclusion false and the premises true: P = T, Q = F, S = F, and T = T.
What would happen if we tried this method on a valid argument? First, let's take note of a difference between what it takes to show that an argument is valid and what it takes to show it is invalid:
To show that an argument is invalid, we only need to find one row of its truth table in which the premises are true and the conclusion false.
To show that an argument is valid, we need to show that there is no row of its truth table in which the premises are true and the conclusion false.
The important difference is that once we have found a single row with true premises and a false conclusion, we can stop (since we know that the argument is invalid), but in order to prove that it is valid we will have to check every row.
As a side note, you may think that the reason proving an argument is valid requires more work than proving it is invalid is that "it is hard to prove a negative." The real reason, however, is that proving validity requires proving something universal: it requires proving, for every possible combination of truth values, that that combination does not make the premises true and the conclusion false. Consider this sentences:
Every member of Congress is either a Democrat or a Republican.
There is nothing negative about this sentence, but in order to prove it you will need to determine, for each and every member of Congress, whether or not that person is either a Democrat or a Republican. On the other hand, to prove it false, all you need to do is find one member of Congress who is neither a Democrat nor a Republican.