Truth tables can be used for other purposes. One is to test statements for certain logical properties. Some sentences have the property that they cannot be false under any circumstances. An example is P v ~P:
| P | ~P | P v ~P |
|---|---|---|
| T | F | T |
| F | T | T |
A sentence with this property is called a tautology. Another example:
(P → Q) ↔ ~(P &~Q)
| P | Q | (P→Q)↔ ~(P &~Q) | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| P→Q | ↔ | ~(P&~Q) | ||||||||
| P | → | Q | ~(P&~Q) | |||||||
| ~ | (P&~Q) | |||||||||
| ~ | P | & | ~Q | |||||||
| ~ | Q | |||||||||
| T | T | T | T | T | T | T | T | F | F | T |
| T | F | T | F | F | T | F | T | T | T | F |
| F | T | T | T | T | T | T | F | F | F | T |
| F | F | T | T | F | T | T | F | F | T | F |
If there are sentences that are always true, then there are sentences that are always false. Such sentences are called INCONSISTENT. One example:
| P | ~P | P & ~P |
|---|---|---|
| T | F | F |
| F | T | F |
We defined an argument as "a set of sentences (the premises) and a sentence (the conclusion)." That definition does not actually say that an argument must have premises, only that it must have a set of premises. However, sets can be empty, that is, they can have no members. We can also have an argument with an empty set of premises. As a sequent, such an argument would be written like this:
|- Φ
If a sequent has an empty set of premises, can it be valid? Yes: if it is a tautology, then it is valid. The definition applies to it because it is impossible for it to have true premises and a false conclusion (since it is impossible for it to have a false conclusion at all).
All other sentences--that is, all those that are neither tautologies nor inconsistent--are called CONTINGENT. A contingent sentence is one that is neither always true nor always false. Equivalently, a contingent sentence is one that is true for at least one combination of truth values and false for at least one combination of truth values.
Here are a few provable truths:
We can extend the notion of inconsistency to sets of sentences. Even if two sentences are both contingent, it may be impossible for them both to be true at the same time. Example:
P→Q and P&~Q
To see this, look at the truth table above for (P → Q) ↔ ~(P &~Q), but note the columns for P→Q and P&~Q:
| P | Q | (P→Q)↔ ~(P &~Q) | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| P→Q | ↔ | ~(P&~Q) | ||||||||
| P | → | Q | ~(P&~Q) | |||||||
| ~ | (P&~Q) | |||||||||
| P | & | ~Q | ||||||||
| ~ | Q | |||||||||
| T | T | T | T | T | T | T | T | F | F | T |
| T | F | T | F | F | T | F | T | T | T | F |
| F | T | T | T | T | T | T | F | F | F | T |
| F | F | T | T | F | T | T | F | F | T | F |
There is no row in which these two wffs have the same truth value. Therefore, it is not possible for them to be true at the same time. We call these two sentences INCOMPATIBLE.
If the premises of an argument are incompatible, then it is impossible for all the premises to be true. If it is impossible for the premises all to be true, then it is impossible for the premises all to be true and the conclusion false. So, an argument with incompatible premises is valid.