In the preceding lesson, we saw that some sentences are true in some truth assignments and false in others. However, this is not always the case. There are sentences that are always true and sentences that are always false as well as sentences that are sometimes true and sometimes false. This leads to a partition of sentences into three disjoint categories.
A sentence is valid if and only if it is satisfied by every truth assignment. For example, the sentence (p ∨ ¬p) is valid. If a truth assignment makes p true, then the first disjunct is true and the disjunction as a whole true. If a truth assignment makes p false, then the second disjunct is true and the disjunction as a whole is true.
A sentence is unsatisfiable if and only if it is not satisfied by any truth assignment. For example, the sentence (p ∧ ¬p) is unsatisfiable. No matter what truth assignment we take, the sentence is always false. The argument is analogous to the argument in the preceding paragraph.
Finally, a sentence is contingent if and only if there is some truth assignment that satisfies it and some truth assignment that falsifies it. For example, the sentence (p ∧ q) is contingent. If p and q are both true, it is true. If p and q are both false, it is false.
In one sense, valid sentences and unsatisfiable sentences are useless. Valid sentences do not rule out any possible truth assignments, and unsatisfiable sentences rule out all truth assignments. Thus, they tell us nothing about the world. In this regard, contingent sentences are the most useful. On the other hand, from a logical perspective, valid and unsatisfiable sentences are useful in that, as we shall see, they serve as the basis for legal transformations that we can perform on other logical sentences.
For many purposes, it is useful to group validity, contingency, and unsatisfiability into two groups. We say that a sentence is satisfiable if and only if it is valid or contingent. In other words the sentence is satisfied by at least one truth assignment. We say that a sentence is falsifiable if and only if it is unsatisfiable or contingent. In other words, the sentence is falsified by at least one truth assignment.