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Evaluation and satisfaction are processes involving sentences and specific truth
assignments. In logic we are sometimes more interested in the properties of
sentences that hold across different truth assignments. In particular, sentential
truth assignment imposes a classification of sentences in a language based on
whether or not there are truth assignments that satisfy that sentence. A sentence is
valid, for example, if and only if it is satisfied by every truth assignment. A
sentence is unsatisfiable if and only if it is satisfied by no truth assignment.
And a sentence is contingent if and only if there is some truth assignment that
satisfies it, and some truth assignment that falsifies it. It is frequently useful
to group these classifications as shown here. A sentence is satisfiable if and
only if it is either valid or contingent. That is, if and only if there is at least
one truth assignment that satisfies it. A sentence is falsifiable if and only if it
is either contingent or unsatisfiable. That is, if and only if there is at least
one truth assignment that falsifies it. We can check whether a sentence is valid or
contingent, or unsatisfiable by evaluating it for all truth assignments. In the
example here, we show that the sentence P implies Q, or Q implies R, is always true.
In other words, it's valid. Looking at P implies Q. We note that it is true except
when P is true and Q is false. Look the sentence Q implies R is true except when Q
is true and R is false. Combining these two, we see that the overall sentence is
true whenever either of these components is true. And in this case we notice that
it's true for every row of the table. Hence the sentence is valid. There are
many examples of validities. Some of them even have names. There's double negation,
p if and only if not, not p; DeMorgan's laws, which allows to distribute negation
over conjunction and disjunction; implication introduction; implication
distribution; and so forth. [sound]. This exercise tests your understanding of these
concepts by ask ing you to say whether various sentences are valid, contingent,
or unsatisfiable.