4.3: Vacuously true statements
We have to be careful with quantified predicates because it is (seemingly) possible to violate the Law of Contradiction (see Basic Tautology 4 in Example 1.4.1 ).
Let \(x\) be a variable in the domain of all living humans. Define predicates
\begin{align*} A(x) &= \text{ “} x \text{ is an Augustana student,”} \\ B(x) &= \text{ “} x \text{ is three hundred years old,”} \\ C(x) &= \text{ “} x \text{ is tall,”} \end{align*}
and consider the statement
\begin{equation*} (\forall x)\{\{A(x) \land B(x)\} \rightarrow C(x)\}, \end{equation*}
which says “every three-hundred-year-old Augustana student is tall”. This statement is true, since a conditional \(p \rightarrow q\) is true when \(p\) is false, and \(A(x) \land B(x)\) is false for each and every \(x\text{:}\) there is no living human who is both three hundred years old and is an Augustana student (issues concerning the existence of vampires notwithstanding). But by the same reasoning, the statement “every three-hundred-year-old Augustana student is not tall” is true. This seems to be a contradiction: how can every three-hundred-year-old Augustana student be both tall and not tall? The answer is that you can say anything you like about things that do not exist and your statement will be true. So you should avoid altogether making claims about things that do not exist.
a statement of the form \((\forall x)\{P(x) \rightarrow Q(x)\}\) where \(P(x)\) is false for every \(x\) in its domain
Check your understanding.
Determine the negation of \((\forall x)\{P(x) \rightarrow Q(x)\}\text{.}\) Is the negation of a vacuously true statement true or false?