Definition 1.2.1.1.
A tautology is a compound proposition that is true no matter what the truth values of the propositional variables that occur in it.
Subsection 1.2.1 Terminology
Some compound propositions are true for a silly reason. Basically, the compound proposition is true, regardless of the truth values of the component propositions. For example, “I’ll get to it when I get to it.” If \(g\) is the proposition “I get to it,”, then this compound proposition is the conditional \(g\implies g\) . Whether \(g\) is true or false, the conditional is true. This is an example of a tautology.
Example 1.2.1.2. Tautology.
Let \(p\) be a proposition. Then \(p \implies p\) is a tautology. To see this, we compute the truth table for \(p\implies p\) .
\begin{equation*}
\begin{array}{|*{1}{c|}|*{1}{c|}}
\hline
p \amp p \implies p \\
\hline
\true \amp \true\\
\false \amp \true\\
\hline
\end{array}
\end{equation*}
Since \(p\implies p\) is always true, \(p\implies p\) is a tautology.
Definition 1.2.1.3.
A contradiction is a compound proposition that is always false no matter what the truth values of the propositional variables that occur in it.
Example 1.2.1.4. Contradiction.
Let \(p\) be a proposition. Then \(\lnot(p \implies p)\) is a contradiction. To see this, we compute the truth table for \(\lnot(p \implies p)\) .
\begin{equation*}
\begin{array}{|*{1}{c|}|*{2}{c|}}
\hline
p \amp p \implies p \amp \lnot(p \implies p)\\
\hline
\true \amp \true \amp \false\\
\false \amp \true \amp \false\\
\hline
\end{array}
\end{equation*}
Since \(\lnot(p \implies p)\) is always false, \(\lnot(p \implies p)\) is a contradiction.
Remark 1.2.1.5.
Definition 1.2.1.6.
A contingency is a compound proposition that is neither a tautology nor a contradiction.
Example 1.2.1.7. Contingency.
Let \(p\) and \(q\) be propositions. The conditional \(p\implies q\) is a contingency. To see this, compute the truth table for \(p\implies q\) .
\begin{equation*}
\begin{array}{|*{2}{c|}|*{1}{c|}}
\hline
p\amp q\amp p \implies q\\
\hline
\true\amp \true\amp \true\\
\true\amp \false\amp \false\\
\false\amp \true\amp \true\\
\false\amp \false\amp \true\\
\hline
\end{array}
\end{equation*}
Since \(p \implies q\) is true for some choices of \(p\) and \(q\) and false for some other choices, \(p\implies q\) is a contingency.
Example 1.2.1.8. Contradiction and Tautology.
Let \(p\) be a proposition. Then \(p\land\lnot p\) is a contradiction and \(p\lor\lnot p\) is a tautology as shown in the table below. Not that the column for \(p\land\lnot p\) is all false, and the column for \(p\lor\lnot p\) is all true.
\begin{equation*}
\begin{array}{|*{1}{c|}|*{3}{c|}}
\hline
p \amp \lnot p \amp p\land\lnot p \amp p\lor\lnot p \\
\hline
\true\amp \false \amp \false \amp \true\\
\false \amp \true\amp \false \amp \true\\
\hline
\end{array}
\end{equation*}
Definition 1.2.1.9.
Two propositions \(p\) and \(q\) are logically equivalent denoted \(p \equiv q\) ,when \(p \iff q\) is a tautology.
Theorem 1.2.1.10. Contrapositive.
Let \(p\) and \(q\) be propositions then the conditional \(p\implies q\) is logically equivalent to its contrapositive:
\begin{equation*}
p\implies q\equiv \lnot q\implies\lnot q
\end{equation*}
and its converse is equivalent to its inverse:
\begin{equation*}
\lnot q \implies\lnot p\equiv \lnot p \implies\lnot q.
\end{equation*}
Proof.
By looking at the truth table in Figure 1.1.4.3, we see that a conditional \(p\implies q\) is logically equivalent to its contrapositive \(\lnot q\implies\lnot q\) , because the truth values in the respective columns are the same, which proves the first equivalence.
Also the truth values in the columns for the converse \(\lnot q \implies\lnot p\) are the same as the truth values in the column of the inverse \(\lnot p \implies\lnot q\) , which proves the second equivalence.
https://xkcd.com/703/). Hey, why do YOU get to be the president of Tautology Clu--- wait, I can guess.
