Tautology/Examples/((not p) implies (q or r)) iff ((not q) implies ((not r) implies p))

Examples of Tautologies

The WFF of propositional logic:

$\paren {\paren {\lnot p} \implies \paren {q \lor r} } \iff \paren {\paren {\lnot q} \implies \paren {\paren {\lnot r} \implies p} }$

is a tautology.

Proof

Proof by truth table:

$\begin{array}{cccccc|c|ccccccc} ((\lnot & p) & \implies & (q & \lor & r)) & \iff & ((\lnot & q) & \implies & ((\lnot & r) & \implies & p)) \\ \hline

T & F & F & F & F & F & T & T & F & F & T & F & F & F \\
T & F & T & F & T & T & T & T & F & T & F & T & T & F \\
T & F & T & T & T & F & T & F & T & T & T & F & F & F \\
T & F & T & T & T & T & T & F & T & T & F & T & T & F \\
F & T & T & F & F & F & T & T & F & T & T & F & T & T \\
F & T & T & F & T & T & T & T & F & T & F & T & T & T \\
F & T & T & T & T & F & T & F & T & F & T & F & T & T \\
F & T & T & T & T & T & T & F & T & T & F & T & T & T \\

\end{array}$

As can be seen by inspection, the truth value under the main connective is true for all rows.

$\blacksquare$

Sources

• 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Exercise $4$