Tautology/Examples/Arbitrary Example 1

Examples of Tautologies

The WFF of propositional logic:

$\paren {\paren {\paren {\lnot p} \implies q} \implies \paren {\paren {\paren {\lnot p} \implies \paren {\lnot q} } \implies p} }$

is a tautology.

Proof

Proof by truth table:

$\begin{array}{|cccc|c|ccccccc|} \hline (((\lnot & p) & \implies & q) & \implies & (((\lnot & p) & \implies & (\lnot & q)) & \implies & p)) \\ \hline \T & \F & \F & \F & \T & \T & \F & \T & \T & \F & \F & \F \\ \T & \F & \T & \T & \T & \T & \F & \F & \F & \T & \T & \F \\ \F & \T & \T & \F & \T & \F & \T & \T & \T & \F & \T & \T \\ \F & \T & \T & \T & \T & \F & \T & \T & \F & \T & \T & \T \\ \hline \end{array}$

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: Example $1.6 \ \text{(d)}$