Question

The following statement $(p \to q) \to \left[ {(\sim p \to q) \to q} \right]$ is
A. Tautology
B. Equivalent to $\sim p \to q$
C. Equivalent to $p \to \sim q$
D. A fallacy

Answer 1

A tautology is a formula or assertion that is true in every possible interpretation. Whereas a fallacy is a condition that is always false in every possible interpretation. Equivalent means similar and $\sim p$ means negation $p$ or not $p$ and $(\sim p \to q)$ means if not $p$ then $q$ that is if not $p$ is true then $q$ is true. For example: The dog is either brown, or the dog is not brown. The sentence is always true because one or the other is so. This is different from a statement that says “the dog is either or the dog is white” because dogs can be black, gray or a mix of colors. Note that when we put both halves of the logical tautology together it feels a bit reluctant, just like a verbal tautology.

Step-by-step solution: The statement is given to us is $(p \to q) \to \left[ {(\sim p \to q) \to q} \right]$ $= x$
Let us draw the truth table

$p$	$q$	$\sim p$	$p \to q$	$\sim p \to q$	$(\sim p \to q) \to q$	$x$
$T$	$T$	$F$	$T$	$T$	$T$	$T$
$T$	$F$	$F$	$F$	$T$	$F$	$T$
$F$	$T$	$T$	$T$	$T$	$T$	$T$
$F$	$F$	$T$	$T$	$F$	$T$	$T$

Hence, the given question which we took as $x$ is $(p \to q) \to \left[ {(\sim p \to q) \to q} \right]$ a tautology. Since all the results in the truth table come out to be true.

Note:
In this type of question students often make mistakes while writing the truth table and end up getting the wrong answer. Students should remember that $\sim p$ is always of $p$ and $p \to q$ means if $p$ is true the $q$ have to be true.