Question

Examine whether the following logical statement pattern is tautology, contradiction or contingency
\[\left[ {\left( {{\rm{p}} \to {\rm{q}}} \right) \wedge {\rm{q}}} \right] \to {\rm{p}}\]

Answer 1

Hint: Here we will be examining the logical statement by using a truth table. A logical statement is called contingency if its truth table contains at least one ‘True’ and at least one ‘False’. Similarly, a logical statement is called tautology if its truth table contains all ‘True’ and logical statement is called contradiction if its truth table contains all ‘False’.

Complete step-by-step answer:
Let’s consider the statement pattern:
 \[\left[ {\left( {{\rm{p}} \to {\rm{q}}} \right) \wedge {\rm{q}}} \right] \to {\rm{p}}\].
Here ‘\[{\rm{p}} \to {\rm{q}}\]’ means If p is true then q is true.
'\[\left( {{\rm{p}} \to {\rm{q}}} \right) \wedge {\rm{q}}\]’ means if anyone of this is false then the statement is false.
We will make a truth table to find out the pattern of the given logical statement:
\[\left[ {\left( {{\rm{p}} \to {\rm{q}}} \right) \wedge {\rm{q}}} \right] \to {\rm{p}}\]

\[{\rm{p}}\]	\[{\rm{q}}\]	\[{\rm{p}} \to {\rm{q}}\]	\[\left( {{\rm{p}} \to {\rm{q}}} \right) \wedge {\rm{q}}\]	\[\left[ {\left( {{\rm{p}} \to {\rm{q}}} \right) \wedge {\rm{q}}} \right] \to {\rm{p}}\]
T	T	T	T	T
T	F	F	F	T
F	T	T	T	F
F	F	T	F	T

From the above truth table we can say that given logical statement: \[\left[ {\left( {{\rm{p}} \to {\rm{q}}} \right) \wedge {\rm{q}}} \right] \to {\rm{p}}\]is contingency as it contains both true and false.
Note: In \[{\rm{p}} \to {\rm{q}}\] , p is called hypothesis (or premise) and q is called conclusion (or consequence). Other meaning of \[{\rm{p}} \to {\rm{q}}\] is ‘if p then q’. Most common mistakes which students make while using \[{\rm{p}} \to {\rm{q}}\] is that they write the statement false even when p is false and q is true which is absolutely wrong. Correct answer in this case is true. Always remember, do not examine the logical statements without using the truth table. Truth table is mandatory in examining logical statements to get the answer correct.