Question

The proposition $(p\Rightarrow p)\wedge (p\Rightarrow p)$ is a
A)Tautology
B)Neither tautology nor contradiction
C)Contradiction
D)None of these

Answer 1

Hint: We know that a tautology is a formula which is "always true" that is, it is true for every assignment of truth values to its simple components. You can think of a tautology as a rule of logic. The opposite of a tautology is a contradiction, a formula which is "always false". A proposition which is neither a tautology nor a contradiction is called contingency. Write a truth table for the given proposition and see whether all are true or all are false or neither of those.

Complete step by step answer:
By first writing the truth table of $(p\Rightarrow p)$, we get it as

p	p	$(p\Rightarrow p)$
T	T	T
F	F	T

Now by writing the truth table of $(p\Rightarrow p)\wedge (p\Rightarrow p)$, we get it as

p	p	$(p\Rightarrow p)$	$(p\Rightarrow p)\wedge (p\Rightarrow p)$
T	T	T	T
F	F	T	T

Truth table for a tautology has T in its every row.
Truth table for a contradiction has F in its every row.
A proposition which is neither a tautology nor a contradiction is called contingency.
Contingency has both T and F in its truth table.
In this resulted table we can see that all the outputs are T i.e. True.
Therefore, it is clearly tautology because all the outputs are T irrespective of the value of p.

So, the correct answer is option A.

Note:
Read the definitions of tautology , contradiction and contingency. Practice more problems to get a hold of these types of problems. Remember that if a composite proposition is contingent then it cannot be tautology and it also cannot be a contradiction.