Reasoning

Logic and Fallacies

1,716 Questions

Understand the fundamentals of propositional logic, logical inference rules, and paradoxes. This set includes identifying logical fallacies, including those found in classical Nyaya logic. Strong grasp of these concepts is crucial for scoring well in the reasoning sections of competitive tests.

Propositional logic modelsLogical inference rulesTypes of logical fallaciesParadoxes and contingent truthsNyaya logic concepts

Logic and Fallacies Questions

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

$p,q,r$ are $3$ statements such that $\left(p\rightarrow q\right)\wedge \left(q\rightarrow r\right)=Rightarrow \left(p\rightarrow r\right)$ is

  1. $Tautology$
  2. $Contradiction$
  3. $P\wedge q$
  4. $p\wede(\sim q)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expression (p -> q) ^ (q -> r) -> (p -> r) is a classic logical law known as the Hypothetical Syllogism. A truth table shows that it evaluates to True for all possible truth values of p, q, and r, making it a tautology.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

The statement  $\sim ( p \wedge q ) \vee q$

  1. is a tautology
  2. is equivalent to $( p \wedge q ) \vee ( - q )$
  3. is equivalent to $p \vee q$

  4. is a contradiction
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expression ~(p ^ q) v q is equivalent to (~p v ~q) v q. By associativity, this is ~p v (~q v q), which is ~p v T, which is T. Thus, it is a tautology.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

Consider :
Statement - I :$(p\wedge \sim q)\wedge (\sim p\wedge q)$ is a fallacy.
Statement - II :$(p\rightarrow q)\leftrightarrow (\sim q\rightarrow \sim p)$ is a tautology.

  1. Statement - I is true: Statement - II is true: Statement - II is a correct explanation for Statement - I.

  2. Statement - I is true: Statement - II is true: Statement - II is not a correct explanation for Statement - I.

  3. Statement - I is true; Statement - II is false.

  4. Statement - I is false; Statement - II is true.
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

The statement $\sim (p \rightarrow q) \leftrightarrow  (\sim p \vee \sim q)$ is 

  1. a tautology

  2. a contradiction

  3. neither a tautology nor a contradiction

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

When $p$ and $q$ both are true then 

$(p \rightarrow  q) and (\sim p \vee \sim q)$ both are false

i.e. $\sim (p \rightarrow  q) \leftrightarrow  (\sim p \vee  \sim q)$ is true when $p$ and $q$ both are false then  

$\sim (p \rightarrow  q)$ is false and $(\sim p \vee \sim q)$ is true

i.e. $\sim (p \rightarrow  q) \leftrightarrow  (\sim p \vee  \sim q)$ is false

Hence $\sim (p \rightarrow  q) \leftrightarrow  (\sim p \vee  \sim q)$ is neither tautology nor contradiction

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

Which of the following statement is a contradiction ?

  1. $(\sim p \vee \sim q) \vee (p \vee \sim q)$
  2. $(p \rightarrow q) \vee (p \wedge \sim q)$
  3. $(\sim p \wedge q) \wedge (\sim q)$
  4. $(\sim p \wedge q) \vee (\sim q)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A contradiction is a statement that is always false. Option C, (~p ^ q) ^ (~q), simplifies to ~p ^ (q ^ ~q), which is ~p ^ F, which is always false.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

If $p$ is any statement, $t$ is a tautology and $c$ is a contradiction, then which for the following is NOT correct?

  1. $p \wedge (\sim c) \equiv p$
  2. $p \vee (\sim t) \equiv p$
  3. $t \vee c \equiv p \vee t$
  4. $(p\wedge t) \vee (p \vee c) \equiv (t \wedge c)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A.   $p \wedge (\sim c )\equiv p \wedge t  \equiv p$

B.   $p \vee (\sim t )\equiv p \vee c \equiv p$

C.   $ t \vee c \equiv t \equiv p \vee t$

Clearly option 'A', 'B', 'C' are correct.

Hence option 'D' is the correct choice.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

If $p$ is any statement, $t$ and $c$ are a tautology and a contradiction respectively, then which of the following is INCORRECT?

  1. $p \wedge t \equiv p $
  2. $ p \wedge c \equiv c$
  3. $p \vee t \equiv p $
  4. $ p \vee c \equiv p$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Truth table,

$p$             $p\wedge t$        $p\wedge c$          $p\vee  t$      $p\vee c$
T T            F               T          T
F F            F            T         F

Hence option 'C' is the correct choice.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

The statement $(p \rightarrow ~p) \wedge (~ p \rightarrow p)$ is

  1. a tautology.

  2. a contradiction.

  3. neither a tautology nor a contradiction.

  4. None of these.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If $p$ is a true statement, then $p \rightarrow p$ is true.

Also, if $p$ is a false statement, then $p \rightarrow p$ is true.
Then, $(p \rightarrow p) \wedge (p \rightarrow p)$ is always true.
Hence, the given statement is a tautology.