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

Which of the following statement is a tautology?

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

$\because (\sim p  \vee \sim q) \vee (p  \vee \sim q) $

$\equiv (\sim p \vee p ) \vee (p \vee \sim q )$    (by distributive law)

$ \equiv t \vee \sim q \equiv t $                      t is a tautology

Hence $(\sim p  \vee \sim, q ) \vee (p  \vee \sim q )$ is a tautology.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

$ p\Rightarrow p \vee q$ is

  1. a tautology.

  2. a contradiction.

  3. a tautology and a contradiction.

  4. neither a tautology nor a contradiction.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$p$       $q$      $p\vee q$       $p\rightarrow (p\vee q)$
T T     T            T
T F    T            T
F T    T              T
F F     F            T

Since, all the entries in the last column has true value. So, the given statement is a tautology

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

$p\Rightarrow \sim p$ is

  1. a tautology.

  2. a contradiction.

  3. a tautology and a contradiction.

  4. neither a tautology nor a contradiction.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
 $p$  $\sim p$  $p\Rightarrow \sim p$
 T  F  F
 F  T  T

So the result of the Truth table show that $p\Rightarrow \sim p$ is neither a tautology not a contradiction

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

$ p\wedge  (\sim p)$ is

  1. a tautology.

  2. a contradiction.

  3. a tautology and a contradiction.

  4. neither a tautology nor a contradiction.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Truth table is,

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

Hence given statement is contradiction.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

Which of the following is a contradiction?

  1. $p\vee q$
  2. $p\wedge q$
  3. $p\vee (\sim p)$
  4. $p\wedge (\sim p)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$p$  $q$  $p\vee q$  $p\wedge q$
$ T$ $ T$ $ T$ $ T$
$ T$ $ F$ $ T$ $ F$
$ F$ $T$ $T$ $F$
$ F$ $F$ $ F$ $ F$
So $p\vee q$ and $p\wedge q$ are not contradiction 

Now check other options:

| $p$ |  $\sim p$ |  $p\vee (\sim p)$ |  $p\wedge (\sim p)$ | | --- | --- | --- | --- | |  $T$ | $ F$ | $ T$ | $ F$ | | $ F$ | $T$ | $T$ | $F$      |
Hence $p\wedge (\sim p)$ is contradiction 

Note: If a compound statement is always False , then it is called contradiction 
Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

The proposition $(p\rightarrow \sim p)\wedge (\sim p\rightarrow q)$ is

  1. a tautology

  2. a contradiction

  3. neither a tautology nor a contradiction

  4. a tautology and a contradiction
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$p$      $q$      $\sim p$      $p\rightarrow \sim p$     $\sim p \rightarrow q$     $(p\rightarrow \sim p)\wedge(\sim p\rightarrow q)$
T T    F      F       T         F
T F    F      F       T         F
F T    T      T       T         T
F F    T      T       F         F
Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

The statement $(p-q)\rightarrow [(\sim p \rightarrow q)\rightarrow q]$ is 

  1. a tautology
  2. equivalent to $\sim p \rightarrow q$
  3. equivalent to $p\rightarrow \sim q$

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

The statement (p -> q) -> ((~p -> q) -> q) is a tautology. It can be verified using a truth table or logical equivalences.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

Which of the following proposition 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

Contradiction is a preposition which is always false(F).
Here $\sim \equiv negation$ and $\wedge \equiv AND$
Let $ x=(\sim p \wedge q) \wedge (\sim q)$
If $p=F$ and $q=F$ then $x=F$
If $p=F$ and $q=T$ then $x=F$
If $p=T$ and $q=F$ then $x=F$
If $p=T$ and $q=T$ then $x=F$
Hence option (c) is correct

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

$p,q,r$ are $3$ statement such that $(p\rightarrow q)\wedge (q\rightarrow r)\Rightarrow (p\rightarrow r)$ is 

  1. Tautology

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

The expression (p -> q) ^ (q -> r) -> (p -> r) is the Law of Hypothetical Syllogism, which is a fundamental tautology in propositional logic.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

$p,q,r$ are $3$ statement such that $(p \rightarrow q)\wedge (q \rightarrow r)\Rightarrow (P \rightarrow r)$ is

  1. Tautology

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

This is identical to the previous question; the expression (p -> q) ^ (q -> r) -> (p -> r) is a tautology.