Reasoning

Logic and Fallacies

1,803 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

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
B Correct answer
Explanation

Statement I is true because the expression contains contradictory sub-parts making it a fallacy or contradiction. Statement II is true because a conditional is logically equivalent to its contrapositive, making it a tautology; however, Statement II does not explain Statement I.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

The statement (p ^ q) ^ (-pv - q) is _______________.

  1. a tautology

  2. a contradiction

  3. a contingency

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

The expression contains a conjunction of a statement and its direct negation through De Morgan expansion of the second term, ensuring it is always false. Thus, the statement is a contradiction.

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.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

Which of the following statement is a contradiction?

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

We check for contradiction for all the given options.

A. $\left( p\wedge q \right) \wedge \left( \sim \left( p\vee q \right)  \right) $

$p$ $q$ $\left( p\wedge q \right)$ $\left( p\vee q \right)$  $\left( \sim \left( p\vee q \right) \right)$  $\left( p\wedge q \right) \wedge \left( \sim \left( p\vee q \right)  \right) $ 
T T  T  T  F  F
T F  F  T  F  F
F T  F  T  F  F
F F  F  F  T  F

All F so this is a contradiction.


B. $p\vee \left( \sim p\wedge q \right) $

 $p$  $q$ $\sim p$ $\sim p\wedge q$  $p\vee \left( \sim p\wedge q \right) $ 
 T  T  F  F  T
 T  F  F  F  T
 F  T  T  T  T
 F  F  T  F  F

So not a contradiction.


C. $\left( p\longrightarrow q \right) \rightarrow p$

 $p$  $q$  $\left( p\longrightarrow q \right) $  $\left( p\longrightarrow q \right) \rightarrow p$
 T  T  T  T
 T  F  F  T
 F  T  T  F
 F  F  T  F

So it is also not a contradiction.


D. $\sim p\vee \sim q$

 $p$  $q$ $\sim p$ $\sim q$ $\sim p\vee \sim q$
 T  T  F  F  F
 T  F  F  T  T
 F  T  T  T  T
 F  F  T  T  T

So it is also not a contradiction.

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

$\sim (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 is a tautology?

  1. $p\implies p\wedge q$
  2. $p\implies p\vee q$
  3. $(p\vee q)\implies(p\wedge q)$

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

A tautology is a statement that is always true.
The statement $ p\Longrightarrow p\vee q $ is read as " if p is true, then either p or q is true. " as the symbol $ \vee  $ denotes OR.
Hence, the given statement is a tautology.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

Which of the following statements is/are true?

  1. $p\wedge (\sim p)$ is a contradiction.
  2. $(p\rightarrow q)\Leftrightarrow (\sim q \rightarrow \sim p)$ is a contradiction.
  3. $\sim(\sim p) \Leftrightarrow p$ is a tautology.
  4. $p\vee (\sim p)$ is a tautology.
Reveal answer Fill a bubble to check yourself
A,C,D Correct answer
Explanation

A,C,D obvious and for B
p $\rightarrow$ q is same as $\sim$ q $\rightarrow$ $\sim$p
$\therefore$ it is tautology not contradiction.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

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

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

Tautology is the preposition which is always true and contradiction is a preposition which is always false.
Here $\wedge \equiv AND, \vee \equiv OR$, $T$=true and $F$=false
Given statement=$p$, tautology=$t$ and contradiction=$c$
a)$p\wedge t\equiv p$ irrespective of value of $p$ as $t=T$ always
And if $p=T$ then $T\wedge T=T$ and if $p=F$ then $F\wedge T=F$
Thus option (a) is correct.
b)$p\wedge c\equiv c$ irrespective of value of $p$ as$ c=F$ always
And if $p=T$ then $T\wedge F=F$ and if $p=F$ then $F\wedge F=F$
Thus option (b) is correct
c)$p\vee t\equiv c$
If $p=T$ then $T\vee T\equiv T\neq c$ and if $p=F$ then $F\vee T\equiv T \neq c$
Thus option (c) is not correct
d)$p\vee c\equiv p$
If $p=T$ then $T\vee F\equiv T$ and if $p=F$ then $F\vee F\equiv F$
Thus it depends on the value of $p$
Hence option (d) is correct