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 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

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

Which one of the following statements is a tautology?

  1. $\left( p\vee q \right) \rightarrow q$
  2. $p\vee (p\rightarrow q)$
  3. $ p\vee (q\rightarrow p)$
  4. $p\rightarrow (p\rightarrow q)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$ p$  $q$  $p\to q$  $q\to p$ $pvq$  $p\to (p\to q)$  $(pvq)\to q$  $pv(p\to q)$   $pv(q\to q$
 T  T  T  T  T  T  T  T  T
 T  F  F  T  T  F  F  T  T
 F  T  T  F  T  T  T  T  F
 F  F  T  T  F  T  T  T  T

$\left( p\vee q \right) \rightarrow q$

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

If $p$ and $q$ are two statement, then $(p  \wedge \sim q) \wedge   (\sim p  \wedge   q)$ is

  1. a fallacy

  2. a tautology

  3. neither tautology nor a fallacy

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$p$         $q$          $p \land \sim q$       $q \land \sim p $          $(p \land \sim q ) \land(q \land \sim p )$
T T     F       F          F
T F     T       F          F
F T      F       T          F
F F     F       F          F


$\therefore\ (p \land \sim q ) \land(q \land \sim p )$ is a fallacy

Multiple choice maths mathematical reasoning special words/phrases duality mathematical logic

The conditional $( p \wedge q ) \Rightarrow p$ is 

  1. neither tautology nor contradiction

  2. a tautology

  3. a contradiction

  4. both tautology and contradiction
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The expression (p and q) implies p is a tautology because if (p and q) is true, then p must be true by definition of the conjunction operator.

Multiple choice maths mathematical reasoning special words/phrases duality mathematical logic

The conditional $( p \wedge q ) \Rightarrow p$ is 

  1. neither tautology nor contradiction

  2. a tautology

  3. a contradiction

  4. both tautology and contradiction
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Constructing a truth table for (p AND q) => p: if (p AND q) is true, then p must be true. If (p AND q) is false, the implication is vacuously true. Thus, it is a tautology.

Multiple choice maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

If point $P$ lies on $AB$, then $AB$ is always greater than $AP$. This concept is on which of the following Euclid's Axioms.

  1. First Axiom

  2. Second Axiom

  3. Third Axoim

  4. Fifth Axiom
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The fifth axiom of Euclid's about geometry is the whole of anything is greater than the part of it.

Here AB is the whole line and AP is the part and according to the fifth axiom we have AB is always greater than AP.
So the given statement is Euclid's fifth axiom.

Multiple choice maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

Axioms are assumed

  1. universal truths in all branches of mathematics

  2. universal truths specific to geometry

  3. theorems

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

From time immemorial, axioms have been acquired by man through the day to day experiences .

No mathematical deduction is needed to prove them.
Practically they are starting points of reasoning.

So axioms are assumed universal truths in all branches of mathematics.

Ans- Option A.

Multiple choice maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

Euclid's fourth axiom says that everything equals itself.

  1. True

  2. False

  3. Ambiguous

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

Euclid's fourth axiom states that "things which coincide with one another are equal to one another."

So the given statement is true by the axiom IV.
Ans- Option A.

Multiple choice maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

Identify the given statement: It is possible to produce a finite straight continuously in a straight line.

  1. theroem

  2. conjectures

  3. operation

  4. postulate
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
It is possible to produce a finite straight continuously in a straight line.
The given statement is postulate. A postulate is a statement that is accepted without proof. Axiom is another name for a postulate. 

For example, if you know that Pam is five feet tall and all her siblings are taller than her, you would believe her if she said that all of her siblings are at least five foot one. Pam just stated a postulate, and you just accepted it without grabbing a tape measure to verify the height of her siblings.
Multiple choice maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

Read the following axioms:
(i) Things which are equal to the same thing are equal to one another.
(ii) If equals are added to equals, the wholes are equal.
(iii) Things which are double of the same thing are equal to one another.
Check whether the given system of axioms is consistent or inconsistent

  1. consistent

  2. inconsistent

  3. Only (i) & (ii) are consistent

  4. Only (iii) is consistent
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

(i) This statement is  Euclid's first Axiom.

(ii) This statement is  Euclid's  second Axiom.
(ii) This statement is true if we apply Euclid's first Axiom.
Let $2a=b$ and $2a=c$.
Then both  b & c are equal to 2a
i.e $ b=c$
which is consistent with the first Axiom of Euclid.
So the given system of axioms is consistent.
ans- Option A.