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

Which of the following is a valid inference rule in epistemic logic?

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

The inference rule (K(p) \rightarrow K(K(p))) is known as the rule of positive introspection. It states that if an agent knows something, then they also know that they know it. This rule is valid in epistemic logic because it is a fundamental property of knowledge that we are aware of what we know.

Multiple choice

What is the formula for the law of necessitation?

  1. $ \vdash \phi \rightarrow \Box \phi $
  2. $ \vdash \Box \phi \rightarrow \phi $
  3. $ \vdash \phi \rightarrow \Diamond \phi $
  4. $ \vdash \Diamond \phi \rightarrow \phi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The law of necessitation states that if a formula is provable, then its necessity is also provable.

Multiple choice

What is the formula for the law of distribution of disjunction over necessity?

  1. $ \Box (\phi \wedge \psi) \leftrightarrow \Box \phi \wedge \Box \psi $
  2. $ \Box (\phi \vee \psi) \leftrightarrow \Box \phi \vee \Box \psi $
  3. $ \Diamond (\phi \wedge \psi) \leftrightarrow \Diamond \phi \wedge \Diamond \psi $
  4. $ \Diamond (\phi \vee \psi) \leftrightarrow \Diamond \phi \vee \Diamond \psi $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The law of distribution of disjunction over necessity states that the necessity of a disjunction is equivalent to the disjunction of the necessities.

Multiple choice

What is the formula for the axiom of reflexivity for belief?

  1. $ \vdash B\phi \rightarrow \phi $
  2. $ \vdash \phi \rightarrow B\phi $
  3. $ \vdash \Diamond B\phi \rightarrow B\phi $
  4. $ \vdash B\phi \rightarrow \Diamond B\phi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The axiom of reflexivity for belief states that if an agent believes a proposition, then the proposition is true.

Multiple choice

What is the formula for the axiom of positive introspection for belief?

  1. $ \vdash B\phi \rightarrow B B\phi $
  2. $ \vdash B B\phi \rightarrow B\phi $
  3. $ \vdash \Diamond B\phi \rightarrow B\phi $
  4. $ \vdash B\phi \rightarrow \Diamond B\phi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The axiom of positive introspection for belief states that if an agent believes a proposition, then the agent believes that they believe the proposition.

Multiple choice

What is the formula for the axiom of negative introspection for belief?

  1. $ \vdash B\phi \rightarrow B B\phi $
  2. $ \vdash B B\phi \rightarrow B\phi $
  3. $ \vdash \Diamond B\phi \rightarrow B\phi $
  4. $ \vdash B\phi \rightarrow \Diamond B\phi $
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

The axiom of negative introspection for belief states that if an agent does not believe a proposition, then the agent believes that they do not believe the proposition.

Multiple choice

What is the formula for the rule of necessitation for belief?

  1. $ \vdash B\phi \rightarrow B \Box B\phi $
  2. $ \vdash B \Box B\phi \rightarrow B\phi $
  3. $ \vdash \Diamond B\phi \rightarrow B\phi $
  4. $ \vdash B\phi \rightarrow \Diamond B\phi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The rule of necessitation for belief states that if an agent believes a proposition, then the agent believes that it is necessary that they believe the proposition.

Multiple choice

What is the formula for the rule of distribution of belief over conjunction?

  1. $ \vdash B(\phi \wedge \psi) \leftrightarrow (B\phi \wedge B\psi) $
  2. $ \vdash B(\phi \vee \psi) \leftrightarrow (B\phi \vee B\psi) $
  3. $ \vdash \Diamond B\phi \rightarrow B\phi $
  4. $ \vdash B\phi \rightarrow \Diamond B\phi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The rule of distribution of belief over conjunction states that an agent believes a conjunction if and only if the agent believes both conjuncts.

Multiple choice

What is the formula for the rule of distribution of belief over disjunction?

  1. $ \vdash B(\phi \wedge \psi) \leftrightarrow (B\phi \wedge B\psi) $
  2. $ \vdash B(\phi \vee \psi) \leftrightarrow (B\phi \vee B\psi) $
  3. $ \vdash \Diamond B\phi \rightarrow B\phi $
  4. $ \vdash B\phi \rightarrow \Diamond B\phi $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The rule of distribution of belief over disjunction states that an agent believes a disjunction if and only if the agent believes at least one disjunct.

Multiple choice

What is the formula for the rule of generalization for belief?

  1. $ \vdash B\phi \rightarrow B \forall x \phi $
  2. $ \vdash B \forall x \phi \rightarrow B\phi $
  3. $ \vdash \Diamond B\phi \rightarrow B\phi $
  4. $ \vdash B\phi \rightarrow \Diamond B\phi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The rule of generalization for belief states that if an agent believes a proposition, then the agent believes the universal generalization of that proposition.

Multiple choice

What is the formula for the rule of instantiation for belief?

  1. $ \vdash B\forall x \phi \rightarrow B\phi [t/x] $
  2. $ \vdash B\phi [t/x] \rightarrow B\forall x \phi $
  3. $ \vdash \Diamond B\phi \rightarrow B\phi $
  4. $ \vdash B\phi \rightarrow \Diamond B\phi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The rule of instantiation for belief states that if an agent believes a universal generalization, then the agent believes the instance of that generalization obtained by substituting any term for the variable.

Multiple choice

What is the formula for the rule of modus ponens for belief?

  1. $ \vdash B(\phi \rightarrow \psi) \wedge B\phi \rightarrow B\psi $
  2. $ \vdash B(\phi \rightarrow \psi) \wedge B\psi \rightarrow B\phi $
  3. $ \vdash \Diamond B\phi \rightarrow B\phi $
  4. $ \vdash B\phi \rightarrow \Diamond B\phi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The rule of modus ponens for belief states that if an agent believes a conditional proposition and the antecedent of that proposition, then the agent believes the consequent of that proposition.

Multiple choice

What are some examples of the principle of identity?

  1. A book is a book.

  2. A tree is a tree.

  3. A person is a person.

  4. All of the above.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The principle of identity states that everything is identical to itself. This means that a book is a book, a tree is a tree, and a person is a person.

Multiple choice

What is the opposite of the principle of identity?

  1. The principle of contradiction

  2. The principle of the excluded middle

  3. The principle of non-identity

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

The opposite of the principle of identity is the principle of non-identity. The principle of non-identity states that something can be both itself and something else at the same time and in the same respect.

Multiple choice

Which of the following is a central concept in epistemic logic?

  1. Knowledge operator

  2. Belief operator

  3. Common knowledge operator

  4. All of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Epistemic logic uses various operators to represent knowledge, belief, and common knowledge.