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Modal Logic and Belief

This quiz covers the basics of Modal Logic and Belief, including the concepts of possibility, necessity, and belief.

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Which of the following is a modal operator?

  1. *
  2. +
  3. !
  4. ?
Question 2 Multiple Choice (Single Answer)

What is the dual of the possibility operator ?

  1. *
  2. +
  3. !
  4. ?
Question 3 Multiple Choice (Single Answer)

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 $
Question 4 Multiple Choice (Single Answer)

What is the formula for the law of distribution of conjunction over possibility?

  1. $ \Diamond (\phi \wedge \psi) \leftrightarrow \Diamond \phi \wedge \Diamond \psi $
  2. $ \Diamond (\phi \vee \psi) \leftrightarrow \Diamond \phi \vee \Diamond \psi $
  3. $ \Box (\phi \wedge \psi) \leftrightarrow \Box \phi \wedge \Box \psi $
  4. $ \Box (\phi \vee \psi) \leftrightarrow \Box \phi \vee \Box \psi $
Question 5 Multiple Choice (Single Answer)

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 $
Question 6 Multiple Choice (Single Answer)

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 $
Question 7 Multiple Choice (Single Answer)

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 $
Question 8 Multiple Choice (Single Answer)

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 $
Question 9 Multiple Choice (Single Answer)

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 $
Question 10 Multiple Choice (Single Answer)

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 $
Question 11 Multiple Choice (Single Answer)

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 $
Question 12 Multiple Choice (Single Answer)

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 $
Question 13 Multiple Choice (Single Answer)

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 $
Question 14 Multiple Choice (Single Answer)

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 $
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