9.6. Modus Ponens and Modus Tollens

Try these first

1. 1.

   What does the nested quantifier ∀x ∃y M(x,y) express?

   Hint

   Think about the relationship between people.

2. 2.

   State the conclusion derived from 'If it is sunny (P), then it is bright (Q)' given that it is sunny.

   Hint

   What happens if the first part is true?

3. 3.

   What is the meaning of the quantifier ∃?

   • For all
   • There exists
   • None

   Hint

   Think about existence vs. all.

4. 4.

   Is Modus Tollens valid under all situations?

   • True
   • False

   Hint

   It’s about validating implications.

5. 5.

   Consider the statement 'If every student studies, they pass the exam.' If one student fails, analyze the implications under Modus Tollens.

   Hint

   Look for logical negation.

6. 6.

   Construct a logical argument using both Modus Ponens and nested quantifiers, and explain your reasoning.

   Hint

   Focus on the connection between submission and outcomes.