Syllogistic forms for validity

Sample Solution

Question 1: Testing Syllogistic Forms for Validity Using Venn Diagrams

To test the validity of syllogistic forms using Venn diagrams, we draw circles to represent the major term (M), the minor term (S), and the middle term (P) in the syllogism. Then, we shade the areas of the circles that represent the propositions in the syllogism.

Syllogism #1

• Premise 1: All M is P (Shade the entire M circle)

• Premise 2: All M is S (Shade the entire M circle)

Full Answer Section

• Conclusion: All S is P (Shade the intersection of the M and P circles)

Syllogism #2

• Premise 1: Some M is P (Shade a portion of the M circle that overlaps with the P circle)

• Premise 2: Some M is not S (Shade a portion of the M circle that does not overlap with the S circle)

• Conclusion: Some S is not P (Shade a portion of the S circle that does not overlap with the P circle)

Syllogism #3

• Premise 1: Some P is M (Shade a portion of the P circle that overlaps with the M circle)

• Premise 2: Some S is not M (Shade a portion of the S circle that does not overlap with the M circle)

• Conclusion: Some S is P (Shade a portion of the S circle that overlaps with the P circle)

Based on the Venn diagrams, all three syllogisms are valid. This means that the conclusions necessarily follow from the premises.

Question 2: Statistical Independence

Formal Definition:

Two propositions are statistically independent if the probability of one proposition does not affect the probability of the other proposition. In other words, the events described by the two propositions do not influence each other.

Intuitive Definition:

Two propositions are statistically independent if knowing the truth or falsity of one proposition does not provide any information about the truth or falsity of the other proposition.

Question 3: Mutual Exclusivity

Formal Definition:

Two propositions are mutually exclusive if they cannot both be true at the same time. In other words, the events described by the two propositions are disjoint.

Intuitive Definition:

Two propositions are mutually exclusive if they are opposites or contradict each other.

Question 4: Mutual Exclusivity and Independence

Two propositions cannot be both mutually exclusive and independent. This is because if two propositions are mutually exclusive, then knowing the truth or falsity of one proposition immediately provides information about the truth or falsity of the other proposition. Therefore, the two propositions are not independent.

Question 5: Drawing Two Kings from a Standard Deck

The probability of drawing a King from a standard fifty-two-card deck is 4/52 = 1/13. After replacing the King and reshuffling, the probability of drawing another King remains the same, 1/13.

Therefore, the probability of drawing a King and then another King is (1/13) x (1/13) = 1/169.