Question

use an euler diagram to determine whether the argument is valid or invalid.

all cats like fish.
henry does not like fish.
therefore, henry is not a cat.

Explanation:

Analyze the premises and conclusion

Premise 1: "All cats like fish." This means the set of "Cats" is entirely contained within the set of things that like "Fish" (\(\text{Cats} \subseteq \text{Fish}\)).
Premise 2: "Henry does not like fish." This means the element representing Henry (\(H\)) must be placed entirely outside the set of things that like "Fish" (\(H
otin \text{Fish}\)).
Conclusion: "Therefore, Henry is not a cat." Since \(H\) is outside the "Fish" set, and the "Cats" set is entirely inside the "Fish" set, \(H\) must also be outside the "Cats" set (\(H
otin \text{Cats}\)).

Evaluate the diagrammatic representations

• Option 1: Shows overlapping sets labeled "Fish" (partially cut off as "Fish") and "Cats" with \(H\) inside "Cats" and labeled "Invalid". This does not represent the premise \(\text{Cats} \subseteq \text{Fish}\).
• Option 2: Shows "Cats" inside "Fish", with \(H\) outside "Fish", but labels the argument as "Invalid".
• Option 3: Shows "Cats" inside "Fish", with \(H\) outside "Fish", and labels the argument as "Valid".
• Option 4: Shows "Fish" inside "Cats", which is the reverse of the first premise.

Determine the correct option

The third option correctly represents the premise \(\text{Cats} \subseteq \text{Fish}\) by placing the "Cats" circle entirely inside the "Fish" circle. It correctly places \(H\) outside the "Fish" circle. Since \(H\) is forced to be outside the "Cats" circle, the conclusion is guaranteed, making the argument "Valid".

Answer:

• (A) An Euler diagram showing overlapping circles labeled "Fish" and "Cats" with point H inside "Cats", labeled "Invalid"
• (B) An Euler diagram showing the "Cats" circle inside the "Fish" circle, with point H outside the "Fish" circle, labeled "Invalid"
• (C) An Euler diagram showing the "Cats" circle inside the "Fish" circle, with point H outside the "Fish" circle, labeled "Valid" (Correct answer)
• (D) An Euler diagram showing the "Fish" circle inside the "Cats" circle, with point H inside "Cats" but outside "Fish", labeled "Valid"