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Botanist: All commercial varieties of the *Amaranth* grain exhibit rapid stalk development. Most grain varieties that exhibit rapid stalk development are highly susceptible to stem rust. Interestingly, no grain varieties that contain the *K-22* genetic sequence are susceptible to stem rust. Yet, some commercial varieties of the *Amaranth* grain contain the *K-22* genetic sequence.

Based on the botanist's statements, which of the following must be true? (Select all that apply.)

  1. Some commercial varieties of the *Amaranth* grain are not highly susceptible to stem rust.Answer
  2. Some grain varieties that exhibit rapid stalk development are not highly susceptible to stem rust.Answer
  3. C
    Most commercial varieties of the *Amaranth* grain are highly susceptible to stem rust.
  4. D
    All grain varieties that exhibit rapid stalk development are commercial varieties of the *Amaranth* grain.
  5. E
    Any grain variety that does not contain the *K-22* genetic sequence is highly susceptible to stem rust.

Answer

The correct statements are that some commercial varieties of the Amaranth grain are not highly susceptible to stem rust, and some grain varieties that exhibit rapid stalk development are not highly susceptible to stem rust.
The correct statements rely on combining the 'some' and 'all/no' premises. Because some commercial Amaranth grains contain K-22, and K-22 guarantees resistance to stem rust, we can definitively say some commercial Amaranth grains are not susceptible. Furthermore, because all commercial Amaranth grains exhibit rapid stalk development, those resistant Amaranth grains also exhibit rapid stalk development, proving that some grain varieties with rapid stalk development are not susceptible.

Step-by-Step Solution

1
Analyze the formal logic of the premises.
Premise 1: Commercial Amaranth (CA) → Rapid Stalk (RS). Premise 2: RS → (most) → Stem Rust Susceptible (S). Premise 3: K-22 → NOT S (~S). Premise 4: CA → (some) → K-22.
Diagramming the relationships ensures we do not conflate sufficient and necessary conditions, or overextend quantifiers like 'most' and 'some'.
2
Combine Premise 4 and Premise 3 to make a deduction.
Since some CA have K-22, and all things with K-22 are ~S, it must be true that some CA are ~S (Some commercial varieties of Amaranth are not susceptible to stem rust).
When a 'some' statement shares a term with the sufficient condition of an 'all' (or 'no') statement, a valid transitive deduction can be drawn.
3
Combine the deduction from Step 2 with Premise 1.
We know some CA have K-22 and are therefore ~S. We also know from Premise 1 that ALL CA have RS. Therefore, those specific CA that are ~S also have RS. This proves that some RS are ~S.
Applying an 'all' rule to an existing subgroup allows us to attribute the property of the 'all' rule to that specific subgroup.

Key Concept

Chaining Conditional Statements with Mixed Quantifiers
Estimated Time:2m 0s
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