Introduction / Context:
This is a classic categorical syllogism. If an individual item belongs to a class, and that entire class has a property, then the individual inherits that property. The question is whether the conclusion necessarily follows from the two given premises without extra assumptions.
Given Data / Assumptions:
- A toothpick ∈ Useful-things.
- All Useful-things ⟶ Valuable-things.
- We assume ordinary logical reading: “are valuable” applies to every member of the set of useful things.
Concept / Approach:
- Syllogistic form: If A is a B, and all B are C, then A is a C.
- No exceptions or qualifiers like “some,” “often,” or “usually” are present; the second premise is universal in scope.
Step-by-Step Solution:
From Premise 1: toothpick ∈ Useful.From Premise 2: Useful ⊆ Valuable.Therefore: toothpick ∈ Valuable. The conclusion follows deductively and must be true.
Verification / Alternative check:
Venn-style thinking: The “Useful” circle sits entirely within the “Valuable” circle. Since the toothpick is inside “Useful,” it must also lie inside “Valuable.”
Why Other Options Are Wrong:
false: Contradicts a valid deductive inference.uncertain: There is no uncertainty; the premises are universal and sufficient.both true and false: Mutually inconsistent; the conclusion has a determinate truth value under the premises.
Common Pitfalls:
Injecting real-world opinions about the value of a toothpick. Deductive questions rely solely on the stated premises, not personal beliefs.
Final Answer:
true
Discussion & Comments