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Difficulty: HardMust Be True / Most Strongly Supported

Archaeological metallurgists studying Early Bronze Age tools have established the following facts regarding a collection of recovered Aegean metalwork:

• Every arsenic-bronze dagger that exhibits a dendritic crystalline microstructure was cast using an open stone mold.
• No artifact cast using an open stone mold was alloyed with imported tin from Central Asia.
• Most arsenic-bronze daggers recovered from the Cycladic burial complex were alloyed with imported tin from Central Asia.
• Any artifact that was not alloyed with imported tin from Central Asia exhibits localized surface oxidation.

If the statements above are true, which one of the following must also be true?

  1. A
    Any arsenic-bronze dagger that does not exhibit a dendritic crystalline microstructure was alloyed with imported tin from Central Asia.
  2. B
    No artifact that exhibits localized surface oxidation was alloyed with imported tin from Central Asia.
  3. At least some arsenic-bronze daggers recovered from the Cycladic burial complex do not exhibit a dendritic crystalline microstructure.Answer
  4. D
    Every artifact recovered from the Cycladic burial complex that exhibits localized surface oxidation was cast using an open stone mold.
  5. E
    Most of the artifacts cast using an open stone mold were recovered from the Cycladic burial complex.

Answer

At least some arsenic-bronze daggers recovered from the Cycladic burial complex do not exhibit a dendritic crystalline microstructure.
The deduction follows strictly from combining the quantifier statement with the conditional contrapositives. We are given that most Cycladic arsenic-bronze daggers contain imported tin. The second premise states that open stone mold casting precludes imported tin; its contrapositive means that any artifact with imported tin was not cast in an open stone mold. The first premise establishes that having a dendritic microstructure requires open stone mold casting; its contrapositive ensures that an arsenic-bronze dagger not cast in an open stone mold cannot exhibit a dendritic microstructure. Combining these steps guarantees that the majority of Cycladic arsenic-bronze daggers lack a dendritic microstructure, which fully validates the claim that at least some do not.

Step-by-Step Solution

1
Symbolize the formal conditional relationships and quantifiers from the premises.
Premise 1: (Arsenic-Bronze Dagger AND Dendritic Microstructure) → Open Stone Mold. Premise 2: Open Stone Mold → NOT Imported Tin. Premise 3: Most Cycladic Arsenic-Bronze Daggers → Imported Tin. Premise 4: NOT Imported Tin → Surface Oxidation.
Clarifies the logical dependencies and chainable terms without extraneous narrative detail.
2
Derive the contrapositive of Premise 2 and connect it to Premise 1.
Imported Tin → NOT Open Stone Mold. By contrapositive of Premise 1: (Arsenic-Bronze Dagger AND NOT Open Stone Mold) → NOT Dendritic Microstructure. Therefore, for any arsenic-bronze dagger: Imported Tin → NOT Dendritic Microstructure.
Allows deduction of microstructural characteristics directly from the presence of imported tin.
3
Apply the derived conditional chain to the quantified subset in Premise 3.
Because 'most' Cycladic arsenic-bronze daggers contain imported tin, that majority subset cannot have been cast in an open stone mold and therefore cannot exhibit a dendritic crystalline microstructure. A majority ('most') strictly entails 'at least some'.
Deduces the necessary inference that is logically guaranteed by the intersection of the conditional chain and the quantifier.

Key Concept

Quantifier Intersection and Conditional Contrapositive Chaining
Estimated Time:2m 0s
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Topics

Must Be True / Most Strongly Supported

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