The Syllogism Answer Key That Needs a Second Look
During a timed CEPTAM STA-B reasoning sectional, a candidate named Farah reaches a syllogism item with four statements: (i) All oscilloscopes in the lab are calibrated instruments. (ii) Some calibrated instruments are imported. (iii) No imported instrument is battery-operated. (iv) All battery-operated devices are portable. The four conclusions are: I. Some oscilloscopes are imported. II. Some calibrated instruments are not battery-operated. III. No oscilloscope is battery-operated. IV. All portable devices being calibrated instruments is a possibility. Farah marks I and III, because she thinks the imported instruments mentioned in (ii) must include oscilloscopes and that a battery-operated oscilloscope would contradict (iii). She has 40 seconds left and does not check the possibility conclusion. The answer key says II and IV are correct. Before accepting the key, you must test each conclusion with a counterexample or a valid Venn arrangement, and decide whether Farah’s reasoning contains a hidden assumption.
1. Which conclusions follow logically, and which of Farah's two choices is invalid? Show the Venn reasoning.
Read the worked answer
Valid conclusions are II and IV. II follows because statement (ii) guarantees at least one calibrated instrument that is imported; statement (iii) says no imported instrument is battery-operated, so that same instrument is a calibrated instrument that is not battery-operated. I does not follow: the imported calibrated instruments need not be oscilloscopes. III does not follow: all oscilloscopes are calibrated instruments, but the calibrated instruments that are not battery-operated could be the imported ones, while a different oscilloscope could still be battery-operated. IV is possible: all battery-operated devices can be placed inside the calibrated-instrument circle provided they stay outside the imported set; no statement forbids this. Farah's hidden assumption is that 'some calibrated instruments are imported' means 'some oscilloscopes are imported'.
2. Construct a concrete counterexample for conclusion III using small sets, and explain what it proves.
Read the worked answer
Let O = {o1, o2, b1}, C = {o1, o2, b1, i1}, I = {i1}, B = {b1}, P = {b1}. All O are C; some C (i1) are I; no I is B; all B are P. Yet b1 is an oscilloscope and is battery-operated, so 'No oscilloscope is battery-operated' is false. One consistent counterexample is enough to prove that III is not a necessary conclusion.
3. Name one single additional premise that would make conclusion III valid, and explain why it is sufficient.
Read the worked answer
Either 'All oscilloscopes are imported' or 'No calibrated instrument is battery-operated' works. If all oscilloscopes are imported, then by (iii) no imported instrument is battery-operated, so no oscilloscope can be battery-operated. If no calibrated instrument is battery-operated, then since all oscilloscopes are calibrated, the same result follows directly. The second premise is the more direct strengthening of the link between calibrated instruments and battery operation.