Youth Parliament Notice Board: Square of Opposition and Obversion
At a district-level youth parliament selection, the student council must display four notices on one board. The notices concern 42 volunteers who worked at the science fair. N1: 'All volunteers who worked at the science fair are students who received service credit.' N2: 'No volunteers who worked at the science fair are students who received service credit.' N3: 'Some volunteers who worked at the science fair are students who received service credit.' N4: 'Some volunteers who worked at the science fair are not students who received service credit.' Attendance records confirm that at least one of the 42 volunteers actually worked at the science fair. The council has two display rules: contradictory notices may not be displayed together; contrary or subcontrary notices may be displayed together only with a caution note. A teacher asks the council to decide the logical relations, to derive the obverse of N1, and to decide whether N3 can be displayed as following from N1. The council must also decide which notices, if any, can be displayed together without a caution note.
1. Classify N1–N4 by type (A, E, I, O) and identify every contradictory pair and every contrary or subcontrary pair. Which pairs may not be displayed together under the first rule?
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N1 is A: All S are P. N2 is E: No S are P. N3 is I: Some S are P. N4 is O: Some S are not P. The contradictory pairs are N1–N4 and N2–N3; in each pair exactly one member must be true, so they may not be displayed together. The contrary pair is N1–N2: both cannot be true, but both can be false if some volunteers received credit and some did not. The subcontrary pair is N3–N4: both cannot be false, but both can be true. Subaltern pairs are N1–N3 and N2–N4. Under the first rule, the barred pairs are therefore N1 with N4 and N2 with N3. N1 with N2 and N3 with N4 would require a caution note under the second rule.
2. Derive the obverse of N1 and state whether it is equivalent. Then decide whether N3 follows from N1 given the attendance record. Explain using subalternation or distribution.
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The obverse of N1 ('All S are P') is 'No S are non-P': 'No volunteer who worked at the science fair is a student who did not receive service credit.' Obversion changes the quality from affirmative to negative and replaces the predicate with its complement; the two statements are equivalent. N3 ('Some S are P') does follow from N1, but only because the attendance record establishes that S is not empty. In the traditional square, A entails I by subalternation when the subject class has at least one member. Without that existence fact, N1 could be true vacuously while N3 is false. Since at least one volunteer worked at the fair, N3 is a valid subalternate inference here.
3. The council wants to display N1 and N3 together without a caution note. Is that allowed? What about displaying N1 and N4?
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N1 and N3 are in a subaltern relation: N1 entails N3, and they are consistent. They are not contradictory, contrary, or subcontrary, so the stated rules do not require a caution note for that pair. N1 and N4 are contradictory: an A proposition and its corresponding O proposition cannot both be true and cannot both be false. Therefore they must not be displayed together at all. The council can display N1 with N3 without caution, but must keep N1 and N4 apart.