The Merit Scholarship Argument That Was True but Invalid
At a Kanpur commerce college, the student welfare committee is reviewing 240 students for a merit scholarship. The posted rule states: 'All merit scholarship recipients must have 75% attendance.' Records show 60 scholarship recipients, all with 75% attendance; 90 hostellers, of whom 50 have 75% attendance; and only 15 of those 50 hostellers are scholarship recipients. In the committee meeting, a member argues: 'All scholarship holders are students with 75% attendance. Some hostellers are students with 75% attendance. Therefore some hostellers are scholarship holders.' The chair likes the conclusion because the records indeed show 15 hostellers who are scholarship holders, but the secretary says the argument is invalid and should not be used even to defend a true conclusion. The committee must decide whether to accept the reasoning, how to classify it, and whether a repair is available without falsifying the data.
1. Put the member's argument into standard categorical form. Identify its mood and figure, and name the specific defect if it is invalid.
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Let S = scholarship holders, H = hostellers, A = students with 75% attendance.
Standard form:
- Major premise: All S are A. (A proposition)
- Minor premise: Some H are A. (I proposition)
- Conclusion: Some H are S. (I proposition)
The middle term is A. In the major premise A is the predicate; in the minor premise A is also the predicate. Therefore the argument is in figure 2. Its mood is AII-2.
Figure 2 has the valid moods EAE, AEE, EIO, and AOO. AII-2 is not among them, so the argument is invalid.
The defect is the fallacy of undistributed middle:
- In 'All S are A', the subject S is distributed, but the predicate A is undistributed.
- In 'Some H are A', neither term is distributed.
- So the middle term A is undistributed in both premises.
Check the other terms: S is distributed in the major premise but not in the conclusion; that is permissible. H is undistributed in both the minor premise and the conclusion. Thus the only defect is the undistributed middle.
2. The chair says: 'The conclusion is true because 15 hostellers are scholarship holders, so the argument must be valid.' Evaluate this claim using the distinction between truth and validity. What do the 15/50/60 figures establish and what do they not establish?
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The chair is confusing the truth of a conclusion with the validity of an argument. Validity concerns whether the conclusion is entailed by the premises; truth concerns whether the premises and conclusion correspond to the facts.
The argument can have true premises and a true conclusion and still be invalid. To see this, imagine a possible arrangement consistent with the two premises in which the conclusion is false:
- 50 hostellers have 75% attendance, but none of them is a scholarship holder.
- 60 scholarship holders all have 75% attendance, but all are non-hostellers. Then 'All S are A' is true and 'Some H are A' is true, but 'Some H are S' is false. That is enough to show the form is invalid.
The figures 60, 50, and 15 establish that the conclusion happens to be true on the actual records: 15 hostellers are scholarship holders. They do not establish that the conclusion follows from the two premises. The extra fact '15 of the 50 are scholarship holders' is not contained in either premise. The chair is therefore using a true conclusion to excuse an invalid inference.
3. A member proposes adding the premise 'All students with 75% attendance are scholarship holders.' Would the argument then be valid? Is the added premise true on the records? What does this show about repairing invalid arguments?
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Adding the premise changes the argument to:
- All A are S.
- Some H are A.
- Therefore some H are S.
This is AII-1 (Darii), which is valid. The middle term A is now distributed in the major premise, so the undistributed-middle defect is removed.
However, the added premise is false on the records. There are 50 hostellers with 75% attendance, but only 15 of them are scholarship holders. So at least 35 students with 75% attendance are not scholarship holders. The added premise contradicts the data.
This shows that an invalid argument can always be made valid by adding a premise strong enough to entail the conclusion, but that repair is worthless if the added premise is false. A valid argument with a false premise does not establish its conclusion. The committee should either find a true premise that genuinely connects attendance to scholarship status, or weaken the conclusion, for example by reporting only the verified fact that 15 hostellers are scholarship holders.