Archive Labels, Conversion, and Existential Import
A district archive has 40 incunabula, 120 manuscripts, and a special-collections reading room. The cataloguer has verified two general statements: (S1) 'All incunabula in the collection are non-loanable.' (S2) 'Some manuscripts in the collection are loanable.' A volunteer prepares a label: 'All non-loanable items are incunabula.' A second volunteer, noting that all 40 incunabula exist and that 25 manuscripts are loanable, proposes: 'Some loanable manuscripts are not incunabula.' A third proposes: 'No loanable item is an incunabulum.' The archivist must decide which labels are logically warranted by S1 and S2, which require the existence information, and which confuse a valid immediate inference with an invalid conversion. The archive uses classical categorical logic and treats 'manuscript' as including incunabula unless stated otherwise.
1. State S1 in standard form. Give its obverse, converse, and contrapositive, and identify which of these are valid immediate inferences from S1 alone.
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S1 is: All I are non-L, where I = incunabula and L = loanable items.
Obverse: No I are L. This is valid by obversion: change the quality from affirmative to negative and replace the predicate with its complement.
Converse: All non-L are I. This is invalid. Converting an A proposition is not generally valid, because the predicate of an A proposition is undistributed while the subject of the converse would be distributed.
Contrapositive: All L are non-I. This is valid for an A proposition: 'All S are P' entails 'All non-P are non-S'. Here P = non-L, so non-P = L; and non-S = non-I.
A further valid step is the conversion of the obverse: from 'No I are L' (E proposition) infer 'No L are I'. That is valid conversion of an E proposition, but it is not the converse of the original A proposition.
2. Volunteer 1's label is 'All non-loanable items are incunabula.' Is it warranted by S1? Name the error and give a counterexample consistent with the archive's situation.
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No. The label is not warranted by S1.
S1 says: All I are non-L. The label says: All non-L are I. This is the converse of S1, and conversion of an A proposition is invalid. The error is illicit conversion, sometimes called conversion by accident.
A counterexample consistent with S1 is a non-loanable atlas or a non-loanable reference set that is not an incunabulum. Such an item is non-loanable, so it satisfies the predicate of S1, but it is not an incunabulum. The archive can have many non-loanable items besides its 40 incunabula. Thus S1 does not license the universal label.
3. Evaluate Volunteer 2's statement and Volunteer 3's statement using S1, S2, and the counts. Show the derivations and say whether the existence information is needed.
Read the worked answer
Volunteer 3: 'No loanable item is an incunabulum.' Derivation from S1 alone:
- All I are non-L. (S1)
- No I are L. (obversion)
- No L are I. (valid conversion of the E proposition in step 2) So Volunteer 3 is correct, and the conclusion follows from S1 alone. The counts are not needed.
Volunteer 2: 'Some loanable manuscripts are not incunabula.' Derivation:
- Some M are L. (S2)
- No L are I. (from S1, as above)
- Therefore some M are non-I. That is, some manuscripts are loanable and are not incunabula. This is valid. S2 already asserts that at least one manuscript is loanable, so no extra existential assumption is needed. The figure 25 confirms S2 but is not required by the form.
The statement 'Some non-loanable items are incunabula' would need an existence premise about incunabula. The archive has 40 incunabula, so it is true in fact, but it is not an immediate inference from S1 alone.