study 3: enthymeme & sorites
Enthymeme
An otherwise perfectly valid categorical syllogism may appear not to be so when one of its propositions is suppressed or understood but not explicitly stated. Such an argument is known as an enthymeme. The first enthymeme has a suppressed major premise, the second, a suppressed minor premise, and the third, a suppressed conclusion.
Suppressed Major Premise
Some NFL quarterbacks are good passers because some NFL quarterbacks have strong throwing arms.
Identify the conclusion first, then classify the premise as either the major or minor. In this case, the premise is the minor premise, since it contains the minor term.
Complete Syllogism: A(ba) I(cb) < I(ca). Valid: AII-1, Darii.
Suppressed Minor Premise
No one in his right mind claims infallibility, for only perfect persons can claim infallibility.
Complete Syllogism: A(ab) E(cb) < E(ca). Valid: AEE-2, Camestres.
Suppressed Conclusion
No fair-minded person is capricious and some capricious people are irresponsible.
Complete Syllogism: E(ab) I(bc) < O(ca). Valid: EIO-4, Fresison.
Phase 2, First Example:
Inv, EAE-4, Rule #2 (The minor term, poorly-constructed-things, is undistributed in the premise but distributed in the conclusion.)
Phase 2, Second Example:
Valid, EIO-1, Ferio-1. The tests of Five Rules are met in this example. To illustrate, let us show that Bokardo (OAO-3) is valid by RAA proof.
O(ba) A(bc) < O(ca) Bokardo-3
1. O(ba) true premise
2. A(bc) true premise Therefore, O(ca)
Option 1:
Option 2:
An otherwise perfectly valid categorical syllogism may appear not to be so when one of its propositions is suppressed or understood but not explicitly stated. Such an argument is known as an enthymeme. The first enthymeme has a suppressed major premise, the second, a suppressed minor premise, and the third, a suppressed conclusion.
Suppressed Major Premise
Some NFL quarterbacks are good passers because some NFL quarterbacks have strong throwing arms.
Identify the conclusion first, then classify the premise as either the major or minor. In this case, the premise is the minor premise, since it contains the minor term.
- Missing Major: All persons with strong throwing arms are good passers. A(ba)
- Minor: Some NFL quarterbacks have strong throwing arms. I(cb)
- Conclusion: .:. Some NFL quarterbacks are good passers. I(ca)
Complete Syllogism: A(ba) I(cb) < I(ca). Valid: AII-1, Darii.
Suppressed Minor Premise
No one in his right mind claims infallibility, for only perfect persons can claim infallibility.
- Major: All persons claiming infallibility are perfect persons. A(ab)
- Missing Minor: No person in his right mind claims to be a perfect person. E(cb)
- Conclusion: No person in his right mind claims infallibility. E(ca)
Complete Syllogism: A(ab) E(cb) < E(ca). Valid: AEE-2, Camestres.
Suppressed Conclusion
No fair-minded person is capricious and some capricious people are irresponsible.
- Major: No fair-minded person is capricious. E(ab)
- Minor: Some capricious people are irresponsible. I(bc)
- Missing Conclusion .:. Some irresponsible people are not fair-minded. O(ca)
Complete Syllogism: E(ab) I(bc) < O(ca). Valid: EIO-4, Fresison.
Phase 2, First Example:
- Major: No German cars are inexpensive.
- Minor: All inexpensive things are poorly constructed.
- .:. No poorly constructed things are German cars.
Inv, EAE-4, Rule #2 (The minor term, poorly-constructed-things, is undistributed in the premise but distributed in the conclusion.)
Phase 2, Second Example:
- Major: No irreplaceable things are deductible.
- Minor: Some of the stolen books are irreplaceable.
- .:. Some of the stolen books are not deductible.
Valid, EIO-1, Ferio-1. The tests of Five Rules are met in this example. To illustrate, let us show that Bokardo (OAO-3) is valid by RAA proof.
O(ba) A(bc) < O(ca) Bokardo-3
1. O(ba) true premise
2. A(bc) true premise Therefore, O(ca)
- Assume: 3. O(ca) is false RAA method
- Then: 4. A(ca) is true contradictory of 3
- Then: 5. A(ca) A(bc) < A(ba) 4 & 2; Barbara-1
- But: 6. A(ba) cannot be true contradictory of 1, O(ba)
- So: 7. A(ba) must be false 1 & 6 contradictory
- But if: 8. A(ba) is false Step 7
- Then: 9 A(ca) or A(bc) is false. 5 & 8; Barbara-1
Option 1:
- Assume: 10. A(ca) is false From Step 9
- Then: 11. O(ca) in 3 can't be false 3 & 10 contradictories
- Then: 13. O(ca) is both true & false 3 & 11; Impossible!
Option 2:
- Assume: 14. A(bc) is false From Step 9
- Then: 15. A(bc) is both true & false 2 & 14; Impossible!
Sorites
Nonstandard categorical syllogisms may contain more than the required three forms. A sorites consists of a series of propositions in which the predicate of each is the subject of the next. The conclusion consists of the first subject and the last predicate. The chain of propositions is arranged in pairs of premises to make explicit the suppressed conclusion, thereby revealing the syllogism. The validity of the entire chain will depend on the validity of each syllogism in the chain. In this example, a = atheists; t = theologians; n = nihilists; s = scholars; and u = unreasonable (people). What can be concluded, given the following four propositions?
One interpretation takes "nihilists" in the first two propositions as the middle term and rearranging the premises yields the first syllogism.
Major: (ii) All nihilists are misologists. A(nm)
Minor: (i) All atheists are nihilists. A(an)
1st Conclusion: All atheists are misologists. A(am) (made explicit)
Using the 1st Conclusion as a premise in conjunction with the third proposition and rearranging the premises yields the second syllogism.
Major: (iii) All misologists are unreasonable. A(mu)
1st Conclusion (Minor): All atheists are misologists. A(am)
2nd Conclusion: All atheists are unreasonable. A(au) (made explicit)
Using the 2nd Conclusion as a premise in conjunction with the fourth proposition and rearranging the premises yields the third syllogism.
Major: (iv) All unreasonable ones are fools. A(uf)
2nd Conclusion (Minor): All atheists are unreasonable. A(au)
3rd Conclusion: All atheists are fools. A(af) (made explicit)
As stated earlier, for a sorites to be valid each syllogism forming a part of the sorites must be valid; otherwise the sorites is invalid. Each syllogism above is an instance of AAA-1, Barbara. Therefore, the sorites as a whole is valid.
In evaluating a sorites, keep in mind these requirements:
Major: No irreplaceable things are deductible.
Minor: Some of the stolen books are irreplaceable.
Therefore, Some of the stolen books are not deductible.
Valid, EIO-1, Ferio-1. The tests of Five Rules are met in this example. To illustrate, let us show that Bokardo (OAO-3) is valid by RAA proof.
O(ba) A(bc) < O(ca) Bokardo-3
1. O(ba) true premise
2. A(bc) true premise Therefore, O(ca)
Nonstandard categorical syllogisms may contain more than the required three forms. A sorites consists of a series of propositions in which the predicate of each is the subject of the next. The conclusion consists of the first subject and the last predicate. The chain of propositions is arranged in pairs of premises to make explicit the suppressed conclusion, thereby revealing the syllogism. The validity of the entire chain will depend on the validity of each syllogism in the chain. In this example, a = atheists; t = theologians; n = nihilists; s = scholars; and u = unreasonable (people). What can be concluded, given the following four propositions?
- (i) All atheists are nihilists. A(an)
- (ii) All nihilists are misologists. A(nm)
- (iii) All misologists are unreasonable. A(mu)
- (iv) All unreasonable ones are fools. A(uf)
One interpretation takes "nihilists" in the first two propositions as the middle term and rearranging the premises yields the first syllogism.
Major: (ii) All nihilists are misologists. A(nm)
Minor: (i) All atheists are nihilists. A(an)
1st Conclusion: All atheists are misologists. A(am) (made explicit)
Using the 1st Conclusion as a premise in conjunction with the third proposition and rearranging the premises yields the second syllogism.
Major: (iii) All misologists are unreasonable. A(mu)
1st Conclusion (Minor): All atheists are misologists. A(am)
2nd Conclusion: All atheists are unreasonable. A(au) (made explicit)
Using the 2nd Conclusion as a premise in conjunction with the fourth proposition and rearranging the premises yields the third syllogism.
Major: (iv) All unreasonable ones are fools. A(uf)
2nd Conclusion (Minor): All atheists are unreasonable. A(au)
3rd Conclusion: All atheists are fools. A(af) (made explicit)
As stated earlier, for a sorites to be valid each syllogism forming a part of the sorites must be valid; otherwise the sorites is invalid. Each syllogism above is an instance of AAA-1, Barbara. Therefore, the sorites as a whole is valid.
In evaluating a sorites, keep in mind these requirements:
- 1. If a conclusion is negative, then one and only one of the premises must be negative.
- 2 If the conclusion is affirmative, all of the propositions must be affirmative.
- 3. If the conclusion is universal, all of the premises must be universal.
- 4. A particular conclusion calls for not more than one particular premise.
Major: No irreplaceable things are deductible.
Minor: Some of the stolen books are irreplaceable.
Therefore, Some of the stolen books are not deductible.
Valid, EIO-1, Ferio-1. The tests of Five Rules are met in this example. To illustrate, let us show that Bokardo (OAO-3) is valid by RAA proof.
O(ba) A(bc) < O(ca) Bokardo-3
1. O(ba) true premise
2. A(bc) true premise Therefore, O(ca)
- Assume: 3. O(ca) is false RAA method
- Then: 4. A(ca) is true contradictory of 3
- Then: 5. A(ca) A(bc) < A(ba) 4 & 2; Barbara-1
- But: 6. A(ba) cannot be true contradictory of 1, O(ba)
- So: 7. A(ba) must be false 1 & 6 contradictory
- But if: 8. A(ba) is false Step 7
- Then: 9 A(ca) or A(bc) is false. 5 & 8; Barbara-1
- Assume: 10. A(ca) is false From Step 9
- Then: 11. O(ca) in 3 can't be false 3 & 10 contradictories
- Then: 13. O(ca) is both true & false 3 & 11; Impossible!
- Assume: 14. A(bc) is false From Step 9
- Then: 15. A(bc) is both true & false 2 & 14; Impossible!
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