2.4 Categorical Syllogisms

In this section, we study categorical syllogisms and learn how to identify their argument forms. A syllogism is an argument with two premises and one conclusion.

2.4.1 Standard Categorical Syllogisms

A standard categorical syllogism is a syllogism that consists of three categorical sentences, in which there are three terms, and each term appears exactly twice.

The three terms in a standard categorical syllogism are the major, the minor and the middle terms. The major term is the predicate term of the conclusion. The minor term is the subject term of the conclusion. The middle term is the term that appears twice in the premises.

A categorical syllogism is presented in standard form when its statements are arranged in the order of the major premise, the minor premise and the conclusion. Here the major premise is the premise that contains the major term, and the minor premise is the premise that contains the minor term.

The mood of a standard categorical syllogism is determined by the types of categorical statements it contains. In the following example, the major premise is an E statement and the minor premise is an I statement. The conclusion is an O statement. So its mood is EIO.

The figure of a standard categorical syllogism is determined by the positions of the two appearances of the middle term. In the above example, the middle term is “spiders.” The relative positions of its two appearances show that its figure is 4.

By combining mood and figure together, we can give the argument form of each standard categorical syllogism a unique name. In the above example, the argument form is EIO-4.

Quite often, categorical syllogisms are not presented in the standard form. To use the Venn Diagram method to decide their validity, we need to first change them into the standard form. This process may involve two steps:

  1. Paraphrase sentences into categorical sentences.
  2. Reduce the number of terms to three.

2.4.2 Paraphrasing Categorical Sentences

The following are some formulas to help you paraphrase sentences. The symbol “⇒” is used here to stand for “... is paraphrased as ...”.

“A few …” and “Few …”

A few S are P.  ⇒  Some S are P

For example, the sentence

A few soldiers are heroes. (A few S are H.)

is paraphrased as

Some soldiers are heroes. (Some S are H.)

It is important not to confuse “A few” with “Few”.

Few S are P.  ⇒  Some S are P and some S are not P.

The key point of the sentence “Few tigers are white” is to stress that white tigers are rare. We need two categorical sentences to capture its meaning:

Some tigers are white animals, but most tigers are not white animals.

But since in Categorical Logic we can use only the universal quantifier “all” or the particular quantifier “some,” we have to replace “most” with “some,” and rewrite the sentence further as

Some tigers are white animals and some tigers are not white animals.

Granted that these two categorical sentences together do not capture all the meanings in the original sentence, but this is the best we can do in Categorical Logic.

“Not every …”, “Not all …” and “Not a …”

Not every S is a P.  ⇒  Some S are not P.

The sentence

Not every swan is white. (Not every S is W.)

is paraphrased as

Some swans are not white birds. (Some S are not W.)

Not all S are P.  ⇒  Some S are not P.

“Not all S are P.” is a variation of “Not every S is a P.” The sentence

Not all apples are red. (Not all A are R.)

is rewritten as

Some apples are not red things. (Some A are not R.)

Notice that the next formula is different from the previous two.

Not an S is a P.  ⇒  No S are P.

The sentence

Not a penguin can fly.

emphasizes that no penguin can fly, and thus is paraphrased as

No penguins are flyers. (No P are F.)

“If …, then …” and “ … if …”

Conditional sentences are fairly common in logic. Here we learn how to paraphrase them as the A statements.

If it is an S, then it is a P.  ⇒  All S are P.

For example,

If a driver is drunk, then he is dangerous.

is paraphrased as an A statement

All drunk drivers are dangerous drivers. (All D1 are D2.)

Here “ D1” stands for “drunk drivers” and “ D2” stands for “dangerous drivers.” Subscripts are used to distinguish the two terms.

Sometimes, the conditional sentences are written with “if” in the middle of the sentence.

It is a P if it is an S.  ⇒  All S are P.

The sentence

A driver is dangerous if he is drunk.

is just a variation of

If a driver is drunk, then he is dangerous.

and hence is also rewritten as

All drunk drivers are dangerous drivers. (All D1 are D2.)

“Only …”, “None except …” and “None but …”

Sentences begin with “only” are paraphrased using the following formula.

Only S are P.  ⇒  All P are S.

When paraphrasing the sentence

Only warm-blooded vertebrates are mammals. (Only W are M.)

it is important to reverse the order of the two terms, and rewrite it as

All mammals are warm-blooded vertebrates. (All M are W.)

It would be a mistake to paraphrase the sentence as “All W are M.” It is not true that all warm-blooded vertebrates are mammals. So we can clearly see that it is not logically equivalent to “Only W are M.”

“None except S are P” and “None but S are P” are just other ways of saying “Only S are P.”

None except S are P.  ⇒  All P are S.
None but S are P.  ⇒  All P are S.

The sentence

None but club members can vote. (None but C are V.)

is transformed as

All the persons who can vote are club members. (All V are C.)

“The only …”

Some sentences start with the phrase “the only.” “The only S are P” is another way of saying “Only P are S.”

The only S are P.  ⇒  All S are P.

The sentence

The only students eligible for sports team tryouts are freshmen. (The only E are F.)

says the same thing as

Only freshmen are eligible for sports team tryouts. (Only F are E.)

and is paraphrased as

All the students eligible for sports team tryouts are freshmen. (All E are F.)

“All except …” and “All but …”

It takes two categorical sentences to paraphrase sentences beginning with “all except” or “all but”

All except S are P.  ⇒  No S are P and all non-S are P.
All but S are P.  ⇒  No S are P and all non-S are P.

For example, it is clear that

All except freshmen are eligible players.

excludes freshmen from eligibility. But in addition, it also says that all non-freshmen (that is, sophomores, juniors and seniors) are eligible. Therefore, it needs to be rewritten as two categorical sentences.

No freshmen are eligible players and all non-freshmen are eligible players. (No F are E and all non-F are E.)

2.4.3 Reducing the Number of Terms

As discussed early, there can only be three terms in a categorical syllogism. However, in the following argument

All mean people are inconsiderate people.

All kind people are considerate people.

All unkind people are mean people.

we have five terms because both “kind people” and its complement “unkind people” appear in the syllogism. So do the terms “considerate people” and “inconsiderate people.” When this happens, we need to use the three operations (conversion, obversion or contraposition) to reduce the number of terms to three. First, we replace each term with a capital letter or its complement.

All M are non-C.

All K are C.

All non-K are M.

Afterwards, we apply contraposition to the minor premise and change it to “All non-C are non-K.”

All M are non-C.
All K are C.
All non-K are M.
 
— Contrapos. →
 
All M are non-C.
All non-C are non-K.
All non-K are M.

As a result, we have three terms: M, non-K and non-C. The argument form is AAA-4.

The following example shows how a categorical syllogism is transformed from a written passage into the standard form through paraphrasing and reducing the number of terms.

Not all religious people are spiritual. If one is spiritual, then one is unprejudiced, but some religious people are prejudiced.

First of all, we paraphrase each sentence into a categorical sentence, and identify the major and the minor premises.

All S are non-P.

Some R are P.

Some R are not S.

Afterwards, we apply obversion to the major premise to reduce the number of terms to three.

All S are non-P.
Some R are P.
Some R are not S.
— Obv. →
 
 
No S are P.
Some R are P.
Some R are not S.

The resulting standard form is EIO-2.

When we try to reduce the number of terms by applying an operation to a sentence, we need to choose an operation so that the resulting sentence is logically equivalent to the original sentence. This way, we make sure we do not change the argument and turn it into a different argument. The following operation is incorrect because contraposition is applied to an E statement. The resulting statement “No H are D” is not logically equivalent to “No non-D are non-H.”

Some H are B.
No non-D are non-H.
Some D are not B.
— Contrapos. →
 
Some H are B.
No H are D.
Some D are not B.

We thus need to use different operations to reduce the number of terms. One way to do so is to apply conversion and then obversion to both the premises.

Some H are B.
No non-D are non-H.
Some D are not B.
— Conv. & Obv. →
— Conv. & Obv. →
 
Some B are not non-H.
All non-H are D.
Some D are not B.

The mood and figure of the argument form is OAO-4.

Exercise 2.4

  1. Use the capital letters provided to paraphrase the following sentences into categorical sentences.
    • Click on Check Answers at the end of the section to see if your answers are correct.
    • You can change the answers marked as incorrect and click on Check Answers again.
    • Reload the page to get rid of answers and correction marks.
  1. None but the logical are rational. (L: logical people, R: rational people)
  2. A few writers win the Nobel Prize in literature. (W: writers, N: winners of the Nobel Prize in literature)
  3. Only accountants will be hired. (A: accountants, H: people who will be hired)
  4. A gas-guzzler is not environment-friendly. (G: gas-guzzlers, E: environment-friendly cars)
  5. Not all movie stars are good actors. (M: movie stars, G: good actors)
  6. All except the company employees can enter the sweepstakes. (C: company employees, E: people who can enter the sweepstakes)
  7. Not a tabloid story is reliable. (T: tabloid stories, R: reliable reports)
  8. Few rumors are true. (R: rumors, T: true assertions)
  9. Only drugs approved by FDA are safe. (D: drugs approved by FDA, S: safe medicines)
  10. Not a child should be without health care. (C: children, H: people who should have health care)
  11. Bats are the only true flying mammals. (B: bats, T: true flying mammals)
  12. If one works hard, then one will succeed. (W: hard-working people, S: people who will succeed)
  13. A few small business owners support living wages. (O: small business owners, S: supporters of living wages)
  14. All but career politicians favor term limits. (C: career politicians, F: people who favor term limits)
  15. None but the wealthy benefit from the tax cut. (W: wealthy people, B: people who benefit from the tax cut)
  16. Whatever is obscene is immoral. (O: things that are obscene, M: things that are moral)
  17. Most drugs have side effects. (D: drugs, S: drugs with side effects)
  18. Few drugs are without side effects. (D: drugs, S: drugs with side effects)
  19. Only open-minded people are reasonable. (O: open-minded people, R: reasonable people)
  20. Open-minded people are the only reasonable people. (O: open-minded people, R: reasonable people)
  1. (1) Reduce the number of terms to three; (2) indicate the operation(s) (Conv., Contrapos. or Obv.) used; and (3) identify the mood and figure.
    • Click on Check Answers at the end of the section to see if your answers are correct.
    • You can change the answers marked as incorrect and click on Check Answers again.
    • Reload the page to get rid of answers and correction marks.

Example:

All non-C are O.
Some V are not C.
Some V are O.
 
— Obv. →
 
All non-C are O.
Some V are non-C.
Some V are O.
 
AII-1
 
  1. Some non-D are B.
    All B are H.
    Some H are not D.
     
     
    —→
    Some non-D are B.
    All B are H.
     
     
  2. All M are S.
    Some non-S are A.
    Some A are non-M.
    —→
     
     
    Some non-S are A.
    Some A are non-M.
     
     
  3. No E are R.
    All non-G are R.
    All G are E.
    —→
    —→
     
    All G are E.
     
     
  4. Some C are non-E.
    No K are non-C.
    Some K are not E.
    —→
    —→
     
    Some K are not E.
     
     
  5. No non-N are non-P.
    All N are non-L.
    All L are non-P.
     
    —→
     
    No non-N are non-P.
    All L are non-P.
     
     
  6. No non-B are D.
    Some non-D are not non-E.
    Some E are not B.
    —→
    —→
     
    Some E are not B.
     
     
  7. Some R are non-D.
    No non-M are D.
    Some M are R.
    —→
    —→
     
    Some M are R.
     
     
  8. All non-G are E.
    Some non-E are not B.
    Some B are G.
    —→
     
    —→
    Some non-E are not B.
     
     
  9. All non-K are non-F.
    All K are C.
    All C are F.
    —→
     
     
    All K are C.
    All C are F.
     
     
  10. Some R are not A.
    No non-D are A.
    Some D are not R.
     
    —→
     
    Some R are not A.
    Some D are not R.
     
     
  11. All N are H.
    All G are N.
    All non-G are non-H.
    —→
     
     
  12. No J are M.
    All non-J are L.
    Some L are M.
    —→
     
     
    All non-J are L.
    Some L are M.
     
     
  13. No B are P.
    Some non-B are non-D.
    Some D are not P.
    —→
    —→
     
    Some D are not P.
     
     
  14. All O are R.
    Some A are not non-R.
    Some A are not non-O.
     
    —→
    —→
    All O are R.
     
     
  15. Some non-D are E.
    Some A are not E.
    Some non-A are D.
    —
     
    —→
     
     

 

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