2.3 Three Operations

A categorical statement such as

Some employed workers are not people with health insurance.

can be rewritten as

Some people without health insurance are not unemployed workers.

to counter the bias that people without health insurance are people without jobs. The logic rules that enable us to do so are called the three transformation rules or operations. They are conversion, contraposition and obversion. When applied to the right kinds of statements, they yield new statements that are logically equivalent to the original statements.

Two statements are logically equivalent if they necessarily have the same truth value. That is, if one of them is true, the other must also be true, and if one of them is false, the other must also be false. We can illustrate that two statements p and q are logically equivalent in terms of the truth table by ruling out the second and the third cases.

2.3.1 Conversion

The operation conversion moves the subject term to the predicate term position, and moves the predicate term to the subject term position. In short, it switches the subject and the predicate terms. If we apply convertion to the A statement “All S are P”, the resulting statement would be “All P are S”. Click on the Conversion button below to see what the operation does when applied to a statement. You can use the dropdown menu to select a different type of categorical statement or a different subject term or predicate term.

Applying an operation to a statement would yield a new statement. To see whether the new statement is logically equivalent to the original statement, we can use the Venn Diagrams. The following diagram shows how sets, their complements and the intersections are represented as different colored areas. To see the areas, move the cursor on top of each term.

Applying conversion to E and I statements will yield logically equivalent statements. From the Venn Diagrams we can see clearly why they are equivalent. Notice that the diagrams on the right-hand side are the mirror images of the ones on the left.

For example, if we apply conversion to the E statement

No ravens are white birds.

we would get a logically equivalent statement

No white birds are ravens.

Applying conversion to the I statement

Some swans are white birds.

would yield a logically equivalent statement

Some white birds are swans.

However, when applied to the A and O statements, conversion does not yield logically equivalent statements.

In either pair, the Venn diagrams are not mirror images of each other.

Applying conversion to the A statement

All tigers are mammals.

would result in a statement

All mammals are tigers.

which is false, and thus cannot be logically equivalent to the original statement. It is also evident that the O statement

Some mammals are not whales.

is not logically equivalent to

Some whales are not mammals.

2.3.2 Contraposition

The operation contraposition has two steps:

  1. Like conversion, it switches the subject and the predicate terms;
  2. Afterwards, it replaces the two terms with their complements.

So if the initial statement is “All S are P”, the operation would turn it into “All non-P are non-S”. Click on the Contraposition button below to see what the operation does when applied to a statement. Use the dropdown menu to select a different type of categorical statement or a different subject term or predicate term.

Applying contraposition to A and O statements will yield logically equivalent statements. Again, this can be seen by comparing the Venn diagrams.

Contraposition would transform the A statement

All animals that can fly are animals with wings.

into

All animals without wings are animals that cannot fly.

To see more clearly the operation at work, we can use “F” for “animals that can fly” and “W” for “animals with wings.” The term “animals that cannot fly” would then be “non-F,” and “animals without wings” would be “non-W.”

All F are W. = All non-W are non-F.

This example illustrates how contraposition helps us recognize that two sentences are logically equivalent. The pair of O statements that we saw at the beginning of this section

Some employed workers are not people with health insurance. (Some E are not H.)

and

Some people without health insurance are not unemployed workers. (Some non-H are not non-E.)

are also logically equivalent because of contraposition. So are the following two O statements

Some useful gadgets are not successful inventions. (Some U are not S.)

Some unsuccessful inventions are not useless gadgets. (Some non-S are not non-U.)

When contraposition is applied to the E and I statements, the new statements would not be logically equivalent to the original ones. We can see this clearly from their Venn Diagrams.

For example, these two E statements are not logically equivalent:

No dead organisms are immortal beings. (No non-L are non-M.)

No mortal beings are living organisms. (No M are L.)

Nor are the following pair of I statements:

Some persuasive arguments are illogical. (Some P are non-L.)

Some logical arguments are unpersuasive. (Some L are non-P.)

2.3.3 Obversion

The operation obversion is quite different from the previous two. It also has two steps:

  1. Change the quality of the statement;
  2. Replace the predicate term with its complement.

The first step means changing an affirmative statement to a negative statement, or a negative statement to an affirmative one. So if the original statement is an A statement “All S are P”, whose quality is affirmative, obversion would turn it into an E statement, whose quality is negative. Next, the predicate term P is replaced with its complement non-P. Click on the Obversion button to see the resulting E statement. Now use the dropdown menu to select the O statement. The statement should now read “Some S are not non-P”. The quality of the O statement is negative. When we apply obversion to an O statement, we would turn it into an I statement, whose quality is affirmative. After replacing the predicate term non-P with P, we would then have “Some S are P”. Click on the Obversion button to see the resulting I statement.

Applying obversion to categorical statements will always yield logically equivalent statements.

All S are P. = No S are non-P.

No S are P. = All S are non-P.

Some S are P. = Some S are not non-P.

Some S are not P. = Some S are non-P.

As we can see in the following examples, obversion is fairly common in daily language. Even though the jargon “obversion” may be new to you, you have used it countless times before.

No genetic diseases are contagious. (No D are C.)

All genetic diseases are non-contagious. (All D are non-C.)

Some medical tests are not accurate. (Some T are not A.)

Some medical tests are inaccurate. (Some T are non-A.)

Some people are irrational. (Some P are non-R.)

Some people are not rational. (Some P are not R.)

After an operation is applied, if the new statement is logically equivalent to the original statement, then their truth values must be the same. But if they are not logically equivalent, then their truth values don’t have to be the same. This means that the truth value of the new statement may or may not be the same as the truth value of the original statement. As a result, the truth value of the new statement would be undetermined.

Exercise 2.3

  1. Give the new statement or determine which operation is used to derive the new statement. Then decide whether the new statement is true (T), false (F) or undetermined (?).
    • 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.
Given Statement (T/F)
Operation
New Statement
T/F/?
  1. All B are non-C. (T)
    Conversion
  2. Some M are not non-A. (F)
    Contraposition
  3. No H are non-G. (F)
    Conversion
  4. Some non-K are H. (T)
    Obversion
  5. No non-R are non-D. (T)
    Contraposition
  6. All non-N are W. (F)
    Obversion
  7. Some S are not non-E. (T)
    Obversion
  8. All non-P are G. (T)
    Contraposition
  9. No D are non-R. (F)
    Contraposition
  10. Some non-S are not T. (F)
    Obversion
  11. Some U are not L. (T)
    Some U are non-L.
  12. No K are non-M. (F)
    No non-M are K.
  13. All non-B are A. (T)
    No non-B are non-A.
  14. Some F are non-D. (F)
    Some D are non-F.
  15. Some non-R are A. (F)
    Some non-R are not non-A.
  16. Some C are not non-H. (F)
    Some non-H are not C.
  17. All T are non-S. (F)
    All S are non-T.
  18. Some non-P are G. (F)
    Some non-P are not non-G.
  19. No non-D are M. (F)
    No non-M are D.
  20. All B are non-C. (T)
    All non-C are B.
  1. Suppose you are stranded on a desert island. In searching for foods, you discover that some bright-colored fungi on the island are edible. Use three operations and the logical relations in the Squares of Opposition to infer whether the following sentences are true (T), false (F) or undetermined (?).
    • 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.
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  1. All bright-colored fungi on the island are edible.
  2. Some bright-colored fungi on the island are not edible.
  3. Some edible fungi on the island are bright-colored.
  4. Some edible fungi on the island are not bright-colored.
  5. No edible fungi on the island are bright-colored.
  6. All bright-colored fungi on the island are inedible.
  7. All edible fungi on the island are non-bright-colored.
  8. Some inedible fungi on the island are bright-colored.
  9. Some non-bright-colored fungi on the island are edible.
  10. All non-bright-colored fungi on the island are edible.

 

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