LESSON # 13

Conversion, Obversion and Contraposition

Reading Assignment: 4.4 (215-222)

Click here to skip the following discussion and go straight to the assignments.

In this lesson we will be learning how to translate categorical propositions (with one or more negated terms) into equivalent categorical propositions that are easier to understand and work with.

The three operations used to accomplish this translation are conversion, obversion, and contraposition. The following is a summary of what you must understand to perform these functions.

Term Compliments

Before we start with the particular operations, a note about term compliments. You will use them for obversion and contraposition. To determine the complement of a term you need to make it into the "opposite" term, or rather, into the class of things that is outside the class of the original term. An example is best here:

TERM

COMPLEMENT

dogs

non-dogs

qualified persons

unqualified persons

persons who like asparagus

persons who do not like asparagus

non-cats

cats

sensible actions

senseless actions

Usually adding or subtracting a "non-" before the term will suffice to make the term a complement.

But in the case of multi-word terms this sometimes doesn't make any sense. In such cases you reduce the scope of the discourse. "Non-persons who like asparagus" would have been a crazy sounding term compliment in the above example, so I reduced the scope to the class of persons. Use your common sense for this and you'll do fine.

Also notice that Hurley makes ample use of prefixes such as "endo-" and "exo-" to denote term complements and you might miss them if you aren't careful.

 

 

Conversion:

Conversion is the simplest operation to perform. All you do is switch the subject and predicate term. You must remember that conversion is only to be used on E and I statements. By this I mean that when you convert an E or an I statement, the new statement will retain the same truth value (T or F) as the original statement (the one you just converted). When you use conversion on an A or O statement, the resulting statement does not necessarily have the same truth value as the original, making it unreliable for drawing inferences.

e.g. No cats are dogs no dogs are cats.

Hint: I remember that conversion is valid for E and I statements by looking at the name of the operation. These are the two vowels following "con": convErsIon.

Obversion:

Obversion is a bit more complicated, but it is useful for all statement types (A, E, I and O). It is a two-step process:

1. You must change the quality (affirmative/negative) of the statement.

2. You must replace the predicate term with its complement.

To change the quality but not the quantity means that A and E statements will switch with one another and that I and O statements will switch with one another, e.g., "All S are P" becomes "No S are P."

Remember that you only replace the predicate term with its complement. (You leave the subject term as it is.)

e.g. No cats are dogs All cats are non-dogs.

Contraposition:

Contraposition is easier to perform than obversion, although it also involves a two-step process. You can only contrapose A and O statements, if you want to preserve truth value.

1. You switch the subject and predicate terms (like conversion).

2. You replace both the subject and predicate terms with their complements.

e.g. Some cats are not black animals Some non-black animals are not non-cats.

Hint: In the word "contraposition," A and an O are the vowels that follow the "con." ContrApOsition.

OPERATION

 

PROCESS

 

USE FOR

ConvErsIon

- Switch subject and predicate -

E & I

Obversion

- Change quality and negate* predicate -

All

ContrApOsition

- Switch subject and predicate and negate* both terms -

A & O

*The word "negate" is an informal but simpler way of saying "replace the term with its complement."

NOTE: If you perform an operation on a type of statement for which it is not valid, the resulting (new) statement will have an undetermined truth value. To make an inference on the basis of such an operation is to commit a logical fallacy.

Venn diagrams

These are used as they were in the last lesson, with some modifications to accommodate term compliments. Hurley mainly uses Venn diagrams in this section to illustrate which operations preserve truth value on the different statements. Unless you want to try your hand at these in the extra credit you don't really have to draw them yourself, just look at his drawings to determine when truth value is preserved. 

If you do want to draw your own, just remember that with term compliments ("non-" terms) your shading or "x" goes outside of the circle you would otherwise (if the term were not negated) put it in. This can be confusing, but if you go slowly and check yourself you'll be ok. Usually your results, no matter how confused you were when you worked the problem, will look normal. There are, however, instances when you get weird results.

Weird results

1. Sometimes, with contraposition, you will be shading completely outside of the set of circles. And yes, this looks weird, but if you are expecting it it and don't let it catch you off guard, it isn't really too complicated. (No non-C are non-D, for example) Usually, if you get this result you have shown that the operation was used illicitly, but this isn't necessarily the case.

similarly,

2. sometimes an "x" will be completely outside the circles. (Some non-C are non-D for example)

Logic Coach Assignment: 4.4 II 1-3; III 1-15

Assignment 1: (six points each)

Perform whichever operations are useful (preserve truth value) for the following statements. (You will perform two operations on each statement; obversion (for all) and one of the other two, either conversion or contraposition) In the statements with words be careful about using term compliments that make sense. Write these sentences out in full.

1. All non-A are B

2. No A are non-B

3. Some non-A are non-B

4. Some non-A are not B

5. Some college football coaches are not people who slip money to their players.

6. No cult leaders are people who fail to brainwash their followers.

7. Some murals by Diego Rivera are works that celebrate the revolutionary spirit.

8. All Grateful Dead concerts are festive spectacles.

9. Some organ transplants are not sensible operations.

10. Some states having limited powers are not slave states.

 

ASSIGNMENT 2:

Use conversion, obversion and contraposition to determine whether the following arguments are valid or invalid. (ten points each.) If the arguments are invalid state the fallacy committed.

1. Some states having limited power are not slave states.
Therefore, some free states are not states having unlimited power.

2. No flat taxes are equitable sources of revenue.
Therefore, all flat taxes are inequitable sources of revenue.

3. Some carnival performers are not jugglers.
Therefore, some jugglers are not carnival performers.

4.Some poisonous mushrooms are inedible fungi. 
Therefore, some edible fungi are nonpoisonous mushrooms.

 

Extra Credit: Check the arguments in assignment 2 using Venn diagrams.

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