## Study 2: conversion, obversion, contraposition

**Conversion**

The converse of a proposition is formed by interchanging the subject and predicate. For example, the converse of "Some men are believers" is "Some believers are men." Application of conversion yields the following:

**5. I(ab) logically implies I(ba)**

Chart 2.1 shows that every time I(ba) is true, so is I(ab)

**6. E(ab) logically implies E(ba)**

Every time E(ba) is true, so is E(ab), as the Chart 2.1 shows.

**7. A(ab) logically implies I(ba)**

- This implication is valid only by conversion per accidens, from "All" to "Some".
- Chart 2.1 shows: I(ab) is true every time A(ab) is true.
- Then by conversion, I(ab) < I(ba) is valid.

The O form has no converse, since to permit it warrants drawing a false proposition from a true one. For example, the converse of the true proposition "Some vegetables are not carrots" would be the false proposition "Some carrots are not vegetables." Moreover, O(ba) is not true every time O(ab) is true; check Chart 2.1 diagrams again. (Note as well that O(ab) < O(ba) is invalid, since a term in the premise which is undistributed, is distributed in the conclusion.)

**Obversion**

To form the obverse of a proposition, change the quality of the proposition and replace the predicate by its complement. For example, the obverse of "All believers are saved" is "No believers are non-saved." A term and its complement are said to exhaust the universe of objects. Thus, if "S" stands for the class Saved, "S' " (read: S-prime or non-S) stands for the complement class, non-Saved. The combined classes SS' totally exhaust the universe of entities, since everything in the universe must fall either into one class or its complement class. Each of the four forms has an obverse.

**8. A(ab) logically implies E(ab'), the obverse**.

- In other words, All a is b logically implies No a is non-b.
- Thus, if All men are sinners, then No men are non-sinners.

**9. E(ab) logically implies A(ab'), the obverse.**

- Or, No a is b logically implies All a is non-b.
- If No atheists are Christian, then All atheists are non-Christian.

**10. I(ab) logically implies O(ab'), the obverse.**

- Rephrasing: Some a is b logically implies Some a is not non-b.
- Thus, if Some men are efficient, then Some men are not inefficient (non-efficient).

**11. O(ab) logically implies I(ab'), the obverse.**

- Rewording: Some a is not b logically implies Some a is non-b.
- Thus, if some things are not excusable, then some things are inexcusable (non-excusable).

**Contraposition**

The contrapositive of a proposition is one in which the complement terms of the subject and predicate are interchanged. Contraposition can be thought as first obverting the original, converting the resulting proposition, then obverting the resultant one. For example, the proposition "All humans are mortal" obverts to "No humans are immortal (non-mortal)" which converts to "No non-mortal persons are human" which when obverted yields "All non-mortals are non-human."

**12. A(ab) logically implies A(b'a'), the contrapositive.**

- Reread the example as shown above.
- Note: the contrapositive of A(b'a') is A(ab).

**13. E(ab) logically implies O(b'a'), the contrapositive.**

- The obverse of an E form yields an A form which when converted yields an I form.
- (Recall, the A form undergoes conversion per accidens, or conversion by limitation of quantity from universal to particular.)
- Obverting this I form results in the contrapositive O form.

Step by step process is shown below.

Form E: "No ungraciousness is excusable.”

- Step 1: Obversion: "All ungraciousness is inexcusable."
- (Changed Quality from Negative to Affirmative and predicate replaced by complement.)
- Step 2: Conversion by Limitation: "Some inexcusable behavior is ungracious."
- (Changed Quantity from Universal to Particular; terms are exchanged.)
- Step 3: Obversion: "Some inexcusable behavior is not gracious."
- (Changed as in Step 1 above.)

**14. O(ab) logically implies O(b'a'), the contrapositive.**

- Some a is not b logically implies Some non-b is not non-a.
- Consider: Some pleasant things are not worthy.
- Does this mean that some unworthy things are not unpleasant?
- Answer: Yes.

The chart below summarizes all three immediate inferences. Remember, the symbol for prime is “ ' “ attached to a letter indicates the complement of the term. In Chart 2.4, each of the immediate inferences has the original in the "Form" column.

Chart 2.4: Summary of 3 Immediate Inferences

*Forms Conversion Obversion Contraposition*

A(ab) I(ba)* E(ab') A(b'a')

E(ab) E(ba) A(ab') O(b'a')

I (ab) I(ba) O(ab') (None)

O (ab) (None) I(ab’) O(b'a')

**Conversion per accidens (by limitation)*

Let's review: Conversion is simple enough being the exchange of the terms of the E and I forms. There is no converse for the O form, and the A form converts by limitation. (We have suggested that immediate inferences can be more properly treated as logical implications. We introduced the symbol "<" to stand for logically implies, and the prime symbol " ' " to designate the complement of whatever form to which it is attached.)

Two previous examples are cited below for review: one for obversion, the other for contraposition.

**OBVERSION: #8**:

A(ab) < E(ab'), is a valid logical implication. The conclusion is obtained by obverting the premise, A(ab) in this manner: change the quality of the premise and replace its predicate (b) by its complement, (b').

**CONTRAPOSITION: #14**:

O(ab) < O(b'a') is a valid logical implication. The conclusion is obtained by interchanging the complements of the terms in the premise, O(ab). Or, using the step-by-step procedure, we obvert, then convert, then obvert as described in what follows:

Premise: O(ab)

Step 1. obvert above to I(ab')

Step 2. convert I(ab') to I(b'a)

Step 3. obvert I(b'a) to O(b'a')

While the formulation of immediate inferences as logical implications using the "<" and " ' " as required, and applying Chart 2.1 features to determine validity, this procedure becomes more complicated and tiresome for obversion and contraposition. For these reasons, we have evaluated these immediate inferences using definitions. Nevertheless, all of the immediate inferences can be tested using Chart 2.1. Should the student be inclined to carry out these procedures, it should be noted that for each case, everything outside of circle a is a'; everything outside of circle b is b', and vice versa. These additional notations must be filled in the 5 Cases of circles correctly for accurate results.

*Return to Study 2*

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