1. Operations (Transformations) On Categorical Sentences
Conversion (Simple & Limited)
Obversion
Contraposition

2. Equivalent Sentences*
Sentences are logically equivalent if (and only if) it is impossible for them to differ in truth value. In every situation they are either both true or both false.
Examples:
Some S are P. Some P are S.
No S are P No P are S.

3. Conversion
To form the converse of a categorical sentence simply exchange its terms.
*** S *** P *** P *** S
Example: Some cats are not males. Some males are not cats.
Are the sentences in this example logically equivalent? Or do they just happen to be both true in virtue of the subject mater ?
Stay tuned.

4. Converse of Universal Affirmative (A) All S are P
S P

5. Converse of Universal Affirmative (A)’
All S are P
S P
S P
All P are S
No Logical Connection. Logically independent.

6. Converse of Universal Negative (E)
No S are P
S P
No P are S
Equivalent

7. Converse of Universal Negative (E)
No S are P
No P are S
Equivalent
S P

8. Converse of Particular Affirmative (I)
Some S are P
Some P are S
Equivalent
S P *

9. Converse of Particular Negative (O)’ Some S are not P Some P are not S
S P S P
* *
Not Equivalent ----
(Truth values not always the same.)
Some cats are not males Some males are not cats.
Some mammals are not cats. ---Some cats are not mammals.

10. LogiCat
Cats ? Males ?
CATegorical Logic !

11. Limited Converse
To form the limited converse of a form A sentence (All S are P) limit the quantity to particular and then convert. This requires an existential assumption that there is at least one S.
Example:
All freshmen are logicians.
Some logicians are freshmen.
(This is implication, not equivalence.)

12. Conversion by Limitation
Converse:
All P are S
All S are P
No Logical Relationship
S P S P * T
Assume S exist..
Implication
S P *
Some S are P
Converse (equivalent)
Some P are S

13. Obversion
To form the obverse of a categorical sentence simply change the quality of that sentence and negate its predicate term.
Example:
No Republicans are Gaulists.
All Republicans are non-Gaulists.

14. Negating a term
To negate a term is to refer to all things other than what the term designates. T
nonT (things other than T).

15. Negating Terms in English
To negate a single-word term, simply prefix it with “non.” To negate a complex term, prefix it with “things other than...”
Examples:
tomatoes --- non-tomatoes
funny stories --- things other than funny stories.

16. Negating English Terms (Further Examples)
1) non-citizen
non-non-citizen
2) onions picked by Miss America
non-onions picked by Miss America
things other than onions picked by Miss America
3) truths told by Bill Clinton
non-truths told by Bill Clinton
things other than truths told by Bill Clinton
4) theme parks
things other than theme parks.

17. Examples of Obversion (1) Some wealthy women are politicians.
Some wealthy women are not non-politicians.
(2) Every U.S. Senator is sworn to uphold the Constitution.
All U.S. Senators are people sworn to uphold the Constitution. (Standard Form)
No U.S. Senators are things other than people sworn to uphold the Constitution.

18. A and Its Obverse A: All S are P (No S are outside P)
S P
A: All S are P
(No S are outside P)
Obverse: No S are non-P
Equivalent ----

19. E and Its Obverse E: No S are P. Obverse: All S are non-P.
S P
E: No S are P.
Obverse: All S are non-P.
(No S are in P)
--- Equivalent ---

20. I and Its Obverse I: Some S are P Obverse: Some S are not non-P
S P *
I: Some S are P
Obverse: Some S are not non-P
(Some S are not things outside P)
--- Equivalent ---

21. O and Its Obverse O: Some S are not P Obverse: Some S are non-P
S P *
O: Some S are not P
Obverse: Some S are non-P
(Some S are things outside P)
--- Equivalent ---

22. Equivalent ?
A E I O
Converse No Yes Yes No
Obverse Yes Yes Yes Yes

23. S P
negate
non-P non-S
Contraposition

24. Contraposition
To form the contrapositve of a categorical sentence simply exchange and negate its terms.
Examples:
No tables are chairs No non-chairs are non-tables.
Some Republicans are not capitalists. ---
Some non-capitalists are not Republicans.

25. Contraposing a Categorical Sentence
**** S ******* P
negate
**** non-P ******* non-S

26. A and Its Contrapositve
S P
A: All S are P.
Contrapositive: All non-P are non-S
S P
(nothing outside P is inside S)
--- Equivalent --- .

27. E and Its Contrapositve
S P
E: No S are P.
Contrapositive: No non-P are non-S
S P
(nothing outside P is outside S)
--- No Logical Relationship ---

28. I and Its Contrapositve
S P
I: Some S are P. *
Contrapositive: Some non-P are non-S
S P
(Something outside P is outside S) *
--- No Logical Relationship ---

29. O and Its Contrapositve
S P
O: Some S are not P. *
Contrapositive: Some non-P are not non-S
S P
(Something outside P is inside S) *
--- Equivalent ---

30. Equivalent ? A E I O Converse No Yes Yes No Obverse Yes Yes Yes Yes
Contrapositive Yes No No Yes