1. Strict Logical Entailments of Categorical Propositions
Immediate Inferences
Conversion
Obversion
Contraposition

2. Conversion
Simply Switch Positions of the Terms
No Cats are Dogs E SP
No Dogs are Cats E PS
Valid operation for E and I statements

3. Obversion 1. Change the Quality 2. Form the complement of the
predicate term
All Popes are Men
No Popes are non-men
women?

4. In stating the complement of the
predicate class, choose in the
way that makes the best sense
for the example.
No sky-divers are people who wear
glasses.
All sky-divers are non-people who wear
glasses ? ?
All sky-divers are people who do not
wear glasses.

5. Obversion is valid for all four forms
(use ~ to mean “non”)
A SP E S~P
E SP A S~P
I SP O S~P
O SP I S~P

6. These operations can be combined
No cats are dogs
No dogs are cats conversion
All dogs are non-cats obversion

7. But some moves are invalid, e.g.,
All dogs are non-cats (true)
All non-cats are dogs (false)
Conversion is not valid for
A or O statements; nothing
follows from converting an A
or an O.

8. Contraposition Obversion of the Converse of an Obverse
Since conversion is not valid for A or O,
contraposition is not valid for E or I
Why is that?
Because if you begin with
an E, obverting makes it
an A, which cannot be
converted validly

9. Obverse: No Popes are Women
Contraposition
Obverse of the converse of an obverse
All Popes are Men
Obverse: No Popes are Women
Converse: No Women are Popes
Obverse: All Women are non-Popes

10. Contraposition
State the complement of each term
and
2. Switch the position of the terms
All dogs are mammals
All non-mammals are non-dogs

11. Switch position of terms
Conversion
Valid for E and I only
Change quality; state
complement of the
predicate
Obversion
valid for all forms
Contraposition
Form the complement
of both terms, and
switch position
Valid: A, O

12. Logical relationships between statement
forms
The Square of Opposition
A E
I O

13. CONTRADICTORIES: opposite in truth-value
A: universal affirmative
O: particular negative
E: universal negative
I: particular affirmative
CONTRADICTORIES: opposite in truth-value

14. CONTRARIES: A, E A: universal affirmative E: universal negative
Cannot both be true, but can both be
false.
No dogs are birds T
All dogs are birds F
No women are presidential candidates
All women are presidential candidates
CONTRARIES: A, E
Both false.

15. I, O: SUBCONTRARIES I: particular affirmative O: particular negative
Cannot both be false, but can both be true
(“some” means “there is at least one”)
Some Senators are Republicans true
Some Senators are not Republicans true
Some cats are Great Danes false
Some cats are not Great Danes true
I, O: SUBCONTRARIES

16. A: universal affirmative
I: particular affirmative
E: universal negative
O: particular negative
If a universal statement is true, the
particular of the same quality is
necessarily true as well.
SUBALTERNATION

17. A: universal affirmative
I: particular affirmative
E: universal negative
O: particular negative
If a particular statement is false, the
universal of the same quality must
also be false –necessarily.
SUPERALTERNATION

18. The Square of Oppositions
A contrary E t
subalternation
contradiction
superalternation f
I subcontrary O

19. Logical relations of Categorical Propositions
Conversion, Obversion, Contraposition
(generate immediate inferences)
Square of Oppositions
Allowed inferences from one propositional
form to another