Strict Logical Entailments of

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 SP
E SP A SP
I SP O SP
O SP I SP
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
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
A universal affirmative O particular negative
E universal negative I particular affirmative
CONTRADICTORIES opposite in truth-value
14
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 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
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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
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The Square of Oppositions
A contrary
E t subalternation contradiction

superalternation
f I
subcontrary O
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Logical relations of Categorical Propositions
Conversion, Obversion, Contraposition
(generate immediate inferences)
Square of Oppositions
Allowed inferences from one propositional form
to another