Transcript Operations-Transformations

Operations (Transformations)
On Categorical Sentences
• Conversion (Simple & Limited)
• Obversion
• Contraposition
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.
No S are P.
Some P are S.
No P are S.
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.
Converse of Universal Affirmative (A)
All S are P
S
P
Converse of Universal Affirmative (A)’
S
P
S
P
All S are P
All P are S
• No Logical Connection. Logically independent.
Converse of Universal Negative (E)
No S are P
S
No P are S
Equivalent
P
Converse of Universal Negative (E)
No S are P
S
P
No P are S
---------- Equivalent ----------
Converse of Particular Affirmative (I)
Some S are P
S
P
*
Some P are S
-------- Equivalent ----------
Converse of Particular Negative (O)’
Some S are not P
S
Some P are not 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.
LogiCat
Cats ?
Males ?
CATegorical Logic !
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:
(This is implication,
not equivalence.)
All freshmen are logicians.
Some logicians are freshmen.
Conversion by Limitation
Converse: All P are S
All S are P
S
P
*
No Logical
Relationship
T
Implication
S
Assume S exist..
P
*
Some S are P
Converse (equivalent)
Some P are S
S
P
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.
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).
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.
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.
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.
A and Its Obverse
S
P
A: All S are P
(No S are outside P)
Obverse: No S are non-P
(No S are outside P)
-----
Equivalent ----
E and Its Obverse
S
P
E: No S are P.
Obverse: All S are non-P.
(No S are in P)
--- Equivalent ---
I and Its Obverse
S
P
*
I: Some S are P
Obverse: Some S are not non-P
(Some S are not things outside P)
--- Equivalent ---
O and Its Obverse
S
P
*
O: Some S are not P
Obverse: Some S are non-P
(Some S are things outside P)
--- Equivalent ---
Equivalent ?
Converse
A
No
E
I
O
Yes Yes No
Obverse
Yes Yes Yes Yes
Contraposition
S
P
negate
non-P non-S
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.
Contraposing a Categorical
Sentence
**** S ******* P
negate
**** non-P ******* non-S
A and Its Contrapositve
S
P
A: All S are P.
Contrapositive: All non-P are non-S
(nothing outside P is inside S)
S
P
--- Equivalent --.
E and Its Contrapositve
S
P
E: No S are P.
Contrapositive: No non-P are non-S
(nothing outside P is outside S)
S
P
--- No Logical Relationship ---
I and Its Contrapositve
S
I: Some S are P.
P
*
Contrapositive: Some non-P are non-S
(Something outside P is
outside S)
S
P
*
--- No Logical Relationship ---
O and Its Contrapositve
S
O: Some S are not P.
P
*
Contrapositive: Some non-P are not non-S
(Something outside P is inside S)
--- Equivalent ---
S
*
P
Equivalent ?
Converse
A
No
E
I
O
Yes Yes No
Obverse
Yes Yes Yes Yes
Contrapositive
Yes
No
No Yes