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Categorical Logic

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Adam Rosenfeld

on 7 February 2018

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Transcript of Categorical Logic

Categorical Logic
Categories &
Categorical Propositions

Category:
Birds
Mammals
People who jog
Red things
Adam Rosenfeld
Categorical Proposition:
No mammals are birds.
Some people who jog
are Adam Rosenfeld.
All cats are mammals.
Some red things are not
fire engines.
A class of things
a statement that relates two
categories to each other
e.g.

e.g.

Standard Form
All categorical propositions have the following form...
Quantifier
-
Subject
-
Copula
-
Predicate
e.g.
All

snakes

are

reptiles
.
No

snakes

are

birds
.
Some

snakes

are

scary things
.
Some

snakes

are not

pets
.
Quality
There are two options for the QUALITY of
a categorical proposition...
Affirmative
Negative
All classes
are
enriching experiences.
Some students
are
tired people.
No
married people are lonely people.
Some bachelors
are not
happy people.
Quantity
There are two options for the QUANTITY
of a categorical proposition...
Universal
Particular
All
classes
are
enriching experiences.
No
married people
are
lonely people.

Some
students
are
tired people.
Some
bachelors
are not
happy people.
A, E, I, & O classification
All snakes are reptiles.
No snakes are birds.
Some snakes are scary things.
Some snakes are not pets.
A
E
I
O
A: Universal Affirmation

E: Universal Negation

A
FF
I
RMO
N
E
G
O
I: Particular Affirmation
O: Particular Negation
A
All S are P
e.g. "All dogs are mammals."
P
S
E
No S are P
e.g. "No dogs are mammals."
P
S
I
Some S are P
e.g. "Some dogs are
mammals."
P
S
*
O
Some S are not P
e.g. "Some dogs are not mammals."
P
S
*
It's false that
all dogs are mammals.
P
S
*
It's false that
some dogs are mammals.
P
S
It's false that
no dogs are mammals.
It's false that
some dogs are not mammals.
P
S
*
P
S
FALSE "A"
FALSE "I"
FALSE "E"
FALSE "O"
Star where you would have shaded!
Star where you would have shaded!
Shade where you would have starred!
Shade where you would have starred!
A
I
E
O
Some S are not P
e.g. "Some dogs are not mammals."
All S are P
e.g. "All dogs are mammals."
Some S are P
e.g. "Some dogs are mammals."
No S are P
e.g. "No dogs are mammals."
Contra- dictories
Subalterns
Contra- dictories
Contraries
Subcontraries
P
P
S
S
P
S
P
S
X
X
The Traditional Square of Oppositions
Subalterns
CONTRADICTORIES
A
I
E
O
Some S are not P
e.g. "Some dogs are not mammals."
All S are P
e.g. "All dogs are mammals."
Some S are P
e.g. "Some dogs are
mammals."
No S are P
e.g. "No dogs are mammals."
P
P
S
S
P
S
P
S
X
X
(Opposite truth values)
CONTRARIES
A
E
All S are P
e.g. "All dogs are mammals."
No S are P
e.g. "No dogs are mammals."
P
P
S
S
(At least one is false)
SUBCONTRARIES
I
O
Some S are not P
e.g. "Some dogs are not mammals."
Some S are P
e.g. "Some dogs are
mammals."
P
S
P
S
X
X
(At least one is true)
SUBALTERNS
A
I
E
O
Some S are not P
e.g. "Some dogs are not mammals."
All S are P
e.g. "All dogs are mammals."
Some S are P
e.g. "Some dogs are
mammals."
No S are P
e.g. "No dogs are mammals."
P
P
S
S
P
S
P
S
X
X
(Truth flows down, Falsity flows up)
TRUTH
FALSITY
Immediate Inferences from Contradictories
(Opposite Truth Values)
It's false that all classes are easy.
Therefore it's true that some classes are not easy.
False "A"
True "O"
Some bicycles have fancy tires.
Therefore it's false that no bicycles have fancy tires.
False "E"
True "I"
_____________________________________
_____________________________________
No murders are legal.
Therefore it's false that some murders are legal.
False "I"
True "E"
_____________________________________
It's false that some horses are not mammals.
Therefore all horses are mammals.
False "O"
True "A"
Immediate Inferences from Contraries
(Can't both be true)
All snakes are animals.
Therefore it is false that
no snakes are animals
VALID
True "A"
False "E"
No snakes are birds.
Therefore it is false that
all snakes are birds
True "E"
False "A"
INVALID
Fallacy of Illicit Contrary
It's false that no snakes
are poisonous.
Therefore all snakes are
poisonous.
True "A"
False "E"
It's false that all snakes
are poisonous.
Therefore no snakes are
poisonous.
True "E"
False "A"
________________ ________________
Immediate Inferences from Subcontraries
(Can't both be false)
Some students are sleepy.
Therefore it is false that
some students are not
sleepy.
VALID
True "I"
False "O"
Some students are not
sleepy.
Therefore it is false that
some students are sleepy.
True "O"
False "I"
Fallacy of Illicit Subcontrary
It's false that some birds
are not flightless.
Therefore some birds are
flightless.
True "I"
False "O"
It's false that some humans
are immortal.
Therefore some humans
are not immortal.
True "O"
False "I"
________________
INVALID
________________
Immediate Inferences from Subalterns
(Truth flows down, falsity flows up)
All lies are immoral.
Therefore some lies are
immoral.
VALID
True "A"
False "E"
Some students are
dilligent.
Therefore all students
are dilligent.
True "I"
True "A"
Fallacy of Illicit Subalternation
It's false that some birds
are not flightless.
Therefore it's false that
no birds are flightless.
True "I"
False "O"
It's false that no cats are
pets.
Therefore it's false that
some cats are not pets.
False "O"
False "E"
________________
INVALID
________________
Categorical Syllogisms
Syllogism: A deductive argument
consisting of two premises
and one conclusion

All soldiers are patriots.
No traitors are patriots.
Therefore, no traitors are soldiers.
Major
Term
Minor
Term
Middle
Term
______
______
______
(the predicate
in the conclusion)
(the subject
in the conclusion)
(what links the major &
minor terms)
Major Premise
Minor Premise
______
______
______
Standard
Syllogistic Form
Major Premise
is first
Minor Premise
is second
Mood & Figure
Mood
What sorts of categorical
propositions make up the
syllogism
All soldiers are patriots.
No traitors are patriots.
Therefore, no traitors are soldiers.
A
E
E
Mood = AEE
Conclusion is last
(and all propositions are in standard form)
Put the following argument in
standard syllogistic form:
Some kindergarten children are AIDS victims. No AIDS victims are persons who pose an immediate threat to the lives of others. Therefore, some kindergarten children are not persons who pose an immediate threat to the lives of others.
No AIDS victims are persons who pose an immediate threat to the lives of others.
Some kindergarten children are AIDS victims. Therefore, some kindergarten children are not persons who pose an immediate threat to the lives of others.
-------------------
-------------------------------------
------------
-------------------
--------------------------------
------------------
All quasars are supernovas. Because all supernovas are objects that emit massive amounts of energy, and all quasars are objects that emit massive amounts of energy.
Put the following argument in
standard syllogistic form:
All soldiers are patriots.
No traitors are patriots.
Therefore, no traitors are soldiers.
Mood = AEE
Figure = 2
AEE-2
Give the mood and figure of the following argument:
No hounds are lapdogs.
Some sporting dogs are hounds.
Therefore, some sporting dogs are not lapdogs.
-----------
-------
-----------
-------
256 possible syllogistic forms
No AIDS victims are persons who pose an immediate threat to the lives of others.
Some kindergarten children are AIDS victims. Therefore, some kindergarten children are not persons who pose an immediate threat to the lives of others.
-------------------
-------------------------------------
------------
-------------------
--------------------------------
------------------
Is the following syllogism valid?
EIO-1
Using Venn Diagrams for
Categorical Syllogisms

Is the following syllogism valid?
All quasars are supernovas. Because all supernovas are objects that emit massive amounts of energy, and all quasars are objects that emit massive amounts of energy.
Figure
The location of the
middle term
All soldiers are patriots.
No traitors are patriots.
Therefore, no traitors are soldiers.
Figure?
Mood = EIO
Figure =1
EIO-1
Give the mood and figure of the following argument:
All quasars are supernovas. This is because all supernovas are objects that emit massive amounts of energy, and all quasars are objects that emit massive amounts of energy.
AAE-3 ?
All ---- are ----
All ---- are ----
Therefore no ----- are -----
AAE-3 ?
All
M
are ----
All
M

are ----
Therefore no ----- are -----
AAE-3 ?
All M are P
All M are S
Therefore no S are P
Barbara, Celarent, Darii, Ferioque prioris
Cesare, Camestres, Festino, Baroco secundae
Tertia grande sonans recitat Darapti, Felapton
Disamis, Datisi, Bocardo, Ferison.
Quartae sunt Bamalip, Calames, Dimatis, Fesapo,
Fresison.
Rule 1: The middle term must be distributed at least once.
(Fallacy of undistriuted middle)
*Rule 5: If both premises are universal, the conclusion cannot be
particular.
(Existential Fallacy)
Rule 4: A negative premise requires a negative conclusion, and a
negative conclusion requires a negative premise.
(Fallacy of affirmative from a negative/ negative from an affirmative)
Rule 3: Can't have more than one negative premise.
(Fallacy of exclusive premises)
Rule 2: If a term is distributed in the conclusion, it must be
distributed in a premise.
(Fallacy of illicit major/minor)
* Could be conditionally valid for Aristotelian standpoint, but always invalid for Boolean standpoint.
________ Middle
________ Term
-------
-------
Give the mood and figure of the following argument:
All supernovas are objects that emit massive
amounts of energy.
All quasars are objects that emit massive
amounts of energy.
All quasars are supernovas.
-------
-------
--------------------
-----------
--------------------
-----------
------
------
AAA-2
"I" & "O" Diagrams
Some S are P.
Some S are not P.
Some M are not S.
When diagramming multiple statements, always do universal statements before particular ones!
Valid
or

Invalid?
Valid
or

Invalid
?
Valid
or

Invalid
?
This syllogism is VALID, since the conclusion is proven to be true by ONLY diagramming the premises.
Valid
or

Invalid?
NEVER ACTUALLY DIAGRAM THE CONCLUSION OF AN ARGUMENT WHEN TESTING FOR VALIDITY!!!
ONLY CHECK TO SEE IF DIAGRAMMING THE PREMISES ALREADY GUARANTEES THE CONCLUSION!
Because that's what "Validity" means, right?
"If the premises are true, the conclusion MUST BE true."
INVALID!
Valid!
Invalid!
Valid!
Invalid!
Conditionally Valid
As long as "mothers" exist
Diagramming premises does not give us the conclusion - YET!
Note that there is only one region where ANY existing unicorns COULD be placed.
We hypothesize an existing unicorn with a circled *.
Conditionally Valid
(as long as unicorns exist)
But unicorns DON'T exist!
so...
Existential Import!
Universal propositions don't necessarily make an existence claim...
but particular ones do.
Many of the inferences we've been discussing only work if the things we're talking about actually exist.
e.g. "All unicorns have one horn." doesn't necessarily mean that "There exists at least one unicorn that has one horn."
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