Transcript A-Claim - Erick Ramirez

“Only,” Categorical Relationships, logical operators
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Sign In!
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Review
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Only vs. Only if
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Categorical Relationships
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The Square of Opposition
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Logical Operators
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Conversion
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For Next time: Read Chapter 8 pages 265-276
Quick Review
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We are now focusing on translating sentences into
categorical claims
We have already had a lot of practice understanding the
differences between the four different types of categorical
claims:
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A-Claim
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E-Claim
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I-Claim
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O-Claim
Quick Review
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Some sentences are harder to translate than others
Indicator words like “most” or “many” or “a few”
seem to be telling us different things about the
relationships between different categories
For our purposes the following A-Claim is true: ALL
of these words Are words that would be translated as
_________?
If you see one of those words you know you are
dealing with an I or O-Claim
Only
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Things are a little more complicated with the indicator
word “only”
“Only students may use the gym”
“Critical Thinking students are the only students who
can avoid cognitive biases”
“Only” can indicate different relationships between
groups depending on whether it is found by itself or
with the word “the”
What is the word “only” saying in the two examples
above?
Only (more)
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Whenever we see “only” by itself in a sentence then
we know we are being introduced to the predicate
term in an A-claim
Only bananas are fed to the monkeys
Whenever we see “the only” in a sentence we know
we are being introduced to the subject term in an Aclaim
Red meat is the only thing fed to the lions
Example
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Franklin Delano Roosevelt once said:
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“The only thing we have to fear is fear itself”
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What are the subject and predicate terms of this
sentence
How would we translate Roosevelt's sentence into a
categorical claim?
Curious facts about Categorical Claim types
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If we diagram the four claim types (A,E,I,O) we can notice
that they are related to one another in different ways
It is impossible for both an A-Claim and an E-Claim to be
true at the same time (about the same groups)
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A-Claim: All Chihuahuas are cute
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E-Claim: No Chihuahuas are cute
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This makes A and E-Claims contraries. These two claim
types cannot both be true together
What about I and O-Claims?
More Curious Facts
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I and O-Claims are related to one another as well
I and O-Claims can both be true together but they can
never both be false
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I-Claim: Some mammals are egg layers
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O-Claim: Some mammals are not egg layers
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This is because we will be working under the assumption
that the groups we use in our claims are not empty
Because of this, I and O-Claims are called subcontrary
claims
One More Curious Fact
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Contrary claims cannot both be true but they can both be false
Subcontrary claims cannot both be false but they can both be
true
But A and O-Claims AND E and I-Claims have a different
relationship to one another: they never have the same truth
value
If an A-Claim is true then the corresponding O-Claim must be
false (and vice-versa)
For this reason A & O-Claims and E & I-Claims are called
contradictory claims
The Square of Opposition
Inferences based on Categorical Claims
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If we keep in mind the relationships between different kinds of
categorical claims then we can infer the truth values of other
claims once we know the truth value of one categorical claim
For example: if we know that the A-Claim “All squirrels are
creatures with bushy tails” is true then we know that the EClaim “No squirrels are creatures with bushy tails” is false
We know this because A and E-Claims are contraries, they
cannot both be true
We also know that the I-Claim “Some squirrels are creatures
with bushy tails” is also true
Inferences based on Categorical Claims
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Similarly, we know that the two subcontrary claim
types (I and O) can never both be false
If we know that the O-Claim “Some mammals are not
cute” is false then we know that its subcontrary I-Claim
“Some mammals are cute” must be true because they
cannot both be false
The same goes for contradictory claims. If we know
that the O-Claim “Some asparagus plants are not green
things” is true then we know that the contradictory AClaim “All asparagus plants are green things” is false
Quick Review
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Knowing the truth value of one of the categorical claim
types helps us to infer the truth values of some of the
other claim types
A-Claim: if true we can infer that the E-Claim and OClaim are false and the I-Claim is true
A-Claim: if false we can infer that the O-Claim is true
but we can't say much more with certainty
The square of opposition can help us see those
relationships and we will need them when we start
doing proofs for validity
Practice
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If we know that the A-claim “all circus clowns are
scary things” is true, what can we infer about the truth
values of the other categorical claims?
It might help to draw the Venn Diagram of the AClaim to see what it rules out or rules in
What if we knew that “Some people are people who
hate pizza”? What truth values could we infer from
that claim?
Here again, it might help to draw the I-Claim diagram
Logical Operators
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A logical operation is a way of translating a claim into
another claim without changing the meaning or truth
value of the claim
There is a corresponding sense of this process in
mathematics
We can say that x + y = z is equivalent to y + x = z, for
example, without changing the meaning of the
expression or the truth value of the equation
In logic, being able to do these operations will be
extremely useful when we construct proofs
Conversion
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The first equivalency we will look at is conversion
To find the converse of a claim all we have to do is switch the
order of a claim's subject and predicate terms
Some claim types are equivalent to their converses. This
means that the truth value and meaning of a claim is not
changed by translating it into its converse
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All E and I-Claims are equivalent to their converses
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For example, diagram these two I-Claims:
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Some elephants Are large animals
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Some large animals Are elephants
Why Should We Care?
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The important thing about logical operations like conversion is
that we can always safely replace any I-Claim in an argument
with its converse claim without changing the argument
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The same goes for any E-Claim
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For example:
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No ghosts Are beings who exist
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No beings who exist Are ghosts
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When we check the Venn diagram we see that both of these
two claims are equivalent as well
Complementary Terms
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A complementary term refers to a group that excludes all
members of another group
For example:
[Students] is a group; its complementary term would be a group
that excludes every member of the [Students] group
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We might call that group [Non-Students]
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Every group in a categorical claim has a complementary group
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[Bananas]/[Non-Bananas]; [Things that eat pizza]/[Things that
do not eat pizza]; [boring things]/[non-boring things]
Obverse Claims
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ALL categorical claims
(A,E,I,O) are equivalent to their
obverse claims
An obverse claim can be found
by first looking at the claim to
the left/right of a claim type on
the Square of opposition
THEN switch the predicate term
with its complementary term
Voila! This gives you the
obverse claim
Examples
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For the following claims, find the obverse claim
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All monkeys Are creatures who eat bananas
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Some ants Are dangerous things
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No circus clowns Are truly scary things
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Here again we can check the Venn diagram to make
sure that each claim and its obverse claim are
equivalent
For Next Time
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Read Chapter 8 pages 265-276