### Categorical Logic

```What is logic?
What is inference?
“Logic is the study of the
methods and principles
used to distinguish correct
from incorrect reasoning.”
“A process of linking
propositions by affirming
one proposition on the
basis of one or more other
propositions.”
• (Copi, Introduction to Logic)
• (Copi, Introduction to
Logic)
What is the value of
reason and logic?
What, if anything, can
logic tell us?
What, if anything, can
logic and reason tell us
about the nature of the
world “out there?”
What, if anything, can
logic and reason tell us
about truth?
Concerned with relations of
inclusion and exclusion
among classes (or
categories).
Propositional logic
(or truth-functional
logic) considers a
statement the
smallest indivisible
unit and studies
ways of joining &
modifying
statements to form
other statements.
Instead of ….
If Simon is a man (S), then he’s going
to die (M).
Or “If S, then M.”
Or “S → M.”
• A: All_____ are _______.
• E: No _____ are _______.
• I: Some _____ are _____.
• O: Some ____are not ___.
Subject Term
Predicate Term
Affirmative Claim
Negative Claim
Equivalent Claims
1. Every salamander is a
lizard.
2. Not every lizard is a
salamander.
3. Only reptiles can be
lizards.
4. Snakes are the only
members of the
suborder Ophidia.
5. The only members of
the suborder Ophidia
are snakes.
1. All salamanders are
lizards.
2. Some lizards are not
salamanders.
3. All lizards are reptiles.
4. All members of the
suborder Ophidia are
snakes.
5. All members of the
suborder Ophidia are
snakes.
6. None of the burrowing snakes is
poisonous.
6. No burrowing snakes are poisonous
snakes.
7. Anything that’s an alligator is a
reptile.
7. All alligators are reptiles.
8. Anything that qualifies as a frog
qualifies as an amphibian.
9. There are frogs wherever there are
snakes.
10. Wherever there are snakes, there
are frogs.
11. Whenever the frog population
decreases, the snake population
decreases.
12. Nobody arrived except the
cheerleaders.
8. All frogs are amphibians.
9. All places there are snakes are
places there are frogs.
10. All places there are snakes are
places there are frogs.
11. All times the frog population
decreases are times the snake
population decreases.
12. All people who arrived are
cheerleaders.
13. Except for vice-presidents, nobody
got raises.
13. All people who got raises are vice
presidents.
14. Unless people arrived early, they
couldn’t get seats.
14. All people who got seats are people
who arrived early.
15. Most home movies are as boring as
dirt.
15. Some home movies are things that are
as boring as dirt.
16. Socrates is a Greek.
16. All people identical with Socrates are
Greeks.
17. The bank robber is not Jane’s fiance.
18. If an automobile was built before
1950, it’s an antique.
19. Salt is a meat preservative.
20. Most corn does not make good
popcorn.
17. No people identical to the bank robber
are people identical to Jane’s fiancé.
18. All automobiles built before 1950 are
antiques.
19. All examples of salt are things that
preserve meat.
20. Some examples of corn are not things
that make good popcorn.
All things not
included in another class.
1. No Sunnis are
Christians.
Students
Non-Students
6. Some Indians are
not Hindus.
Switch S & P term positions
and replace with complementary terms.
But…while all A- and O-claims are
equivalent to their contrapositives, no Eand I-claims are equivalent to their
contrapositives.
8. All Catholics are
Christians.
All
things not included in another
class.
Students
Students
Non-
Change claim from
affirmative to negative (of viceversa) and replace the predicate
term with its complementary
term.
All A-, E-, I-, O- claims are
equivalent to their obverses.
2. Some Arabs are Christians.
3. All Sunnis are Muslims.
10. No Muslims are Christians.
Switch
positions of S & P terms.
4. Some Kurds are not
Christians.
But….while all E- and Iclaims are equivalent to
5. No Hindus are Muslims.
their converse claims, no Aand O-claims are
equivalent to their
7. All Shiites are Muslims.
converses.
9. All Protestants are
Christians.
A & E claims can both
be false but cannot
both be true.
I & O claims can both
be true but cannot
both be false.
A & O and E & I claims
can never have the
same truth value.
There’s not such thing as a
harmless drug. (translate
to standard form?)
A or E is true, then we
can infer the remaining
truth values.
I or O is false, then we
can infer the remaining
truth values.
No drugs are drugs that
are completely harmless.
(True)
• A-claim is________;
• I-claim is ________;
• O-claim is _______.
E-claim is________;
I-claim is ________;
Logic exercises are easy.
(translate to standard form)
All logic exercises are
exercises that are easy.
(false)
O-claim is _______.
1. Whenever the battery is
dead, the screen goes
blank; that means, of
course, that whenever
the screen goes blank,
the battery is dead.
Valid?
How was the 2nd claim
derived from the first:
conversion, obversion, or
contraposition?
Requires the
of an A-claim: invalid.
2. For a while there, some What kind of claim is the
students were desperate
first claim?
for good grades, which
meant some weren’t right? What does a “true” I-claim
allow us to infer about A-,
Valid?
E-, and O-claims?
3. Some players in the last
election weren’t members
of the Reform Party.
Obviously, therefore, some
members of the Reform
Party weren’t players in the
last election.
What kind of claim is the
first claim?
Valid?
Can an O-claim be
converted?
How was the 2nd claim
derived from the first:
conversion, obversion, or
contraposition?
4. Since some of the
students who failed the
exam were students who
didn’t attend to review
session, it must be that
some students who weren’t
at the review session failed
the exam.
What kind of claim is the
first claim?
Valid?
Can an I-claim be
converted?
How was the 2nd claim
derived from the first:
conversion, obversion, or
contraposition?
5. None of the people who
arrived late were people
who got good seats, so
none of the good seats
were occupied by late
comers.
Valid?
What kind of claim is the
first claim?
How was the 2nd claim
derived from the first:
conversion, obversion, or
contraposition?
Can an E-claim be
converted?
6. Everybody who arrived on
time was given a box lunch, so
the people who did not get a
box lunch were those who
didn’t there on time.
Valid?
What kind of claim is the
first claim?
How was the 2nd claim
derived from the first:
conversion, obversion, or
contraposition?
Can an A-claim be
contraposed?
7. None of the people who
gave blood are people who
were tested, so everybody
who gave blood must have
been untested.
Valid?
What kind of claim is the
first claim?
How was the 2nd claim
derived from the first:
conversion, obversion, or
contraposition?
Can an E-claim be obverted?
8. Some of the people who
were not tested are people
who were allowed to give
blood, from which it
follows that some of the
people who were not
allowed to give blood must
have been people who
were tested.
Valid?
What kind of claim is the
first claim?
How was the 2nd claim
derived from the first:
conversion, obversion, or
contraposition?
Can an I-claim be
contraposed?
9. Everybody who was in
uniform was able to play, so
nobody who was out of
uniform must have been
able to play.
Valid?
What kind of claim is the
first claim?
How was the 2nd claim
derived from the first:
conversion, obversion, or
contraposition?
Can an A-claim be
converted?
10. Not everybody in uniform
was allowed to play, so some
people who were not allowed
to play must have not been
people in uniform.
What kind of claim is the
first claim.
Valid?
Does a “true” value of an Ior O-claim imply the truth
value of any other claims?
What kind of claim is the
second claim?
Indeterminate.
Students who wrote poor
exams didn’t get admitted
to the program.
No students who wrote
poor exams are students
who were admitted to the
program.
• All students who wrote
poor exams are
students who were not
admitted to the
program. (by obversion)
If you live in the dorms, you
can’t own a car.
No people who live in the
dorms are people who own
cars.
All people who live in the
dorms are people who do
not own cars. (by obversion)
All people who own cars are
people who do not live in
the dorms. (by
contraposition)
There are a few left-handed
students in this class.
Some students in this class
are students who are lefthanded.
Some students in this class
are students who are not
right-handed (or non lefthanded). (by obversion)
• Some students who are
left-handed are students
who are in this class. (by
conversion)
I’ve had days like this
before.
Some days I’ve had are
days like this day.
Some days I’ve had are
not days unlike this day.
(by obversion)
Some days like this day
are days I’ve had. (by
conversion)
A two-premise deductive
argument whose every
claim is standard form
categorical claim.
Three terms occur
exactly twice in exactly
two of the claims.
Which of the following is
a categorical syllogism?
All PHIL 1 students are honest.
Some PHIL 1 students are hardworking.
Therefore, some hard-working
students are honest.
All PHIL 1 students are honest.
Many PHIL 1 students are hardworking.
Therefore, many PHIL 1 students
are honest and hard-working.
Validity: not possible for
Some CR students are men.
premises to be true while the Some CR students are women.
conclusion is false.
Some men are women.
Major term is predicate term
of conclusion.
Minor term is subject term of
conclusion.
Middle term is term of
premises but not conclusion.
All CR students are men.
Some CR students are superheroes.
Some men are superheroes.
Some CR students are zombies.
All zombies are dead.
Some CR students are dead.
Is the argument valid?
Why would AT&T leave
the premise unstated?
What is the conclusion AT&T
is hoping we’ll supply?
Translate premises to
standard form.
How could we make this
into a valid argument?
All A’s are X.
All B’s are X.
Therefore, all B’s are A’s.
Some X’s are not Y’s.
Therefore, some Y’s are not X’s.
All A’s are X.
No A’s are Y.
Therefore, no X’s are Y’s.
Some X’s are Y’s.
Therefore, some X’s are not Y’s.
All X’s are Y’s.
Therefore, all Y’s are X’s.
Some X’s are not Y’s.
Therefore, some X’s are Y’s.
All business executives have
accounting experience, and
some business executives are
not economists.
A. Some economists do not have
accounting experience.
B. Some people with accounting
experience are not
economists.
C. All people with accounting
experience are business
executives.
D. More than one of these.
E. None of these.
Coffee is a stimulant, since
coffee contains caffeine.
A. All substances that
contain caffeine are
stimulants.
B. All stimulants are
substances that contain
caffeine.
C. Neither creates validity.
D. Both create validity.
All physicians own mutual
funds, from which it
follows that no professors
are physicians.
Pornography violates women’s rights. It carries
a demeaning message about a woman’s worth
and purpose and promotes genuine violence.
This is, indeed, a violation of women’s civil rights
and justifies the Minneapolis City Council in
attempting to ban pornography.
All instances of pornography
are things that demean
women’s worth and
promote violence.
[Unstated:] All things that
demean women’s worth
and promote violence are
things that violate
women’s rights.]
All instances of pornography
are things that violate
women’s rights.
All instances of pornography
are things that violate
women’s rights.
[Unstated:] All things that
violate women’s rights are
things that are justifiably
banned by the
Minneapolis City Council.
All instances of pornography
are things that are
justifiably banned by the
Minneapolis City Council.
```