CRITICAL INQUIRY
PART FOUR
CHAPTER 8
CATEGORICAL LOGIC
• Students will learn to:
• Recognize the four types of categorical claims and the
Venn diagrams that represent them
• Translate a claim into standard form
• Use the square of opposition to identify logical relationships
between corresponding categorical claims
• Use conversion, obversion, and contraposition with
standard form to make valid arguments
• Recognize and evaluate the validity of categorical
syllogisms
CHAPTER 8
CATEGORICAL LOGIC
• Introduction
• Categorical Logic
•
•
•
•
All Xs are Ys
No Xs are Ys
Some Xs are Ys
Some Xs are not Ys.
• Examples & Applications
• Categorical Claims
• Introduction
•
•
•
•
•
Categorical Claim
Standard form categorical claim
Term
Predicate Term
Noun/Noun Phrase
CHAPTER 8
CATEGORICAL LOGIC
• Venn Diagrams
S
P
S
P
• Venn Diagrams
• Affirmative claim
• Negative Claim
A-Claim: All S are P
S
P
X
O-Claim: Some S are not P
E-Claim: No S are P
S
P
X
I-Claim: Some S are P
CHAPTER 8
CATEGORICAL LOGIC
• Translation into Standard Form
• Equivalent Claim
• “Only”: introduces the predicate of an A claim.
• Only sophomores are eligible candidates.
• All eligible candidates are sophomore.
• “The Only”: introduces the subject of an A claim.
• Bats are the only true flying mammals.
• All true flying mammals are bats.
• Time & Space
• “Whenever”: often indicates an A or E claim.
• I always get nervous whenever I take logic exams.
• All times I take logic exams are times I get nervous.
• “Wherever”: often indicates an A or E claim.
• He makes trouble wherever he goes.
• All places he goes are places he makes trouble.
CHAPTER 8
CATEGORICAL LOGIC
• Claims about single individuals
• Translated to claims about classes.
• A or E claim.
• A claim about an X of type Y becomes All/No Ys identical to X are Ps
• Aristotle is a logician=All people identical to Aristotle are logicians.
• Tallahassee is in Florida=All cities identical to Tallahassee are cities
in Florida.
• Claims involving mass nouns
• Treated as claims about examples of the kind of stuff.
• Gold is a heavy metal=All examples of gold are heavy metal.
CHAPTER 8
CATEGORICAL LOGIC
• The Square of Opposition
•
•
•
•
•
•
•
•
The Square
Contrary Claims
Subcontrary Claims
Contradictory Claims
Logical Relations
Empty Subset Classes
Assumption
Use
CHAPTER 8
CATEGORICAL LOGIC
• Three Categorical Operations
• Conversion
•
•
•
•
•
•
•
Switching the subject and predicate terms.
(A) All S are P: All P are S
(E) No S are P: No P are S
(I) Some S are P: Some P are S
(O)Some S are Not P: Some P are not S
E and I claims are equivalent to their converses.
A and O claims are not.
CHAPTER 8
CATEGORICAL LOGIC
• Obversion
• 1)Replace the claim with the claim directly across from it on the
square or opposition and 2) change the predicate to its
complement.
• (A) All S are P: No S are non-P
• (E) No S are P: All S are non-P
• (I) Some S are P: Some S are not non-P
• (O)Some S are Not P: Some S are non-P
• Complementary Class
• Complementary Term
• All categorical claims are equivalent to their obverses.
CHAPTER 8
CATEGORICAL LOGIC
• Contraposition
• 1)Switch the subject and predicate terms 2) replaces both
terms with their complements
• (A) All S are P: All non-P are non-S
• (E) No S are P: No non-P are non-S
• (I) Some S are P: Some non-P are non-S
• (O)Some S are Not P: Some non-P are not non-S
• Complementary Class
• Complementary Term
• All categorical claims are equivalent to their obverses.
• A and O claims are equivalent to their contrapositions.
• E and I claims are not.
CHAPTER 8
CATEGORICAL LOGIC
• Categorical Syllogisms
• Syllogism: an argument with 2 premises and 1 conclusion.
• Categorical Syllogism
• 1. All Americans are consumers.
• 2. Some consumers are not democrats.
• C. Therefore, some Americans are not Democrats.
• Terms of a syllogism
• Major term (P): the term that occurs as a predicate term of the
syllogism’s conclusion.
• Minor term (S): the term that occurs as the subject term of the
syllogism
• Middle term (M): the term that occurs in both of the premises but
not in the conclusion.
• Validity & the relation between the terms.
CHAPTER 8
CATEGORICAL LOGIC
• The Venn Diagram Method of Testing For Validity
• Steps
• Diagram premise 1
• Diagram premise 2
• Determine if the conclusion can be read from the diagram (valid)
or not (invalid).
CHAPTER 8
CATEGORICAL LOGIC
• Example
• 1. No Republicans are collectivists.
• 2. All socialists are collectivists.
• C. Therefore, no socialists are Republicans.
CHAPTER 8
CATEGORICAL LOGIC
• Example
• 1. Some S are not M
• 2. All P are M
• C. Some S are not P
CHAPTER 8
CATEGORICAL LOGIC
• Example
• 1. All P are M
• 2. Some S are M
• C. Some S are P
CHAPTER 8
CATEGORICAL LOGIC
• Categorical Syllogisms With Unstated Premises
• Example: You shouldn’t give chicken bones to dogs. They
could choke on them.
• 1. All chicken bones are things dogs could choke on.
• 2. (No things dogs could choke on are things you should give dogs.
• C. No chicken bones are things you should give dogs.
• Real Life Syllogisms
• It can be useful to replace long phrases with letters.
• Example
• All C are D
• No D are S
• No C are S
CHAPTER 8
CATEGORICAL LOGIC
• Rules Method for Testing Validity
• Distribution
Claim Distribution ( )
A-claim All (S) are P
I-claim
E-claim
O-claim Some S are not (P)
No (S) are (P)
Some S are P
CHAPTER 8
CATEGORICAL LOGIC
The Rules
The number of negative claims in the premises must
be the same as the number of negative claims in the
conclusion.
At least one premises must distribute the middle term
Rule #1
Rule #2
Rule #2
Any term that is distributed in the conclusion must be
distributed in its premise.
• Examples
• Breaks Rule #1
• 1. No dogs up for adoption at the animal shelter are pedigreed
dogs.
• 2. Some pedigreed dogs are expensive dogs.
• C. Some pedigreed dogs up for adoption at the animal shelter are
expensive dogs.
CHAPTER 8
CATEGORICAL LOGIC
• Breaks Rule #2
• 1. All pianists are keyboard players.
• 2. Some keyboard players are percussionists.
• C. Some pianists are not percussionists.
• Breaks Rule #3
• 1. No mercantilists are large land owners.
• 2. All mercantilists are creditors.
• C. No creditors are large landowners.
CHAPTER 8
CATEGORICAL LOGIC
• Recap
• 1. The four types of categorical claims include A, E, I, and O.
• 2. There are Venn diagrams for the four types of claims.
• 3. Ordinary English claims can be translated into standard form
categorical claims. Some rules of thumb for such translations are as
follows:
•
•
•
•
a. “Only” introduces the predicate of an A-claim.
b. “The only” introduces the subject term of an A-claim.
c. “Whenever” means times or occasions.
d. “whenever” means places or locations.
• 4. Square of opposition displays contradictions, contrariety, and
subcontrariety among corresponding standard-form claims,
• 5. Conversion, obversion, and contraposition are three operations
that can be performed on standard-form claims; some are equivalent
to the original and some or not.
• 6. Categorical syllogisms are standardized deductive arguments; we
can test them for validity by the Venn diagram method or by the rules
method-the latter relies on the notions of distribution and the
affirmative and negative qualities of the claims involved.
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Students will learn to:
• Understand the basics of truth tables and truth-functional
symbols
• Symbolize normal English sentences with claim letters and
truth-functional symbols
• Build truth tables for symbolizations with several letters
• Evaluate truth-functional arguments using common
argument forms
• Use the truth-table and short truth-table methods to
determine whether an argument is truth-functionally valid
• Use elementary valid argument forms and equivalences to
determine the validity of arguments.
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Introduction
• Basic Concepts
• Truth functional logic
• Truth functional claims
• Applications
•
•
•
•
Set theory
Foundation of mathematics
Electronic circuits
Analysis of arguments
• Benefits of learning truth functional logic
• Learning about the structure of language.
• Learning what it is like to work in a precise, nonmathematical
system of symbols.
• Learning how to communicate better.
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Truth Tables and Truth-Functional Symbols
• Claims & Claim Variables
• Claim variable
• Any claim is either true or false (but not both).
• Truth Tables
• One variable table & Two Variable Table
P
P
Q
T
T
T
F
T
F
F
T
F
F
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Negation
• A negation is false when the claim being negated is true,
otherwise it is true.
• Corresponds with “not” and is symbolized by ~
• Claim variable
• Any claim is either true or false (but not both).
• Truth Table for Negation
P
~P
T
F
F
T
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Conjunction
• A conjunction is true only if both of its conjuncts are true,
otherwise it is false.
• Corresponds with “and” and is symbolized by &.
• “But’, “while”, “even though” and other phrases also form
conjunctions.
• Truth Table for Conjunction
P
T
Q
T
P&Q
T
T
F
F
F
T
F
F
F
F
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Disjunction
• A disjunction is false only if both of its disjuncts are false,
otherwise it is true.
• Corresponds with “or” and is symbolized by v.
• Truth Table for Disjunction
P
T
Q
T
PvQ
T
T
F
T
F
T
T
F
F
F
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Conditional Claim
• Antecedent: the “A” in “If A then B.”
• Consequent: The “B” in “If A then B.”
• A conditional claim is false if any only if its antecedent is true
and its consequent is false.
• A conditional corresponds to “if…then…” and is symbolize by
“”.
P
T
Q
T
PQ
T
T
F
F
F
T
T
F
F
T
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Combinations
P
T
Q
T
~P
F
~PQ
T
T
F
F
T
F
T
T
T
F
F
T
F
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Constructing Tables
• Formula for determining the number of rows: r=2N, where r is
the number of rows in the table and n is the number of
claims.
• Constructing at table
• Alternate Ts and Fs in the right most column.
• Alternate pairs of Ts and Fs in the next column to the left.
• Alternative sets of four Ts and four Fs in the next column to the
left .
• Alternate sets of 8 Ts and 8 Fs and so on until all rows for the
claim variables are filled
• The top half of the left most column will always be all s and the
bottom half will be all Fs
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Three Variable Table
P
Q
R
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• More on Constructing Tables
• Parentheses
• Example: If Paula goes to work, then Quincy and Rogers get
the day off.
• Symbolized as P  (Q&R).
• The parentheses are needed
• The truth value of a compound claim depends entirely
upon the truth of its parts.
• If the parts are themselves compounded, their truth values
depends on the truth value of the parts, and so on.
• Constructing the table
• The reference columns are those for variables.
• The table provides a truth functional analysis of the claim.
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Three Variable Example Table
P
Q
R
Q&R P-->(Q&R)
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
F
F
F
T
F
F
F
T
F
F
F
T
T
T
T
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Truth Functional Equivalent
• Defined
• Example
P
Q
~P PQ ~P v Q
T
T
F
F
T
F
T
F
F
F
T
T
T
F
T
T
T
F
T
T
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Symbolizing Compound Claims
• Truth functional structure
• Truth functionally equivalent
• “If” and “only if”
• “If” introduces the antecedent of a conditional.
• Sam will buy the popcorn if Sally buys the tickets
• If Sally buys the tickets, then Sam will buy the popcorn.
• P, if Q = Q  P
• “Only if” introduces the consequent of a conditional.
• Sam will buy the popcorn only if Sally buys the tickets.
• If Sam buys the popcorn, then Sally buys the tickets.
• P only if Q = P  Q
• “If and only If” combines “if” and “only if”
• Sam will go if and only if Sally goes.
• If Sam goes, then Sally will go and if Sally goes, then Sam will go.
• P if and only if Q = (P Q) & (Q P)
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Necessary & Sufficient Conditions
• Necessary Condition
• A is necessary for B= “If A is the case, then B can be the case” or “if
A is not the case, then B cannot be the case.”
• The necessary condition is the consequent of the conditional.
• Oxygen is necessary for human life=If there is human life, then there
is oxygen.
• P is necessary for Q = Q P
• “Only if” introduces the necessary condition.
• Sufficient Condition
• A is sufficient for B= “If A is the case, then B must be the case.”
• Earning a 60 or better is sufficient to pass this class = if a person
earns a 60 or better, then they pass the class.
• P is sufficient for Q = P Q
• Sufficient conditions are not necessary conditions, and vice versa.
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Necessary and sufficient Condition
• If A is necessary and sufficient for B, then B cannot occur
without A and if A occurs, then B must occur.
• “If and only if”
• A person is a bachelor if and only if he is an unmarried man=if a
person is a bachelor then he is an unmarried man and if a
person is an unmarried man, then he is a bachelor.
• P is necessary and sufficient for Q = (PQ) & (Q P)
• Ordinary Language
•
•
•
•
Fast & Loose
You can watch television only if you clean your room.
Intended: If you clean your room, then you can watch TV.
Actual: If you watch TV, then you have cleaned your room.
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Unless
• P unless Q = if not Q, then P = ~Q  P= P v Q
• Bill will go unless Sally goes= If Sally does not go, then Bill will
go=Sally will go or Bill will go.
• Either
• Either indicates a disjunction.
• Either P and Q or R= (P&Q) v R
• P and either Q or R = P & (Q v R)
• Truth Functional Arguments
• Validity
• An argument is valid if and only if the truth of the premises
guarantees the truth of the conclusion.
• It does not matter whether the premises are actually true or
not.
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Valid Truth Functional Argument Patterns
• Modus Ponens (Valid)
• If P, then Q
• P
• Therefore Q
• Modus Tollens (Valid)
• If P, then Q
• Not Q
• Therefore not P
• Chain Argument (Valid)
• If P, then Q
• If Q, then R
• Therefore If P, then R
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Invalid Truth Functional Argument Patterns
• Affirming the Consequent (Invalid)
• If P, then Q
• Q
• Therefore P
• Denying the Antecedent(Invalid)
• If P, then Q
• Not P
• Therefore Not Q
• Undistributed Middle(Invalid)
• If P, then Q
• If R, then Q
• Therefore If P, then R
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Truth Table Test for Validity
• Present all the possible circumstances for an argument by
building a truth table for it.
• Look to see if there are any circumstances in which all the
premises are true and the conclusion is false.
• If there is even a single row in which all the premises are true
and the conclusion is false, then the argument is invalid.
• Otherwise the argument is valid.
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Example
• Argument: If the Saints beat the Forty-Niners, then the Giants
will make the playoffs. But the Saints won’t beat the FortyNiners. So the Giants won’t make the play-offs.
• Symbolized:
• P -->Q
• ~P
• ~Q
P
T
T
F
F
Q
T
F
T
F
~P
F
F
T
T
PQ
T
F
T
T
~Q
F
T
F
T
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Example
• Argument: We’re going to have large masses of arctic air (A)
flowing into the Midwest unless the jetstream (J) moves south.
Unfortunately, there’s no chance of the jet stream going south. So
you can bet there’ll be arctic air flowing into the Midwest.
• Symbolized
• AvJ
• ~J
• A
A
J
AvJ
~J
T
T
F
F
T
F
T
F
T
T
T
F
F
T
F
T
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Example
•
Argument: If Scarlet is guilty of the crime, then Ms. White must have left the back door
unlocked and the colonel must have retired before ten o’clock. However, either Ms.
White did not leave the back door unlocked, or the colonel did not retire before ten.
Therefore, Scarlet is not guilty of the crime.
•
•
•
•
S= Scarlet is guilty of the crime.
W= Ms. White left the back door unlocked.
C=The colonel retired before ten o’clock.
Symbolization
S-->(W&C)
~W v ~C
~S
S
T
T
T
T
F
F
F
F
WC
T T
T F
F T
F F
T T
T F
F T
F F
~W
F
F
T
T
F
F
T
T
~C
F
T
F
T
F
T
F
T
W&C
T
F
F
F
T
F
F
F
S--> (W&C)
T
F
F
F
T
T
T
T
~W v ~C
F
T
T
T
F
T
T
T
~S
F
F
F
F
T
T
T
T
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• Short Truth Table Method
•
•
•
The idea behind this method is that if an argument is invalid, then the argument must have at
least one row in which all the premises are true and the conclusion is false.
The method is to look directly for such a row by trying to make all the premises true and the
conclusion false at the same time.
In some cases neither the conclusion nor the premises forces an assignment.
•
•
•
•
•
•
•
•
•
In such cases trial and error must be used.
It must be kept in mind that it only takes one row in which the premises are all true and the
conclusion is false to make an argument invalid.
To be valid, an argument must have a true conclusion in every row in which the premises are all
true.
Example
Argument:
P-->Q
~Q-->R
~P-->R
For ~P -->R to be false, ~P must be true (P must be false) and R must be false.
Assuming P is false, P-->Q is true when Q is true or false.
Assuming R is false, ~Q-->R is true when ~Q is false, so Q must be assumed to be true.
This row makes the premises all true and the conclusion false, which proves the argument to be
invalid.
P Q R P-->Q
F T F T
~Q
F
~Q-->R
T
~P
T
~P-->R
F
P
F
CHAPTER 09
TRUTH FUNCTIONAL LOGIC
• The Method
• Try to assign Ts and Fs to the letters in the symbolization so that all the
premises come out true and the conclusion comes out false.
• There may be more than one way to do this, any one will do to prove
the argument to be invalid.
• If it is impossible to do this, the argument is valid.
CHAPTER 8 EXAMPLES
CATEGORICAL LOGIC
• 8-11
• 5. Every voter is a citizen, but some citizens are not residents. Therefore, some
voters are not residents.
•
• 1. All voters are citizens.
• 2. Some citizens are not residents.
• C. Some voters are not residents.
• Invalid.
•
CHAPTER 8 EXAMPLES
CATEGORICAL LOGIC
•
•
•
•
•
•
•
•
•
•
•
•
8-12
5. A few compact disc players use 24X sampling, so some of them must cost at least fifty
dollars, because you can’t buy a machine with 24X sampling for less than $50.
1:Some compact disc players are players that use 24x sampling.
2: No players that use 24x sampling are players that cost under $50
C: Some compact disc players are not players that cost under $50.
Valid
Or
P1: Some compact disc players are players that use 24X sampling.
P2: All players that use 24X sampling are players that cost more that $50.
C: Some compact disc players are players that cost more than $50.
CHAPTER 8 EXAMPLES
CATEGORICAL LOGIC
I was talking to Bill the other day and he told me
that he is a runner. People who run, at least if they
have any sense, own at least one pair of running
shoes. So, I’m sure that Bill has a pair of running
shoes.
• P1: All people identical to Bill are people who run.
• P2: All people who run are people who have/own
running shoes.
• C: All people identical to Bill are people who
have/own running shoes.
CHAPTER 8 EXAMPLES
CATEGORICAL LOGIC
• P1: All people identical to Bill are people who run.
• P2: All people who run are people who have/own
running shoes.
• C: All people identical to Bill are people who
have/own running shoes.
CHAPTER 8 EXAMPLES
CATEGORICAL LOGIC
• It is often said that all creatures with blood are either
cold-blooded or warm-blooded. It is well known that
every non-mammal is a non-cat. Of course, it is also
known that All mammals are non cold-blooded things.
So, it must be concluded that not a single cat is cold
blooded. The same is true of dogs.
•
•
•
•
•
•
P1 (before contraposition): All non-mammals are non-cats.
P2 (before obversion): All mammals are non cold-blooded things.
P1: All cats are mammals.
P2: No mammals are cold-blooded things.
C: No cats are cold-blooded things.
CHAPTER 8 EXAMPLES
CATEGORICAL LOGIC
• P1: All cats are mammals.
• P2: No mammals are cold-blooded things.
• C: No cats are cold-blooded things.
CHAPTER 8 EXAMPLES
CATEGORICAL LOGIC
• It is well known from biology that not a single mammal is a
creature that lacks a developed spine. Spines are, of course,
composed of bone and contain an important part of the
nervous system. So, it is obvious that all creatures with spines
have some sort of nervous system. It can be concluded that
each mammal has some sort of nervous system.
• P1 (before obversion): No mammals are creatures without developed
spines.
•
• P1: All mammals are creatures that have developed spines.
• P2: All creatures that have developed spines are creatures that have
some sort of nervous system.
• C: All mammals are creatures that have some sort of nervous system.
CHAPTER 8 EXAMPLES
CATEGORICAL LOGIC
• P1: All mammals are creatures that have developed spines.
• P2: All creatures that have developed spines are creatures
that have some sort of nervous system.
• C: All mammals are creatures that have some sort of nervous
system.
CHAPTER 9 EXAMPLES
TRUTH FUNCTIONAL LOGIC
• Translations
• #1. If the first party fails to fulfill the contract, then the second party is
entitled to a refund or a replacement product of equivalent value. The
first party failed to fulfill the contract, so either the second party will
receive a refund or a replacement product.
• P= The first party fails to fulfill the contract.
• Q= The second party is entitled to a refund.
• R= The second party is entitled to a replacement product.
•
•
•
•
Translation
P1: P-->(Q vR)
P2: P
C: Q v R
CHAPTER 9 EXAMPLES
TRUTH FUNCTIONAL LOGIC
• #2. The payment of fees is sufficient to become a member of the
club. Either Bill will pay his fees or he will not and he will do
something else. Unless he becomes a member of the club, he will
do something else. Bill didn’t do something else, so he is in the
club.
•
• P= Payment of fees.
• Q= Become a member of the club
• R= Do something else.
•
•
•
•
•
P1: P-->Q
P2: P v (~P & R)
P3: ~Q-->R
P4: ~R
C: Q
CHAPTER 9 EXAMPLES
TRUTH FUNCTIONAL LOGIC
• P1: PvQ
• P2: ~P
• C: Q-->P
P
T
T
F
F
Q
T
F
T
F
~P
F
F
T
T
PvQ
T
T
T
F
Q-->P
T
T
F
T
CHAPTER 9 EXAMPLES
TRUTH FUNCTIONAL LOGIC
• P1: P-->Q
• P2: ~Q
• C: ~P
P
T
T
F
F
Q
T
F
T
F
~P
F
F
T
T
~Q
F
T
F
T
P-->Q
T
F
T
T
CHAPTER 9 EXAMPLES
TRUTH FUNCTIONAL LOGIC
• P1: (P v Q) -->P
• P2: Q
• C: P&Q
P
T
T
F
F
Q
T
F
T
F
PvQ
T
T
T
F
P&Q
T
F
F
F
(P v Q )-->P
T
T
F
T