1Logic

Logic is, basically, the study of valid reasoning. When searching the internet, we use

Boolean logic – terms like “and” and “or” – to help us find specific web pages that fit in the

sets we are interested in. After exploring this form of logic, we will look at logical

arguments and how we can determine the validity of a claim.

Boolean Logic

We can often classify items as belonging to sets. If you went the library to search for a book

and they asked you to express your search using unions, intersections, and complements of

sets, that would feel a little strange. Instead, we typically using words like “and”, “or”, and

“not” to connect our keywords together to form a search. These words, which form the basis

of Boolean logic, are directly related to our set operations.

Boolean Logic

Boolean logic combines multiple statements that are either true or false into an

expression that is either true or false.

In connection to sets, a search is true if the element is part of the set.

Suppose M is the set of all mystery books, and C is the set of all comedy books. If we search

for “mystery”, we are looking for all the books that are an element of the set M; the search is

true for books that are in the set.

When we search for “mystery and comedy”, we are looking for a book that is an element of

both sets, in the intersection. If we were to search for “mystery or comedy”, we are looking

for a book that is a mystery, a comedy, or both, which is the union of the sets. If we searched

for “not comedy”, we are looking for any book in the library that is not a comedy, the

complement of the set C.

Connection to Set Operations

A and B

elements in the intersection A ⋂ B

A or B

elements in the union A ⋃ B

Not A

elements in the complement Ac

Notice here that or is not exclusive. This is a difference between the Boolean logic use of the

word and common everyday use. When your significant other asks “do you want to go to the

park or the movies?” they usually are proposing an exclusive choice – one option or the

other, but not both. In Boolean logic, the or is not exclusive – more like being asked at a

restaurant “would you like fries or a drink with that?” Answering “both, please” is an

acceptable answer.

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Example 1

Suppose we are searching a library database for Mexican universities. Express a reasonable

search using Boolean logic.

We could start with the search “Mexico and university”, but would be likely to find results

for the U.S. state New Mexico. To account for this, we could revise our search to read:

Mexico and university not “New Mexico”

In most internet search engines, it is not necessary to include the word and; the search engine

assumes that if you provide two keywords you are looking for both. In Google’s search, the

keyword or has be capitalized as OR, and a negative sign in front of a word is used to

indicate not. Quotes around a phrase indicate that the entire phrase should be looked for.

The search from the previous example on Google could be written:

Mexico university -“New Mexico”

Example 2

Describe the numbers that meet the condition:

even and less than 10 and greater than 0

The numbers that satisfy all three requirements are {2, 4, 6, 8}

Sometimes statements made in English can be ambiguous. For this reason, Boolean logic

uses parentheses to show precedent, just like in algebraic order of operations.

Example 3

The English phrase “Go to the store and buy me eggs and bagels or cereal” is ambiguous; it

is not clear whether the requestors is asking for eggs always along with either bagels or

cereal, or whether they’re asking for either the combination of eggs and bagels, or just cereal.

For this reason, using parentheses clarifies the intent:

Eggs and (bagels or cereal) means Option 1: Eggs and bagels, Option 2: Eggs and cereal

(Eggs and bagels) or cereal means Option 1: Eggs and bagels, Option 2: Cereal

Example 4

Describe the numbers that meet the condition:

odd number and less than 20 and greater than 0 and (multiple of 3 or multiple of 5)

The first three conditions limit us to the set {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}

The last grouped conditions tell us to find elements of this set that are also either a multiple

of 3 or a multiple of 5. This leaves us with the set {3, 5, 9, 15}

Notice that we would have gotten a very different result if we had written

(odd number and less than 20 and greater than 0 and multiple of 3) or multiple of 5

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The first grouped set of conditions would give {3, 9, 15}. When combined with the last

condition, though, this set expands without limits:

{3, 5, 9, 15, 20, 25, 30, 35, 40, 45, …}

Be aware that when a string of conditions is written without grouping symbols, it is often

interpreted from the left to right, resulting in the latter interpretation.

Conditionals

Beyond searching, Boolean logic is commonly used in spreadsheet applications like Excel to

do conditional calculations. A statement is something that is either true or false. A

statement like 3 < 5 is true; a statement like “a rat is a fish” is false. A statement like “x < 5”
is true for some values of x and false for others. When an action is taken or not depending on
the value of a statement, it forms a conditional.
Statements and Conditionals
A statement is either true or false.
A conditional is a compound statement of the form
“if p then q” or “if p then q, else s”.
Example 5
In common language, an example of a conditional statement would be “If it is raining, then
we’ll go to the mall. Otherwise we’ll go for a hike.”
The statement “If it is raining” is the condition – this may be true or false for any given day.
If the condition is true, then we will follow the first course of action, and go to the mall. If
the condition is false, then we will use the alternative, and go for a hike.
Example 6
As mentioned earlier, conditional statements are commonly used in spreadsheet applications
like Excel. If Excel, you can enter an expression like
=IF(A1 5, 2*A1, 3*A1) is used. Find the result if A1 is 3, and the result if
A1 is 8
This is equivalent to saying
If A1 >5, then calculate 2*A1. Otherwise, calculate 3*A1

If A1 is 3, then the condition is false, since 3 > 5 is not true, so we do the alternate action,

and multiple by 3, giving 3*3 = 9

If A1 is 8, then the condition is true, since 8 > 5, so we multiply the value by 2, giving

2*8=16

Example 8

An accountant needs to withhold 15% of income for taxes if the income is below $30,000,

and 20% of income if the income is $30,000 or more. Write an expression that would

calculate the amount to withhold.

Our conditional needs to compare the value to 30,000. If the income is less than 30,000, we

need to calculate 15% of the income: 0.15*income. If the income is more than 30,000, we

need to calculate 20% of the income: 0.20*income.

In words we could write “If income < 30,000, then multiply by 0.15, otherwise multiply by
0.20”. In Excel, we would write:
=IF(A1 100” in Excel, you would need to enter
“AND(A1100)”. Likewise, for the condition “A1=4 or A1=6” you would enter
“OR(A1=4, A1=6)”
Example 10
In a spreadsheet, cell A1 contains annual income, and A2 contains number of dependents.
A certain tax credit applies if someone with no dependents earns less than $10,000 and has
no dependents, or if someone with dependents earns less than $20,000. Write a rule that
describes this.
There are two ways the rule is met:
income is less than 10,000 and dependents is 0, or
income is less than 20,000 and dependents is not 0.
Informally, we could write these as
(A1 < 10000 and A2 = 0) or (A1 < 20000 and A2 > 0)

Notice that the A2 > 0 condition is actually redundant and not necessary, since we’d only be

considering that or case if the first pair of conditions were not met. So this could be

simplified to

(A1 < 10000 and A2 = 0) or (A1 < 20000)
In Excel’s format, we’d write
= IF ( OR( AND(A1 < 10000, A2 = 0), A1 < 20000), “you qualify”, “you don’t qualify”)
Truth Tables
Because complex Boolean statements can get tricky to think about, we can create a truth
table to keep track of what truth values for the simple statements make the complex
statement true and false
Truth table
A table showing what the resulting truth value of a complex statement is for all the
possible truth values for the simple statements.
Example 11
Suppose you’re picking out a new couch, and your significant other says “get a sectional or
something with a chaise”.
This is a complex statement made of two simpler conditions: “is a sectional”, and “has a
chaise”. For simplicity, let’s use S to designate “is a sectional”, and C to designate “has a
chaise”. The condition S is true if the couch is a sectional.
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A truth table for this would look like this:
S
C
S or C
T
T
T
T
F
T
F
T
T
F
F
F
In the table, T is used for true, and F for false. In the first row, if S is true and C is also true,
then the complex statement “S or C” is true. This would be a sectional that also has a chaise,
which meets our desire.
Remember also that or in logic is not exclusive; if the couch has both features, it does meet
the condition.
To shorthand our notation further, we’re going to introduce some symbols that are commonly
used for and, or, and not.
Symbols
The symbol ⋀ is used for and:
The symbol ⋁ is used for or:
The symbol ~ is used for not:
A and B is notated A ⋀ B.
A or B is notated A ⋁ B
not A is notated ~A
You can remember the first two symbols by relating them to the shapes for the union and
intersection. A ⋀ B would be the elements that exist in both sets, in A ⋂ B. Likewise, A ⋁ B
would be the elements that exist in either set, in A ⋃ B.
In the previous example, the truth table was really just summarizing what we already know
about how the or statement work. The truth tables for the basic and, or, and not statements
are shown below.
Basic truth tables
A B A⋀B
T T
T
T F
F
F T
F
F
F
F
A
T
T
F
F
B
T
F
T
F
A⋁B
T
T
T
F
A
T
F
~A
F
T
Truth tables really become useful when analyzing more complex Boolean statements.
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Example 12
Create a truth table for the statement A ⋀ ~(B ⋁ C)
It helps to work from the inside out when creating truth tables, and create tables for
intermediate operations. We start by listing all the possible truth value combinations for A,
B, and C. Notice how the first column contains 4 Ts followed by 4 Fs, the second column
contains 2 Ts, 2 Fs, then repeats, and the last column alternates. This pattern ensures that all
combinations are considered. Along with those initial values, we’ll list the truth values for
the innermost expression, B ⋁ C.
A B C B⋁C
T T T
T
T T F
T
T F T
T
T F F
F
F T T
T
F T F
T
F F T
T
F F F
F
Next we can find the negation of B ⋁ C, working off the B ⋁ C column we just created.
A B C B ⋁ C ~(B ⋁ C)
T T T
T
F
T T F
T
F
T F T
T
F
T F F
F
T
F T T
T
F
F T F
T
F
F F T
T
F
F F F
F
T
Finally, we find the values of A and ~(B ⋁ C)
A B C B ⋁ C ~(B ⋁ C)
A ⋀ ~(B ⋁ C)
T T T
T
F
F
T T F
T
F
F
T F T
T
F
F
T F F
F
T
T
F T T
T
F
F
F T F
T
F
F
F F T
T
F
F
F F F
F
T
F
It turns out that this complex expression is only true in one case: if A is true, B is false, and
C is false.
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Try it Now 1
Create a truth table for this statement: (~A ⋀ B) ⋁ ~B
When we discussed conditions earlier, we discussed the type where we take an action based
on the value of the condition. We are now going to talk about a more general version of a
conditional, sometimes called an implication.
Implications
Implications are logical conditional sentences stating that a statement p, called the
antecedent, implies a consequence q.
Implications are commonly written as p → q
Implications are similar to the conditional statements we looked at earlier; p → q is typically
written as “if p then q”, or “p therefore q”. The difference between implications and
conditionals is that conditionals we discussed earlier suggest an action – if the condition is
true, then we take some action as a result. Implications are a logical statement that suggest
that the consequence must logically follow if the antecedent is true.
Example 13
The English statement “If it is raining, then there are clouds is the sky” is a logical
implication. It is a valid argument because if the antecedent “it is raining” is true, then the
consequence “there are clouds in the sky” must also be true.
Notice that the statement tells us nothing of what to expect if it is not raining. If the
antecedent is false, then the implication becomes irrelevant.
Example 14
A friend tells you that “if you upload that picture to Facebook, you’ll lose your job”. There
are four possible outcomes:
1) You upload the picture and keep your job
2) You upload the picture and lose your job
3) You don’t upload the picture and keep your job
4) You don’t upload the picture and lose your job
There is only one possible case where your friend was lying – the first option where you
upload the picture and keep your job. In the last two cases, your friend didn’t say anything
about what would happen if you didn’t upload the picture, so you can’t conclude their
statement is invalid, even if you didn’t upload the picture and still lost your job.
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In traditional logic, an implication is considered valid (true) as long as there are no cases in
which the antecedent is true and the consequence is false. It is important to keep in mind that
symbolic logic cannot capture all the intricacies of the English language.
Truth values for implications
p
q
p→q
T T
T
T F
F
F T
T
F
F
T
Example 15
Construct a truth table for the statement (m ⋀ ~p) → r
We start by constructing a truth table for the antecedent.
m p ~p m ⋀ ~p
T T
F
F
T F
T
T
F T
F
F
F
F
T
F
Now we can build the truth table for the implication
m
T
T
F
F
T
T
F
F
p
T
F
T
F
T
F
T
F
~p
F
T
F
T
F
T
F
T
m ⋀ ~p
F
T
F
F
F
T
F
F
r
T
T
T
T
F
F
F
F
(m ⋀ ~p) → r
T
T
T
T
T
F
T
T
In this case, when m is true, p is false, and r is false, then the antecedent m ⋀ ~p will be true
but the consequence false, resulting in a invalid implication; every other case gives a valid
implication.
For any implication, there are three related statements, the converse, the inverse, and the
contrapositive.
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Related Statements
The original implication is
The converse is:
The inverse is
The contrapositive is
“if p then q”
“if q then p”
“if not p then not q”
“if not q then not p”
p→q
q→p
~p → ~q
~q → ~p
Example 16
Consider again the valid implication “If it is raining, then there are clouds in the sky”.
The converse would be “If there are clouds in the sky, it is raining.” This is certainly not
always true.
The inverse would be “If it is not raining, then there are not clouds in the sky.” Likewise,
this is not always true.
The contrapositive would be “If there are not clouds in the sky, then it is not raining.” This
statement is valid, and is equivalent to the original implication.
Looking at truth tables, we can see that the original conditional and the contrapositive are
logically equivalent, and that the converse and inverse are logically equivalent.
p
T
T
F
F
q
T
F
T
F
Implication
p→q
T
F
T
T
Converse
q→p
T
T
F
T
Inverse
~p → ~q
T
T
F
T
Contrapositive
~q → ~p
T
F
T
T
Equivalent
Equivalence
A conditional statement and its contrapositive are logically equivalent.
The converse and inverse of a statement are logically equivalent.
We have now examined several truth-functional connectives, three of which are two-place
connectives – conjunction, disjunction, and implication (conditional) – and one of which is a
one-place connective (negation). There is one remaining connective that is generally studied
in sentential logic, the biconditional, which corresponds to the English “if and only if”.
Like the implication, the biconditional is a two-place connective; if we fill the two blanks
with statements, the resulting expression is also a statement. For example, we can begin with
11
the statements “I am happy” and “I am relaxed” and form the compound statement “I am
happy if and only if I am relaxed”.
Truth values for the biconditional
p
q
p↔q
T T
T
T F
F
F T
F
F
F
T
Arguments
A logical argument is a claim that a set of premises support a conclusion. There are two
general types of arguments: inductive and deductive arguments.
Argument types
An inductive argument uses a collection of specific examples as its premises and uses
them to propose a general conclusion.
A deductive argument uses a collection of general statements as its premises and uses
them to propose a specific situation as the conclusion.
Example 17
The argument “when I went to the store last week I forgot my purse, and when I went today I
forgot my purse. I always forget my purse when I go the store” is an inductive argument.
The premises are:
I forgot my purse last week
I forgot my purse today
The conclusion is:
I always forget my purse
Notice that the premises are specific situations, while the conclusion is a general statement.
In this case, this is a fairly weak argument, since it is based on only two instances.
Example 18
The argument “every day for the past year, a plane flies over my house at 2pm. A plane will
fly over my house every day at 2pm” is a stronger inductive argument, since it is based on a
larger set of evidence.
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Evaluating inductive arguments
An inductive argument is never able to prove the conclusion true, but it can provide
either weak or strong evidence to suggest it may be true.
Many scientific theories, such as the big bang theory, can never be proven. Instead, they are
inductive arguments supported by a wide variety of evidence. Usually in science, an idea is
considered a hypothesis until it has been well tested, at which point it graduates to being
considered a theory. The commonly known scientific theories, like Newton’s theory of
gravity, have all stood up to years of testing and evidence, though sometimes they need to be
adjusted based on new evidence. For gravity, this happened when Einstein proposed the
theory of general relativity.
A deductive argument is more clearly valid or not, which makes them easier to evaluate.
Evaluating deductive arguments
A deductive argument is considered valid if all the premises are true, and the
conclusion follows logically from those premises. In other words, the premises are
true, and the conclusion follows necessarily from those premises.
Example 19
The argument “All cats are mammals and a tiger is a cat, so a tiger is a mammal” is a valid
deductive argument.
The premises are:
All cats are mammals
A tiger is a cat
The conclusion is:
A tiger is a mammal
Both the premises are true. To see that the premises must
logically lead to the conclusion, one approach would be use
a Venn diagram. From the first premise, we can conclude
that the set of cats is a subset of the set of mammals. From
the second premise, we are told that a tiger lies within the
set of cats. From that, we can see in the Venn diagram that
the tiger also lies inside the set of mammals, so the
conclusion is valid.
Mammals
Cats
Tiger
x
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Analyzing arguments with Venn diagrams1
To analyze an argument with a Venn diagram
1) Draw a Venn diagram based on the premises of the argument
2) If the premises are insufficient to determine what determine the location of an
element, indicate that.
3) The argument is valid if it is clear that the conclusion must be true
Example 20
Premise:
Premise:
Conclusion:
All firefighters know CPR
Jill knows CPR
Jill is a firefighter
From the first premise, we know that firefighters all lie
inside the set of those who know CPR. From the second
premise, we know that Jill is a member of that larger set, but
we do not have enough information to know if she also is a
member of the smaller subset that is firefighters.
Know CPR
Jill x?
x?
Firefighters
Since the conclusion does not necessarily follow from the
premises, this is an invalid argument, regardless of whether
Jill actually is a firefighter.
It is important to note that whether or not Jill is actually a firefighter is not important in
evaluating the validity of the argument; we are only concerned with whether the premises are
enough to prove the conclusion.
Try it Now 2
Determine the validity of this argument:
Premise:
No cows are purple
Premise:
Fido is not a cow
Conclusion: Fido is purple
In addition to these categorical style premises of the form “all ___”, “some ____”, and “no
____”, it is also common to see premises that are implications.
Example 21
Premise:
Premise:
Conclusion:
If you live in Seattle, you live in Washington.
Marcus does not live in Seattle
Marcus does not live in Washington
Washington
x?
Marcus x?
Technically, these are Euler circles or Euler diagrams, not Venn diagrams, but for the sake of simplicity we’ll
Seattle
continue to call them Venn diagrams.
1
14
From the first premise, we know that the set of people who live in Seattle is inside the set of
those who live in Washington. From the second premise, we know that Marcus does not lie
in the Seattle set, but we have insufficient information to know whether or not Marcus lives
in Washington or not. This is an invalid argument.
Example 22
Consider the argument “You are a married man, so you must have a wife.”
This is an invalid argument, since there are, at least in parts of the world, men who are
married to other men, so the premise not insufficient to imply the conclusion.
Some arguments are better analyzed using truth tables.
Example 23
Consider the argument
Premise:
If you bought bread, then you went to the store
Premise:
You bought bread
Conclusion: You went to the store
While this example is hopefully fairly obviously a valid argument, we can analyze it using a
truth table by representing each of the premises symbolically. We can then look at the
implication that the premises together imply the conclusion. If the truth table is a tautology
(always true), then the argument is valid.
We’ll get B represent “you bought bread” and S represent “you went to the store”. Then the
argument becomes:
Premise:
B→S
Premise:
B
Conclusion: S
To test the validity, we look at whether the combination of both premises implies the
conclusion; is it true that [(B→S) ⋀ B] → S ?
B
T
T
F
F
S
T
F
T
F
B→S
T
F
T
T
(B→S) ⋀ B
T
F
F
F
[(B→S) ⋀ B] → S
T
T
T
T
Since the truth table for [(B→S) ⋀ B] → S is always true, this is a valid argument.
Try it Now 3
Determine if the argument is valid:
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Premise:
Premise:
Conclusion:
If I have a shovel I can dig a hole.
I dug a hole
Therefore I had a shovel
Analyzing arguments using truth tables
To analyze an argument with a truth table:
1. Represent each of the premises symbolically
2. Create a conditional statement, joining all the premises with and to form the
antecedent, and using the conclusion as the consequent.
3. Create a truth table for that statement. If it is always true, then the argument is
valid.
Example 24
Premise:
Premise:
Conclusion:
If I go to the mall, then I’ll buy new jeans
If I buy new jeans, I’ll buy a shirt to go with it
If I got to the mall, I’ll buy a shirt.
Let M = I go to the mall, J = I buy jeans, and S = I buy a shirt.
The premises and conclusion can be stated as:
Premise:
M→J
Premise:
J→S
Conclusion: M → S
We can construct a truth table for [(M→J) ⋀ (J→S)] → (M→S)
M
T
T
T
T
F
F
F
F
J
T
T
F
F
T
T
F
F
S M→J J→S
T
T
T
F
T
F
T
F
T
F
F
T
T
T
T
F
T
F
T
T
T
F
T
T
(M→J) ⋀ (J→S)
T
F
F
F
T
F
T
T
M→S
T
F
T
F
T
T
T
T
From the truth table, we can see this is a valid argument.
The previous problem is an example of a syllogism.
Syllogism
[(M→J) ⋀ (J→S)] → (M→S)
T
T
T
T
T
T
T
T
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A syllogism is an implication derived from two others, where the consequence of one
is the antecedent to the other. The general form of a syllogism is:
Premise:
p→q
Premise:
q→r
Conclusion:
p→r
This is sometime called the transitive property for implication.
Example 25
Premise:
Premise:
Conclusion:
If I work hard, I’ll get a raise.
If I get a raise, I’ll buy a boat.
If I don’t buy a boat, I must not have worked hard.
If we let W = working hard, R = getting a raise, and B = buying a boat, then we can represent
our argument symbolically:
Premise
H→R
Premise
R→B
Conclusion: ~ B → ~ H
We could construct a truth table for this argument, but instead, we will use the notation of the
contrapositive we learned earlier to note that the implication ~ B → ~ H is equivalent to the
implication H → B. Rewritten, we can see that this conclusion is indeed a logical syllogism
derived from the premises.
Try it Now 4
Is this argument valid?
Premise:
If I go to the party, I’ll be really tired tomorrow.
Premise:
If I go to the party, I’ll get to see friends.
Conclusion: If I don’t see friends, I won’t be tired tomorrow.
Lewis Carroll, author of Alice in Wonderland, was a math and logic teacher, and wrote two
books on logic. In them, he would propose premises as a puzzle, to be connected using
syllogisms.
Example 26
Solve the puzzle. In other words, find a logical conclusion from these premises.
All babies are illogical.
Nobody is despised who can manage a crocodile.
Illogical persons are despised.
Let B = is a baby, D = is despised, I = is illogical, and M = can manage a crocodile.
Then we can write the premises as:
B→I
M → ~D
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I→D
From the first and third premises, we can conclude that B → D; that babies are despised.
Using the contrapositive of the second premised, D → ~M, we can conclude that
B → ~M; that babies cannot manage crocodiles.
While silly, this is a logical conclusion from the given premises.
Logical Fallacies in Common Language
In the previous discussion, we saw that logical arguments can be invalid when the premises
are not true, when the premises are not sufficient to guarantee the conclusion, or when there
are invalid chains in logic. There are a number of other ways in which arguments can be
invalid, a sampling of which are given here.
Ad hominem
An ad hominem argument attacks the person making the argument, ignoring the
argument itself.
Example 27
“Jane says that whales aren’t fish, but she’s only in the second grade, so she can’t be right.”
Here the argument is attacking Jane, not the validity of her claim, so this is an ad hominem
argument.
Example 28
“Jane says that whales aren’t fish, but everyone knows that they’re really mammals – she’s
so stupid.”
This certainly isn’t very nice, but it is not ad hominem since a valid counterargument is made
along with the personal insult.
Appeal to ignorance
This type of argument assumes something it true because it hasn’t been proven false.
Example 29
“Nobody has proven that photo isn’t Bigfoot, so it must be Bigfoot.”
Appeal to authority
These arguments attempt to use the authority of a person to prove a claim. While often
authority can provide strength to an argument, problems can occur when the person’s
opinion is not shared by other experts, or when the authority is irrelevant to the claim.
18
Example 30
“A diet high in bacon can be healthy – Doctor Atkins said so.”
Here, an appeal to the authority of a doctor is used for the argument. This generally
would provide strength to the argument, except that the opinion that eating a diet high
in saturated fat runs counter to general medical opinion. More supporting evidence
would be needed to justify this claim.
Example 31
“Jennifer Hudson lost weight with Weight Watchers, so their program must work.”
Here, there is an appeal to the authority of a celebrity. While her experience does provide
evidence, it provides no more than any other person’s experience would.
Appeal to consequence
An appeal to consequence concludes that a premise is true or false based on whether
the consequences are desirable or not.
Example 32
“Humans will travel faster than light: faster-than-light travel would be beneficial for space
travel.”
False dilemma
A false dilemma argument falsely frames an argument as an “either or” choice, without
allowing for additional options.
Example 33
“Either those lights in the sky were an airplane or aliens. There are no airplanes scheduled
for tonight, so it must be aliens.”
This argument ignores the possibility that the lights could be something other than an
airplane or aliens.
Circular reasoning
Circular reasoning is an argument that relies on the conclusion being true for the
premise to be true.
19
Example 34
“I shouldn’t have gotten a C in that class; I’m an A student!”
In this argument, the student is claiming that because they’re an A student, though shouldn’t
have gotten a C. But because they got a C, they’re not an A student.
Straw man
A straw man argument involves misrepresenting the argument in a less favorable way
to make it easier to attack.
Example 35
“Senator Jones has proposed reducing military funding by 10%. Apparently he wants to
leave us defenseless against attacks by terrorists”
Here the arguer has represented a 10% funding cut as equivalent to leaving us defenseless,
making it easier to attack.
Post hoc (post hoc ergo propter hoc)
A post hoc argument claims that because two things happened sequentially, then the
first must have caused the second.
Example 36
“Today I wore a red shirt, and my football team won! I need to wear a red shirt every time
they play to make sure they keep winning.”
Correlation implies causation
Similar to post hoc, but without the requirement of sequence, this fallacy assumes that
just because two things are related one must have caused the other. Often there is a
third variable not considered.
Example 37
“Months with high ice cream sales also have a high rate of deaths by drowning. Therefore
ice cream must be causing people to drown.”
This argument is implying a causal relation, when really both are more likely dependent on
the weather; that ice cream and drowning are both more likely during warm summer months.
Try it Now 5
Identify the logical fallacy in each of the arguments
a. Only an untrustworthy person would run for office. The fact that politicians are
untrustworthy is proof of this.
20
b. Since the 1950s, both the atmospheric carbon dioxide level and obesity levels have
increased sharply. Hence, atmospheric carbon dioxide causes obesity.
c. The oven was working fine until you started using it, so you must have broken it.
d. You can’t give me a D in the class – I can’t afford to retake it.
e. The senator wants to increase support for food stamps. He wants to take the taxpayers’
hard-earned money and give it away to lazy people. This isn’t fair so we shouldn’t do it.
Try it Now Answers
1.
A B ~A
~A ⋀ B
T T
F
F
T F
F
F
F T
T
T
F F
T
F
~B
F
T
F
T
(~A ⋀ B) ⋁ ~B
F
T
T
T
2. Since no cows are purple, we know there is no
overlap between the set of cows and the set of
purple things. We know Fido is not in the cow
set, but that is not enough to conclude that Fido
is in the purple things set.
Purple
things
Cows
x?
Fido x?
3. Let S: have a shovel, D: dig a hole
The first premise is equivalent to S→D. The second premise is D. The conclusion is S.
We are testing [(S→D) ⋀ D] → S
S D S→D (S→D) ⋀ D
[(S→D) ⋀ D] → S
T T
T
T
T
T F
F
F
T
F T
T
T
F
F F
T
F
T
This is not a tautology, so this is an invalid argument.
4. Letting P = go to the party, T = being tired, and F = seeing friends, then we can represent
this argument as P:
Premise:
P→T
Premise:
P→F
Conclusion: ~ F → ~T
21
We could rewrite the second premise using the contrapositive to state ~F → ~P, but that does
not allow us to form a syllogism. If we don’t see friends, then we didn’t go the party, but
that is not sufficient to claim I won’t be tired tomorrow. Maybe I stayed up all night
watching movies.
5.
a.
b.
c.
d.
e.
Circular
Correlation does not imply causation
Post hoc
Appeal to consequence
Straw man
Exercises
Compute the truth values of the following symbolic statements, supposing that the truth value
of A, B, C is T, and the truth value of X, Y, Z is F.
1. ~A ⋁ B
2. ~B ⋁ X
3. ~Y ⋁ C
4. ~Z ⋁ X
5. (A ⋀ X) ⋁ (B ⋀ Y)
6. (B ⋀ C) ⋁ (Y ⋀ Z)
7. ~(C ⋀ Y) ⋁ (A ⋀ Z)
8. ~(A ⋀ B) ⋁ (X ⋀ Y)
9. ~(X ⋀ Z) ⋁ (B ⋀ C)
10. ~(X ⋀ ~Y) ⋁ (B ⋀ ~C)
11. (A ⋁ X) ⋀ (Y ⋁ B)
12. (B ⋁ C) ⋀ (Y ⋁ Z)
13. (X ⋁ Y) ⋀ (X ⋁ Z)
14. ~(A ⋁ Y) ⋀ (B ⋁ X)
15. ~(X ⋁ Z) ⋀ (~X ⋁ Z)
16. ~(A ⋁ C) ⋁ ~(X ⋀ ~Y)
17. ~(B ⋁ Z) ⋀ ~(X ⋁ ~Y)
18. ~[(A ⋁ ~C) ⋁ (C ⋁ ~A)]
19. ~[(B ⋀ C) ⋀ ~(C ⋀ B)]
20. ~[(A ⋀ B) ⋁ ~(B ⋀ A)]
21. [A ⋁ (B ⋁ C)] ⋀ ~[(A ⋁ B) ⋁ C]
22. [X ⋁ (Y ⋀ Z)] ⋁ ~[(X ⋁ Y) ⋀ (X ⋁ Z)]
23. [A ⋀ (B ⋁ C)] ⋀ ~[(A ⋀ B) ⋁ (A ⋀ C)]
24. ~{[(~A ⋀ B) ⋀ (~X ⋀ Z)] ⋀ ~[(A ⋀ ~B) ⋁ ~(~Y ⋀ ~Z)]}
25. ~{~[(B ⋀ ~C) ⋁ (Y ⋀ ~Z)] ⋀ [(~B ⋁ X) ⋁ (B ⋁ ~Y)]}
26. A → B
27. A → X
28. B → Y
29. Y → Z
22
30. (A → B) → Z
31. (X → Y) → Z
32. (A → B) → C
33. (X → Y) → C
34. A → (B → Z)
35. X → (Y → Z)
36. [(A → B) → C] → Z
37. [(A → X) → Y] → Z
38. [A → (X → Y)] → C
39. [A → (B → Y)] → X
40. [(X → Z) → C] → Y
41. [(Y → B) → Y] → Y
42. [(A → Y) → B] → Z
43. [(A ⋀ X) → C] → [(X → C) → X]
44. [(A ⋀ X) → C] → [(A → X) → C]
45. [(A ⋀ X) → Y] → [(X → A) → (A → Y)]
46. [(A ⋀ X) ⋁ (~A ⋀ ~X)] → [(A → X) ⋀ (X → A)]
47. {[A → (B → C)] → [(A ⋀ B) → C]} → [(Y → B) → (C → Z)]
48. {[(X → Y) → Z] → [Z → (X → Y)]} → [(X → Z) → Y]
49. [(A ⋀ X) → Y] → [(A → X) ⋀ (A → Y)]
50. [A → (X ⋀ Y)] → [(A → X) ⋁ (A → Y)]
Construct the complete truth table for each of the following formulas:
51. (P ⋀ Q) ⋁ (P ⋀ ~Q)
52. ~(P ⋀ ~P)
53. ~(P ⋁ ~P)
54. ~(P ⋀ Q) ⋁ (~P ⋁ ~Q)
55. ~(P ⋁ Q) ⋁ (~P ⋀ ~Q)
56. (P ⋀ Q) ⋁ (~P ⋀ ~Q)
57. ~(P ⋁ (P ⋀ Q))
58. ~(P ⋁ (P ⋀ Q)) ⋁ P
59. (P ⋀ (Q ⋁ P)) ⋀ ~P
60. ((P Q) → P) → P
61. ~(~(P → Q) → P)
62. (P → Q) ↔ ~P
63. P → (Q → (P ⋀ Q))
64. (P ⋁ Q) ↔ (~P → Q)
65. ~(P ⋁ (P → Q))
66. (P → Q) ↔ (Q → P)
67. (P → Q) ↔ (~Q → ~P)
68. (P ⋁ Q) → (P ⋀ Q)
69. (P ⋀ Q) ⋁ (P ⋀ R)
70. [P ↔ (Q ↔R)] ↔ [(P ↔Q) ↔R]
71. [P → (Q ⋀ R)] → [P → R]
72. [P → (Q ⋁ R)] → [P → Q]
73. [(P → Q) → R] → [P → R]
23
74. [(P ⋀ Q) → R] → [P → R]
75. [(P ⋀ Q) → R] → [(Q ⋀ ~R) → ~P]

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