MAT 110 - Chapter 1 1 MAT 110 Chapter 1 Notes Logic & Reasoning David J. Gisch Definitions • Logic is the study of the methods and principles of reasoning. • An argument uses a set of facts or assumptions, called premises, to support a conclusion. ▫ Premises are facts ▫ Conclusions are supported by premises Definitions uses a set of facts or assumptions, called Logic Argument Fallacy Fallacy Structures Appeal to Popularity False Cause Appeal to Ignorance Hasty Generalization Limited Choice Appeal to Emotion Personal Attack Circular Reasoning Diversion -Red Herring Straw Man
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MAT 110Chapter 1 NotesLogic & ReasoningDavid J. Gisch
Definitions• Logic is the study of the methods and principles of
reasoning.
• An argument uses a set of facts or assumptions, called premises, to support a conclusion.▫ Premises are facts▫ Conclusions are supported by premises
Definitions• Logic is the study of the methods and principles of
reasoning.
• An argument uses a set of facts or assumptions, called premises, to support a conclusion.
• A fallacy is a deceptive argument—an argument in which the conclusion is not well supported by the premises.
Logic
Argument
Fallacy
Fallacy Structures
Appeal to Popularity
False Cause
Appeal to Ignorance
Hasty Generalization
Limited Choice
Appeal to Emotion
Personal Attack
Circular Reasoning
Diversion -Red Herring
Straw Man
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Appeal to Popularity• Ford makes the best pickup trucks in the world. After
all, more people drive Ford pickups than any other light truck.
• Analysis:▫ Identify premises and conclusions▫ Simply because it is popular doesn’t mean it is the best.▫ Fallacy: Many people believe it so it’s true.
False Cause• I placed the quartz crystal on my forehead, and in five
minutes my headache was gone. The crystal made my headache go away.
• Analysis:▫ Identify premises and conclusions▫ Fallacy: One thing happened before another, but that doesn’t
prove a connection between them
Appeal to Ignorance
• Scientists have not found any concrete evidence of aliens visiting Earth. Therefore, anyone who claims to have seen a UFO must be hallucinating. (Aliens have not visited Earth)
• Analysis:▫ Identify premises and conclusions▫ Fallacy: Lack of knowledge. An absence of evidence is not evidence
of absence.
Hasty Generalization
• Two cases of childhood leukemia have occurred along the street where the high-voltage power lines run. The power lines must be the cause of theses illnesses.
• Analysis:▫ Identify premises and conclusions▫ Fallacy: Conclusion is drawn from an inadequate number of cases
that have not been sufficiently analyzed.
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Limited Choice
• You don’t support the President, so you are not a patriotic American.
• Analysis:▫ Identify premises and conclusions▫ Fallacy: Artificially limits choice. There could be many other
reasons you don’t support the President.
Appeal to Emotion
• In a commercial for Michelin tires, a picture of a baby is shown with the words “because so much is riding on your tires.”.
• Analysis:▫ Identify premises and conclusions▫ Fallacy: No appeal to logic, simply attempts to evoke emotion.
Personal Attack
• Gwen: Merle, you should stop smoking because it’s hurting your health, and endangering people around you.
• Merle: I’ve seen you smoke on occasion yourself.
• Analysis:▫ Identify premises and conclusions▫ Gwen makes a logical argument.▫ Merle simply attacks Gwen rather than making logical argument.▫ Fallacy: I have a problem with the person or group claiming
something, so it can’t be true.
Circular Reasoning
• Society has an obligation to shelter the homeless because the needy have a right to the resources of the community.
• Analysis:▫ Identify premises and conclusions▫ Fallacy: The premise and conclusion say the same
thing.
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Diversion (Red Herring)
• We should not continue to fund cloning research because there are so many ethical issues involved. Ethics is at the heart of our society, and we cannot afford to have too many ethical loose ends.
• Analysis:▫ Identify premises and conclusions▫ Fallacy: Premises are all about ethics not cloning.
Attempts to divert attention from real issue, cloning.
Straw Man
• The mayor of a large city has proposed decriminalizing drug possession in order to reduce overcrowding in jails. His challenger in the upcoming elect says, “The mayor doesn’t think there’s anything wrong with drug use, but I do.”
• Analysis:▫ Identify premises and conclusions▫ Fallacy: The mayor has said nothing about his view on
drugs. He has proposed a solution to another problem. Distorting another’s views is called Straw Man.
Fallacy Structures
Appeal to Popularity Everybody does it. Many people believe p is true, therefore p is true.
False Cause The premise and conclusion are not connected. A came before B, therefore A caused B.
Appeal to Ignorance Lack of knowledge. There is no proof that p is true, therefore p is false.
Hasty Generalization Conclusion is drawn from inadequate number of cases. A and B are linked one or a few times, therefore A causes B or vice versa.
Fallacy Structures
Limited Choice All the possible choices have not been considered. Make a statement and draw a conclusion, but the conclusion isn’t supported by the facts. p is false, therefore only q can be true.
Appeal to Emotion Evoke an emotional response. pis associated with a positive emotional response, therefore p is true.
Personal Attack Attack a person rather than an issue. I have a problem with the person or group claiming p, therefore p is not true.
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Fallacy Structures
Circular Reasoning Both the premise and conclusion say the same thing. p is true. p is restated in different words.
Diversion (Red Herring) Divert attention from real issue. p is related to q and I have an argument concerning q; therefore p is true.
Straw Man Distort another views. I have an argument concerning a distorted version of p; therefore I hope you are fooled into concluding I have an argument concerning the real version of p.
Propositions and Truth Values
Definitions
• A proposition makes a claim (either an assertion or a denial) that may be either true or false. It must have the structure of a complete sentence.
• Any proposition has two possible truth values:T = true or F = false.
• A truth table is a table with a row for each possible set of truth values for the propositions being considered.
Propositions and Truth Values• It is a proposition if:
▫ It is a complete sentence▫ It makes a claim▫ The claim can be true or false
• It will not be a proposition if:▫ It is a question▫ Does not assert or deny anything▫ Is not a complete sentence
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Are they propositions?• Joan is sitting in a chair.
• I did not take the pen.
• Are you going to the store?
• Three miles south of here
• 7 + 9 = 2
Truth Tables• A truth table is a table that shows all of the possible
logical outcomes for given propositions.• We use the letters p, q, and r to represent propositions.• If we have a proposition p, then it is either true or false,
so our truth table would have 2 rows (one for each possibility).
PTF
Truth Tables• Suppose we had TWO propositions.
p qT TT FF TF F
T or F T or FChoices for P Choices for Q
2 x 2Choices for P Choices for Q
=4 combinations
So if we have two starting propositions then there are 4 rows, one for each possible combination?
What if we had 3 propositions p, q, and r?
Negation (Opposites)
p not p T F F T
The negation of a proposition p is another proposition that makes the opposite claim of p.
← If p is true (T), not p is false (F).← If p is false (F), not p is true (T).
Symbol: ~
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What would the negative be?• Joan is sitting in a chair.
• I took the pen.
• Betsy is the fastest runner on the team.
• 7 + 9 = 2
Double Negation
• Double negation has the same truth value as the original proposition
• It’s like turning over a coin▫ Start with heads▫ Turn it over once, tails▫ Turn it over again, back
to heads
p not p not not p T F T F T F
The double negation of a proposition p, not not p, has the same truth value as p.
Propositions are often joined with logical connectors—words such as and, or, and if…then.
Example:p = I won the game.q = It was fun.
Logical Connectors
Logical Connector
and
or
if…then
New Proposition
I won the game and it was fun.
I won the game or it was fun.
If I won the game, then it was fun.
p q p and q T T T T F F F T F F F F
Given two propositions p and q, the statement p and qis called their conjunction. It is true only if p and q are both true.
Symbol:
And Statements (Conjunctions)
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NOTE
•A conjunction is only true if both p and q are true.
p q p and q T T T T F F F T F F F F
• An inclusive or means “either or both.”
• An exclusive or means “one or the other, but not both.”
The word or can be interpreted in two distinct ways:
In logic, assume or is inclusive unless told otherwise.
Or Statements (Disjunctions)
Or Statement ( Disjunctions) • Example: INCLUSION
• A health insurance policy covers hospitalization in cases of illness or injury.▫ Covers illness
Or▫ Covers injury
Or ▫ Both
Or Statement (Disjunction)• Example:EXCLUSION
• A restaurant offers soup or salad.▫ Offers soup
Or▫ Offers salad
NOT▫ Both
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p q p or q T T T T F T F T T F F F
Or Statements (Disjunctions)Given two propositions p and q, the statement p or qis called their disjunction. It is true unless p and qare both false.
Symbol:
NOTE
•A disjunction is true unless both p and q are false.
p q p or q T T T T F T F T T F F F
If …Then Statement (Conditional)• If all politicians are liars then Representative Smith is a
liar. • Conditional propositions
▫ p is called the� Hypothesis or� Antecedent
▫ q is called the� Conclusion� Consequence
▫ q is true on the condition that p is true.
A statement of the form if p, then q is called a conditional proposition (or implication). It is true unless p is true and q is false.
p q if p, then q T T T T F F F T T F F T
n Proposition p is called the hypothesis.n Proposition q is called the conclusion.
If… Then Statements (Conditionals)
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• Think of if-then statements as a “rule.”▫ Row 1, you did the condition and got the reward (conclusion). No
problem.▫ Row 2, you did the condition but did not get the reward
(conclusion). PROBLEM▫ Row 3 & 4, you did not do the condition so the rule no longer
applies and whatever happens does not conflict with the rule.
p q if p, then q T T T T F F F T T F F T
If…ThenThink of If-Then statements as a rule.
RULE: If you clean your room we will go get ice cream.
1. Sally cleans here room and gets ice cream.
2. Sally cleans her room but does not get ice cream.
3. Sally does not clean her room, but still gets ice cream.
4. Sally does not clean her room and does not get ice cream.
If…ThenRULE: On many of your Quizzes (projects) it says NO LATE WORK WILL BE ACCEPTED. (i.e. if it is turned in on time I will grade it.)
1. You turn it in on time, I grade it.
2. You turn it on time, but I refuse to grade it.
3. You turn it in on time, but I decide to grade it and give you half credit.
4. You turn it in late, and I refuse to grade it.
� � �⋀� � ∨ �
Truth Tables
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� � ∼ � � ∨∼ �
Truth Tables Truth Table Practice
Note: ~ signifies NEGATION ∧ signifies AND ∨ signifies OR
Practice by writing the truth values of each row in the table above.
p q ~ p ~ q p ∧ q ~ p ∨ ~ q ~ ( p ∧ q)
T T T F F T F F
Truth Table Practice
Note: ~ signifies NEGATION ∧ signifies AND ∨ signifies OR
Practice by writing the truth values of each row in the table above.
p q ~ p ~ q p ∧ q ~ p ∨ ~ q ~ ( p ∧ q)
T T F F T F F T F F T F F
Truth Table Practice
Note: ~ signifies NEGATION ∧ signifies AND ∨ signifies OR
Practice by writing the truth values of each row in the table above.
p q ~ p ~ q p ∧ q ~ p ∨ ~ q ~ ( p ∧ q)
T T F F T F F T F F T F T T F T F F
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Truth Table Practice
Note: ~ signifies NEGATION ∧ signifies AND ∨ signifies OR
Practice by writing the truth values of each row in the table above.
p q ~ p ~ q p ∧ q ~ p ∨ ~ q ~ ( p ∧ q)
T T F F T F F T F F T F T T F T T F F T T F F
Truth Table Practice
Note: ~ signifies NEGATION ∧ signifies AND ∨ signifies OR
Practice by writing the truth values of each row in the table above.
p q ~ p ~ q p ∧ q ~ p ∨ ~ q ~ ( p ∧ q)
T T F F T F F T F F T F T T F T T F F T T F F T T F T T
Sets and Venn Diagrams
Definition• A set is a well-defined collection of objects.
▫ We denote sets with capital letters▫ We write sets with brackets as follows
3, 4, 5▫ This is referred to as roster form of a set.
• Any item belonging to a set is called an element or member of that set.▫ We denote elements of a set as follows
3 ∈ 3, 4, 57 ∉ 3, 4, 5
Why well-defined?Give me the set of people in this room who are nice.
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Ellipses• Use three dots, …, to indicate a continuing pattern if
there are too many members to list. For example,
{1945, 1946, 1947 . . . 1991}
{ 6, 7, 8 . . .}
{ . . ., -3, -2, 1, 0, 1, 2, . . .}
You need to list three items to establish a clear pattern!
Definition• Repetitions of elements do not matter. Whether it is
listed once or twice it is still a member of the set and that is all that matters.
• Order also does not matter in sets, unless it is used to establish a pattern.
Definition• The set of all things being discussed is referred to as the
universal set. We denote the universal set as set �.
• For example, if we were discussing arithmetic in third grade we might use the universal set of whole numbers. In college algebra the universal set would be all real numbers.
The Real NumbersExample: Let the universal set be the set of real numbers.
Natural = {
Whole = {
Integers = {
Rational = {
Irrational = {
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SetsExample: Write out each of the following sets in roster form.
(a) The set of all numbers (integers) between 2 and 7.
(b) The set days of the week that begin with the letter S.
(c) The set of planets in our solar system that begin with the letter C.
Sets and Propositions• Now that we’ve been introduced to sets and have studied
a little bit about set we are ready to discuss propositions that make claims about sets.
• As you know, Propositions are in the form of complete sentences.
• The sets referenced in a proposition can be identified as follows:▫ one set appears in the subject of the sentence▫ one set appears in the predicate of the sentence.
Sets and Propositions
• There are four standard categorical propositions– All Subject are Predicate
– No Subject are Predicate
– Some Subject are Predicate
– Some Subject are not Predicate
– Note:
• S (propositions in the subject)• P (propositions in the predicate)
Venn Diagram• Venn diagrams are pictorial representations of sets.
▫ The box represents all things in the universal set.▫ Circles within the box represent a set of things.▫ Note that there is a dichotomy in the diagram. Once you draw a
circle (set) the part inside represents that set but it also implies the part outside is NOT that set.
▫ So you do not draw two circles.
MenWomen
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Sets and Propositions
Example: All whales are mammals.
All S are P
Non-mammals
Sets and Propositions
Example: No fish are mammals.
No S are P
Sets and PropositionsExample: Some doctors are women.
Some S are P
Sets and PropositionsExample: Some teachers are not men.
Some S are not P
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Set Relations• Sets can be
▫ Proper subsets (one is contained in the other).� � � �|������� ��� � � �|����� �� � ��
▫ Have some overlap (called the intersection).� � � �|������� ������� � � �|�������������
▫ Have no overlap (called disjoint sets).� � � �|������������ � � �|����� �� � ��
Venn DiagramsExample: Eight hundred students were surveyed and the results of the campus blood drive survey are shown below.
(a) How many students were willing to donate blood or serve breakfast?
(b) How many were willing to do neither?
18042 51
Donated Breakfast
Venn DiagramsExample: A survey of 120 college students was taken at registration. Of those surveyed, 75 students registered for a math course, 65 for an English course, and 40 for both math and English. Of those surveyed,
(a) How many registered only for a math course?
(b) How many registered only for an English course?
(c) How many registered for a math course or an English course?
(d) How many did not register for either a math course or an English course?
Venn DiagramsExample: Use the Venn Diagram below to answer the following.
(a) How many students read none of the publications?
(b) How many read Business Week and Fortune but not the Journal?
