Diagram Flashcards
D is inspected two days before E
G marches three groups behind F
There are three spaces between L and M, and M is examined before L
How do you diagram
J and H cannot be selected together?
It depends on whether it is a linear or a group game
Double Not Arrow means the two terms can never be together
(Grouping Question)
inference – double not - important for grouping
Attach two statements
Arun will be selected if and only if Bai is selected
if and only if creates two seperate conditional statements
- A if B
- A only if B
Only two scenarios are possible under a double arrow
- A and B are selected
- Neither A nor B is selected
Double Arrow
Only two scenarios are possible under a double arrow
- A and B are selected
- Neither A nor B is selected
Introduced in three ways
- If and only if - or its variants - if but only if - or - if yet only if
- Vice Versa (as in “If A is selected then B is selected and vice versa”)
- By repeating and reversing the terms (“If A is selected then B is selected, and if B is selected then A is selected”)
(Can be made through if and only if)
if and only if creates two separate conditional statements
- A if B
- A only if B
S is not selected if R is not selected
(And contrapositive)
There are two days between the day Q is inspected and the day R is inspected
There are two spaces between Q and R
F is displayed immediately prior to G, and is displayed at some point before I
Whatever term is modified by unless, except, without, until becomes the ….
Necessary Condition
The other term is negated and becomes the sufficient condition
Unless a person studies, they will not recieve an A
study –> A
S is not fourth unless T is fifth
S4 –> T5
Whatever term is modified by unless, except, without, until becomes the Necessary Condition
The other term is negated and becomes the sufficient condition
Unless a person studies, they will not recieve an A
study –> A
Laron performs second only if Nancy performs sixth
If V is displayed immediately before W, then S must be displayed third
If K is first then M is before P
G does not speak fourth unless Q speaks second
Only desginates a
necessary condition… just like only if
Only zombies eat brains
Eat brains > Zombie
-
only designates necessary condition
Only if designates
necessary condition
Only if one is starving it is acceptable to eat a slim jim
n and s?
slim jim > starving
-
only if desginates necessary
With “If and only if” statements what are the only two conditions
Either both conditions occur or neither one does
Diagram
No man wears perfume
man > -perfume-
no always negates the second term
Diagram
No A’s are B’s
A > -B-
no always negates the second term
Diagram and Contra
If the professional wrestler is in the ring, then he is wearing knee pads or a spandex body suit, but not both
Diagram and Contra
If the professional wrestler is in the ring, then he is wearing knee pads or a spandex body suit, but not both
-
-
contra
If the wrester is wearing neither knee pads nor a spandex suit, or if he is wearing both knee pads and a spandex body suit, then he is not in the ring
What deductions can you make from two most statements
- Most NFL players are incredibly fast
- Most NFL players weigh over 250 pounds
The only deduction you can make is if two most statements share a suffiient condition, in which case a some connection is produced
-
-
- Most NFL players are incredibly fast
- Most NFL players weigh over 250 pounds
-
Some of those who are incredibly fast weigh over 250 pounds
Deductions
-
- Most NFL players are incredibly fast
- Most NFL players weigh over 250 pounds
The only deduction you can make is if two most statements share a suffiient condition, in which case a some connection is produced
-
-
- Most NFL players are incredibly fast
- Most NFL players weigh over 250 pounds
-
Some of those who are incredibly fast weigh over 250 pounds
Deductions?
-
All of the members of the circus perform high-wire acts,
Some members of the circus do not have life insurance
Some people who perform high-wire acts without life insurance
-
With All + Some statements, the shared term must be the sufficient condition of the all statement
With All + Some statements ….
To make a deduction with All + Some statements, the shared term must be the suffifient condition of the all statement
If all A’s are B’s and Some A’s are C’s, then Some B’s are C’s
If all A’s are B’s and Some A’s are C’s, then ….
If all A’s are B’s and Some A’s are C’s, then Some B’s are C’s
To make a deduction with All + Some statements, the shared term must be the sufficient condition of the all statement
With All + Most statements…
There are two ways of combining All + Most statements,
-
If the necessary condition of the Most statement is the sufficient condition of the All statement, then you can produce a Most statement as the conclusion
If they share the same sufficient condition, then you can only produce a some statement
When combining an All + Most statements,
If the necessary condition of the Most statement is the sufficient condition of the All statement, then…
There are two ways of combining All + Most statements,
-
If the necessary condition of the Most statement is the sufficient condition of the All statement, then you can produce a Most statement as the conclusion
If they share the same sufficient condition, then you can only produce a some statement
When combining an All + Most statements,
If they share the same sufficient condition, then you can only produce a some statement
-
If the necessary condition of the Most statement is the sufficient condition of the All statement, then you can produce a Most statement as the conclusion
Most Hannah Montana fans wear skinny jeans.
All Hannah Montana fans are 12 year old girls
Combine..
Some of those who wear skinny jeans are 12 year old girls
When combining All + Most:
If they share the same sufficient condition, then you can only produce a some statement
If the necessary condition of the Most statement is the sufficient condition of the All statement, then you can produce a Most statement as the conclusion
Combining two Most statements
The only time you can derive a conclusion from two most statements is when they share the sufficient condition
–
If most A’s are B’s and most A’s are C’s, then some B’s are C’s
If most A’s are B’s and most A’s are C’s, then ..
The only time you can derive a conclusion from two most statements is when they share the sufficient condition
–
If most A’s are B’s and most A’s are C’s, then some B’s are C’s
Combining two Some statements
Two Some statements can never lead to a valid conclusion
also
A Most and a Some can never lead to a valid conclusion
Combining a Most and a Some Statement
A Most and a Some can never lead to a valid conclusion
also
Two Some statements can never lead to a valid conclusion
Principle of Distribution
In Ordering games, a Principle of Distribution restricts how many slots a player can fill or how many players can be assigned to a slot
In Ordering games, a ………… restricts how many slots a player can fill or how many players can be assigned to a slot
In Ordering games, a Principle of Distribution restricts how many slots a player can fill or how many players can be assigned to a slot
Family #1, Must Be or PROVE Family
-
- Must Be True
- Main Point
- Point at Issue
- Method of Reasoning
- Flaw in the Reasoning
- Parallel Reasoning
-
The First Question Family is based on the principle of using the information in the stimulus to prove that one of the answer choices must be true
-
Stimulus accepted as given while the answer choices are “under suspicion,” and the information in the stimulus is used to prove one of them.
-
Rules
1. The information in the stimulus is suspect. There are often reasoning errors present, and you will further weaken the argument in some way.
-
2. The answer choices are accepted as given, even if they include “new” information. Your task is to determine which answer choice best attacks the argument in the stimulus.
Family #2, Help Family
-
- Assumption
- Justify the Conclusion
- Strengthen/Support
- Resolve the Paradox
-
Rules
1. The information in the stimulus is suspect. There are often reasoning errors present, and depending on the question, you will help shore up the argument in some way.
-
- The answer choices are accepted as given, even if they include “new” information. Your task is to determine which answer choice best meets the question posed in the stem.
Family #3, Hurt Family
-
consists of one question type:
Weaken
-
Compared to the Second Question Family, the Help family, the only difference between the diagrams is that the third family diagram has a bar across the arrow. This bar signifies a negative: instead of strengthening or helping the argument, you attack or hurt the argument.
-
Rules:
- ## The information in the stimulus is suspect. There are often reasoning errors present, and you will further weaken the argument in some way.
- The answer choices are accepted as given, even if they include “new” information. Your task is to determine which answer choice best attacks the argument in the stimulus.
consists of one question type:
Cannot Be True
-
The 4th question family is based on the principle that you must use the information in the stimulus to prove that one of the answer choices cannot occur. Compared to the First Question Family, the only difference in the diagram is that the Fourth Family diagram has a bar across the arrow. Again, this bar signifies a negative
-
Rules
- ## You must accept the stimulus information—even if it contains an error of reasoning—and use it to prove that one of the answer choices cannot occur.
- Any information in an answer choice that does not appear either directly in the stimulus or as a combination of items in the stimulus will be incorrect. The correct answer choice will directly disagree with the stimulus or a consequence of the stimulus.
Family #1
Family #1, Must Be or PROVE Family
-
- Must Be True
- Main Point
- Point at Issue
- Method of Reasoning
- Flaw in the Reasoning
- Parallel Reasoning
-
The First Question Family is based on the principle of using the information in the stimulus to prove that one of the answer choices must be true
-
Stimulus accepted as given while the answer choices are “under suspicion,” and the information in the stimulus is used to prove one of them.
-
Rules
1. The information in the stimulus is suspect. There are often reasoning errors present, and you will further weaken the argument in some way.
-
2. The answer choices are accepted as given, even if they include “new” information. Your task is to determine which answer choice best attacks the argument in the stimulus.
Family #2
Family #2, Help Family
-
- Assumption
- Justify the Conclusion
- Strengthen/Support
- Resolve the Paradox
-
Rules
1. The information in the stimulus is suspect. There are often reasoning errors present, and depending on the question, you will help shore up the argument in some way.
-
- The answer choices are accepted as given, even if they include “new” information. Your task is to determine which answer choice best meets the question posed in the stem
Family #3
Family #3, Hurt Family
-
consists of one question type:
Weaken
-
Compared to the Second Question Family, the Help family, the only difference between the diagrams is that the third family diagram has a bar across the arrow. This bar signifies a negative: instead of strengthening or helping the argument, you attack or hurt the argument.
-
Rules:
- ## The information in the stimulus is suspect. There are often reasoning errors present, and you will further weaken the argument in some way.
- The answer choices are accepted as given, even if they include “new” information. Your task is to determine which answer choice best attacks the argument in the stimulus.
Family #4
consists of one question type:
Cannot Be True
-
The 4th question family is based on the principle that you must use the information in the stimulus to prove that one of the answer choices cannot occur. Compared to the First Question Family, the only difference in the diagram is that the Fourth Family diagram has a bar across the arrow. Again, this bar signifies a negative
-
Rules
- ## You must accept the stimulus information—even if it contains an error of reasoning—and use it to prove that one of the answer choices cannot occur.
- Any information in an answer choice that does not appear either directly in the stimulus or as a combination of items in the stimulus will be incorrect. The correct answer choice will directly disagree with the stimulus or a consequence of the stimulus.
The …. Question Family is based on the principle of using the information in the stimulus to prove that one of the answer choices must be true
The First Question Family (Must Be or PROVE Family)is based on the principle of using the information in the stimulus to prove that one of the answer choices must be true
-
- Must Be True
- Main Point
- Point at Issue
- Method of Reasoning
- Flaw in the Reasoning
- Parallel Reasoning
-
Stimulus accepted as given while the answer choices are “under suspicion,” and the information in the stimulus is used to prove one of them.
-
Rules
1. The information in the stimulus is suspect. There are often reasoning errors present, and you will further weaken the argument in some way.
-
2. The answer choices are accepted as given, even if they include “new” information. Your task is to determine which answer choice best attacks the argument in the stimulus.
The ……. Question Family is based on the principle of assisting or helping the author’s argument or statement in some way, whether by revealing an assumption of the argument, by resolving a paradox, or in some other fashion
The Second Question Family (Help Family) is based on the principle of assisting or helping the author’s argument or statement in some way, whether by revealing an assumption of the argument, by resolving a paradox, or in some other fashion
-
- Assumption
- Justify the Conclusion
- Strengthen/Support
- Resolve the Paradox
-
Rules
1. The information in the stimulus is suspect. There are often reasoning errors present, and depending on the question, you will help shore up the argument in some way.
-
- The answer choices are accepted as given, even if they include “new” information. Your task is to determine which answer choice best meets the question posed in the stem
The …. Question Family asks you to attack the author’s argument.
The Third Question Family (Hurt Family) consists of only one question type—Weaken and it asks you to attack the author’s argument.
-
consists of one question type:
Weaken
-
Compared to the Second Question Family, the Help family, the only difference between the diagrams is that the third family diagram has a bar across the arrow. This bar signifies a negative: instead of strengthening or helping the argument, you attack or hurt the argument.
-
Rules:
- ## The information in the stimulus is suspect. There are often reasoning errors present, and you will further weaken the argument in some way.
- The answer choices are accepted as given, even if they include “new” information. Your task is to determine which answer choice best attacks the argument in the stimulus.
The ….. question family is based on the principle that you must use the information in the stimulus to prove that one of the answer choices cannot occur.
The Fourth Question Family (Hurt Family) consists of only one question type—Cannot Be True. As such, this question family is based on the principle that you must use the information in the stimulus to prove that one of the answer choices cannot occur.
-
consists of one question type:
Cannot Be True
-
The 4th question family is based on the principle that you must use the information in the stimulus to prove that one of the answer choices cannot occur. Compared to the First Question Family, the only difference in the diagram is that the Fourth Family diagram has a bar across the arrow. Again, this bar signifies a negative
-
Rules
- ## You must accept the stimulus information—even if it contains an error of reasoning—and use it to prove that one of the answer choices cannot occur.
- Any information in an answer choice that does not appear either directly in the stimulus or as a combination of items in the stimulus will be incorrect. The correct answer choice will directly disagree with the stimulus or a consequence of the stimulus.
How to approach stimulus of a Main Point Question
- Look for words that indicate Premises (since, because) and Conclusions (therefore, thus, hence, so, it follows that)
-
- Look for words that indicate a Shift in Attitude (but, however)
-
- Look for Evaluative Statements that indicate the author’s attitude
- Why Test
Quantity Indicators
Quantity Indicators
-
all
every
most
many
some
several
few
sole
only
not all
none
Probability Indicators
must
will
always
not always
probably
likely
should
would
not necessarily
could
rarely
never
Introduced by sufficient and necessary words such as: if…then, when, all, every, and only, where both elements are positive or both elements are negative
Introduced by conditional statements where exactly one of the terms is negative, or by statements using words such as “no” and “none” that imply the two variables cannot “go together.”
-
Example Statement: No X’s are Y’s
Relationships involving Some
Row V Column
column is vertical
Columns are..
Vertical.
Think of roman columns
Rows are
Horizontal
Columns are Vertical
There are 4 people and each of them must drive at least once in the course of 6 days.
What is the greatest number of days each person can drive?
One person could drive three times with the other three driving once
3+1+1+1=6
-
The other possiblity is
Two people drive twice and two people drive once
2+2+1+1
Bob has a date with 7 girls on 5 days - Monday through Friday. Every girl gets a date and at least one date is scheduled each day.
What is the distribution
2 possibiltiies
3 on one day and 1 on the remaining 4
3 1 1 1 1 = 7
or
2 on two days and 1 on the remaining three
2 2 1 1 1 = 7
4 boys and 3 girls will be assigned to a row of 5 adjacent lockers.
Each locker must be assigned to either one or two children.
Each shared locker must be assigned to one boy and one girl.
What is the distribution
2 2 1 1 1
One girl will get her own locker
BG, BG, B, B, G
Most Vs The Most
Most - a majority = 50% +
The most can be a plurality = ie more than any other
reciprical?
Some is reciprical = create a line
most is not… its an arrow
A –all–> B
B — some–> C
Cant get anything from this
A –most–> B
B—most–> C
nothing
if it were Alls you could use the transative
A > B
B > C
A > C
All A’s are B’s
Some B’s are C’s
Nothing
Most A’s are B’s
Most A’s are C’s
Some B’s are C’s
Common Error/Flaw Type
1) Uncertain Use of a Term or Concept
2) Ad Hominem - flawed argument attacks the person (or source) instead of the argument they advance
3) Circular Reasoning - the author assumes as true what is supposed to be proved
4) Errors of Conditional Reasoning
5) Mistaken Cause and Effect (temporal relationship, correlation, alternate cause, reversal)
6) Straw Man - when an author attempts to attack an opponent’s position by ignoring the actual statements made by the opposing speaker and instead distorts and refashions the argument, making it weaker in the process
7) Lack of Relevant Evidence for the Conclusion
8) Internal Contradiction
9) Appeal Fallacies
a. Authority
b. Popular Opinion/Appeal to Numbers
c. Emotion
- Survey Errors
- Exceptional Case/Overgeneralization
- Errors of Composition and Division
- False Analogy
- False Dilemma
- Errors in the Use of Evidence
- Time Shift Errors
The Seperation Principle
Applies when variables involved in blocks/not-blocks are placed in a limited number of spaces resulting in the rigid allocation of the variables that can be exploited. EX: Rule: B__B__B and only 5 spaces available…so it would be, B1, B3, B5
Overlap Principle
When members of two separate variable sets are both assigned into a fixed number of spaces, there will be an overlap between the groups IF the sum of the two groups is greater than the total number of spaces. EX: There are three chairs in a classroom. Two of the chairs are green and two boys sit in the chairs. SO–Two variable sets are green chairs and boys…SUM of these two groups is 4, TOTAL number of spaces is 3 chairs…MEANS–at least 1 boy will sit in 1 green chair.
If Jays, Martins, or both are in the forest, then so are Harriers
J -> H
M -> H
G <–II–> H
implied by..
A) If G speaks at the fundraiser then H cannot speak
or
B) If G does not speak at the fundraiser, H speaks
A)
it implies that the two cannot be together
.
A) If G speaks at the fundraiser then H cannot speak
G > =H= contra is H > =G=
-
-
-
-
this is opposed to B
B) If G does not speak at the fundraiser, H speaks
=G= -> H
No G implies H. But we do not know whether G can be with H.
No G is simply sufficient to know H is in
What does this imply
P departs later than R only if Q departs later than R
R>P -> R>P/Q
contra
Q>R -> P/Q>R
When do you use the Double Not Arrow
A <–II–> B
Only when the Neccesarry condition is itself negative
No A’s are B’s = A > ~B
If You are an A you are not a B
If not A then B
situation of One or the Other
be careful not to be tricked by single negation of some necessary conditions.. work it out
Double Not Arrow
The double not arrow is a shorthand method of showing that two elements cannot be in the same group together.
A –> ~B can be expressed A <–|–> B because the original statement and its contrapositive (B –> ~A) rule out the possibility that A and B can be in the same group.
This is not true of ~A –> B. This statement and its cp (~B –> A) indicate only that if one is absent, the other must be there. They do not rule out the possibility that both A and B can be in the same group.
the negated term has to be the neccessary condition to have a double not arrow, and all the double not arrow is, is a simplification of writing the original diagram and its contrapositive
~A <–II–> ~F
which
A) If A is not included in the competition F cannot be included
D) A cannot be included in the competition unless F is also included
E) Either A or F must be included in the competition
E
a double arrow between Not A and Not F tells us that the two conditions - absent A and absent F - cannot simultaneously occur.
Therefore, either A or F, or both, must be present
Either A is painted and B is painted, or A is not painted and B is not painted
J is painted if and only if K is not painted
J (double arrow) ~K
J is painted if and only if K is not painted
J (double arrow) ~K
~F –> H
which
A) F and H cannot be selected together
B) At least one of F or H must be selected
C) If H is seleccted, then F is selected
B)
This conditional relationship provides that at least one of F or H must be selected in any valid template
What is implied by this statement
F –> G~
and
~G –> F
Which?
A) If Fran joins the team, Greg will not join tthe team
B) Fran will not join the team only if Greg does not join
C) Greg will not join the team if Frans joins the team, and Frans will join if Greg does not
C
F –> ~G
and
~G –> F
C) Greg will not join the team if Frans joins the team, and Frans will join if Greg does not
If Q is on the stage, then R is on the stage
R is not on the stage unless P is not on the stage
Q > R
R> ~P
P is negated and Necessary, – Double not arrow
R<–II–>P
Q<–II–>P
Diagram
If A is not interviewed then B must be interviewed, and
There is a group (and an out group)
If R is not sselected, then S must be selected
~R –> S
produces a
~R ~S
-
If R is not selected then S is selected
+
If S is not selected then R is selected
=
Therefore there is no situation in which they are both not selected. Either onee or both are not selected.
M –> O
One of them must be hired since there is only one space in the out group.
The original statement signifies that it is possible for one (M) not to be hired; however, if O isnt hired, M isnt hired, which would violate the game’s setup.
O must always be hired
If you are not T, then you are not V (T and V both negative)’
You can, of course take the contrapositive of this diagram and force both terms to be positive:
Some X’s are Y’s
X some Y
X some Y
Some X’s are Y’s
Some W’s are not Z’s
Some W’s are not Z’s
same as
Not all W’s are Z’s
“Some are not” can be defined as at least one is not, possibly all are
not. Thus, if I say, “Some of my friends are not present,” it could be true that none of my friends are present.
One of the most popular ways to introduce the idea that some are not is to use the phrase not all, which is functionally equivalent to some are not.
When diagramming statements involving some are not, simply place the word some between the two elements and negate the second element:
Not all T’s are V’s
Not all T’s are V’s – same as– Some T’s are not V’s
“Some are not” can be defined as at least one is not, possibly all are
not. Thus, if I say, “Some of my friends are not present,” it could be true that none of my friends are present.
One of the most popular ways to introduce the idea that some are not is to use the phrase not all, which is functionally equivalent to some are not.
When diagramming statements involving some are not, simply place the word some between the two elements and negate the second element:
Not all T’s are V’s – same as– Some T’s are not V’s
“Some are not” can be defined as at least one is not, possibly all are
not. Thus, if I say, “Some of my friends are not present,” it could be true that none of my friends are present.
One of the most popular ways to introduce the idea that some are not is to use the phrase not all, which is functionally equivalent to some are not.
When diagramming statements involving some are not, simply place the word some between the two elements and negate the second element:
Most X’s are Y’s
- most
- a majority
- more than half
- almost all
- usually
- typically
Most W’s are not Z’s
When “most” appears as “most are not,” the interpretation changes due to the not. “Most are not” can be defined as at majority are not, possibly all are not. Thus, if I say, “Most of my friends are not present,” it could be true that none of my friends are present.
When diagramming statements involving most are not, simply place the word most between the two elements, place an arrow under the most pointing at the second element, and negate the second element:
Most are not
When “most” appears as “most are not,” the interpretation changes due to the not. “Most are not” can be defined as at majority are not, possibly all are not. Thus, if I say, “Most of my friends are not present,” it could be true that none of my friends are present.
When diagramming statements involving most are not, simply place the word most between the two elements, place an arrow under the most pointing at the second element, and negate the second element:
Most W’s are not Z’s
Most W’s are not Z’s
When “most” appears as “most are not,” the interpretation changes due to the not. “Most are not” can be defined as at majority are not, possibly all are not. Thus, if I say, “Most of my friends are not present,” it could be true that none of my friends are present.
When diagramming statements involving most are not, simply place the word most between the two elements, place an arrow under the most pointing at the second element, and negate the second element:
Radio airplay restrictions are nationally imposed regulations. The City Club has compiled a guide to all nationally imposed regulations except those related to taxation or to labor law. Radio airplay restrictions are related neither to taxation nor to labor law, so the City Club’s guide covers radio airplay restrictions.
not taxation and not labor -> in guide
not taxation and not labor -> therefore
in guide
Diagram
All too many weaklings are also cowards, and few cowards fail to be fools. Thus there must be at least oneperson who is both a weakling and a fool.
From a flaw question
All too many weaklings are also cowards, and few cowards fail to be fools. Thus there must be at least oneperson who is both a weakling and a fool.
W-some-C
C-some-F
W-some-F
Diagram:
Few cowards fail to be fools
C -some- F
could be most but take the weaker of the two interpretations - be mindful though
could be either
Diagram
and Contra
Either Patrick will win or Miranda will win
P ~ → M
or
M ~ → P
The formulation “Either A or B” essentially means “If not A, then B” and “If not B, then A”
This situation does not prevent both from wining, it just means that at least one must win
Tax payers must either declarer all recieved income or face a penalty
either or turns into “If not A, then B” - ~A → B
~ declare all → face penalty
~ face penalty → declare all
Diagram
No one without sufficient exposure to linear algebra can enroll in this class
~ Suff Exp → ~ Can Enroll
Can Enroll → Suff Exp
Diagram
Suspects shall be presumed innocent until proven guilty
~ Presumed Innocent → Proven Guilty
~ Proven Guilty → Presumed Innocent
Diagram and Contra
You cannot lose if you do not play
USE IF to diagram
If you do not play, you cannot lose
~Play → ~Lose
Lose → Play
diagram
Except for Mary, everyone came to the party
~ Come to the Party → Mary
~ Mary → Come to the party
Diagram
Either Scott or Michael will drive
~ S → M
~ M → S
At least one may drive, unless told otherwise both might be able to. Either or tells us means that one not being present means the other must be
How many ways can you combine all statements
2 Ways
More common is classic linking
S → M
M → P
S → P
Other way is if two all statements share a Sufficient Assumption
similar two two mosts - you have to know they exist
A → B
A → C
B some C
When should you diagram?
- The statements are conditional statements, not just a description of a state of affairs and
- There is at least one term in common between the statements so you might be able to combine them.
( ~ Y → R ) + (R → ~ W)
Inferences?
( ~ Y → R ) + (R → ~ W)
Transitive allows for connection
~ Y→ R → ~ W
~ Y → ~ W
W → Y
9 forms of Reasoning
- Rejecting Alternatives
- Applyiing a General Principle
- Appealing to an Authorority
- Using a Counterexample
- Making an Analogy
- Using a Line of Reasoning to draw an Absurd Conclusion Reductio ad absurdum (not a fallacy - slippery slope is)
- Proposing an Alternate Cause for an Observed Effect
- Underinmining a Premise or Conclusion (generally 2nd perspective)
- Offering New Evidence / Challenging an Assumption (generally second perspective)
Assume P is true. From this assumption, deduce that Q is true. Also deduce that Q is false.
Thus, P implies both Q and not Q (a contradiction, which is necessarily false).
Therefore, P itself must be false.
Called?
Reductio ad absurdum
Wrong answer characteristics for Assumption Questions
Premise Boosters - answers that try to convince us that the premise is true
Conclusion Redundancy - answers that simply restate the conclusion
Opposites - answers that actually hurt the argument
Out of Scope - answers that fall outside the scope of the argument
Diagram
None of the Pencils have Erasers
P → ~E
If N is assigned, then neither P nor S can be assigned
N → ~ P + ~ S
N → ~ P
N → ~ S
P or S (or both) → ~ N
Diagram
Neither W nor X can be in the same boat as Y
Y → ~ W + ~ X
W or X → ~ Y
Lance: If experience teaches us nothing else, it teaches us that every general rule has at least one excpetion.
Frank: What you conclude is itself a general rule. If we assume that it is true, then there is at least one general rule that has no exceptions. Therefore, you must withdraw your conclusion.
Franks argument is an attempt to counter Lance’s conclusion by
Answer is
B) Showing that Lance’s conclusion involves him in a contradiction
This method of reasoning is formally known as
Reductio ad absurdum
Often used as a method of reasoning against a first perspective or speaker
Using a line of reasoning to draw an absurd conclusion
you take a statement which you wish to prove false, and show that if you assume it to be true, then it necessarily leads to a logical contradiction. Or to put it a bit more formally, If statement A is true, then statement B is true, and statement B is false. Therefore, statement A is not true.
A
~B
——
A - B
B + ~B .: ~A
Arguments which use universals such as, “always”, “never”, “everyone”, “nobody”, etc., are prone to being reduced to absurd conclusions. The fallacy is in the argument that could be reduced to absurdity – so in essence, reductio ad absurdum is a technique to expose the fallacy.
Assume P is true. From this assumption, deduce that Q is true. Also deduce that Q is false.
Thus, P implies both Q and not Q (a contradiction, which is necessarily false).
Therefore, P itself must be false.
Since the demands of a possible suffieicient assumpton tend to be great, the correct answers to a sufficient assumption question are often…
strong
Remember on Sufficient Assumption questions, if you have a new term in the conclusion…..
Remember on Sufficient Assumption questions, if you have a new term in the conclusion, it must be in the answer choices.
Differences between Conditional and Causal statements
In Conditional statements, there is no implied temporal relationship: The sufficient assumption can happen before, at the same time, or after the necessary condition
In Causal statements there is is an implied temporal relationship: The cause must happen first and the effect must happen at some point in time after the cause
In Causal statements, the events are related in a direct way
In Conditional statements, the sufficient and necessary conditions are often related directly, but they do not have to be
“Before the war can end, I must eat this ice cream cone.” The sufficient condition does not make the necessary condition happen, it just indicates that it must occur
The the words that introduce the statements are very different
Causal indicators are active, almost powerful words,
whereas most conditional indicators do not possess those traits.
Causality in the Conclusion versus Causality in the Premises
Causal statements can be found in the premise or conclusion of an argument. If
the causal statement is the conclusion, then the reasoning is flawed. If the causal statement is the premise, then the argument may be flawed, but not because of the causal statement
On the LSAT, when a cause and effect statement appears as the conclusion, the conclusion is….
On the LSAT, when a cause and effect statement appears as the conclusion, the conclusion is flawed
ex. (of an argument with a causal conclusion)
Premise: In North America, people drink a lot of milk.
Premise: There is a high frequency of cancer in North America.
Conclusion: Therefore, drinking milk causes cancer.
Post Hoc
A Post Hoc is a fallacy with the following form:
A occurs before B.
Therefore A is the cause of B.
Post hoc ergo propter hoc - “After this, therefore because of this.”
This fallacy is committed when it is concluded that one event causes another simply because the proposed cause occurred before the proposed effect. More formally, the fallacy involves concluding that A causes or caused B because A occurs before B and there is not sufficient evidence to actually warrant such a claim.
When two events occur simultaneously….
Dont assume one caused the other.
While one event could have caused the other, the
two events could be the result of a third event, or the two events could
simply be correlated but one does not cause the other.
When an LSAT speaker concludes that one
occurrence caused another, that speaker also assumes
the stated cause is the only possible cause of the effect and that consequently the stated cause will always produce the effect.
This assumption is incredibly extreme and farreaching,
and often leads to surprising answer choices that would appear
incorrect unless you understand this assumption.
How to Attack a Causal Conclusion
Diagram
If Abe joins, then Bill joins, and vice versa
A → B
B → A
A ⇔ B
Either A and B both join or neither does
If Abe is in the committee then Bill is in the committee.
If Bill is in the committee then Abe also is
A → B
B → A
A ⇔ B
Either A and B both join or neither does
Diagram:
A and B cannot both be selected
A ⇔//⇔ B
~ A → B → C → D
A or D
Role Questions
(argument part)
The stimuli that accompany Method-AP questions tend to be more complex than the average LSAT stimulus. Some problems feature two conclusions (one is the main conclusion, the other is a subsidiary conclusion), and often the stimulus includes two different viewpoints or the use of counterpremises. Thus, the ability to identify argument parts using indicator words is important.
If you do see the main conclusion at the end of a Method-AP problem, be prepared to answer a question about a part of the argument other than the conclusion. The test makers do this because they know students are very good at identifying the conclusion when it appears in the last sentence.
These questions often feature two conclusions—a main conclusion and subsidiary conclusion—where the main conclusion is typically placed in the first or second sentence, and the last sentence contains the subsidiary conclusion. In addition, the subsidiary conclusion is often preceded by a conclusion indicator such as “thus” or “therefore” while the main conclusion is not prefaced by an indicator. This is an intentional stimulus formation designed to trick many students into erroneously believing that the last sentence contains the main point.
Often, the identifiers used before the subsidiary conclusion are dramatic and somewhat misleading, such as “clearly” and “obviously.” In these cases, the conclusion is neither clear nor obvious, and those words are used to lead the reader into thinking that the conclusion should simply be accepted without further analysis.
Common incorrect answers for describe Reasoning questions
- “New” Element Answers
- Half Right, Half Wrong Answers
- Exaggerated Answers
- The Opposite Answer
- The Reverse Answer
Role Questions
*conclusions*
The stimuli that accompany Role questions tend to be more complex than the average LSAT stimulus.
Often feature two conclusions—a main conclusion and subsidiary conclusion—where the main conclusion is typically placed in the first or second sentence, and the last sentence contains the subsidiary conclusion. In addition, the subsidiary conclusion is often preceded by a conclusion indicator such as “thus” or “therefore” while the main conclusion is not prefaced by an indicator.
One trick used by the test makers in Role questions is to create wrong answers that describe parts of the argument other than the part named in the question stem
Synthesis Reading Passage Structure
Synthesis consists of three different points of view. For a large part of it, the passage may sound like a antithesis - an argument - with 2 conflicting POVs. However, synthesis passages are marked by the emergence of a third party who reconciles or resolves the dispute
In synthesis passages, it is very likely the author will advocate the third poinnt of view
Synthesis: Present Author (typically aligned with synthesis)
- Main idea - Look for answer choices that best capture the author’s point
- Purpose - Look for answer choices that best describe the author’s positition relative to the views presented
- Author’s Attitude - Look for answer choices that align with the author’s opinion on any of the theses
- Question anticipation - Be ready for questions about the author’s view regarding the theses (viewpoint, agree/disagree)
Synthesis: Absent Author (RARE)
- Main idea - look for answer choices that best encompass the synthesis
- Purpose - Look for an answer choice that best describes the convergence of the thesis and the antithesis in a neutral manner
- Author Attitude: Look for a neutral answer (objective, impartial)
- Be ready for questions about argument advocacy (viewpoint, disagree/agree)
For Synthesis passages, what to answer for Main Idea Questions
Present Author (typically aligned with synthesis)
- Look for answer choices that best capture the author’s point
Absent Author (Rare)
- Look for answer choices that best capture the synthesis
For Synthesis passages what to answer for Purpose questions
Present Author (typically aligned with synthesis)
- Look for answer choices that best describe the author’s position relative to the views preseted
Absent Author (Rare)
- Look for answer choices that best describe the convergence of the thesis and antithesis in a neutral manner
Use of Examples in Reading Passages
Examples support a general claim. In this way they are like premises
If an example is used extensively: (Extensive) - there will typically be several questions that reference it. Expect a specific reference as well as more general questions referencing the example
If it is a localized example, be ready for a specific reference question
Diagram
If Dixie is not assigned to team B, then Candy is assigned to team B
DB OR CB
~D → B
~B → D
At least one
Common Correct vs Incorrect answers for Disagree questions
Common Correct Answers
- Statements that both speakers have an opinion about
- Disagreement about truth of a premise or conclusion
- Disagreement about type of evidence or the importance of evidence used by speaker
- Disagreement about causes for an observed effect
Common Incorrect Answers
- Statements for which one or both speakers do not have an opinion
- Statements for which the speakers do or could agree
Common Correct vs Incorrect answers for Agree questions
Common Correct Answers
- Statements that both speakers must have an opinion about
- Agreement about the truth of a premise or conclusion
- Agreement that a phenomenon is occuring (even if they disagree about the cause or extent)
Common Incorrect Answers
- Statements for which one or both speakers do not necessarily have an opinion
- Statements for which the speakers do or could agree
How to approach stimulus of Agree/Disagree questions
- Identify the conclusion and premise(s) of each argument
- Do not incorporate any additional knowledge or information
- Paraphrase the point(s) of agreement/contention
When the structure of a reading passage is question and answer, what will the main point be?
The main point of a question and answer passage will always be the answer to the question
further, the primary structure (thesis, antithesis, synthesis) will be determined by how many answers are posed
How to determine the primary structure of a question and answer reading passage?
The primary structure (thesis, antithesis, synthesis) will be determined by how many answers are posed
The main point of a question and answer passage will always be the answer to the question
How to weaken a Cause and Effect relationship
-
Cause Without Effect
- Show that the claimed cause was not followed by the claimed effect in a similar situation
-
Effect Without Cause
- Show that the claimed effect was achieved despite the abscence of the claimed cause in a similar situation
- Alternate Cause
How to strengthen a Cause and Effect relationship
-
Same Cause and Same Effect
- Show that the claimed cause was followed by the claimed effect in a similar situation
-
No Cause No Effect
- Show thaat the abscence of the claimed cause was not followed by the claimed effect in a similar situation
- Eliminate Alternate Causes
What is a Crux question
A question that asks for information that would be valuable in evaluating an argument
I call them Evaluation questions
ex. In evaluating Josh’s argument it would be most helpful to know whether
- The correct answer will be a piece of information that will critically affect the validity of the argument
- Look for causal relationships - answers will either strengthen or weaken this relationship if it is present
- If no causal relationship present, look for an answer whose truth or falsisity will either strengthen the argument - could be the assumption etc
How to approach Sufficient Assumption questions
- Locate the conclusion and the premises that support it
- Anticipate a logical bridge between the two
All sixth graders are insecure. Therefore, Mark must be insecure
SA: Mark is in the sixth grade
Correct answers to Sufficient Assumption questions are often strong and take the form of conditional assumptions
Sufficient Assumption Answers
Correct answers to Sufficient Assumption questions are often strong and take the form of conditional assumptions
They must gaurantee the conclusion will be properly inferred
In comparison to answers for Sufficient Assumption questions, Necessary Assumption answers tend to be…
Weak.
The correct answer does not guarantee that the argument is valid, but that rather, without it, the argument cant be valid.
It is common for necessary assumptions to be weaker than what is required to prove the conclusion
Answer to a primary purpose question
Correct answer choices wll either describe
- the author’s attitutde
- the structure of the passage
If the author is present, look for a description of the passage from the author’s point of view.
If the author is not present, look for a description of the structure of the passage
Answer to main point reading passage questions
Main point … duh
If the author is present, this is their view
What is wrong with this answer to the flaw question
Statistician: Changes in the Sun’s luminosity correlate exceedingly well with with average land temperatures on Earth. Clearly, and contrary to accepted opinion among meteorologists - the Sun’s luminosity essentially controls the temperature on Earth.
Meteorologist: I disagree. Any professional meteorologist will tell you that in a system as complicated as that giving rise to the climate, no significant aspect can be controlled by a single variable
The Reasoning in the Meteorologist’s counterargument is questionable because that argument:
A) rejects a partial explanation, not because it is incorrect, but only because it is not complete
(A) fails to utilize the meteorologist’s evidence. On Flaw questions, it’s really important that the flaw describe a reasoning error committed between the evidence and the conclusion. Nowhere does the author say that the explanation is incomplete.
Correct answer:
E) appeals to the authoritiveness of an opinion without evaluating the merit of a putative counterexample
The meteorologist’s conclusion is that the sun’s luminosity does not essentially control land temperatures on Earth. What does the meteorologist offer in support for this conclusion? Not much! Just that no meteorologist would tell you that this could happen.
So the meteorologist is essentially saying that something is true, because many other people would say that it’s true - this is an appeal argument. I wouldn’t say this is an appeal to an inappropriate authority, since meteorologists are experts in the field of weather patterns - the topic at hand.
I will do either A or B, not both and not none
Exactly one outcome
Diagram
A → ~ B (Not both)
~ A → B (At least one)
Exactly one outcome
Ann will either take a leave of absence from Technocomp or else she will quit her job there
Diagram
Q → ~ LA
~ Q → LA
where (Q=quit) and (LA=Leave of Abscense)
She will either quit or take a leave of absence, exactly one outcome will occur of the two
Diagram
Technocomp will allow Ann to take a leave of absence if it does not find out she has been offered the fellowship, but not otherwise.
Technocomp will allow Ann to take a leave of absence if it does not find out she has been offered the fellowship, but not otherwise.
~ FO → ALA (if doesn’t find out, allow leave)
ALA → ~ FO (Not otherwise - ie. Allowing leave of absence requires them not finding out. Otherwise will not be allowed.)
What two types of Equivocation occur on the LSAT
- Shift in Meaning
- Related but Distinct Concepts
Shift in Meaning - If the meanings of any key wordshift in meaning during the course of an argument
Related but Distinct Concepts - Treating two unlike concepts as though they are the same
Perception vs. Reality Flaw
Arguments that invalidly base conclusions on what a person or group of people say or believe
Innapropriate Authority - When an agument relies on an opinion’s opinion or testimony, its important to establish that the topic is within that person’s area of expertise. Otherwise, the conclusion will not be adequatel supported and the argument is flawed.
Irrelevant Opinions - Arrguments will sometimes rely on the opinions of large groups of people to support conclusions. There are times when this is acceptable. For instance, if a company is running a taste test on consumers. However, if you are attempting to measure the profitability of the same company, the opinions of consumers would be irrelevant.
It is a fallacy to rely on the opinions of people when their opinions are irrelevant to the truth of the matter at hand
16) What is the role played by the claim that the obligation to express gratitude cannot be fuffilled anonymously?
Psychologist: The obligation to express gratitude cannot be fufilled anonymously. However much society may have changed over the centuries, human psychology is still driven primarily by personal interaction. Thus, the important social function of positively reinforcing those behaviors that have beneficial consequences for others can be served only if the benefactor knows the source of the gratitude.
The word “THUS” is used to begin the final sentence. Language such as “thus/therefore/hence/so” always indicates a conclusion, but it doesn’t necessarily indicate the main conclusion.
Main Conclusion and Describe the Role questions are notorious for NOT putting the main conclusion as the last sentence and NOT making it obvious by using a term like “thus/therefore/hence/so”.
(The main task in Main Conclusion and Describe the Role is to find the conclusion, so LSAT needs to make that task trickier than usual)
Does the 1st sentence support the 3rd, or does the 3rd sentence support the 1st? We can use the Therefore Test to see which ordering makes more sense.
You shouldn’t express gratitude anonymously.
THEREFORE
The social function of reinforcing good behavior only works if the benefactor knows the source of gratitude.
or
The social function of reinforcing good behavior only works if the benefactor knows the source of gratitude.
THEREFORE
You shouldn’t express gratitude anonymously.
The second ordering is much more logical. This means that the 3rd sentence is designed to support the 1st sentence.
Our whole argument breaks down like this:
MAIN CONC:
Obligation to express gratitude can’t be satisfied anonymously.
(why?)
SUBSIDIARY/INTERMEDIATE CONC:
[because] The social function of reinforcing good behavior only works if the benefactor knows the source of gratitude.
(why?)
PREMISE:
[because] Human psych is still driven mainly by personal interaction.
Now that we’ve deconstructed the argument, we know the claim the question asks about (the 1st sentence) is our Main Conclusion. So we can scan the answers looking for applicable wording.
The use of the phrase “directly proportional” in Sufficient Assumption / Justify Question answer choices.
The LSAT uses “directly proportional” in wrong answers fairly frequently. Directly proportional means: if one thing goes up 10%, then another thing also goes up by exactly 10%.
Direct proportionality is almost never relevant in an argument
example:
In successful plays, audiences must care what happens to at least some characters. But in Brecht’s plays, it’s hard to discern any (any=all) of the characters’ personalities.
Correct Answer: An audience that cannot readily discern a character’s personality will not take any interest in that character
Trick Answer: The extent to which a play succeeds is directly proportional to the extent which the play’s audiences care about its character
A conclusion isn’t false just because the argument is BLANK
Inadaquete… bad
Suppose I make the following argument: “The moon is made of green cheese, therefore it’s interesting.”
That’s a stupid argument. My evidence is wrong: the moon is not made of green cheese. But does that mean the moon is boring?
Of course not. A conclusion isn’t false just because someone makes a bad argument.
Example:
Critic: An art historian says 15th Century painters were better because they were more planimateric than were 16th Century Painters. But this is wrong. 15th century painters were not more masterful, for being better at planemateric is irrelevant to mastery.
The author showed the planimetric argument is not convincing. So the argument for the conclusion is wrong. But that doesn’t mean the conclusion (15th century painters are better) is wrong.
Rejecting a position on the grounds that an inadequate argument has been made for it IS A FLAW
Rejecting a position on the grounds that an inadequate argument has been made for it….
IS A FLAW
Suppose I make the following argument: “The moon is made of green cheese, therefore it’s interesting.”
That’s a stupid argument. My evidence is wrong: the moon is not made of green cheese. But does that mean the moon is boring?
Of course not. A conclusion isn’t false just because someone makes a bad argument.
Example:
Critic: An art historian says 15th Century painters were better because they were more planimateric than were 16th Century Painters. But this is wrong. 15th century painters were not more masterful, for being better at planemateric is irrelevant to mastery.
The author showed the planimetric argument is not convincing. So the argument for the conclusion is wrong. But that doesn’t mean the conclusion (15th century painters are better) is wrong.
Rejecting a position on the grounds that an inadequate argument has been made for it IS A FLAW
Numerical Distributions
Ten drinks are served to two bar patrons. Each patron is served at least one drink but no more than 7.
7-3
6-4
5-5
Numerical Distributions
Six bones are given to three dogs - a greyhound, a mastiff, and a terrier. Each dog is given at least one bone, and the terrier is given exactly one less bone than the mastiff.
G - M - T
3 - 2 - 1
1 - 3 - 2
Numerical Distributions
Eleven cookies are placed in four jars. Each jar contains at least one cookie but no more than five cookies.
5-4-1-1
5-3-2-1
5-2-2-2
4-4-2-1
4-3-3-1
4-3-2-2
3-3-3-2
Numerical Distributions
Eight tables are assigned to four different servers - A, B, C and D
Each server is assigned to at least one table. Server A is assigned exactly twice the number of tables as Server B.
A - B - C - D
4 - 2 - 1 - 1
2 - 1 - 4 - 1
2 - 1 - 1 - 4
2 - 1 - 3 - 2
2 - 1 - 2 - 3
Numerical Distributions
Twelve Students are assigned to five floors of a dormitory. At least two students are assigned to each floor
4 - 2 - 2 - 2 - 2
3 - 3 - 2 - 2 - 2
Numerical Distributions
Seven animals are placeed into three cages. Each cage contains at most five animals.
5 - 2- 0
5 - 1 - 1
4 - 3 - 0
3 - 3 - 1
3 - 2 - 2
Numerical Distributions
Thirteen toys are given to four children - W, X, Y, and Z. Each child is given at least two toys and Y is given exactly three times as many toys as W.
W X Y Z
2 3 6 2
3 2 6 3
Numerical Distributions
Twenty-one pills are placed into six bottles. At least three pills are placed in each bottle.
6 3 3 3 3 3
5 4 3 3 3 3
4 4 4 3 3 3
Numerical Distributions
Seven Appointments with a doctor are scheduled over four days - Monday, Tuesday, Wednesday, Thursday. At least one appointment must be scheduled on each day, and there is exactly one more appointment scheduled Wednesday than Thursday.
M-T-W-Th
1 - 1 - 3 - 2
3 - 1 - 2 - 1
2 - 2 - 2 - 1
1 - 3 - 2 - 1
Numerical Distributions
Each of seven lawyers is assigned to represent exactly one of four defendents. Each defendent is represented by at least one but no more than three attorneys.
3 2 1 1
2 2 2 1
Numerical Distributions
Each of four horses is treated by exactly one of three veterinarians. No veterenerian treats more than three horses.
Four horses distributed three vets
3 1 0
2 2 0
2 1 1
Numerical Distributions
At a historically presevered home with seven rooms, there is exactly one computer terminal to each room. To avoid circuit overload, at most twice as many terminals can be turned on as turned off at any given time. At least two terminals must be turned on at any given time.
Seven computer terminals distributed to two positions (on/off)
On - Off
2 - 5
3 - 4
4 -
Negate
If Smith gets elected, he will serve only one term as mayor
turns into
If Smith gets elected, he might not serve only one term as mayor
You can negate the necessary condition using wont necessarily or might not
Negate
You cannot enter unless you pay admission
You can enter even if you do not pay admission
Negate
Early to bed and early to rise makes a person healthy, wealthy and wise
Early to bed and early to rise does not necessarily makes a person healthy, wealthy and wise
Diagram
No recording that is not played on the radio is one that record companies believe would be profitable if transferred to CD.
No recording that is not played on the radio is one that record companies believe would be profitable if transferred to CD.
V
All recordings that are not played on the radio, are those the record companies believe will Not be profitable if transferred to CD
V (contrapositive) V
To be considered profitable to transfer to CD, a recording must be played on the radio
Negate
Becoming aware of individuals who have won a major jackpot leads at least some people to incorrectly estimate their chances for winning.
Becoming aware of individuals who have won a major jackpot leads at least some people to incorrectly estimate their chances for winning.
VV
Becoming aware of individuals who have won a major jackpot leads at least No One to incorrectly estimate their chances for winning.
What to look out for when combining a All and Some statement
Just that the Sufficient of the All appears in the Some
REMEMBER - what position in the some does not matter
A -some- B
is the exact same thing as B -some- A
G is assembled later than both H and I if and only if H and I are both assembled later than J
H+I > G ⇔ J > H+I
Contrapositive
H or I > J ⇔ G > H or I
Diagram
If V is more popular than S then neither W nor T is more popular than V
V > S ⇒ V > W + T (and S)
W or T > V ⇒ S > V
Diagram
If neither N nor M is bigger than L, then both G and J are bigger than L
L - N+M ⇒ G+J - L - N+M
L - G or J (L before G or J) ⇒ N or M - L (No or M before L)
Either the K exhibit closes earlier than the M exhibit, or the M exhibit closes earlier than the L exhibit, but not both
~ (K - M) → (M - L)
(K - M) → ~ (M - L)
K-M or M-L (Not Both)
M is written earlier than each L and P, or else M is written later than each of L and P
~ (M - L+P) ⇒ (L+P - M)
