Conditional reasoning Flashcards
Formal Logic indicators
all
none
some
most
How do you represent this in a diagram?
All workaholics are happy
W —–> H
How do you represent this in a diagram?
All workaholics are unhappy
W ——> H/ (H is negated)
use those same letters to represent the group throughout your diagram and inferences.
For example,
if you represent “happy” with “H” as you
begin your diagram, and later you are presented with a seemingly new element, “unhappy,”** do not create a new variable, “UH**.” Instead, simple negate “happy” and use “H” .
When do you use this single arrow?
——->
Introduced by sufficient and necessary words such as:
if…then
when
all
every
only,
where both elements are positive or both elements are negative.
How do you represent this in a diagram?
All X’s are Y’s
(X and Y both positive)
Diagram: X—–> Y
How do you represent this in a diagram?
If you are not T, then you are not V
(T and V both negative)
Diagram: T/ ——-> V/
Contrapositive of
O/ ——-> P/
is?
P ——-> O
When do you use this double arrow?
<———–>
When conditions are Introduced by;
- if and only if
- vice versa
- repeating and reversing the terms (as in “If A attends then B attends, and if B attends then A attends”).
- or by any other situations where the author implies that the arrow goes “both ways .
Double-arrow statements allow for only two possible outcomes:
1. both occur
or
2. neither occur
How do you represent this in a diagram?
X if and only if Y
X <——–> Y
How do you represent this in a diagram?
All W’s are Z’s, and all Z’s are W’s
W <——> Z
When do you use this double-not arrow?
<——-/——–>
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.”
How do you represent this in a diagram?
No X’s are Y’s
X <——/—–> Y
How do you represent this in a diagram?
If you are a T, then you are not a V
T ———> V/
or
T <—–/—-> V
Some
Can also be defined as?
At least one
Possibly all
at least some
a few
a number
several
part of
a portion
How do you represent this in a diagram?
Some X’s are Y’s
X some Y
How do you represent this in a diagram?
“Some X’s are not Y’s”
X some Y/
“Some are not”
Can also be defined as?
At least one is not…
Not all …
Possibly all are not…
..
Most
Can also be defined as?
Majority
Possibly all
Usually
Typically
More than half
almost all
How do you represent this in a diagram?
Most X’s are Y’s
X most Y
—–>
Most are not
Can also be defined as?
Majority are not..
Possibly all are not..
more than half are not
almost all are not
usually not
typically not
T/F?
“Some” & “Most” statements have contrapositive
False
Only the arrow statements like “all” have contrapositives; some and most do not because they do not necessarily encompass an entire group.
How do you represent this in a diagram?
Not all of the Smallville roads are safe.
SR some S/
How do you represent this in a diagram?
Most W’s are not Z’s
W most Z/
—–>
What is the numerical estimate?
All
100%
What is the numerical estimate?
None
0%
What is the numerical estimate?
Some
1-100 (at least one)
What is the numerical estimate?
Some are not
0-99 (Not all)
What is the numerical estimate?
Most
> 50% (majority)
What is the numerical estimate?
Most are not
< 50% (minority)
Reversibility
means?
means that the relationship between the two variables has exactly the same meaning regardless of which “side” of the relationship is the starting point of your analysis.
T/F ?
A —–> B = B ——> A
False
no single arrow is reversible. but we can infer that some B’s are A’s
T/F ?
A some B = B some A
True
List the reversible relationships
Double arrow < ——>
Double-not (None) <—-/—–>
some (some)
List the non-reversible relationships
all ——>
most
—–>
T/F ?
A some B/ = B some A/
False
we know that some A’s are not B’s.
Correctly reversed, the relationship reads, “Some things that are not B are A’s.”
B/ some A ✅
Key: “Some” do not have contrapositive, but they are reversible
Thus, you can reverse a some are not statement, but you must be careful when doing so in order to avoid
accidentally moving the “not.”
Reverse this statement
G some H/
H/ some G
Thus, you can reverse a some are not statement, but you must be careful when doing so in order to avoid
accidentally moving the “not.”
Reverse this statement
C most D
—–>
This statement is irreversible
T/F ?
A —–> B <—-/—–> C =
A <—-/—–> C
True
This is an additive inference, so-called because it comes from “adding” two statements together to make the inference.
T/F ?
A ——–> B ,
It must be true that….
* Most A’s are B’s
* Some A’s are B’s
* Some B’s are A’s
True
This is an inherent inference: inferences that are known to be true simply from the relationship between the two variables.
Outline the inherent relationship between
Most, Some, and All
Logic ladder
All —> Most —-> Some
Deals only with positive terms
if you have an all relationship, you automatically know that the most and some relationships for that same statement are true.
So, if a statement is made that “All waiters
like wine,” then you immediately know that “Most waiters like wine,” and “Some waiters like wine.”
If “Most waiters like wine,” then you
automatically know that “Some waiters like wine.”
T/F ?
if most people like wine , we can infer that all people like whine
False
But, because “most” is below “all” on the Logic Ladder, you do not know with certainty that “All waiters like wine” (it is possibly true, but not known for certain).
This reveals a truth about the Logic Ladder: the upper rungs automatically imply the lower rungs, but the
lower rungs do not automatically imply the upper rungs. In other words, as you go down the rungs the
lower relationships must be true, but as you go up the rungs the higher relationships might be true but are not certain.
T/F ?
“All doctors are lawyers”
we can infer that some lawyers are doctors
True
from the Logic Ladder we know that “Some
doctors are lawyers.” And, because some is a reversible term, we then know that “Some lawyers are doctors.”
The presence of inherent inferences in non-reversible terms such as all and most helps to make
complex inferences easier to follow.
What 3 inferences can you draw?
A < ———> P
- Most of A is P
- Most of P is A
- A some P
This is a very tricky problem. Because the arrow between AFR and RP goes in both directions,
we can infer that the most relationships are inherent in both directions.
Arrange the logic between..
Some are not, none, most are not
None —> Most are not —–> Some are not
This is the logic ladder for negative terms
So, if a statement is made that “None of the waiters like wine,” then you immediately know that “Most waiters do not like wine,” and “Some waiters do not like wine.”
just as in the positive Logic Ladder, in the negative
Logic Ladder the upper rungs automatically imply the lower rungs, but the lower rungs do not
automatically imply the upper rungs.
1 rules of formal logic
Always combine common terms
A some B, B —-> C =
A some B —–> C
T/F ?
Sufficient is not = necessary
True
if a sufficient condition occurs,
you automatically know that the necessary condition also occurs.
If a necessary condition occurs, then it is possible but not certain that the sufficient
condition will occur.
3 logical features of conditional statements
- the sufficient condition does not actively cause the necessary condition to happen. Instead, the occurrence of the sufficient condition is a sign or indicator that the necessary condition will occur, is occurring,
or has already occurred. In - Temporally speaking, either condition can occur first, or the two conditions can occur at the same time.
- The conditional relationship stated by the author does not have to reflect reality. Your job is not to figure out what sounds reasonable, but rather to perfectly capture the meaning of the author’s sentence.
Which of the following options are valid or invalid?
If someone gets an A+ on a test, then they must have studied for the
test.
- John received an A+ on the test, so he must have studied for the test.
- John studied for the test, so he must have received an A+ on the test.
- John did not receive an A+ on the test, so he must not have studied on the test.
- John did not study for the test, so he must not have received an A+ on the test.
Stem: A —> S
- A ——> S ✅ - repeat inference
- S ——> A ❌ - Mistaken reversal
- A/ —–>S/ ❌ - Mistaken negation
- S/ —–>A/ ✅ - Contrapositive of stem
Remember A —->S is not reversible
But for A —-S > we can derive a contrapositive, which states that when the necessary condition fails to
occur, then the sufficient condition cannot occur.
To get the contrapositive of a statement …
simply reverse and negate the two terms.
Contrapositive of
A —–> B
B/ ——> A/
T/F ?
C —–> D/ =
D —-> C/
True
the second is a contrapositive of the first, which states that because D occured, C did not occur.
Sufficient indicators
- If
- When
- Whenever
- Every
- All
- Any
- People who …
- In order to …
- To get …
- No
- None
Necessary indicators
- Then
- Only
- Only if
- Must
- Required
- Unless
- Except
- Until
- Without
Which is the necessary and the Sufficient condition?
Unless a person studies, he or she will not receive an A+.
Unless Equation is applied to the diagram:
- Whatever term is modified by “unless,” “except,” “until,” or “without”
becomes the necessary condition. - The remaining term is negated and becomes the sufficient condition.
Since “unless” modifies “a person studies,” “Study” becomes the necessary
condition. The remainder, “he or she will not receive an A+,” is negated by
dropping the “not” and becomes “he or she will receive an A+.” Thus, the sufficient condition is “A+,” and the diagram is as follows:
A+ ——-> S
Which is the sufficeint & necerssary?
The park closes when the sun goes down.
The park closes if the sun goes down.
S —–> PC
The term “when” typically introduces a sufficient condition, and so “sun goes down” is the sufficient condition.
where a word such as “if” or “when” appears in the second half of a sentence it introduces the sufficient condition — the first part of the stimulus is the necessary condition.
The fastest approach is a mechanistic one:
observe the form of the problem by identifying the indicators (like “when”), then diagram the
problem using those as your guides.
Only resort to weighing the merits of each condition if no indicators are present to assist you.
In this way you can avoid over-thinking a problem and introducing errors into the system.
Which is the sufficeint & necerssary?
The only way to achieve success is to work hard.
This is another tricky problem.
As usual, “only” introduces the necessary condition, but the test
makers use a deceptive device: in this sentence “only” modifies “way,” and the “only way” refers to working hard.
Thus, “work hard” is in fact the necessary condition.
This type of construction appears with enough frequency that you should be familiar with it
Which is the sufficeint & necerssary?
No citizen can be denied the right to vote.
C —-> DV/
You should convert to am if /then statement
When you encounter a stimulus that contains conditional reasoning and
a Must Be True question stem. …
look for a contrapositive or a
repeat form in the answer choices.
This may be the correct answer
avoid mistaken reversals or mistaken negation, which are contrapositives of each other.
Commom incorrect answer traps in conditional questions
- Mistaken reversal
- Mistaken negation
- Exagerated (out of scope)
4.
When considering the likelihood that one of two
possible answers will appear, you can expect that
the test makers will typically choose the answer that
requires the most work to reach the correct answer.
Either / OR
in the test means?
“at least one of the two,
possibly both.”
In its everyday use outside of the LSAT, “either/
or” has come to mean “one or the other, but not both,” but this usage is incorrect on the LSAT.
Draw the logic diagram for this
Either John or Jack will attend the party.
John——–> Jack/
or
Jack ———> John/
Note: second diagram is the contrapositive of the first diagram
The diagrams reflect the fact that if one of the two fails to attend, then the other must attend in order to satisfy the “at least one of the two” condition imposed by the “either/or” term.
Note that neither of these two diagrams preclude both John and Jack from attending the party. Meaning it is possible that both also attend.
Either Cindy or Clarice will attend the party, but not both.
Cindy ——-> Clarice/
Clarice —— > Cindy/
Note: both cannot occur together here because the passage clarified this
Contrapositive T/F ?
To graduate from Throckmorton College you must be both smart and
resourceful.
Graduate ——-> Smart and Resourceful =
Not Smart and not resourceful –> Not graduate
False
Not smart or Not resourceful ——-> Not graduate
In this case, if either one of the two necessary conditions is not met, then you cannot graduate
Whenever you take a contrapositive of a statement
with multiple terms in the sufficient or necessary
condition, “and” turns into “or,” and “or” turns into “and.”
Graduate ——–> Smart or
Resourceful
Contrapositive is?
Not smart and not resourcefull ———> not graduate
note that as the contrapositive occurred the “or” joining the original necessary conditions changed to “and.”
Rich and Famous —–> Happy
Contrapositive is?
Not happy —–> not rich or not famous
Convert to 3 diagrams ;
Ann will attend if and only if Basil attends.
- “A if B”
B ——> A - “A only if B”
A ——–> B - A <——–> B
A <———> B
how many scenarious can be inferred here and what are they?
- A and B both occur
- Neither A nor B occur
A/ <——-> B/
Remember: double arrow is reversible
In double arrow: Any scenario where one of the two attends occurs but the other does not is impossible.
Draw the diagram and its contrapositive of:
Either Jones or Kim will win the election.
JW ———> KW/
KW ———-> JW/
Depending on the type of election, there may or may not be only one “winner” of the election.
Look for CLUES in the passage to determine if there is a posibility of both occuring
What should you note about “until”
“Until” is the time condition of “unless.”
Because “until” indicates a time condition (before/after), there cannot be a simultaneous occurrence between the necessary and sufficient condition.
T/F?
if A → B and C = B → C.
False
if A → B and C, that doesn’t mean that if B → C.
what can/cannot be inferred from this?
Either A & B or C will win. However both cannot win together.
- A &B → C/
- C → A/ or B/
Because both cannot occur together, we use the ←/→ that indicates that 2 items cannot occur together i.e. when one is +ve(occurs), the other is -ve (cannot occur).
We also know the ←/→ is reversible
Therefore;
* A&B ←/→ C/
* C ←/→ A/ or B/
Inferrences;
* if A and B occured, then C must not occur
* if C did not occur, then A and B must occur
* if C occured, then A/ occured or B/ occured. ie atleast one out of the 2 occured, possibly both.
* If A/ occured, C occured
* If B/ occured, C occured
Which of the following inferences is valid /invalid and why?
A&B ←/→ C/
From above we can inferr that;
A. if C happened, then A did not happen
B. If C happened, then B did not happen
C. If A happened, then C happened
D. If B happened, then C happened
E. If A did not happen, then C happened
F. If B did not happen, then C happened
G. If C happened, then both did not happen
H. If C happened, then Either A or B did not happen, possibly both did not happen
Firstly contrapositive - C ←/→ A/ or B/
A. if C happened, then A did not happen ❌
Reason: The necessary condition is not dependednt only on absence of A alone.
It is possible that B was absent instead while A was present. The correct answer should have been “either A was absent or B was Absent. In other words, atleast one of the two (A or B) was absent”
B. If C happened, then B did not happen ❌
Reason: same as A
C. If A happened, then C happened
❌
Reason: Mistaken negation
D. If B happened, then C happened
❌
Reason: Mistaken negation
E. If A did not happen, then C happened
✅
Reason: we need at least one not to happen as a necessary condition for C to occur
F. If B did not happen, then C happened
✅
Reason: we need at least one not to happen as a necessary condition for C to occur
G. If C happened, then both did not happen ❌
Reason: either or means at least one, possibly both
H. If C happened, then Either A or B did not happen, possibly both did not happen
✅
Reason- A paraphrase of the exact contrapositive.
