Conditional Logic - Basics

Question 1

DEFINE:

Conditional relationship

Answer 1

A relationship between two conditions where the truth of one is SUFFICIENT for the truth of the other

OR

A relationship between two conditions where the truth of one is NECESSARY for the truth of the other

Question 2

DEFINE:

Sufficient condition

Answer 2

The condition that, if true, GUARANTEES the truth of the other condition

Question 3

DEFINE:

Necessary condition

Answer 3

The condition that is REQUIRED in order for the other condition to be true

Question 4

FILL IN THE BLANKS:

___________ condition —> ____________ condition

Answer 4

SUFFICIENT condition —> NECESSARY condition

Question 5

VISUALIZE:

Venn diagram form of “If A, then B”

Answer 5

A is the SUFFICIENT condition and B is the NECESSARY condition.

The SUFFICIENT condition circle falls entirely within the NECESSARY condition circle.

Question 6

DEFINE:

Valid inference / Valid conclusion

Answer 6

Something that MUST BE TRUE based on the truth of another statement.

Question 7

DEFINE:

Invalid inference / Invalid conclusion

Answer 7

Something that does NOT have to be true based on the truth of another statement.

Question 8

DEFINE:

Contrapositive

Answer 8

Valid inference that because the necessary condition is false, the sufficient condition must be false.

Examples:

Premise: A -> B

Conclusion: /B -> /A

Premise: A -> B

Premise: /B

Conclusion: /A

Question 9

How do we take the CONTRAPOSITIVE of a conditional statement?

Answer 9

SWITCH both sides of the arrow AND NEGATE each term

A -> B

Contrapositive: /B -> /A

Question 10

DEFINE:

Converse fallacy

Answer 10

Invalidly inferring that because the necessary condition is true, the sufficient condition must be true.

Examples:

Premise: A -> B

Conclusion: B -> A

Premise: A -> B

Premise: B is true.

Conclusion: A is true.

Question 11

DEFINE:

Inverse fallacy

Answer 11

Invalidly inferring that because the sufficient condition is false, the necessary condition must be false.

Examples:

Premise: A -> B

Conclusion: /A -> /B

Premise: A -> B

Premise: /A

Conclusion: /B

Question 12

TRUE OR FALSE:

We should diagram most conditional statements that we read in the Logical Reasoning section.

Answer 12

FALSE.

Diagramming conditionals is the most useful when the problem is about CONNECTING multiple conditional statements OR when there is difficult conditional language/structure.

Read more of the stimulus first before deciding whether and what to diagram.

Question 13

DIAGRAM:

X if Y.

Answer 13

Y -> X

/X -> /Y

IF introduces a SUFFICIENT condition.

Question 14

DIAGRAM:

L only if S.

Answer 14

L -> S

/S -> /L

ONLY IF introduces a NECESSARY condition.

Question 15

DIAGRAM:

Only J are F.

Answer 15

F -> J

/J -> /F

ONLY introduces a NECESSARY condition.

Question 16

DIAGRAM:

The only A are Q.

Answer 16

A -> Q

/Q -> /A

THE ONLY introduces a SUFFICIENT condition.

Question 17

DIAGRAM:

R if and only if L.

Answer 17

R L

/L /R

IF AND ONLY IF creates a BICONDITIONAL relationship. Which side the concepts are on doesn’t matter, since BOTH concepts are SUFFICIENT and NECESSARY.

Question 18

DIAGRAM:

Y unless R.

Answer 18

/R -> Y

/Y -> R

UNLESS introduces SUFFICIENT, BUT NEGATED.

Alternate method: UNLESS introduces NECESSARY, but the NEGATE OTHER CONCEPT.

Question 19

DIAGRAM:

No Z is W.

Answer 19

Z -> /W

W -> /Z

NO introduces SUFFICIENT, but NEGATE OTHER CONCEPT.

Alternate method: NO introduces NECESSARY, BUT NEGATED.

Question 20

DIAGRAM:

T is essential for F.

Answer 20

F -> T

/T -> /F

Something that is ESSENTIAL is NECESSARY.

Question 21

DIAGRAM:

Y is a precondition for E.

Answer 21

E -> Y

/Y -> /E

A PRECONDITION is NECESSARY.

Question 22

DIAGRAM:

Every K is an N.

Answer 22

K -> N

/N -> /K

EVERY introduces a SUFFICIENT condition.

Question 23

DIAGRAM:

Only if there is W can there be Q.

Answer 23

Q —> W

/W —> /Q

ONLY IF introduces the NECESSARY condition.

Question 24

DIAGRAM:

V is the only way to A.

Answer 24

A -> V

/V -> /A

THE ONLY introduces the SUFFICIENT condition.

Question 25

DIAGRAM: T cannot occur without J.

Answer 25

/J —\> /T T —\> J **WITHOUT** is like **UNLESS:** introduces **SUFFICIENT, BUT NEGATED**. Alternate method: WITHOUT introduces NECESSARY, but NEGATE THE OTHER CONCEPT.

Question 26

TRUE OR FALSE: A conditional statement and its contrapositive are logically equivalent to each other.

Answer 26

TRUE. The contrapositive is just another way to express the exact same conditional relationship. (So if you diagram the contrapositive of a statement when an explanation diagrammed it the other way, that's OK.)

Question 27

DIAGRAM: G requires H.

Answer 27

G —\> H /H —\> /G **REQUIRES** introduces a **NECESSARY** condition. (Note that "G requires H" = "H is required for G.")

Question 28

DIAGRAM: T, except L.

Answer 28

/L —\> T /T —\> L **EXCEPT** is like **UNLESS**. It introduces **SUFFICIENT, BUT NEGATED**. Alternate method: EXCEPT introduces NECESSARY, but NEGATE THE OTHER CONCEPT.

Question 29

DIAGRAM: Y are B.

Answer 29

Y —\> B /B —\> /Y A basic sentence can be diagrammed with the **SUBJECT** as the **SUFFICIENT** condition and the **PREDICATE** as the **NECESSARY** condition.

Question 30

DEFINE: Subject (grammar)

Answer 30

The noun or noun phrase that is doing the action expressed by the verb. Example: "The [hard-working students who mastered these flashcards] felt their brains get bigger." The bracketed phrase is the SUBJECT which is doing the action: feeling their brains get bigger.

Question 31

DEFINE: Predicate (grammar)

Answer 31

The part of a statement that describes a property of the subject or that characterizes the subject. Example: "The hard-working students who mastered these flashcards [felt their brains get bigger.]" The bracketed phrase is the PREDICATE: it is something that is true about the subject of the sentence.

Question 32

DIAGRAM: Tim will go to the party only if his friends go.

Answer 32

Tim goes to party —\> His friends go to the party His friends do NOT go to the party —\> Tim does NOT go to the party. **ONLY IF** introduces the **NECESSARY** condition (the "then" part of an If-then statement.)

Question 33

DIAGRAM: The plague will not stop unless we sacrifice 20 goats.

Answer 33

If we do NOT sacrifice 20 goats —\> Plague will NOT stop Plagus stops —\> Sacrifice 20 goats **UNLESS** introduces the **SUFFICIENT, BUT NEGATED**. *Alternate method*: UNLESS introduces NECESSARY but NEGATE THE OTHER CONCEPT.

Question 34

DIAGRAM: Colonizing the moon is the only way to prevent overpopulation of the Earth.

Answer 34

Prevent overpopulation of Earth —\> Colonize the moon If we do NOT colonize the moon —\> CANNOT prevent overpopulation of Earth **THE ONLY** introduces the **SUFFICIENT** condition.

Question 35

DIAGRAM: None of the students who were late were qualified for extra credit.

Answer 35

If student was late —\> NOT qualified for extra credit If qualified for extra credit —\> NOT student who was late **NONE** is like **NO.** It intoduces the **SUFFICIENT, BUT NEGATE OTHER CONCEPT.** *Alternate method*: NONE introduces NECESSARY BUT NEGATED.

Question 36

DIAGRAM: Only candidates who pander to the masses can win an election.

Answer 36

If win election —\> Pander to masses If NOT pander to masses —\> NOT win election **ONLY** introduces a NECESSARY condition.

Question 37

DIAGRAM: Technological development will not slow until society changes its values.

Answer 37

If society does NOT change its value —\> Tech. dev. will NOT slow down If tech dev. slows down —\> Society changed its values **UNTIL** is like **UNLESS**. It introduces **SUFFICIENT, BUT NEGATED**. *Alternate method*: Until introduces NECESSARY, but NEGATE THE OTHER CONCEPT.

Question 38

DIAGRAM: I cry when I am sad.

Answer 38

Sad —\> Cry NOT Cry —\> NOT Sad **WHEN** introduces the **SUFFICIENT** condition.

Question 39

DIAGRAM: Mastering conditional logic is essential to a high LSAT score.

Answer 39

If High LSAT score —\> Master conditional logic If NOT master conditional logic —\> NOT high LSAT score Something that is **ESSENTIAL** is the **NECESSARY** condition.

Question 40

DIAGRAM: Without adding more butter, the cake will not have a strong taste.

Answer 40

If NOT add more butter —\> Cake will NOT have strong taste If cake has strong taste —\> Added more butter **WITHOUT** is like **UNLESS**. It introduces the **SUFFICIENT, BUT NEGATED**. *Alternate method*: WITHOUT introduces NECESSARY, but NEGATE THE OTHER CONCEPT.

Question 41

DIAGRAM: The illness affected all those who ate the raw fish.

Answer 41

If ate raw fish —\> Illness If NOT illness —\> did NOT eat raw fish **ALL** introduces a **SUFFICIENT CONDITION**.

Question 42

DIAGRAM: To develop a healthy immune system, one must take vitamin supplements daily.

Answer 42

If healthy immune system —\> Takes vitamin supplements daily If NOT takes vitamin supplements daily —\> NOT healthy immune system **MUST** introduces a **NECESSARY** condition.

Question 43

DIAGRAM: Ellen will appear in the show if and only if the lead actress gets injured.

Answer 43

Ellen appears in show \<—\> Lead actress injured Ellen is NOT in show \<—\> Lead actress NOT injured **IF AND ONLY IF** introduces a **BICONDITIONAL**. **BOTH** concepts are **SUFFICIENT** and **NECESSARY**. (You don't have to switch the sides for the contrapositive of a biconditional, because the arrow goes both ways.)

Question 44

VALID ARGUMENT? Our profits will increase only if we raise our prices. We just raised our prices. Therefore, our profits will increase.

Answer 44

INVALID **ONLY IF** introduces the **NECESSARY** condition. Raising prices is NECESSARY in order to increase profits. But it doesn't guarantee that profits will increase. Maybe we raise our prices but there's a global pandemic that strikes and hurts our business.

Question 45

VALID ARGUMENT? Raymond will not enjoy steak unless it is well-done. This steak is well-done. So, Raymond will enjoy it.

Answer 45

INVALID The first premise: If steak is NOT well-done —\> Raymond will NOT enjoy it. But even if it IS well-done, he still might not enjoy it. (What if it's too small or spoiled or has bad seasoning, etc.?) (You can also view the first premise like this: If Raymond enjoys steak —\> it IS well-done.)

Question 46

VALID ARGUMENT? Daily flossing is required in order to maintain healthy gums. The feral child raised by wolves does not floss daily. Hence, he must not have healthy gums.

Answer 46

VALID — CONTRAPOSITIVE The first premise tells us that daily flossing is NECESSARY in order to maintain healthy gums. So if the child doesn't do what's necessary, he won't have healthy gums.