Conditional Logic - Basics Flashcards
Develop deep understanding of the fundamental of conditional logic and diagramming.
DEFINE:
Conditional relationship
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
DEFINE:
Sufficient condition
The condition that, if true, GUARANTEES the truth of the other condition
DEFINE:
Necessary condition
The condition that is REQUIRED in order for the other condition to be true
FILL IN THE BLANKS:
___________ condition —> ____________ condition
SUFFICIENT condition —> NECESSARY condition
VISUALIZE:
Venn diagram form of “If A, then B”
A is the SUFFICIENT condition and B is the NECESSARY condition.
The SUFFICIENT condition circle falls entirely within the NECESSARY condition circle.
DEFINE:
Valid inference / Valid conclusion
Something that MUST BE TRUE based on the truth of another statement.
DEFINE:
Invalid inference / Invalid conclusion
Something that does NOT have to be true based on the truth of another statement.
DEFINE:
Contrapositive
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
How do we take the CONTRAPOSITIVE of a conditional statement?
SWITCH both sides of the arrow AND NEGATE each term
A -> B
Contrapositive: /B -> /A
DEFINE:
Converse fallacy
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.
DEFINE:
Inverse fallacy
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
TRUE OR FALSE:
We should diagram most conditional statements that we read in the Logical Reasoning section.
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.
DIAGRAM:
X if Y.
Y -> X
/X -> /Y
IF introduces a SUFFICIENT condition.
DIAGRAM:
L only if S.
L -> S
/S -> /L
ONLY IF introduces a NECESSARY condition.
DIAGRAM:
Only J are F.
F -> J
/J -> /F
ONLY introduces a NECESSARY condition.
DIAGRAM:
The only A are Q.
A -> Q
/Q -> /A
THE ONLY introduces a SUFFICIENT condition.
DIAGRAM:
R if and only if L.
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.
DIAGRAM:
Y unless R.
/R -> Y
/Y -> R
UNLESS introduces SUFFICIENT, BUT NEGATED.
Alternate method: UNLESS introduces NECESSARY, but the NEGATE OTHER CONCEPT.
DIAGRAM:
No Z is W.
Z -> /W
W -> /Z
NO introduces SUFFICIENT, but NEGATE OTHER CONCEPT.
Alternate method: NO introduces NECESSARY, BUT NEGATED.
DIAGRAM:
T is essential for F.
F -> T
/T -> /F
Something that is ESSENTIAL is NECESSARY.
DIAGRAM:
Y is a precondition for E.
E -> Y
/Y -> /E
A PRECONDITION is NECESSARY.
DIAGRAM:
Every K is an N.
K -> N
/N -> /K
EVERY introduces a SUFFICIENT condition.
DIAGRAM:
Only if there is W can there be Q.
Q —> W
/W —> /Q
ONLY IF introduces the NECESSARY condition.
DIAGRAM:
V is the only way to A.
A -> V
/V -> /A
THE ONLY introduces the SUFFICIENT condition.
DIAGRAM:
T cannot occur without J.
/J —> /T
T —> J
WITHOUT is like UNLESS: introduces SUFFICIENT, BUT NEGATED.
Alternate method: WITHOUT introduces NECESSARY, but NEGATE THE OTHER CONCEPT.
TRUE OR FALSE:
A conditional statement and its contrapositive are logically equivalent to each other.
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.)
DIAGRAM:
G requires H.
G —> H
/H —> /G
REQUIRES introduces a NECESSARY condition.
(Note that “G requires H” = “H is required for G.”)
DIAGRAM:
T, except L.
/L —> T
/T —> L
EXCEPT is like UNLESS. It introduces SUFFICIENT, BUT NEGATED.
Alternate method: EXCEPT introduces NECESSARY, but NEGATE THE OTHER CONCEPT.
DIAGRAM:
Y are B.
Y —> B
/B —> /Y
A basic sentence can be diagrammed with the SUBJECT as the SUFFICIENT condition and the PREDICATE as the NECESSARY condition.
DEFINE:
Subject (grammar)
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.
DEFINE:
Predicate (grammar)
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.
DIAGRAM:
Tim will go to the party only if his friends go.
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.)
DIAGRAM:
The plague will not stop unless we sacrifice 20 goats.
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.
DIAGRAM:
Colonizing the moon is the only way to prevent overpopulation of the Earth.
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.
DIAGRAM:
None of the students who were late were qualified for extra credit.
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.
DIAGRAM:
Only candidates who pander to the masses can win an election.
If win election —> Pander to masses
If NOT pander to masses —> NOT win election
ONLY introduces a NECESSARY condition.
DIAGRAM:
Technological development will not slow until society changes its values.
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.
DIAGRAM:
I cry when I am sad.
Sad —> Cry
NOT Cry —> NOT Sad
WHEN introduces the SUFFICIENT condition.
DIAGRAM:
Mastering conditional logic is essential to a high LSAT score.
If High LSAT score —> Master conditional logic
If NOT master conditional logic —> NOT high LSAT score
Something that is ESSENTIAL is the NECESSARY condition.
DIAGRAM:
Without adding more butter, the cake will not have a strong taste.
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.
DIAGRAM:
The illness affected all those who ate the raw fish.
If ate raw fish —> Illness
If NOT illness —> did NOT eat raw fish
ALL introduces a SUFFICIENT CONDITION.
DIAGRAM:
To develop a healthy immune system, one must take vitamin supplements daily.
If healthy immune system —> Takes vitamin supplements daily
If NOT takes vitamin supplements daily —> NOT healthy immune system
MUST introduces a NECESSARY condition.
DIAGRAM:
Ellen will appear in the show if and only if the lead actress gets injured.
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.)
VALID ARGUMENT?
Our profits will increase only if we raise our prices. We just raised our prices. Therefore, our profits will increase.
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.
VALID ARGUMENT?
Raymond will not enjoy steak unless it is well-done. This steak is well-done. So, Raymond will enjoy it.
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.)
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.
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.
