Chapter 6: Conditional Reasoning

Question 1

Conditional reasoning

Answer 1

• A broad name given to logical relationships composed of sufficient and necessary conditions
  -Consists of at least one sufficient condition and at least one necessary condition
  -Brought up using if…then construction, and conditional statements can always be reduced to an if…then form

Question 2

Sufficient condition

Answer 2

-an event or circumstance whose occurrence indicates that a necessary condition must also occur
-If a sufficient condition occurs, you automatically know that the necessary condition also occurs

Question 3

Necessary condition

Answer 3

-An event or circumstance whose occurrence is required in order for a sufficient condition to occur
-If a necessary condition occurs, then it is possible but not certain that the sufficient condition will occur

Question 4

Name the sufficient and necessary conditions in: If someone gets an A+ on a test, then they must have studied for the test

Answer 4

Sufficient condition: Get an A+ (Getting an A+ automatically indicates that someone must have studied)

Necessary condition: Must have studied

Question 5

1st logical feature of conditional reasoning:

Answer 5

1. The sufficient condition does not always make the necessary condition occur
   -In a conditional statement the occurrence of the sufficient condition is a sign or indicator that the necessary condition will occur, is occurring, or has already occurred.

Question 6

2nd logical feature of conditional reasoning:

Answer 6

2. Either condition can occur first, or the two conditions can occur at the same time

Question 7

3rd logical feature of conditional reasoning:

Answer 7

3. 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 sentences

Question 8

Using the initial statement: If someone gets an A+ on a test, then they must have studied for the test

Is this statement valid: John received an A+ on the test, so he must have studied for the test

Answer 8

-Valid, Repeat Form
-The statement basically repeats the parts of the original statement and applies them to the individual in question, John.

• Sufficient condition: A+
• Necessary condition: Study

Question 9

Using the initial statement: If someone gets an A+ on a test, then they must have studied for the test

Is this statement valid: John studied for the test, so he must have received an A+ on the test

Answer 9

• Invalid, Mistaken Reversal
• The attempted inference looks like the reverse of the original statement
• A mistaken reversal switches the elements in the sufficient and necessary conditions creating a statement that does not have to be true

-Sufficient condition: Study
-Necessary condition: A+
-Just because the necessary condition has been fulfilled does not mean that the sufficient condition must occur

Just because John studied for the test does not mean he actually received an A+. He may have only received a B, or possibly failed.

Question 10

Using the initial statement: If someone gets an A+ on a test, then they must have studied for the test

Is this statement valid: John did not receive an A+ on the test, so he must not have studied for the test

Answer 10

• Invalid, Mistaken Negation
• Mistaken negation negates both conditions creating a statement that does not have to be true
• Sufficient condition: A+ (negated with a slash)
• Necessary condition: Study (negated with a slash)
• Just because the sufficient condition has not been fulfilled does not mean that the necessary condition has not occurred

Just because john did not receive an A+ does not mean he did not study. He may have studied but did not happen to receive an A+. Perhaps he received at B+

Question 11

Using the initial statement: If someone gets an A+ on a test, then they must have studied for the test

Is this statement valid: John did not study for the test, so he must not have received an A+ on the test

Answer 11

-Valid, contrapositive
- A contrapositive both reverses and negates. When the necessary condition fails to occur, then the sufficient condition cannot occur
-There is a contrapositive for every conditional statement, and if the initial statement is true, then the contrapositive is also true

-Sufficient condition: Study (negated)
-Necessary condition: A+ (negated)

If studying is the necessary condition for getting an A+, and John did not study, then according to the original statement there is no way John could have received an A+

Question 12

Necessary or sufficient condition indicator: If

Answer 12

Sufficient

Question 13

Necessary or sufficient condition indicator: When

Answer 13

Sufficient

Question 14

Necessary or sufficient condition indicator: whenever

Answer 14

Sufficient

Question 15

Necessary or sufficient condition indicator: Every

Answer 15

Sufficient

Question 16

Necessary or sufficient condition indicator: All

Answer 16

Sufficient

Question 17

Necessary or sufficient condition indicator: Any

Answer 17

Sufficient

Question 18

Necessary or sufficient condition indicator: Each

Answer 18

Sufficient

Question 19

Necessary or sufficient condition indicator: In order to

Answer 19

Sufficient

Question 20

Necessary or sufficient condition indicator: People who

Answer 20

Sufficient

Question 21

Necessary or sufficient condition indicator: Then

Answer 21

Necessary

Question 22

Necessary or sufficient condition indicator: Only

Answer 22

Necessary

Question 23

Necessary or sufficient condition indicator: Only if

Answer 23

Necessary

Question 24

Necessary or sufficient condition indicator: Must

Answer 24

Necessary

Question 25

Necessary or sufficient condition indicator: Required

Answer 25

Necessary

Question 26

Necessary or sufficient condition indicator: Unless

Answer 26

Necessary

Question 27

Necessary or sufficient condition indicator: Except

Answer 27

Necessary

Question 28

Necessary or sufficient condition indicator: Until

Answer 28

Necessary

Question 29

Necessary or sufficient condition indicator: Without

Answer 29

Necessary

Question 30

Necessary or sufficient condition indicator: Precondition

Answer 30

Necessary

Question 31

1st critical rule of conditional reasoning:

Answer 31

1. Either condition can appear first in the sentence

The order of presentation of the sufficient and necessary conditions is irrelevant

Question 32

2nd critical rule of conditional reasoning:

Answer 32

2. A sentence can have one or two indicators

Sentences do not need both a sufficient condition indicator and a necessary condition indicator in order to have a conditional reasoning present

Once you have established that one of the conditions is present, you can examine the remainder of the sentence to determine the nature of the other condition

Question 33

Unless equation

Answer 33

• In cases of “unless”, “except”, “until”, and “without”, the unless equation is applied

1. Whatever term is modified by “unless”, “except”, “until”, or “without” becomes the necessary condition
2. The remaining term is negated and becomes the sufficient condition

Ex: Unless a person studies, he or she will not receive an A+

“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+”.

Sufficient condition: A+
Necessary condition: Study

Question 34

Conditional Linkage

Answer 34

-if an identical condition is sufficient in one statement and necessary in another, the two can be linked to create a chain
-Test makers can link two sufficient conditions or two necessary conditions

Statement 1: A –> B
Statement 2: B –> C
Chain: A —> B —> C
Inference: A —-> C
Contrapositive: C (negated) –> A (negated)

Question 35

Either/Or

Answer 35

• At least one of the two, possibly both
• Since at least one of the terms must occur, if one fails to occur then the other must occur

Ex: Either John or Jack will attend the party

Diagram: John (negated) –> Jack
Contrapositive: Jack (negated) –> John

Question 36

Than either

Answer 36

• When used either translates to both

Ex: Desmond likes biology better than either Chemistry or Physics

The meaning of this statement is that Desmond likes biology better than BOTH chemistry and physics

Question 37

“Only, “Only if”, “ The Only”

Answer 37

• “Only, “Only if”, and “The Only” all work in the same way: they are all necessary condition indicators
• In terms of physical placement within a given sentence, the first two directly precede a necessary condition whereas “the only” directly precedes a sufficient condition

Question 38

in the contrapositive “and” when there are multiple terms in the sufficient or necessary conditions turns into ___

Answer 38

“Or”

Question 39

in the contrapositive “Or” when there are multiple terms in the sufficient or necessary conditions turns into ___

Answer 39

“And”

Question 40

Double arrow

Answer 40

-Indicate that each term is both sufficient and necessary for the other (The two terms must always occur together, or both do not occur)

Ex: Ann will attend if and only if Basil attends

Two separate conditional statements are created:
1. A if B (“if” introduces a sufficient condition)
2. A only if B

“A if B” = B —-> A
“A only if B” = A—> B
Combined the two = A <—-> B

Only two scenarios are possible:
1. A and B both attend (A and B)
2. Neither A nor B attend (A and B both negated)

Any scenario where one of the two attends but the other does not is impossible

Question 41

Terms that introduce double arrows:

Answer 41

1.”if and only if” and any synonymous phrase such as:
“if but only if”
“then and only then”
“then but only then”

2.”vice versa”

3. By repeating and rephrasing the terms

Question 42

The double not arrow

Answer 42

-indicates that two terms cannot occur together

Ex: If Gomez runs for president, then Hong will not run for president

G <–|–> H (the two terms cannot occur together)

Possible scenarios that can occur:
1. G runs for president, H does not ( G and H negated)
2. H runs for president, G does not (G negated and H)
3. Neither G nor H runs for president ( G and H both negated)

The only scenario that cannot occur is both G and H running for president

Question 43

Nested conditionals

Answer 43

• Occur when an entire conditional relationship is used as a complete condition inside another conditional statement

Ex: If you want a table at this restaurant you have to wait, unless you have a reservation

Three indicators (If, have to, unless), two are necessary condition indicators (have to, unless)

First clause is it’s own conditional statement:

If you want a table at this restaurant you have to wait,

Unless you have a reservation

The first clause would be diagrammed as:
Table —> wait

The clause was modified by “unless” which introduces a necessary condition (“reservation), and the remainder is negated

(Table —> Wait) both negated —-> Reservation

Since only one condition needs to be satisfied it can be rewritten as:
Table —> Wait or reservation

Can be simplified further:
Wait (negated) –> Reservation
Reservation (negated) —> Wait