Chapter 6: Conditional Reasoning Flashcards
Conditional reasoning
- 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
Sufficient condition
-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
Necessary condition
-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
Name the sufficient and necessary conditions in: If someone gets an A+ on a test, then they must have studied for the test
Sufficient condition: Get an A+ (Getting an A+ automatically indicates that someone must have studied)
Necessary condition: Must have studied
1st logical feature of conditional reasoning:
- 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.
2nd logical feature of conditional reasoning:
- Either condition can occur first, or the two conditions can occur at the same time
3rd logical feature of conditional reasoning:
- 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
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
-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
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
- 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.
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
- 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+
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
-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+
Necessary or sufficient condition indicator: If
Sufficient
Necessary or sufficient condition indicator: When
Sufficient
Necessary or sufficient condition indicator: whenever
Sufficient
Necessary or sufficient condition indicator: Every
Sufficient
Necessary or sufficient condition indicator: All
Sufficient
Necessary or sufficient condition indicator: Any
Sufficient
Necessary or sufficient condition indicator: Each
Sufficient
Necessary or sufficient condition indicator: In order to
Sufficient
Necessary or sufficient condition indicator: People who
Sufficient
Necessary or sufficient condition indicator: Then
Necessary
Necessary or sufficient condition indicator: Only
Necessary
Necessary or sufficient condition indicator: Only if
Necessary
Necessary or sufficient condition indicator: Must
Necessary
Necessary or sufficient condition indicator: Required
Necessary
Necessary or sufficient condition indicator: Unless
Necessary
Necessary or sufficient condition indicator: Except
Necessary
Necessary or sufficient condition indicator: Until
Necessary
Necessary or sufficient condition indicator: Without
Necessary
Necessary or sufficient condition indicator: Precondition
Necessary
1st critical rule of conditional reasoning:
- Either condition can appear first in the sentence
The order of presentation of the sufficient and necessary conditions is irrelevant
2nd critical rule of conditional reasoning:
- 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
Unless equation
- In cases of “unless”, “except”, “until”, and “without”, the unless equation is applied
- Whatever term is modified by “unless”, “except”, “until”, or “without” becomes the necessary condition
- 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
Conditional Linkage
-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)
Either/Or
- 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
Than either
- 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
“Only, “Only if”, “ The Only”
- “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
in the contrapositive “and” when there are multiple terms in the sufficient or necessary conditions turns into ___
“Or”
in the contrapositive “Or” when there are multiple terms in the sufficient or necessary conditions turns into ___
“And”
Double arrow
-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
Terms that introduce double arrows:
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”
- By repeating and rephrasing the terms
The double not arrow
-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
Nested conditionals
- 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
