conditional statements and...
TRANSCRIPT
Conditional statements and arguments
Phil 12: Logic and Decision MakingFall 2010
UC San Diego10/1/2010
Friday, October 1, 2010
Checking your progress on Inquiry
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Friday, October 1, 2010
Checking your progress on Inquiry
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Friday, October 1, 2010
Review
• Conditional statements (IF A, THEN B)
• Truth conditions:
- false when antecedent is T, consequent is F
- true otherwise
• IF A, THEN B = IF NOT-B, THEN NOT-A.
Friday, October 1, 2010
Conditional statements: IF, THEN vs. ONLY IF
• Compare:
- If you trespass, then you will be arrested
- Only if you trespass will you be arrested
• B only if A is equivalent to If B, then A.
• What follows only if is the consequent
- If you were arrested, then you trespassed
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• False if you don’t trespass and are arrested
• False if you trespass and are not arrested
Friday, October 1, 2010
Conditional statements: unless
Unless can also be used to assert conditional relations
Rule: Unless = If not
Unless you complete the assignment, you will not get promoted
says the same thing as
If you do not complete the assignment, then you will not get promoted
and
If you get promoted, then you completed the assignment.
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Friday, October 1, 2010
Clicker question 1
Which statement is not equivalent to the others?
A. “If there is a storm, the dogs will bark”
B. “Only if the dogs bark is there a storm”
C. “Only if there is a storm will the dogs bark”
D.They’re all equivalent
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Friday, October 1, 2010
Clicker question 2Assume:
What is the truth value of the statement:
If we do not need to build a new plant, then our sales are not increasing.
A.True
B. False
Sales are increasing = T Our sales force is less effective = F
We need to build a new plant = F We have excess production capacity = T
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Friday, October 1, 2010
Clicker question 3Assume:
What is the truth value of the statement:
Only if sales are increasing do we need to build a new plant.
A.True
B. False
Sales are increasing = T Our sales force is less effective = F
We need to build a new plant = F We have excess production capacity = T
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Friday, October 1, 2010
Clicker question 4Assume:
What is the truth value of the statement:
Unless we have excess production capacity, we do not need to build a new plant
A.True
B. False
Sales are increasing = T Our sales force is less effective = F
We need to build a new plant = F We have excess production capacity = T
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Friday, October 1, 2010
Sufficient Conditions• When a conditional statement uses general terms (e.g., dog,
mammal) it expresses relations between categories of things that satisfy those terms
If something is a dog, then it is a mammal
Presents a relation between being a dog and being a mammal
It asserts that meeting the first condition (being a dog) suffices for meeting the second condition (being a mammal)
If _________, then__________
suffices for11
Friday, October 1, 2010
Necessary Conditions• Remember that a true conditional statement cannot have
a true antecedent and a false consequent
• This means the consequent of a conditional expresses something that is necessary if the antecedent is true
If something is a dog, then it is a mammal
Asserts that meeting the second condition (being a mammal) is necessary for meeting the first condition (being a dog)
If _________, then__________
necessary for12
Friday, October 1, 2010
If vs. Only If again• What follows the if of a conditional is a sufficient condition
• What follows only if is a necessary condition
- You can vote only if you are at least 18 years old
Being 18 is a necessary condition for voting
- If you are able to vote, then you are at least 18 years old
Being able to vote is sufficient (evidence) that you are at least 18 years old
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Friday, October 1, 2010
Clicker question 5
Consider the statement:
Unless you go to medical school, your parents won't be happy.
You go to medical school is:
A. a necessary condition for your parents' being happy
B. a sufficient condition for your parents' being happy
C. neither necessary nor sufficient for your parents' being happy
D. both necessary and sufficient condition for your parents' being happy
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Friday, October 1, 2010
The statement:
Taking the SAT is a sufficient condition for being admitted to UCSD
is false if:
A. Someone takes the SAT and is admitted to UCSD
B. Someone takes the SAT and is not admitted to UCSD
C. Someone does not take the SAT and is admitted to UCSD
D. Someone does not take the SAT and is not admitted to UCSD
Clicker question 6
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Friday, October 1, 2010
Using conditionals in argumentsThere are two ways to use a conditional statement in a valid inference:
If A, then BA ∴B Modus ponens
• The first, obvious way:
If A, then BA B
- Why?• If B weren’t true, and A is true, then If A, then B
would be rendered false
• So the following argument form is valid:
Friday, October 1, 2010
Using conditionals in arguments
• The second, less obvious way:
From If A, then B, what happens if B is denied?
If B is false and A is true, then what is the truth value of If A, then B?
It is false. Thus A cannot be true when the whole conditional is true; we know Not A is true.
There are two ways to use a conditional statement in a valid inference:
If A, then BNot B ∴Not A Modus tollens
• So the following argument form is valid:
Friday, October 1, 2010
Uses of conditional arguments inscientific reasoning
• Modus ponens is most commonly invoked to make predictions from a hypothesis
If malaria is transmitted by mosquitoes and we eliminate the mosquitoes, then malaria will decline.
Malaria is transmitted by mosquitoes and we are eliminating the mosquitoes.
∴ Malaria will decline.
• Modus tollens is most commonly invoked to confirm or falsify a hypothesis based on the truth of falsity of a prediction
Friday, October 1, 2010
Invalid conditional arguments• What can you conclude (validly) from:
If A, then BNot A ?
• Remember, to be valid, it must be that if the premises were true, the conclusion would also have to be true
• What can we know about the truth of the conclusion?
- Both B and not B are compatible with the premises
- There is no valid argument here!
Denying the AntecedentINVALIDX
Friday, October 1, 2010
Invalid conditional arguments
• What about this:
If A, then BB ∴ A
• If the premises were true, must the conclusion be true in this case?
- No. Both A and Not A are compatible with these premises
- There is no valid argument here either!
Affirming the ConsequentINVALIDX
Friday, October 1, 2010
Summary: Conditional Arguments
• Valid argument forms:
If A, then B If A, then BA Not B ∴B ∴Not AModus ponens Modus tollens
• Invalid argument forms:
If A, then B If A, then BNot A B ∴Not B ∴ADenying the antecedent Affirming the consequent
Friday, October 1, 2010
Clicker question 7
• What form does this argument have?
I know I passed since I took the test, and if I took the test, I passed.
A. Modus ponens
B. Affirming the consequent
C. Modus tollens
D. Denying the antecedent
Friday, October 1, 2010