Necessary and Sufficient Conditions

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Necessary and Sufficient Conditions
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Conditional Sentences

• Most conditional statements take the following
  form

if then
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Examples

• If I win the lottery, then I will retire to the
  Bahamas.

• If you think Cabin Boy is a good movie,
• then you have no taste at all.

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Conditional Statement

• One in which it is claimed that something is or
  will be the case provided that some other
  situation obtains. Also sometimes called
  hypothetical statements.

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• Crucial to understanding how conditional
  statements work is an understanding of necessary
  and sufficient conditions.

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Sufficient Condition

• A state of affairs that, once true, is enough for
  something else to be true. If X is sufficient for
  Y, then if X is true, Y is true.

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Necessary Condition

• A state of affairs that must be true in order for
  something else to be true, but is not itself
  enough to make something else true. If X is
  necessary for Y, then if X is not true, Y is not
  true.

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Example

• Being a bachelor is a sufficient condition for
  being a man.

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• Male
• Bachelor
•  

Here we would say that being a bachelor is
sufficient for being a man, and that being a
man is necessary for being a bachelor.
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All and Only

• The words All and Only also express necessary
  and sufficient conditions.

• All As are Bs A is sufficient for B
• B is necessary for A

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All bachelors are male.

• To say all bachelors are male is to say that
  there are no bachelors that arent male, or that
  the class of bachelors is contained within the
  class of men.

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Only men are bachelors.

• Only As are Bs A is necessary for B
• B is sufficient for A

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Conditional Claims

• If you look directly at Medusa, then you will
  turn into stone.
• you look directly at Medusa

• Antecedent (Sufficient Condition)

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Medusa Example

• Looking at Medusa is sufficient for turning into
  stone. It is enough for the consequent.

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Medusa Example

• you will turn into stone

Consequent (Necessary Condition)
Turning into stone is a necessary consequence of
looking at Medusa, and is therefore a necessary
condition.
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If it rains, then the streets will be wet

• Antecedent

It rains (Sufficient Condition)

• Consequent

The streets will be wet (Necessary Condition)
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Other forms of conditional claims

• Ill tear you a new one if you keep talking.
• Antecedent

You keep talking (Sufficient Condition)
Consequent

• Ill tear you a new one
• (Necessary Condition)

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Only if

• We also see conditional claims that include only
  if instead of just if.

Only if always introduces a necessary condition.
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Example

• Ill laugh only if you wear those pants.

Necessary Condition
Your wearing those pants is necessary for my
laughing
Sufficient Condition
My laughing is sufficient for your wearing those
pants
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Unless

• If the premise of an argument employs unless,
  the easiest way to identify the necessary and
  sufficient conditions is to translate the
  statement into standard conditional form.

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Unless

• Take what follows unless, negate it and make it
  the antecedent in the conditional. The rest of
  the original claim takes the place of the
  consequent.

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Translating for Unless

• 1. A unless B

If not B then A
2. A unless not B
If B then A
3. Not A unless B
If not B then not A
If B then not A
4. Not A unless not B
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Translating for Unless

• 5. Unless A then B

If not A then B
6. Unless A then not B
If not A then not B
7. Unless not A then B
If A then B
8. Unless not A then not B
If A then not B
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Example

• Unless John does not hurl, well be thrown out of
  the bar.
• Put this into standard conditional form and then
  identify the necessary and sufficient conditions.

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Unless John does not hurl, well be thrown out of
the bar

• If John does hurl, then we will be thrown out of
  the bar.

Johns hurling is sufficient for our being
thrown out of the bar.
Our being thrown out of the bar is necessary for
John hurling
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Practice

• You cant keep your kneecaps unless you pay your
  debt.
• Put the sentence into standard conditional form
  and identify the necessary and sufficient
  conditions.

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You cant keep your kneecaps unless you pay your
debt

• If you do not pay your debt, then you cant keep
  your kneecaps.

• Your not paying your debt is sufficient for not
  keeping your kneecaps.

• Your not keeping your kneecaps is
• necessary for not paying your debt.

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A test question

• You cant have any dessert unless you eat your
  meat.
• Not having any dessert is __________ for not
  eating your meat.

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You cant have any dessert unless you eat your
meat

• If you dont eat your meat, then you cant have
  any dessert.

Not having any dessert is ________ for not
eating your meat.
necessary
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Test question

• You dont have to go to the doctor unless you are
  sick.
• Your not going to the doctor is _________ for
  your not being sick.

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You dont have to go to the doctor unless you are
sick

• If you are not sick, then you dont have to go to
  the doctor.

Your not going to the doctor is _______ for your
not being sick.
necessary
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Definition and Identity

• Sometimes a single condition can be both
  necessary and sufficient.

• Being water is both necessary and
• sufficient for being H2O.

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Definition and Identity

• In this case A is both Necessary and Sufficient
  for B, and B is both Necessary and Sufficient for
  A.

This is because whatever is water is H2O and
whatever is H2O is water.
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Definitions and Identities

• If Batman is identical to Bruce Wayne, then being
  Bruce Wayne is both necessary and sufficient for
  being Batman, and being Batman is both necessary
  and sufficient for being Bruce Wayne.

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If and only if

• These relations are often characterized in terms
  of the following

if and only if which is often put in the short
form iff.
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Iff

• X is water if and only if X is H2O.
• X is Batman iff X is Bruce Wayne.

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Break Time

• 10 Minutes

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Valid Conditional Arguments

• 1. Modus Ponens

Latin for mood that affirms. It is called that
because the second premise and the conclusion
are both affirmations. (Affirming the sufficient
condition)
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Modus Ponens

• The argument takes this form

If p then q
p
Therefore, q
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Example

• If Janet gets the job, then Ill resign from the
  board.
• Janet got the job.
• Therefore, Ill resign.

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Standardizing Modus Ponens
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Example

• Ill buy your dinner if you can eat the whole
  thing.
• You ate the whole thing.
• Therefore, Ill buy you dinner.

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Example
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Example

• Jill will only win the award if she doesnt
  offend the judges.
• Jill won the award.
• Therefore, Jill didnt offend the judges.

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Jill Example

• In premise 2 we are affirming the sufficient
  condition. Remember that in conditional
  statements with only if the sufficient
  condition follows the then rather than the if.

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Valid Conditional Arguments

• 2. Modus Tollens

Latin for mood that denies. Called that
because both the second premise and the
conclusion are negations. (Denying the necessary
condition).
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Modus Tollens

• The argument takes this form

If p then q
Not q
Therefore, not p
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Example

• If you work out, then you will feel tired.
• You dont feel tired.
• Therefore, you didnt work out.

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Standardizing Modus Tollens
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Example

• I can only go to the mountains if I can get a
  ride.
• I didnt get a ride.
• Therefore, I cant go to the mountains.

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Recognizing Valid Conditional Arguments

• Affirm the Sufficient Condition.

• Deny the Necessary Condition.

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This clip contains examples of
Modus Ponens
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Example 1

• If your captain were victor, he would not want me
  and I would have Stonn.

Your captain was the victor.
He does not want me and I have Stonn.
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Invalid forms of Conditional Arguments

• 1. Denying the Antecedent
• (denying the sufficient condition)

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Denying the Antecedent

• This has the following form

If p then q
Not p
Therefore, not q
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Example

• If you take off your clothes, Ill laugh.
• You did not take off your clothes.
• Therefore, I didnt laugh.

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Example

• If it rains, then the streets will be wet.
• It didnt rain.
• Therefore, the streets arent wet.

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Example

• You cant go outside unless you put on your
  shoes.
• You didnt put on your shoes.
• Therefore, you didnt go outside.

Restate the argument with premise one in
conditional form.
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You cant go outside unless you put on your shoes

• If you put your shoes on, then you can go outside.

You didnt put on your shoes. Therefore, you
cant go outside.
Invalid. Denying the antecedent.
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Invalid Conditional Arguments

• 2. Affirming the consequent
• (affirming the necessary condition)

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Affirming the consequent

• This argument has the following form

If p then q
q
Therefore, p
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Example

• If it rains then the streets will be wet.
• The streets are wet.
• Therefore, it rained.

Invalid Affirming the consequent
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Example

• If you bring some chicken, then well have a
  barbeque.
• We had a barbeque.
• Therefore, you brought some chicken.

Invalid Affirming the consequent
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Example

• If you make it, they will come.
• They came.
• Therefore, you made it.

Invalid Affirming the consequent
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Example

• If you pee into the wind, youll get your shoes
  wet.
• You didnt pee into the wind.
• Therefore, you didnt get your shoes wet.

Invalid denying the antecedent
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Example

• Unless he tells me otherwise, Ill paint his room
  pink.
• I painted his room pink.
• Therefore, he didnt tell me otherwise.

• Restate the argument with the first
• premise as a conditional

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Unless he tells me otherwise, Ill paint his room
pink

• If he doesnt tell me otherwise, Ill paint his
  room pink

I painted his room pink. Therefore, he didnt
tell me otherwise.
Invalid. We have affirmed the consequent
(necessary condition) in premise 2.
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Example

• Only if you paid me would I go to see Patch
  Adams.
• I didnt go to see Patch Adams.
• Therefore, You didnt pay me.

Invalid. This denies the sufficient condition.
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Example

• If Tamara doesnt call her mother, then her
  mother will cry.
• Tamara calls her mother.
• Therefore, her mother doesnt cry.

Invalid. We have denied the sufficient condition.