Conditional Statements

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Conditional Statements
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This is VERY IMPORTANT!

• We will show the failure of the fairness criteria
  by showing the first part of the conditional is
  true and the second part is false.
• For example consider the conditional statement
• If you buy a lottery ticket then you will win
  a million dollars.
• To show this is not true somebody has to buy a
  lottery ticket and not win a million dollars.

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Proving a Conditional is False

• In the example If you buy a lottery ticket then
  you will win a million dollars.
• The first part you buy a lottery ticket is
  called the antecedent.
• The second part you will win a million dollars
  is called the consequent.
• To show a conditional statement is false we must
  show the antecedent is true and the consequent is
  false.
• In any other case the conditional statement is
  logically true.

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Logic Puzzle

• Consider the 4 boxes below as cards lying on a
  table.
• You can only see one side of the card but it is
  given that on one side there is a number and on
  the other side is a letter.

A
3
D
4
Consider the conditional If there is a D on
one side then there is a 3 on the other
side. Which cards must be flipped over to reveal
the other side so as to verify that the
conditional is actually true
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Logic Puzzle

• Consider the 4 boxes below as cards lying on a
  table.
• You can only see one side of the card but it is
  given that on one side there is a number and on
  the other side is a letter.

A
3
D
4
The given conditional was If there is a D on
one side then there is a 3 on the other
side. The answer is you must flip over the D
and the 4 cards. (1) You have to check if the
D card has a 3 on the other side And (2) you
also have to check if the 4 card has a D on
the other side Because both of those
possibilities would prove the conditional to be
false.
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Another Logical Question
Now consider the following situation Suppose
you are the owner of a bar on South Beach. The
police pull up outside and tell you there was a
report of under-age drinking in your
establishment. You tell the police If there
is anyone inside drinking then they are of legal
drinking age. Suppose the boxes below are again
cards but this time represent the 4 people in the
bar. On one side of the card is persons age and
on the other side of the card is the type of
refreshment that person is drinking. Which of
these cards must the police flip over to verify
that you are telling the truth
Coke
33
Vodka
15
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Logic Does Not Come Naturally
You tell the police If there is anyone inside
drinking then they are of legal drinking age.
Which of these cards must the police flip over to
verify that you are telling the truth
Coke
33
Vodka
15
Now the question is very easy! Only the second
two cards matter those are the ones that must
be flipped to check if the statement is false.
Those are the only cases were it is possible to
prove the statement is false. In this situation
the logic is easy because it is familiar and
its possible that something in our brains is also
very good at finding someone who is somehow
cheating at something- but the logic is exactly
the same as the previous but more abstract
example. In the pure abstract case you might
have more trouble with the logic it really
doesnt come naturally so be very careful!
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Fairness Criteria

• All Fairness Criteria can be written in the form
  of a conditional.
• To prove a method of voting fails one of these
  fairness criteria we must find an example where
  the first part of the criterion (the antecedent)
  is satisfied and the second part (the consequent)
  is not satisfied.

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Proving the Failure of Fairness Criteria

• For example consider the majority criterion
  which states
• If a candidate has a majority of first place
  votes then that candidate should win.
• To prove that a certain method of voting (like
  Borda Count) can sometimes fail the majority
  criterion will must find an example where a
  candidate has a majority of first places votes
  and does not win the election.
• As another example suppose we want to show the
  plurality method fails the Condorcet Winner
  Criterion which states
• If a candidate beats all other candidates in
  one-on-one contests then that candidate should
  win the election.
• To prove plurality fails this criterion we must
  find an example where a certain candidate beats
  all other candidates in one-on-one contests and
  does not win the election.