Lecture 3: Necessary

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Lecture 3 Necessary Sufficient Conditions
Formal Fallacies

• University of Wyoming
• 05, September 2007

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Gratuitous Monty Python Vid

• http//www.youtube.com/watch?vzrzMhU_4m-g.

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Understanding the conditional

• Suppose I claim that if she is a witch, then she
  is made of wood.
• In symbols
• She is a witch ? She is made of wood.

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• What I am saying is that her being a witch is
  enough to make it true that she is made of wood
  or that being a witch is sufficient for her being
  made of wood.

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The Sufficient Condition

• For this reason we will say that the antecedent
  of a conditional is a sufficient condition for
  its consequent.

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• Notice also that, if my conditional statement
  about is true, then if she is a witch, she must
  be made of wood (her being made of wood is
  required).

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The Necessary Condition

• In general, the consequent of a conditional is a
  necessary condition for its antecedent.
• Thus, her being made of wood is a necessary
  condition of her being a witch.

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Alternative English Phrases

• q, if p (e.g., She is made of wood, if she is a
  witch).
• Write as p ? q
• p only if q (e.g., She is a witch only if she is
  made of wood)
• Write as p ? q
• Note only if is very confusing.

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• Given that
• p, if q means if q, then p and
• p only if q means if p, then q
• What does p if and only if q mean?

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• Answer p if and only if q means if p, then q
  AND if q, then p.
• This is, of course, just the biconditional, p ? q.

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Necessary and Sufficient

• So the biconditional means that q is both
  necessary (p ? q) and sufficient (q ? p) for p.
• Warning keeping this straight is much harder
  than it seems, study carefully!

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What is a fallacy?

• A fallacy is a type of argument that may seem to
  be correct, but that proves, on examination, to
  involve an error of reasoning.

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Affirming the Consequent

• The fallacy of affirming the consequent is an
  argument in which one asserts a conditional,
  asserts the consequent of the conditional, and
  then concludes that the antecedent of the
  conditional is true.

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In Symbols

• p ? q
• q_______
• Therefore, p

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Counterexamples

• How do we decide whether or not a given argument
  form is valid or fallacious?
• Answer find a situation in which all of the
  premises are true and the conclusion false. This
  is a counterexample.

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Provide a Counterexample

• If George Bush is president, then he lives in the
  White House.
• George Bush lives in the White House.
• Thus, George Bush is president.

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Explanation

• We can see why affirming the consequent is a
  fallacy by thinking in terms of necessary and
  sufficient conditions. The first premise of the
  argument (i.e., if p, then q) asserts that p is
  a sufficient condition for q. But no claim is
  made to the effect that p is a necessary
  condition for q?there might well be other
  conditions that are sufficient for q. The
  fallacy of affirming the consequent, therefore,
  goes wrong because it mistakenly treats p as a
  necessary condition, rather than merely a
  sufficient condition.

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Denying the Antecedent

• The fallacy of denying the antecedent, occurs
  when one asserts a conditional, denies the
  antecedent of the conditional, and concludes by
  denying the consequent.

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In Symbols

• p ? q
• ?p_______
• Therefore, ?q

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Provide a Counterexample

• If Hillary Clinton is president, then she lives
  in the White House.
• Hillary is not president.
• She doesnt live in the White House.

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Explanation

• Like the fallacy of affirming the consequent, the
  fallacy of denying the antecedent confuses
  necessary and sufficient conditions. When
  someone commits this fallacy they are mistakenly
  treating p as a necessary condition for q. But
  the first premise asserts only that p is a
  sufficient condition for q.