Conditional Statements and Material Implication

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Conditional Statements and Material Implication
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The Conditional The Fourth Connective

• Conditional statement when two statements are
  combined by placing the word if before the
  first and then before the second
• Ifthen
• We use the arrow ? or horseshoe to represent
  the if-then phrase
• Also called a hypothetical, an implication, or an
  implicative statement
• The component statement that follows the if is
  called the antecedent
• The component statement that follows the then
  is called the consequent
• If (antecedent), then (consequent)

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The Conditional

• A conditional statement asserts that (in any
  case) if its antecedent is true, then its
  consequent is also true
• But as in disjunction, there are a few different
  senses in which a conditional can be interpreted

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The Four Types of Implication

• 1. If all humans are mortal and Socrates is a
  human, then Socrates is mortal.
• Logical Implication the consequent follows
  logically from its antecedent
• 2. If Leslie is a bachelor, then Leslie is
  unmarried.
• Definitional Implication the consequent follows
  the antecedent by definition
• 3. If I put X in acid, then X will turn red.
• Causal Implication The connection between
  antecedent and consequent is discovered
  empirically
• 4. Is we lose the game, then Ill eat my hat.
• Decisional Implication no logical connection nor
  one by definition between the consequent and
  antecedent. This is a decision of the speaker to
  behave in the specified way under the specified
  circumstances.

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Which Sense of Implication Do We Use?

• We must try to find a sense that is at least a
  part of the meaning of all four different types
  of implication
• No matter what type of implication is asserted by
  a conditional statement, part of its meaning is
  the negation of the conjunction of its antecedent
  with the negation of its consequent
• For a conditional to be true (e.g. If p then
  q), (p q) must be true
• Think pA piece of blue litmus paper is placed
  in that solution.
• qThe piece of blue litmus paper will
  turn red.
• If p then q false if paper is placed in
  solution, but doesnt turn red
• The horseshoe symbol does not stand, therefore,
  for all the meanings of if-then there are
  several meanings
• p q abbreviates (p q), whose meaning is
  included in the meanings of each kind of
  implication

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What this means
p q q p q (p q)
T T F F T
T F T T F
F T F F T
F F T F T
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Abbreviated Truth Table for the Conditional
p q p q
T T T
T F F
F T T
F F T
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The only time a conditional is FALSE is when the
antecedent is trueand the consequent is false.
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Material Implication

• represents the material implication
• A fifth type of implication
• E.g. If Hitler was a military genius, then Im a
  monkeys uncle.
• No real connection between antecedent and
  consequent
• This kind of relationship is what is meant by
  material implication
• It just asserts that it is not the case that the
  antecedent is true when the consequent is false.
• Many arguments contain conditional statements of
  various kinds of implication, but the validity of
  all valid arguments (of the general type with
  which we will be concerned) is preserved, even if
  the additional meanings of their conditional
  statements are ignored.

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Some If Indicator Words

• If can be replaced by such phrases as
• in case
• provided that
• given that
• on condition that
• Some indicator words for then include
• implies...
• entails

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Necessary Conditions

• For a normal car to run, it is necessary that
  there is fuel in its tank, its spark plugs
  properly adjusted, its oil pump working, etc.
• If the car runs, then every one of the conditions
  necessary for its occurrence must be fulfilled
• That there is fuel in its tank is a necessary
  condition for the car to run
• The car runs only if there is fuel in its
  tank
• If the car runs then there is fuel in its
  tank
• All these are R F

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Hints

• p is a necessary condition for q
• q p
• only if
• p only if q p q

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Sufficient Conditions

• For a purse to contain over a dollar, it would be
  sufficient for it to contain 101 pennies, 21
  nickels, 11 dimes, 5 quarters, etc.
• If any one of these circumstances obtains, the
  specified situation will be realized
• That a purse contains 5 quarters is a sufficient
  condition for it to contain over a dollar
• If the purse contains 5 quarters then it
  contains over a dollar

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Hints

• p is a sufficient condition for q
• p q (Note here q is a necessary condition
  for p)
• Compare p is a necessary condition for q
• q p (Note here q is a sufficient condition
  for p)
• Formula to help you remember, given any
  sentence saying something is a necessary or
  sufficient condition
• p q

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Necessary condition
Sufficient condition
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Example

• A, B, C are true statementsX, Y, Z are false
  statements
• Determine whether the following is true or false
• (X Y) Z

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Solution

• (X Y) Z
• 1. Main connective is the second horseshoe
  (conditional)
• 2. Look at antecedent (X Y)
• -X and Y are false, so this makes this
  conditional true
• -we know this by using our knowledge of the
  conditional (i.e. truth table for the material
  implication)
• 3. We know Z (the consequent) is also false
• 4. Therefore, a conditional with a true
  antecedent and false consequent is false

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Example

• A, B, C are true statementsX, Y, Z are false
  statements
• Determine whether the following is true or false
• (A X) v (A X) (A X) (X
  A)

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