Conditional Statements

1
Conditional Statements Material Implication

• Kareem Khalifa
• Department of Philosophy
• Middlebury College

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Overview

• Why this matters
• Anatomy of a conditional statement
• Some nuances in translating conditionals
• Truth-conditions for ?
• Weirdness with ?
• Possible solutions Your first foray into
  philosophy of logic!
• Sample Exercises

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Why this matters

• Conditional statements are the most fundamental
  logical connectives, so understanding their
  truth-conditions is necessary for analyzing and
  criticizing many arguments.
• A cheap trick for making any argument valid.
• Sally is under 18.
• If Sally is under 18, then shes not allowed on
  the premises.

• So shes not allowed on the premises.

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Anatomy of conditionals

• If you study hard, then you will pass PHIL0180.
• If p, then q.

CONSEQUENT
ANTECEDENT
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Some nuances in translating conditionals

• If p then q can also be expressed in the
  following ways
• If p, q
• q, if p
• p only if q
• p is sufficient for q
• q is necessary for p
• p requires q

• p entails q
• p implies q
• p renders (yields, produces, etc.) q
• In case of p, q
• Provided that p, q
• Given that p, q
• On the condition that p, q

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Examples

• If Khalifa is human, then Khalifa is a mammal.
• Khalifas being human suffices for his being a
  mammal.
• Khalifas being a mammal is necessary for his
  being human.
• Khalifas humanity requires that he be a mammal.
• Khalifas humanity entails that he is a mammal.

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More examples

• Washing your hands decreases the chance of
  infection.
• If you wash you your hands, then the chance of
  infection decreases.
• Paying off the professor will produce the desired
  effect.
• If the professor is paid off, then the desired
  effect will be produced.

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Truth conditions for conditionals

• Recall A logical connective is a piece of
  logical syntax that
• Operates upon propositions and
• Forms a larger (compound) proposition out of the
  propositions it operates upon, such that the
  truth of the compound proposition is a function
  of the truth of its component propositions.
• Today, were looking at IFTHEN...
• The truth of the whole if-then statement is a
  function of the truth/falsity of the antecedent
  and consequent.

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Truth-conditions for ?

• In logic, we represent if p then q as p ? q.
  This is called material implication.
• Alternatively, ? may be represented as ?.
• p?q is false if antecedent p is true and
  consequent q is false otherwise, true.

p q p ? q
T T
T F
F T
F F
T
F
T
T
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Intuitive examples of ?

• True antecedent, true consequent
• If Khalifa is human, then Khalifa is a mammal.
• False antecedent, true consequent
• If Khalifa is a dog, then Khalifa is a mammal.
• False antecedent, false consequent
• If Khalifa is a dog, then Khalifa is a canine.

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Weirdness with ?

• True antecedent, true consequent
• If 224, then Middlebury is in VT.
• False antecedent, true consequent
• If the moon is made of green cheese, then 22 4.
• False antecedent, false consequent
• If Khalifa is a dog, then the moon is made of
  green cheese.

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More weirdness the paradoxes of material
implication

• The following are both valid arguments
• B, so A ? B
• Ex. 224, so if unicorns exist, then 224.
• A, so A ? B
• Ex. The moon is not made of green cheese, so if
  the moon is made of green cheese, then Khalifa is
  a lizard.

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Different responses to the weirdness

• Response 1 Logic must be revised!
• The English If p then q is just elliptical for
  Necessarily, if p then q. 22 4 doesnt
  necessitate anything about Middlebury, nor does
  the moons green cheesiness necessitate anything
  about arithmetic, etc.
• Ex. Although it is actually the case that 22 4
  and Midd is in VT, it is possible that 224 and
  Midd is not in VT.
• Thus it is not necessary that this conditional be
  true.

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Response 2 (Copi Cohens)

• If then statements in English express
  several different relationships
• Logical If either Pat or Sam is dating Chris and
  Sam is not dating Chris, then Pat is dating
  Chris.
• Definitional If a critter is warm-blooded, then
  that critter has a relatively high and constant
  internally regulated body temperature relatively
  independent of its surroundings.
• Causal If I strike this match, then it will
  ignite.
• Decisional If the median raw score on the exam
  is 60, then I should institute a curve.
• Each of these if-then statements is false when
  the antecedent is true and the consequent is
  false.
• This is exactly what material conditionals state,
  and thus they capture the core of all
  conditional statements. The rest is an issue of
  context.

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Response 3

• Suppose that the English If p then q is true.
• Either p is true or p is true.
• In the first case, p v q is true.
• In the second case, q is true by modus ponens.
• Thus, in either case p v q is true.
• Since p v q is equivalent to p ? q, the latter
  is a fair interpretation of If p then q.

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More on Response 3
p q p p v q p ? q
T T T
T F F
F T T
F F T
F
T
F
F
T
T
T
T
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Exercise A6

• (X ? Y) ? Z
• (F ? F) ? F
• (T) ? F
• F

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Exercise A22

• A? (B?C)? (AB)? C? (Y?B)? (C?Z)
• T? (T?T)? (TT)? T? (F?T)? (T?F)
• T? (T)? (T)? T? (T)? (F)
• T? (T)? (T)? T? F
• T? T? F
• T? F
• F

STOP THINK!
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Exercise B11

• (P ? X) ? (X ? P)
• (P ? F) ? (F ? P)
• (P ? F) ? (T)
• T

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Exercise B24

• P ? (A v X) ? (P ? A) ? X
• P ? (T v F) ? (P ? T) ? F
• P ? (T) ? (T) ? F
• T ? F
• F

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Exercise C22

• Argentinas mobilizing is a necessary condition
  for Chile to call for a meeting of all the Latin
  American states.
• C ? A

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Exercise C25

• If neither Chile nor the DR calls for a meeting
  of all the Latin American states, then Brazil
  will not protest to the UN unless Argentina
  mobilizes.
• (C D) ? (B v A)