Deduction, Induction,

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Deduction, Induction, Truth

• Kareem Khalifa
• Department of Philosophy
• Middlebury College

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Overview

• The Central Issue
• Deductive Validity
• Inductive Strength
• Deductive Validity vs. Inductive Strength
• Validity vs.Truth
• Exercises

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The Central Issue

• Recall an argument is a set of propositions such
  that one member of that set, the conclusion, can
  be affirmed on the basis of the others, the
  premises.
• What does it mean for a proposition to be
  affirmed on the basis of other propositions?

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Deductive Validity The Gold Star

• If the premises guarantee the truth of the
  conclusion, then the conclusion can always be
  affirmed on the basis of the premises.
• In other words, there is no way that the premises
  could be true and the conclusion could be false.
  Thats a guarantee!

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Example of a deductively valid argument

• Premise If Khalifa is a mammal, then Khalifa is
  warm-blooded.
• Premise Khalifa is a mammal.
• Conclusion Khalifa is warm-blooded.

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Deductive Validity The Official Definition

• A deductive argument is valid when, if its
  premises are true, its conclusion must be true.
• This is the SINGLE MOST IMPORTANT CONCEPT IN THE
  CLASS!!!!!!!!!
• Failure to define validity properly is an
  automatic 5 point penalty on anything you do!
  Youll also be very confused if you dont get
  this concept.

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Inductive Strength

• So, deductive validity describes one way in which
  a conclusion can be affirmed on the basis of its
  premises the iron-clad guarantee.
• However, we have many good arguments that do not
  provide such guarantees, for example

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• All observed peaches have pits. So all peaches
  have pits.
• Previously, when I flip the switch, the light
  goes on. So the next time I flip the switch, the
  light will go on.
• My parents have told me my name is Kareem
  Khalifa. So my name really is Kareem Khalifa.
• There is a strong correlation between smoking and
  lung cancer. So smoking causes lung cancer.
• Can you think of ways in which the premises can
  be true and the conclusions false, i.e., examples
  illustrating that the arguments are invalid?

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Deductive validity versus inductive strength

• Recall A deductive argument is valid when, if
  its premises are true, its conclusion must be
  true.
• Inductive strength An inductive argument is
  strong when, if its premises are true, its
  conclusion is probably true.

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Why would we ever settle for inductive arguments?

• Deductive arguments require certainty but we
  often operate with incomplete information.
• Conclusions of deductive arguments contain no new
  information over and above their premises
  inductive arguments do contain new information.
• Much of our reasoning is sensitive to background
  knowledge. Only inductive reasoning allows us to
  adapt our reasons to changes in our background
  knowledge.
• These considerations need to be weighed against
  the inherent risk involved in inductive inference.

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Invalid vs. Inductively strong arguments

• Deductive validity no way the premises are true
  and the conclusion is false.
• Deductive invalidity some way the premises are
  true and the conclusion is false.
• Inductively strong unlikely that the premises
  are true and the conclusion is false.
• So inductively strong arguments are deductively
  invalid arguments. How do we distinguish them?

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Possible replies

• Copi Cohen Deductive arguments (valid or
  invalid) claim to be valid inductive arguments
  (strong or weak) claim to be strong. (30)
• My preference It doesnt matter. If the argument
  is invalid, be aware of
• How it is possible that conclusion is false when
  premises are true AND
• How probable it is that conclusion is false when
  premises are true.
• You should be considering this regardless of what
  the argument claims to do!

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Validity versus truth

• Recall a deductive argument is valid when, if
  its premises are true, its conclusion must be
  true.
• This does not say that valid arguments actually
  have true premises or true conclusions.
• Validity only concerns the connection between
  premises and conclusion. But weak things can be
  connected by something strong.

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Implications for logic

• Propositions are true/false arguments are
  valid/invalid.
• This is an important conceptual point.
• Deductive logic can tell us if a conclusion
  necessarily follows from a set of premises, but
  it cannot tell us if the premises and/or
  conclusions are true/false.
• Thats why there are disciplines other than
  logic!
• There can be valid arguments with false premises
  and/or false conclusions.
• We saw this last class.

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Exercise 1

• Valid, 1 true prem, 1 false prem, false concl.
• If Khalifa is lizard, then Khalifa is a reptile.
• Khalifa is a lizard.
• Khalifa is a reptile.

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Exercise 2

• Valid, 1 true prem, 1 false prem, true concl
• If Khalifa is a koala, then Khalifa is a mammal.
• Khalifa is a koala.
• So Khalifa is a mammal.

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Exercise 3

• Invalid, two true prems, false concl
• If Khalifa is a human, then Khalifa is a mammal.
• If Khalifa is a mammal, then Khalifa is
  warm-blooded.
• So, Khalifa is not a human.

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Exercise 4

• Invalid, two true prems, true concl
• If Khalifa is a human, then Khalifa is a mammal.
• If Khalifa is a mammal, then Khalifa is
  warm-blooded.
• So, 224.
• Or less radically, Khalifa is a human.

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Exercise 5

• Valid, with 2 false prems, true concl
• If Khalifa is an amoeba, then Khalifa is a
  vertebrate.
• Khalifa is an amoeba.
• So Khalifa is a vertebrate.

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Exercise 6

• Invalid, two false prems, true concl
• There are exactly two students in PHIL0180.
• Middlebury tuition costs two dollars.
• 224.

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Exercise 7

• Invalid, 1 true prem, 1 false prem, true concl
• If Khalifa is a reptile, then Khalifa is a
  vertebrate.
• If Khalifa is a vertebrate, then Khalifa is a
  warm-blooded.
• So Khalifa is a vertebrate.

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Exercise 8

• Valid, true prems, true conclcalled a SOUND
  argument
• If Khalifa is a human, then Khalifa is a mammal.
• Khalifa is a human.
• So Khalifa is a mammal.