Introduction to Logic

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Introduction to Logic
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Purpose of Logic

• To formally specify when an argument is sound.
• The form of the argument is sufficient to
  determine its soundness - content is not
  considered.
• Therefore, usually more convenient to remove the
  content and analyze the form independently.
• Historically, people would study logic to improve
  their mind.

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Example 1

• 1. If it rains tomorrow, then the game
• will be canceled.
• 2. If the game is canceled, then our
• team will surely lose the pennant.
• 3. It will rain tomorrow.

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Conclusions from Example 1

• 4. The game will be cancelled.
• (see 1 and 3)
• 5. Our team will surely lose the
• pennant.
• (see 2 and 4)

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Example 2

• 1. If it snows tomorrow, then we
• will go skiing.
• 2. If we go skiing, then we will be
• happy.
• 3. We are not going to be happy.

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Conclusions from Example 2

• 4. We will not be going skiing. (see 1 and
  3)
• 5. It is not going to snow
• tomorrow
• (see 1 and 4)

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Basic methods of reasoning

• Deduction
• Induction
• Abduction

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Deduction

• Applies the principles of logic in a way that
  preserves the truth.
• Move from a general premise, to a more specific
  observation.
• Example
• All ravens are blacks
• Toby is a raven
• Toby is black

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Induction

• Not truth preserving
• cannot be certain if the conclusion is true, or
• if some other conclusion would be better
• A way to generalize based on specific
  observations.
• Example
• I have seen 10 ravens
• All the ravens I have seen are black
• All ravens are black

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Problem of Induction

• A fundamental problem of epistemology and
  philosophy of science.
• Future observations could negate this
  "induction, so
• At what point can one make an induction that is
  logically valid?
• how many ravens must we observe to conclude that
  all ravens are black, in logically justifiable
  way?

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Problem of Induction (cont.)

• To date, there is no logical justification of an
  induction.
• The Logical Positivist movement in the beginning
  of the 20th century tried to address this
  problem.
• This a major issue in machine learning techniques.

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Abduction

• Does not produce a logically valid argument. The
  conclusion is possible, but that's all!
• Example
• If it is raining, then the streets are wet.
• The streets are wet.
  
• It is raining.

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Studies of Conditional Reasoning

• Cognitive psychologists study the conditional
  reasoning capabilities of human beings.
• Research shows that humans are not necessarily
  very good at conditional reasoning!

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Notation
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In Class Logic Experiment
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Rules of Logic

• These rules are used to draw valid conclusions.
• Modus Ponens
• P --gt Q
• P
• Q
• Modus Tolens
• P --gt Q
• Q
• P

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Rules of Logic

• These rules are used when building complex
  logical statements
• Eliminate AND
• P Q
• P
• Q
• Introduce AND
• P
• Q
• P Q

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Answers to Logical Syllogisms
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Common Errors in Logical Reasoning

• Denial of the Antecedent
• see syllogisms 3 4
• Affirmation of the Consequent
• see syllogisms 5 6

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Reasons for Errors In Reasoning

• Humans tend to interpret conditional statements
  as bidirectional.
• If and only if (lt--gt) is not the same as If then
  (-- gt)
• Humans have difficulty applying Modus Tolens
• syllogisms 7 8

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Example 1

• 1. If it rains tomorrow, then the game
• will be canceled.
• 2. If the game is canceled, then our
• team will surely lose the pennant.
• 3. It will rain tomorrow.

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Conclusions from Example 1

• 4. The game will be cancelled
• (see 1 and 3)
• 5. Our team will surely lose the
• pennant
• (see 2 and 4)
• Are these examples of Modus Tolens or Modus
  Ponens?

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Example 2

• 1. If it snows tomorrow, then we
• will go skiing.
• 2. If we go skiing, then we will be
• happy.
• 3. We are not going to be happy.

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Conclusions from Example 2

• 4. We will not be going skiing. (see 1 and
  3)
• 5. It is not going to snow
• tomorrow
• (see 1 and 4)
• Are these examples of Modus Ponens or Modus
  Tolens?

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Logical Reasoning Example
E
K
4
7
If a card has a vowel on one side, then it has an
even number on the other side?
Which cards would you turn over to test the
validity of the above statement?
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Experimental Results
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Reason for Error

• Problems applying Modus Tolens
• Must realize that
• P --gt Q
• is equivalent to
• Q --gt P

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Logical Reasoning Example

• Imagine you are a postal clerk sorting mail and
  that sealed letters must have an extra 10-lire of
  postage.
• If a letter is sealed, then it has a 50-lire
  stamp on it.
• Which envelopes must be turned over?

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• People are better at applying Modus Tolens to
  catch cheaters!