Logical Inferences

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Logical Inferences
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DeMorgans Laws

• (p ? q) ? (p ? q)
• (p ? q) ? (p ? q)
• The law of the contrapositive
• (p ?q) ? (q ?p)

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What is a rule of inference?

• A rule of inference allows us to specify which
  conclusions may be inferred from assertions
  known, assumed, or previously established.
• A tautology is a propositional function that is
  true for all values of the propositional
  variables (e.g., p? p).

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Modus ponens

• A rule of inference is a tautological
  implication.
• Modus ponens ( p ? (p ? q) ) ? q

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Modus ponens An example

• Suppose that we know that the following 2
  statements are true
• If it is 11am in Miami then it is 8am in Santa
  Barbara.
• It is 11am in Miami.
• By modus ponens, we infer that it is 8am in Santa
  Barbara.

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Other rules of inference

• Other tautological implications include
• p ? (p ? q)
• (p ? q) ? p
• q ? (p ? q) ? p
• (p ? q) ? p ?q
• (p ? q) ? (q ? r) ? (p ? r) hypothetical
  syllogism
• (p ? q) ? (r ? s) ? (p ? r) ? (q ? s)
• (p ? q) ? (r ? s) ? (q ? s) ? (p ? r)

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Memorize understand

• DeMorgans laws
• The law of the contrapositive
• Modus ponens
• Hypothetical syllogism

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Common fallacies

• 3 fallacies are common
• Affirming the converse
• (p ? q) ? q ? p
• If Socrates is a man then Socrates is mortal.
• Socrates is mortal.
• Therefore, Socrates is a man.

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Common fallacies ...

• Assuming the antecedent
• (p ? q) ? p ? q
• If Socrates is a man then Socrates is mortal.
• Socrates is not a man.
• Therefore, Socrates is not mortal.

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Common fallacies ...

• Non sequitur
• p ? q
• Socrates is a man.
• Therefore, Socrates is mortal.
• On the other hand (OTOH), this is valid
• If Socrates is a man then Socrates is mortal.
• Socrates is a man.
• Therefore, Socrates is mortal.
• The form of the argument is what counts.

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Examples of arguments

• Given an argument whose form isnt obvious
• Decompose the argument into assertions
• Connect the assertions according to the argument
• Check to see that the inferences are valid.
• Example argument
• If a baby is hungry then it cries.
• If a baby is not mad, then it doesnt cry.
• If a baby is mad, then it has a red face.
• Therefore, if a baby is hungry, it has a red face.

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Examples of arguments ...

• Assertions
• h a baby is hungry
• c a baby cries
• m a baby is mad
• r a baby has a red face
• Argument
• ((h ? c) ? (m ? c) ? (m ? r)) ? (h ? r)
• Valid?

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Examples of arguments ...

• Argument
• Gore will be elected iff California votes for
  him.
• If California keeps its air base, Gore will be
  elected.
• Therefore, Gore will be elected.
• Assertions
• g Gore will be elected
• c California votes for Gore
• b California keeps its air base
• Argument (g ?c) ? (b ? g) ? g (valid?)

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Characters

• ? ? ?
• ? ? ? ? ? ? ?
• ? ? ?
• ? ?
• ? ? ? ? ?
• ? ? ? ? ? ? ? ?