Logical Argument An Example

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Logical Argument An Example

• We considered P1, P2, and Q under a particular
  (common sense) interpretation
• P1 If Socrates is human then Socrates is
  mortal true
• P2 Socrates is human true
• Q Socrates is mortal true
• Thus, they were merely logical constants to us
• P1true
• P2true
• Qtrue

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Logical Argument An Example

• Consider the two arguments
• P1 If Socrates is human then Socrates is
  mortal If J.B. broke his leg then J.B. is
  in pain
• P2 Socrates is human J.B. broke
  his leg
• Therefore
• Q Socrates is mortal J.B. is in pain
• Both arguments share the same structure
• P1 If X then Y
• P2 X
• Therefore
• Q Y
• Then for any interpretation I, as long as I
  satisfies P1 and P2, interpretation I must
  satisfy Q.

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Modus Ponens

• The generalized argument
• P1 X ? Y
• P2 X
• Therefore
• Q Y
• Because it captures the essence of both arguments
  and can be used for infinitely many more.

method of affirming (Lat.)
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Valid Arguments (Revisited)

• Suppose someone makes an argument
• P1,...,PN therefore Q
• The argument is called valid iff
• P1,,PN logically imply Q
• That is
• For any interpretation I that satisfies all Pj,
  interpretation I must necessarily satisfy Q
• Usually Pj and Q are somehow related statements
  and P1 PN can be true or false depending on
  the interpretation I.

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Propositional Logic

• Method 1
• Go through all possible interpretations and check
  the definition of valid argument
• Method 2
• Use inference rules to get from the premises to
  the conclusion in a logically sound way
• derive the conclusions from premises

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Method 1

• Section 1.3 in the text proves many
  arguments/inference rules using truth tables.
• Suppose the argument is
• P1, ,PN therefore Q
• Create a truth table for statement
• F (P1 PN ? Q)
• Check if F is a tautology.

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But Why? Recall

• Statement A implies statement B iff (A?B) is a
  tautology.
• In general
• premises P1, ,PN imply Q
• iff
• Statement F (P1 PN ? Q)
• is a tautology.

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Example 1

• P v (Q v R)
• R
• Therefore
• P v Q
• valid/invalid?
• (example 1.3.2 in the book, p. 31)

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Example 2

• P v Q v R
• R
• Therefore
• Q
• valid/invalid?

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Example 3

• P?Q
• P
• Therefore
• Q
• valid/invalid?
• (Modus ponens)

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Example 4

• P ? Q
• Q
• Therefore
• P
• valid/invalid?

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Example 5

• P ? Q
• Q
• Therefore
• P
• valid/invalid?
• (Modus tollens)

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Example 6

• P ? Q
• Therefore
• Q ? P
• valid/invalid?
• In fact, we proved earlier that
• (P ? Q) ? (Q ? P)

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Example 7

• P v Q
• P Q
• Therefore
• P Q
• valid/invalid?
• Any argument with a contradiction in its premises
  is valid by default

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Pros Cons

• Method 1
• Pro straight-forward, not much creativity ?
  machines can do
• Con the number of interpretations grows
  exponentially with the number of variables ?
  cannot do for many variables
• Con in predicate and some other logics even a
  small formula may have an infinite number of
  interpretations

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Method 2 Inference

• To prove that an argument is valid
• Begin with the premises
• Use valid/sound inference rules
• Arrive at the conclusion

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Inference Rules

• But what are these inference rules?
• They are simply
• valid arguments!
• Example
• X Y
• X Y ? Z W
• therefore
• Z W by modus ponens

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Derivations

• The chain of inference rules that starts with the
  premises and ends with the conclusion
• is called a derivation
• The conclusion is derived from the premises.
• Such a derivation makes a proof of arguments
  validity.

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Example 1

• (XY ? ZW) K
• XY
• Therefore
• ZW
• How?
• (XY ? ZW) K
• XY ? ZW by conjunctive simplification
• XY
• ZW by modus ponens

derivation
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Pros Cons

• Method 2
• Pro often can get a dramatic speed-up over truth
  tables.
• Con requires creativity and intuition (harder to
  do by machines).
• Con semi-decidable there is no algorithm that
  can prove any first-order predicate logic
  argument to be valid or invalid.

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Fallacies

• An error in derivation leading to an invalid
  argument
• Vague formulations of premises/conclusion
• Missing steps
• Using unsound inference rules, e.g.
• Converse error
• Inverse error

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Converse Error

• If John is smart then John makes a lot of money
• John makes a lot of money
• Therefore
• John is smart
• Tries to use this unsound inference rule
• A?B
• B
• Therefore
• A

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Inverse Error

• If John is smart then John makes a lot of money
• John is not smart
• Therefore
• John doesnt make a lot of money
• Tries to use this unsound inference rule
• A?B
• A
• Therefore
• B

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Truth of facts vs. Validity of Arguments

• The premises are assumed to be true ONLY in the
  context of the argument.
• The following argument is valid
• If John Lennon was a rock star then he was a
  woman.
• John Lennon was a rock star.
• Therefore
• John Lennon was a woman.
• But the 1st premise doesnt hold under the common
  sense interpretation.

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Summary

• Equivalence
• A ? B
• A holds iff B holds
• A is a criterion for B
• B is a criterion for A
• A logically implies (entails) B
• B logically implies (entails) A
• A and B are equivalently strong
• Statement F(A?B) is a tautology

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Summary

• Implication (Entailment)
• A entails (logically implies) B
• B follows from A
• A?B is a valid argument
• A is a sufficient condition for B
• B is a necessary condition for A
• If A holds then B holds
• A may be stronger than B
• Statement F(A?B) is a tautology

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The Big Picture

• Logic is used to verify validity of arguments.
• An argument is valid iff its conclusion logically
  follows from the premises.
• Derivations are used to prove validity.
• Inference rules (Table 1.3.1, p40) are used as
  part of derivations