Lecture Notes 8

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Lecture Notes 8

• CS1502

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Example Proof

• A ? (B ? C)
• (A ? B) ? (A ? C)

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(No Transcript)
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Valid Argument

• P1 P2 Pn Q
• Q is a tautological (logical) consequence of P1,
  P2, , Pn
• (P1 ? P2 ? ? Pn) ?Q is a tautology (logical
  necessity). NEW IDEA

Valid Argument
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Example

• Show ?P is a tautological consequence of ?(P ?
  Q).
• Methods of attack
• Boole
• Show ?P is a tautological consequence of ?(P ?
  Q).
• Show ?(P ? Q) ? ?P is a tautology.
• Fitch
• Show ?(P ? Q) is a valid
  argument

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Tautological Consequence
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Tautology
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Using Fitch
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Example

• Show ?P is not a tautological consequence of
  ?(P ? Q).
• Method of attack
• Boole
• Show ?P is not a tautological consequence of ?(P
  ? Q).
• Show ?(P ? Q) ? ?P is not a tautology.
• Build a world
• Show ?(P ? Q) is an invalid
  argument

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Not a Tautological Consequence
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Not a Tautology
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Build a World

• Let P be assigned true and Q false. ?(P ? Q) is
  true while ?P is false.

conclusion
premises
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Example

• Show the following argument is valid.
  Cube(b) ?(Cube(c) ? Cube(b)) ?Cube(c)

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Logical Consequence
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Logical Necessity
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Fitch
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Non-consequence

• Show the following argument is invalid.
  Cube(a) ? Cube(b) ?(Cube(c) ? Cube(b))
  ?Cube(c)

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Counterexample
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Inference Patterns

• Modus Ponens P ? Q P Q

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Tautological Consequence
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Tautology
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? Elimination

• P ? Q P Q

? Elim
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? Introduction

• P Q P ? Q

? Intro
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? Elimination

• P ? Q P Q

? Elim
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? Introduction

• P Q Q
  P P ? Q

? Intro
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Inference Patterns

• Modus Tollens P ? Q ?Q ?P

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Modus Tollens