Intro' Formal Logic Mainly Sec' 1'1

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Intro. Formal LogicMainly Sec. 1.1
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A Classic Example

• All men are mortal,
• Socrates is a man,
• so Socrates is mortal.
• True or False? Prove it

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A more complex example

• blood on the glove matched the murder victims'
  DNA
• a trail led from the crime scene to a man's
  bedroom
• the gloves, allowing for shrinkage, fit
• no one else had entered the bedroom
• the man is therefore guilty of the murder.

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Definitions

• Statement A sentence that is either true or
  false.
• Logical connective
• binary connective
• conjunction -- (AB)
• disjunction -- (A U B)
• unary connective
• negation -- A
• Implication -- A?B
• antecedent (A)
• consequent (B)
• Equivalence
• A?B (A?B B?A)

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Well-formed formula (wff)

• Well-formed formula (wff)
• An expression that is a legitimate string.
• 1. connectives within parentheses, innermost
  parentheses first.
• 2.
• 3. ,V
• 4. ?
• 5. ?
• order of precedence is top to bottom.

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Truth table
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Logic connectives in English word

• It is a warm-up for your GRE logic section
• A is a sufficient condition for B ?
• B is a necessary condition of A
• ??? A only if B ???
• Answer A?B
• ?? What is the Negation ??
• Peter is tall and thin

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Tautologies

• Tautology A wff whose truth value is always
  true.
• Contradiction A wff whose truth value is always
  false.
• Equivalent wffs P,Q P?Q P is Qs dual and
  vice versa
• Some Tautological equivalences (P8)
• Verify the equivalence (truth table).

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Some Tautological equivalences

• Commutative Properties
• AVB? BVA, AB?BA
• Associative Properties
• (AVB)VC?AV(BVC),(AB)C?A(BC)
• Distributive Properties
• AV(BC)?(AVB)(AVC),A(BVC)?(AB)V(AC)
• Identity properties
• AV0A, A1?A
• Complement properties
• AVA1, AA0

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De Morgans Law

• Some information about De Morgan if you are
  interested (not required for test).
• ?? Find the negation of the statement??
• I am not on my knees, and never will be
  again
• De Morgans Law will help
• (AVB)?AB , (AB)?AVB

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Logical Connectives in the Real World

• Web Search Engine
• Programming language

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How to test the tautology?

• Truth table
• Algorithm (Tautology Test)
• Formal logic --derivation rules (will be covered
  later)

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Algorithm

• An algorithm is a set of instructions that can be
  mechanically executed in a finite amount of time
  in order to solve some problem.
• Psudocode The compromise form to describe
  algorithms
• Algorithm Tautology Test (See book, p12).

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Old Assignment I

• Excercises1.1
• 2,5,6(c,d), 7(a,c), 14(e,h), 17(e), 26(c),
• Use Algorithm Tautology Test to prove the
  proposition
• (AB)(AVB)?B