Introduction to Logic

• Logic is the study of arguments.
• It has been studied for a very long time (your book has some history)
• But we are interested in symbolic Logic
  • Gottfried Wilheim Leibniz (1646-1716) did early work
  • George Bool (1815-1864) did much to form a discipline
  • We describe this as boolean logic sometimes.
• Logic is important
  • It is the foundation for many legal documents and arguments
  • It is how computers work.
  • Spock
• A statement is a sentence that can be judged to be true or false
  • Today is Monday
  • The sun is a star
• We define a number of words in logic to connect statements
  • and indicates two statements must both be true for the compound statement to be true.
    • Today is Monday and the sun is a star. False
    • The sun is a star and the earth is a planet.
  • or either one statement or the other (or both) must be true for the statement to be true.
    • Today is Monday or the sun is a star. True
  • not negates the statement.
    • Today is not Monday
    • The sun is not a star.
  • if then
    • If today is Monday then the sun is a star.
  • if and only if
    • Today is Monday if and only if the sun is a star.
  • Sometimes, but, however and nevertheless are equivalent to and.
• To make life easier, we represent statement symbolically
  • Simple statements, those without and, or, not, if-then, if and only if are represented as a single letter
  • The sun is a star. Today is Monday

  • p: The sun is a star.
    q: Today is Monday.
    	

  • We can then write the letter in place of the words.
• Not is negation
  • Simple statements are no problem.
    • Today is Monday, Today is not Monday
    • The sun is a star, the sun is not a star.
  • But statements involving all and none tricky
    • "All frogs are green"
    • Is not negated by "No frogs are green"
    • It is negated by "Some frogs are not green"
    • "No cars are fast"
    • "Some cars are fast"

  • Form of Statement	Form of Negation
    All are	Some are not
    None are	Some are
    Some are	None Are
    Some are not	All are

    
    
  • We use the symbol ~ for negation
  • So ~p from above is : The sun is not a star.
  • We should represent all statements as positive
  • So "It is not snowing" should be

    r: It is snowing.
    The statement becomes ~r
    	

  • We build Truth Tables to represent our logical operations.
  • Negation is easy

    p	~p
    T	F
    F	T

• And statements are called Conjunctions
  • The symbol is ∧
  • p ∧ q
  • The truth table is

    p	q	p∧q
    T	T	T
    T	F	F
    F	T	F
    F	F	F

• The Or statement is called the Disjunction
  • The symbol is ∨
  • p ∨ q
  • The truth table is

    p	q	p∨q
    T	T	T
    T	F	T
    F	T	T
    F	F	F

  • Note: this is the inclusive or
    • A or B or Both
    • The exclusive or ⨁
    • The truth table is

      p	q	p⊕q
      T	T	F
      T	F	T
      F	T	T
      F	F	F

    • But we will not discuss xor
• Complex compound statements
  • We use a comma to indicate where () should be.
  • The simple statements on the same side of the comma should be grouped together within parentheses
  • The dinner includes soup, and salad or vegetable of the day

    p: the dinner includes soup
    q: the dinner includes salad
    r: the dinner includes vegetable  of the day
    
    p ∧  (q ∨ r)
    	

• The if then statement is called a conditional
  • Symbol →
  • If the day is Friday then I stay home.

    p: the day is Friday
    q: I stay home
    
    p → q
    	

  • The truth table is

    p	q	p→q
    T	T	T
    T	F	F
    F	T	T
    F	F	T

  • This is a bit strange.
  • Consider "If you get an A then I will buy you an ice cream cone"
    • If you get and A and I buy you an ice cream cone everything is true.
    • If you get an A and I don't buy you an ice cream cone, I lied
    • If you don't get an A, it doesn't matter what I do.
  • Note this is the only one where the order matters
• If and only if is a biconditional
  • Symbol: ↔
  • I stay at home if and only if the day is Sunday.

    p: I stay at home
    q: The day is Sunday
    
    p ↔ q
    	

  • The truth table is

    p	q	p↔q
    T	T	T
    T	F	F
    F	T	F
    F	F	T

• Exercises
  • P 99 17-26
  • 27- 38
  • 39-48
  • P 100 49-56
  • 57-66
  • P 111 5-18
  • 20-24
  • P 112 27-36
  • P113 53-56
• A compound statement that is always false is a self contradiction
• A Tautology is a compound statment that is always true.
• An implication is a conditionial statement that is a tautology
• P 121 7-24
• 25-30
• 31-36