Statements and Logical Connectives

• We begin our brief study of logic.
• Aristotle taught a form of logic called Aristotelian logic.
  • We will look at this in section 3.6
• Leibniz (1646 - 1716)
  • And my Academic Ancestor!
  • Symbolic Logic foundations
• Boole (1815-1864)
  • Built the subject of symbolic logic
  • Boolean Algebra
• Symbolic Logic is
  • The foundation of mathematical proof.
  • Allows you to detect fallacies (sort of)
  • And is the foundation of modern electronic computing devices
• There are five operations we are worried about
  • and - set intersection
  • or - set union
  • not - set compliment
  • conditional - if then
  • biconditional - if and only if
• A note on or
  • We are discussing the inclusive or (union), A or B or both
  • Not the exclusive or , A or B but not both.
• A statement is a sentence that can be judged true or false.
  • The sky is blue
  • It is Tuesday and it is snowing.
  • Disney World is in Idaho.
  • Elephant jokes are funny.
• A statement is a simple statement if it only conveys one thought
• If a statement conveys multiple thoughts it is a compound statement
• The simplest operation is negation
  • For some statements it is just adding a not.
    • Today is Tuesday, is negated to be Today is not Tuesday
    • It is cold negated is
  • But if the words all, some ,none are included it becomes more difficult
    • All men are tall - this excludes short men
      • Not all men are tall - this is not correct
      • Some men are not tall - this is correct.
    • Negating some, all, none
      • Some are becomes None are
      • All are becomes Some are not
    • Negate the following
      • All butterflies are insects
      • All houses are connected to the internet.
      • Some turtles do note have claws
      • No teachers made the roster
      • Some pedestrians are in the crosswalk
• Representing a simple statement
  • Today is Tuesday can be represented as a single letter.
  • p: Today is Tuesday
  • we normally choose lower case letters p, q,r, s, ...
  • We only represent positive statements.
  • Represent the following
    • Math is fun.
    • Elephant jokes are really funny.
• Representing not statements
  • Use the ~ symbol (or perhaps draw a line over it, or ¬)
  • It is not cold outside
  • represent this as a positive statement
  • not (it is cold outside)
  • p: It is cold outside
  • ~p
  • Represent the following
    • The sun is not visible.
    • Today is not Tuesday
• Representing AND statements
  • And is called the conjunction, symbolized by ∧ (looks like an A)
  • ∧ also looks like ∩ for a good reason.
  • Today is Tuesday and it is cold.

  • p: Today is Tuesday
    q: it is cold
    
    p ∧ q
    	

  • Later we will find this statement is only true if both simple statements are also true.
  • Represent the following
    • You will study your notes and you will do well on the test.
    • You will hear an elephant joke and you will laugh.
    • It is not Sunday and I am going shopping.
• Representing OR statements
  • Or is called the disjunction, symbolized by ∨
  • Unless otherwise stated, we mean inclusive or
  • ∨ looks like ∪ for a good reason.
  • You will participate in class or I will tell elephant jokes

  • p: You will participate in class
    q: I will tell elephant jokes
    
    p ∨ q
            

  • Represent the following
    • It is Tuesday or I am going to eat cake.
    • It is not warm or the dog is happy.
• For compound statements with multiple operations, the comma represents where we should place ()
  • The heater is on and we are not out of fuel, or we will be cold

  • p: the heater is on
    q: we are not out of fuel
    s: we will be cold
    
    (p ∧ q) ∨ s
    	

  • Represent each:
    • You may have soup or you may have salad, and you may have a sandwich.
    • You may have soup, or you may have salad and you may have a sandwich.
• Representing Conditionals
  • If - then statements, represented by →
  • If you answer questions then I will not tell elephant jokes

  • p: you answer questions
    q: I will tell elephant jokes
    
    p → ~q
           

  • Represent the following
    • If you eat your vegetables then you will grow strong.
    • If you do not study then you will not do well on the test.
• Biconditional
  • If and only if
  • ⇔
  • I will tell elephant jokes if and only if you do not participate

  • p: I will tell elephant jokes
    q: you participate
    
    p ⇔ ~q
    	

  • Represent the following
    • A bobcat can can purr if and only if the panther has a long tail.
    • School is delayed if and only if the temperature is below 20°
• Some special problems:
  • it is false that ... is the same as ~( expression)
  • it is not true that ... ~(expression)
  • It is false that the sun is shining and the day is warm.

  • p: the sun is shining
    q: the day is warm
    
    ~(p ∧ q)
           

  • neither-nor is the same as not both simple statements and connect with an and
  • John is neither handsome nor rich

  • p: John is handsome
    q: John is rich
    
    ~p ∧ ~q
           

  • but, however, nevertheless are equivalent to and
  • The sun is shining but it is not warm

  • p: The sun is shining
    q: it is warm
    
    p ∧ ~q
           

  • The sun is shining however it is cold

  • p: the sun is shining
    q: it is cold
    
    p ∧ q