Statements and Logical Connectives

• Logic is the study of arguments.
• In Symbolic Logic we use symbols to represent written statements
• A statement is a sentence that can be judged to be true of false.
  • It is snowing.
  • Edinboro is in Pennsylvania
  • The earth is flat.
• A simple statement conveys only one idea.
• A compound statement conveys two or more ideas.
  • It is snowing and the sun is shining.
  • The earth is flat or Edinboro is in Pennsylvania.
  • It will snow if the temperature is below 32
  • You will get an A if and only if you have an average of 90%
• Connectives are words that link statements.
  • They include, and, or, if then, if and only if
• Not
  • Not changes the truthfulness of a statement.
  • I am a college student
  • I am not a college student.
  • if p is a statement, ~p is the negation of p
  • Both a statement and the negation of a statement can not be true.
  • If p is "I have a dog", state ~p
  • Negation of compound statements is more difficult, and we will deal with this later.
• Quantifiers
  • These are words like some, none or all
  • Quantifiers make negations more difficult.
  • if p is "All dogs are beagles", ~p is "Some dogs are not beagles"
  • if p is "Some classes are easy", ~p is "All classes are not easy"
  • In general,
    • "some are", is negated with "none are"
    • "All are", is negated with "some are not"
• Conjunctions
  • A conjunction is a statement that includes and
  • The test will cover probability and the test will cover logic
  • Or sometimes it includes but, however, or nevertheless
  • It is daytime, but the sun is not shining
  • if p and q are statements, p and q is written p ∧ q
  • p is "Frogs are green", q is "Trees are tall", Represent this as a logical expression:
    • Frogs are green and trees are tall.
    • Trees are tall but frogs are not green.
    • Trees are not tall and frogs are not green.
• Disjunctions
  • Statements involving the inclusive or from disjunctions.
  • Inclusive or means one, or the other or both.
    • Sue will go to the movies or dinner
  • Exclusive or means one or the other, but not both.
    • Sue will go to the movies or dinner
  • if p and q are statements, p or q is written p ∨ q
  • Express each:
    • Frogs are green or trees are tall.
    • Trees are tall or frogs are not green.
    • Trees are not tall or frogs are not green.
• When a compound statement involves more than two simple statements, we use ()
  • The need for () is often indicated by commas.
  • You can have soup, and salad or vegetables
  • p ∧(q ∨r)
• ~p & q only negates the p.
  • It is not true that frogs are green and trees are tall
  • ~(p∧q)
• Conditional
  • Statements containing if - then
  • If it rains then I will get wet.
  • p → q
  • p, it rains is the antecedent
  • q, I will get wet, is the consequent
• Biconditional
  • I involves the phrase if and only if
  • I will go to college if and only if I can afford the tuition.
  • p ↔ q

Truth Tables for Negation, Conjunction and Disjunction.

• Give truth tables for not, and and or
• Do 122 5, 9, 10, 21, 23, 24