Statements and logical connectives.

• We will study just a bit of symbolic logic.
• In symbolic logic symbols are used to represent written statements.
• Words like and, if, or and if .. then, if and only if are called connectives.
• For this section,
  • and will mean intersection
  • or will mean union. (inclusive or)
  • We will not deal with exclusive or
• A sentence which can be judged true or false is a statement.
  • The sun is shining.
  • The snow is falling.
  • I live in Edinboro and I like snow.
  • Each evening I teach class or I watch tv.
  • If I eat dinner then I go to sleep.
  • The first two are simple statements.
  • The last three are compound statements.
  • We represent simple statements with a single letter.
• Qualifiers are words which modify a statement.
  • All, some or no (none)
  • Normally negation is easy
  • Today is Monday. negated becomes Today is not Monday.
  • It becomes more difficult when all and some are involved.
  • All frogs are green.
    • Does not become No frogs are green.
    • It does become Some frogs are not green.
  • Some are changes to None are
  • Some are not changes to All are.
• Compound statements revisited.
  • Two or more simple statements connected with a connective.
  • Not is represented symbolically with ∼
    • p: It is Monday.
    • ∼p: It is not Monday.
    • q: No lakes are blue.
    • ∼q: Some lakes are blue
  • And: Conjunction
    • Symbol: ∧
    • It is Monday and I am teaching.
    • p: It is Monday.
    • q: I am teaching
    • p ∧ q
    • but, however and nevertheless are also and worlds.
  • Or: Disjunction
    • Symbol: ∨
    • It is Monday or I watch TV.
    • p: It is Monday.
    • q: I am watch TV
    • p ∨ q
  • We use parenthesis as normal, to group the order.
  • In a complex sentence, the comma tells us where to put the ()
  • It is Monday, or I am watching TV and eating Dinner.
    • p: it is Monday.
    • q: I am watching TV.
    • r: I am eating Dinner
    • p ∨(q ∧ r)
  • You can have soup, and salad or vegetable.
  • You can have soup and salad, or vegetable.
  • It is not true that Today is Monday and I am watching TV
  • ~(p ∧ q)
  • If then : Conditional
    • If I live in Edinboro then I like snow.
    • p → q
  • If and only If: biconditional
    • I live in Edinboro if and only if I like snow.
    • p ↔ q