Introduction to Logic

• The Greeks studied a t type of logic called Aristotelian or syllogistic logic.
  • If we have time we will look at this in 3.6
• In the 18^th and 19^th century the study switched to symbolic logic.
  • In attempt to form a foundation for mathematics and science.
  • And this is the basis for building computers.
• George Boole
  • Foundations for Boolean Algebra
• Two values: true, false
• A sentence which can be judged to be true or false is a statement
• A simple statement conveys one idea.
  • Today is Wednesday.
  • Snow is falling.
• A compound statement is composed of simple statements and connectives
  • not, and, or, if then, if and only if
  • Today is not Wednesday.
  • Today is Wednesday and snow is not falling.
  • Snow is falling if and only if today is Wednesday.
  • If today is Wednesday then snow is falling.
• Representing statement symbolically
  • We use a lower case letter to represent a simple statement.
  • We only represent positive statements, if there is a not, remove it.
  • p: Today is Wednesday
  • q: Snow is falling
• Not
  • Called negation
  • The symbol is ~
  • The statement "Today is not Wednesday"
    • p: Today is Wednesday
    • The statement becomes : ~p
  • If the statement is "Snow is falling"
    • q: Snow is falling
    • What is ~q?
• Quantifiers make negation more difficult.
  • All, Some, None are the problem.
  • Consider: All trees are tall.
    • This statement is false, bonsai trees.
    • But the statement "No trees are tall" is false as well, consider the Giant Redwood trees.
    • So the negation is "Some trees are not tall."
  • Notice we did not have the problem with "Today is Wednesday".
  • Just learn the following:
    • All Are becomes Some Are Not
    • None Are becomes Some Are
  • Negate "No frogs are green"
  • This is close to the syllogistic logic mentioned above.
• And
  • Called a conjunction
  • represented by ∧
  • Today is Wednesday and snow is falling.
    • p: Today is Wednesday
    • q: Snow is falling
    • The statement becomes: p ∧ q
  • Snow is not falling and it is Wednesday
    • Becomes ~q ∧ p
• OR
  • Called disjunction
  • Symbol ∨
  • Today is Wednesday or snow is falling
  • Becomes: p ∨ q
  • Today is not Wednesday or snow is not falling
  • Becomes: ~p ∨ ~q
• Simple statements on the same side of a comma re grouped together with parentheses.
  • Snow is falling, and today is Wednesday or the month is October.
  • r: the month is October
  • Becomes q ∧ (p ∨ r)
  • Snow is falling and today is Wednesday, or the month is October.
  • Becomes: (q ∧ p) ∨ r
• Negation again
  • ~p is the negation of p: Today is not Wednesday
  • ~p ∧ q: Today is not Wednesday and snow is falling.
  • ~(p ∨ q) : It is not true that today is Wednesday or snow is falling.
• If - Then
  • Called a conditional
  • written →
  • If today is Wednesday then snow is falling
  • p → q
  • It is not true that if snow is falling then the month is October.
  • ~(q → r)
• If and only if
  • Called biconditional
  • Written ↔
  • Snow is falling if and only if today is Wednesday
  • q ↔ p
  • The month is October, if and only if Today is Wednesday and snow is falling.
  • r ↔ (p ∧ q)