Boolean Logic

1
Boolean Logic

• Brian McLeish

2
Introduction

• History
• Method
• Operations
• Denial
• Conjunction
• Disjunction
• Distribution

• Syllogisms
• Applications
• Search
• Argue
• Surf
• Examples

3
History

• George Boole
• English Mathematician
• 1854 The Laws of Thought
• An attempt to mathematize thought
• De Morgan
• Algebraic rules
• Distribution

4
Unique Method

• Truth-Functional
• Each statement either true or false
• Arguments imply conclusion
• Deduction
• No y2 x

• Binary Arguments
• Range x 0,1
• Domain y 0,1
• 0 False
• 1 True
• Follows most algebraic rules

5
Operators Denial

• English not (it is not the case)
• For x
• 1 - x 1
• The denial of x is true
• x 0
• The assertion of x is false
• 1 - x y
• Everything that is not x is y.
• Modern notation -x replaces 1-x
• Not equivalent by normal algebra, but
  logically equivalent

6
Operators Conjunction

• English and
• Symbolized as multiplication
• For x and y
• xy 1
• Both x and y are true
• xy 0
• It is not the case that both x and y are true
• xx x
• x and x identical to x Domain x 0,1

7
Operators Disjunction

• English or
• Symbolized as addition
• For x and y
• x y 1
• Something is either x or y, but not both
• x y 0
• Nothing is either x or y
• Notice that does not cancel -
• Modern notation allows for both (inclusive
  or)

8
Distribution

• De Morgans Laws
• 1-(xy) (1-x) (1-y)
• 1-(x y) (1-x)(1-y)
• Remember binary values, exclusive or
• In modern notation
• -(x y) ? -x v -y
• -(x v y) ? -x -y

9
Syllogisms

• Use the following equations, then solve for x
• All x are y x(1-y) 0
• Some x are y xy v
• No x are y xy 0
• Some x are not y x(1-y) v

10
Applications

• Search Engines
• Use not, and, or to filter
• Argumentation
• Formal logic
• Based on De Morgan and Booles work
• Used to analyze arguments
• Computers
• Binary programming languages
• Digital music, video, et cetera