The Arithmetic of Reasoning
LOGIC AND BOOLEAN ALGEBRA
Chessa Horomanski
&
Matt Corson
Computers
 Ask us questions,
correct our grammar,
calculate our taxes
But…
 Misunderstand what we’re sure we told them, lose
our work, refuse to respond to our commands!
Gottfried Wilhelm Leibniz
 German mathematician
 1694 – mechanical calculating device that could +,-,x,÷

Stepped Reckoner
Used the binary numeration system
in its calculation
 Expressed all numbers as sequences
of 1s and 0s

 An improvement on the first known mechanical adding
machine

Pascal’s Pascaline (could only + and -)
Universal System of Reasoning
 Leibniz set out to create a universal system of
reasoning
 Wanted his system to work “mechanically” according
to a simple set of rules for deriving new statements
from ones already known
 Saw that statements would have to be represented
symbolically

Sought to develop to a universal characteristic (a universal
symbolic language of logic)
Augustus De Morgan



British mathematician
Believed that the 19th century separation
between math and logic was artificial and
harmful
Worked to put many mathematical concepts
on a firmer logical basis to make logic more
mathematical
George Boole


English mathematician
Published The Mathematical Analysis of Logic
and An Investigation of the Laws of Thought
Approached logic in a new way reducing it to a
simple algebra, incorporating logic into
mathematics
 Pointed out the analogy between algebraic symbols
and those that represent logical forms


Boolean algebra (the basis for modern computer logic
systems)
Key Element of Boole’s Work
 His systematic treatment of statements as objects
whose truth values can be combined by logical
operations in much the same way as numbers are
added or multiplied
 Example:
If each of two statements P and Q can be either true
or false, then there are only four possible truth-value
cases to consider when they are combined.
Q
P
T
T
F
T
F
F
F
F
“P and Q”

Q
P
T
T
T
T
F
F
T
F
P
~P
T
F
F
T
True if and only if both of
those statements are true
“P or Q”

True whenever at least one
of the 2 component
statements is true
not
 A statement and its negation
always have opposite truth
values
Logical Operation Tables
 Easy step to translate T and F into 1 and 0
and
1
1
0
1
0

or
0
0
0
1
0
1
1
1
not
0
1
0
1
0
0
1
Became a workable arithmetic system (with
many of the same algebraic properties as
addition, multiplication and negation of
numbers)
Use of Symbols
 Use of symbols for the basic logical connectives can
be used in translating logical arguments into a
language that a machine can understand
Example
 “If the sun is not shining, then I’ll go either to the
mall or to the movies.”
P: “If the sun is shining”
Q: “I’ll go the mall”
R: “I’ll go to the movies”
Written as:
You Try it!
Directions: Assign variable names to the simple
phrases, and write the statement using those
variables along with the logical connectives
I ate my lunch but I did not eat breakfast.
2. It is false that this triangle has both a 30° angle and
a 60° angle.
1.
De Morgan
 An influential, persuasive proponent of the algebraic
treatment of logic
 Helped to refine, extend and popularize the system
by Boole
 2 laws, now named for him


not-(P and Q) ↔ (not-P) or (not-Q)
not-(P or Q) ↔ (not-P) and (not-Q)
Charles Sanders Pierce
 Resurrected and extended De Morgan’s important
contributions
 Mathematicians – want to get to their conclusions as
quickly as possible and so are willing to jump over
steps when they know where an argument is leading
 Logicians – want to analyze deductions as carefully
as possible, breaking them down into small, simple
steps
Logic in Technology
 Reduction of mathematical reasoning to long
strings of tiny, mechanical steps was a critical
prerequisite for the “computer age”
 20th century advances in the design of electrical
devices


Electronic calculators that more than fulfilled the promise of
Leibniz’s Stepped Reckoner
Standard codes for keyboard symbols allowed such
machines to read and write words
Boolean Algebra in the Real World
 1950s – used for telephone switching units
 Now – up and coming electronic computers
 Used in natural sciences, and in disciplines such as
linguistics, law, and computer technology
 Today – used everyday to help people when doing
searches on the Internet


Commonly referred to as a Boolean search
The three Boolean operators used today are as follows: AND,
OR, NOT.
Conclusion
 Work of Boole, De Morgan, C.S. Pierce and others in
transforming reasoning from words to symbols and
then to numbers that has led to the modern
computers

Modern computers – use rapid calculations of long strings of
1s and 0s which empowers the computers to “think”
01010100 01101000 01100101
01000101 01101110 01100100
Activity
 Complete the following truth tables.
P ~ Q
~(P  Q)
~Q
P
F
T
T
T
T
F
F
F
~P
T
~Q
T
F
F
F
T
F
T
Timeline
 1642 - Blaise Pascal and his Pascaline
 1694 - Leibniz creates the Stepped Reckoner
 1847 - Boole publishes The Mathematical Analysis of Logic
 1853 - De Morgan introduced his laws of negation in logic
 1854 - Boole publishes An Investigation of the Laws of
Thought
 1880 - C. S. Pierce led movement to put more logic in math
and soon Boolean algebra turns to strings of binary
 1900s - 1s and 0s turn on and off electrical circuits
 Present day - logic can be seen in our technology.
References
 Berlinghoff, William P. and Fernando Q. Gouvea. Math Through the




Ages. Oxton House Publishers, Maine, 2002.
“De Morgan’s laws.”
http://en.wikipedia.org/wiki/De_Morgan%27s_laws
Ensley, Douglas E. and J. Winston Crawley. Discrete Mathematics:
Mathematical Reasoning with Proof with Puzzles, Patterns, and
Games. John Wiley & Sons, New Jersey, 2006.
“George Boole.” http://en.wikipedia.org/wiki/George_Boole
“George Boole.” MacTutor Math History Archive. http://wwwhistory.mcs.st-andrews.ac.uk/Biographies/Boole.html