7. 7
The parts of syllogistic logic, also known by the name term logic, are the
analysis of the judgments into propositions consisting of two terms that are related
by one of a fixed number of relations, and the expression of inferences by means of
syllogisms that consist of two propositions sharing a common term as premise, and
a conclusion which is a proposition involving the two unrelated terms from the
premises.
Major premise: Eating sweets every day,
is placing yourself at risk for diabetes.
Minor premise: John eat sweats everyday
Conclusion: Therefore John is placing himself
at risk for diabetes
ARISTOTLE (384 BC – 322 BC)
8. 8
The propositions of a syllogism can be any of 4 types
S = Subject of Conclusion
P = Predicate of Conclusion
SaP All S are P All humans are mortal
SeP No S are P No humans are perfect
SiP Some S are P Some humans are healthy
SoP Some S are not P Some humans are not clever
ARISTOTLE (384 BC – 322 BC)
9. 99
English mathematician and a founder of
the algebraic tradition in logic
• He revolutionized logic by
applying methods from symbolic
algebra to logic.
• Traditional (Aristotelian) logic
relied on cataloging the valid
syllogisms of various simple forms.
• Boole sought to give symbolic
form to Aristotle's system of logic.
GEORGE BOOLE (1815 - 1864)
11. His methods were outlined in two major works,
The Mathematical Analysis of Logic (1847) and
An Investigation into the Laws of Thought (1854)
11
GEORGE BOOLE (1815 - 1864)
12. http://www.sciencephoto.com/image/223560/large/H4020347-George_Boole-SPL.jpg
An investigation into the Laws of Thought,
on Which are founded the Mathematical Theories of Logic and Probabilities
12
GEORGE BOOLE (1815 - 1864)
Boolean algebra
Relationship between two possible values: true or false, 1 or 0
Boolean numbers are not the same as binary numbers
Both Boolean math and binary notation use the same two ciphers: 1 and 0
The difference is that Boolean quantities are restricted to a single bit (either 1 or 0),
whereas binary numbers may be composed of many bits adding up in place-
weighted form to a value of any finite size
Operators:
13. 0 1
0 1
1 1
0 1
0 0
0 1
0 1
1 0
AND is represented as multiplication
OR is represented as addition
Not is represented as complement or negation
0 X 0 = 0
0 X 1 = 0
1 X 0 = 0
1 X 1 = 1
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1
In the world of Boolean algebra, there are only two possible
values for any quantity and for any arithmetic operation: 1 or 0
13
GEORGE BOOLE (1815 - 1864)
the opposite of its value
14. 14
The Father Of The Digital Computer
invented the first automatic electronic digital computer
John Vincent Atanasoff (19
15. John Vincent Atanasoff (19
1903 Hamilton, New York
Senior Electrical EngineerMathematics School Teacher
15
16. John Vincent Atanasoff (19
Monroe calculator
Clifford E. BerryJohn Atanasoff
Associate-professor in
physics and mathematics
16
17. 17
John Vincent Atanasoff (19
His machine would have to
be electronic-the
mathematical operations
would occur by changes in
electrical charges rather than
mechanical movements
Use digital, system with base
two, or binary, mathematics
that would ensure precision
and be compatible with the
on-off nature of electronics.
18. John Vincent Atanasoff (19
Atanasoff-Berry Computer:
ABC
18
• Binary system, Boolean logic and
a regenerative capacitor
memory
• Designed to solve up to 29 linear
equations
• Had no CPU, but using vacuum
tubes for digital computation,
used punched cards for input
Computability logic is opposed to classical logic
Computations would be done by what Atanasoff termed
“direct logical action”
19. Lotfali Askar Zadeh (1921)
American, Russian, Iranian,
Azerbaijani
Mathematician, Electrical
Engineer, Computer Scientist
and Professor of Computer
Science
19
20. Lotfali Askar Zadeh (1921)
20
Qualita
tive
Subjecti
ve
http://www.maycomplicateeverything.com/files/gimgs/8_oven.png
Real world is very
complex
Qualitative Measure
22. Lotfali Askar Zadeh (1921)
The law of the excluded
middle
p or ~p must be true.
no middle true proposition
between them
Aristotle’s day: precise as
possible
Soft
unsharp
blurred
elastic
Fuzzyrejects the law of the excluded middle and allows as a truth value any
real number between 0 and 1
22
23. Lotfali Askar Zadeh (1921)
Fuzzy Logic = Computing with Word
The basis for fuzzy logic is the
basis for human communication.
Because fuzzy logic is
built on the structures of
qualitative
description used in everyday
language.
Fuzzy logic is
designed to solve
problems in the
same way that
humans do
Machines can
be
programmed
to process
approximate
data and deal
with the gray
areas of life.
Fuzzy logic's approach to control
problems mimics how a person would
make decisions, only much faster.
Real world is very complex
23
25. 25
Lotfali Askar Zadeh (1921)
How fast they should shift from one setting to another
"If the temperature is cool, then set the motor speed on slow"
"If the temperature is just right, then set the motor speed on medium."
IF...THEN
convert inputs to outputs-one fuzzy set into another
Temperature sets (cold, cool, just right, warm and hot)
Motor speed sets (very slow , slow , medium, fast and very fast)
Fuzzy Inputs
Fuzzy Outputs
sensors
measure changing input conditions
microprocessors
store and process fuzzy rules
26. 26
Output:motorspeed
Input : temperature sets
Cold Cool Just Right Warm Hot
Very Fast
Fast
Medium
Slow
Very Slow
IF Warm
THEN Fast
IF Just Right
THEN Medium
IF Cool
THEN Slow
Lotfali Askar Zadeh (1921)
30. 30
Subtraction implies the existence of negative numbers: 5 - 3 is the same thing
as 5 + (-3), and in Boolean algebra negative quantities are forbidden.
There is no such thing as division in Boolean mathematics, either, since
division is really nothing more than compounded subtraction.
GEORGE BOOLE (1815 - 1864)
To calculate the 2's complement of an integer, invert the binary equivalent of the
number by changing all of the ones to zeroes and all of the zeroes to ones (also
called 1's complement), and then add one.
For example,
0001 0001(binary 17) 1110 1111(two's complement -17)
NOT(0001 0001) = 1110 1110 (Invert bits)
1110 1110 + 0000 0001 = 1110 1111 (Add 1)
The most significant (leftmost) bit indicates the sign of the integer;
therefore it is sometimes called the sign bit.
If the sign bit is zero, then the number is greater than or equal to zero, or positive.
If the sign bit is one, then the number is less than zero, or negative.
2's Complement
31. 31
John Vincent Atanasoff (19
Atanasoff did not
recognize the application
of Boolean algebra to his
problem
He devised his own direct
logical action by trial and
error
He was unaware that in
1938, Claude Shannon
proved that two-valued
Boolean algebra could
describe the operation of
two-valued electrical
Claude Shannon
Notes de l'éditeur
First of all, ARISTOTLE , he talked about the Laws of Logical Thought which is the starting point of the Logic.
This is an Apple (If) this a fresh apple.This cannot be both fresh apple and rot apple in the same timeAnd this apple is fresh apple is True
Logic was established as a discipline by Aristotle, who gave it a fundamental place in philosophy.He asserted that “…”
A All S are P universal affirmatives All humans are mortal.E No S are P universal negatives No humans are perfect.I Some S are P particular affirmatives Some humans are healthy.O Some S are not P particular negatives Some humans are not clever.(ใช้ตัวสีแดง เพื่อให้ได้รู้ว่า a, e, I,oคือ อะไร)มีข้อสงสัยว่า เมื่อ s แทน humans ในทุกๆ type แล้ว ทำไม p ในแต่ละ type ถึงไม่เหมือนกันคะ?????
George Boole Born in the English industrial town of LincolnHis parents were Mary Ann Joyce and John Boole. John made shoes & Mary Ann was a lady's maid.Boole was lucky enough to have a father who interested in science and in particular the application of mathematics to scientific instruments. Young George took to learning like a politician to a pay rise and, by the age of eight, had outgrown his father's self-taught limits. >>growing up to be a mathematical genius
1844 he was concentrating on the uses of combined algebra and calculus to process infinitely small and large figures,It was published by Boole in the “Transactions of the Royal Society”.In the same year, he received the Society's Royal Medal 1854 Boole's published his most important work“An investigation into the Laws of Thought of Aristotle, on Which are founded the Mathematical Theories of Logic and Probabilities”
http://www.allaboutcircuits.com/vol_4/chpt_7/2.htmlHe codified several rules of relationship between mathematical quantities limited to one of two possible values:true or false, 1 or 0.It should be clearly understood that Boolean numbers are not the same as binary numbers. Whereas Boolean numbers represent an entirely different system of mathematics from real numbers, binary is nothing more than an alternative notation for real numbers. The two are often confused because both Boolean math and binary notation use the same two ciphers: 1 and 0. The difference is that Boolean quantities are restricted to a single bit (either 1 or 0), whereas binary numbers may be composed of many bits adding up in place-weighted form to a value of any finite size. The binary number 100112 ("nineteen") has no more place in the Boolean world than the decimal number 210 ("two") or the octal number 328 ("twenty-six").There are three common Boolean operatorsMost search tools offer some type of Boolean search option as a way of constructing your search, and indicating the relationships between your search terms.
The first three sums make perfect sense to anyone familiar with elementary addition. The last sum, though, is quite possibly responsible for more confusion than any other single statement in digital electronics, because it seems to run contrary to the basic principles of mathematics. Well, it does contradict principles of addition for real numbers, but not for Boolean numbers. Remember that in the world of Boolean algebra, there are only two possible values for any quantity and for any arithmetic operation: 1 or 0. There is no such thing as "2" within the scope of Boolean values. Since the sum "1 + 1" certainly isn't 0, it must be 1 by process of elimination.
