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 19 th 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.
P P T F Q T F T T F F F F Q T F T T T F T F P ~P F T P and Q 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 or not 1 0 1 1 0 0 0 0 1 0 1 1 1 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 1. 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.
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 20 th 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) F ~Q T F ~Q T P T F T F T T ~P F T F F 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