George boole and boolean logic

[jackson6612]

Were computers or any such devises there during George Boole's times who lived through 1815-1864? I don't think there were any such things, then what did motivate or make Boole to invent such a new field? Please help me with this. Thanks.

 

[micromass]

No, there were no computers during the life of Boole.

He formulated the concept of Boolean algebra to formalize the rules of formal logic. He thought (correctly), that all the logical laws could be derived from a small set of axioms. And then he came up with Boolean algebra.

He even had the (crazy) idea, that his Boolean algebra should be though in law school and to jurors. In that way, it was certain that they would all reason logically.

 

[jackson6612]

No, there were no computers during the life of Boole.

He formulated the concept of Boolean algebra to formalize the rules of formal logic. He thought (correctly), that all the logical laws could be derived from a small set of axioms. And then he came up with Boolean algebra.

He even had the (crazy) idea, that his Boolean algebra should be though in law school and to jurors. In that way, it was certain that they would all reason logically.

Hi Micro

What is formal logic, by the way? If computers weren't there and Boolean algebra (I suppose an algebra about 'yes' and 'no' statements) had nothing to do with future invention of such machine, then wasn't 'regular' algebra there for all that formal logic?

And what are those logical laws?

Please I'm not a math or science student - quite a layman in these areas - so please try to explain in as much detail as possible.

I thank you for all the help and your time.

 

[micromass]

Logic is the study of statements such as "If it rains AND I'm going inside, THEN I bring an umbrella". Specifically, we want to see the truth values of this statement.

Formal logic tries to replace logic statements with symbols. For example, if we write the statement above in formal logic, then it becomes $$p\wedge q\rightarrow r$$ (where $$\wedge$$ is a symbol for AND, and were $$\rightarrow$$ is a symbol for THEN[/tex].

Now, if we study such symbolic statements, then we see that some statements are always true. For example $$p\rightarrow p$$ and $$p\wedge q\rightarrow q\wedge p$$. But there are possibly infinite many such statements. A natural question is, is there a collection of statements which are always true, such that any other true statement is derivable from our collection. A positive answer to this, is given by Boole.

The idea of Boole, was to replace logic statements by algebraic symbols. For example "NOT p" is represented by "-p". p AND q is represented by "p.q". Then he gave 10 statements (which are always true), and every other statement can be derived from these 10 statements.

So, the original idea of a Boolean algebra was only applicable to formal logic. Fortunately, other mathematicians have seen that the concept of a Boolean algebra was also applicable to other occasions. And even later than that, the concept of Boolean algebra is used in computer science.

 

[jackson6612]

Thank you, Micro. You have really answered very well. Please remember that I'm not a science or math student so many of the questions may seem very silly to you so please be calm and don't get frustrated! Thank you.

Logic is the study of statements such as "If it rains AND I'm going inside, THEN I bring an umbrella". Specifically, we want to see the truth values of this statement.

Formal logic tries to replace logic statements with symbols. For example, if we write the statement above in formal logic, then it becomes $$p\wedge q\rightarrow r$$ (where $$\wedge$$ is a symbol for AND, and were $$\rightarrow$$ is a symbol for THEN[/tex].

Now, if we study such symbolic statements, then we see that some statements are always true. For example $$p\rightarrow p$$ and $$p\wedge q\rightarrow q\wedge p$$. But there are possibly infinite many such statements. A natural question is, is there a collection of statements which are always true, such that any other true statement is derivable from our collection. A positive answer to this, is given by Boole.

The idea of Boole, was to replace logic statements by algebraic symbols. For example "NOT p" is represented by "-p". p AND q is represented by "p.q". Then he gave 10 statements (which are always true), and every other statement can be derived from these 10 statements.

So, the original idea of a Boolean algebra was only applicable to formal logic. Fortunately, other mathematicians have seen that the concept of a Boolean algebra was also applicable to other occasions. And even later than that, the concept of Boolean algebra is used in computer science.

So simple logic is study of statements of the type "IF", "AND", and "THEN" where each is dependent/conditional on the other.

And formal logic is the study those 'verbose' statements of 'simple' logic using symbols.

Is there some mathematical logic as compared to formal logic?

"But there are possibly infinite many such statements. A natural question is..." - There are only three statements possible: $$p\rightarrow p$$, $$p\wedge q\rightarrow q\wedge p$$ and $$q\rightarrow q$$. Can we make more than three statements of out it?

And what answer was given by Boole?

What were those 10 statements?Thanks a lot for all the help and your time.

Best wishes
Jack

 

[CRGreathouse]

Is there some mathematical logic as compared to formal logic?

I don't really understand the question. Formal logic is a part of mathematics. There are different kinds (propositional, first-order, second-order, etc.) and different rules (classical, intuitionistic, etc.).

 

[jackson6612]

Help, please!

 

[CRGreathouse]

Help, please!

You need to clarify most of what you wrote if you're to get any help, I'm afraid.

"But there are possibly infinite many such statements. A natural question is..." - There are only three statements possible: $$p\rightarrow p$$, $$p\wedge q\rightarrow q\wedge p$$ and $$q\rightarrow q$$. Can we make more than three statements of out it?

We can make lots. $$p\rightarrow p\rightarrow p$$, for example -- whether this is $$(p\rightarrow p)\rightarrow p$$ or $$p\rightarrow(p\rightarrow p)$$ depends on exactly how we define things.

I don't know what statements Boole referred to, and the definition of what they are is unclear to me so I can't just come up with them on my own. See, for example, Sloane's A101273.

 

[micromass]

Well, to see what 10 statements Boole came up with, we first need to fix some terminology:

- $$p\wedge q$$ means "p AND q"
- $$p\vee q$$ means "p OR q"
- $${}^\neg p$$ means "NOT p"
- $$p \rightarrow q$$ means "IF p THEN q"
- $$p\leftrightarrow q$$ means "p IF AND ONLY IF q"
- 0 means "FALSE", that is: 0 is the statement which is always false
- 1 means "TRUE", that is: 1 is the statement which is always true

The 10 statements of Boole are the following:
ASSOCIATIVITY:
- $$(p\wedge q)\wedge r~\leftrightarrow p\wedge (q\wedge r)$$
- $$(p\vee q)\vee r~\leftrightarrow p\vee (q\vee r)$$
COMMUTATIVITY:
- $$p\wedge q~\leftrightarrow~q\wedge p$$
- $$p\vee q~\leftrightarrow~q\vee p$$
ABSORPTION:
- $$p\vee (p\wedge q)~\leftrightarrow~ p$$
- $$p\wedge (p\vee q)~\leftrightarrow~ p$$
DISTRIBUTIVITY:
- $$p\vee (q\wedge r)~\leftrightarrow~(p\vee q)\wedge (p\vee r)$$
- $$p\wedge (q\vee r)~\leftrightarrow~(p\wedge q)\vee (p\wedge r)$$
COMPLEMENTATION:
- $$p\vee ({}^\neg p)~\leftrightarrow~1$$
- $$p\wedge ({}^\neg p)~\leftrightarrow~0$$

These 10 logic statements are now known as the axioms of a Boolean algebra.

 

[JeremyHorne]

Several items:

There is a 16 column Table of Functional Completeness that contains not only Boole's four traditional operators, functions, or connectives - however you want to term them, but 12 others, such exclusive or, nor, nand, tautology, reverse implication, and contradiction. Each of these can be a function, AS WELL as a result of a computation, thus demonstrating that binary logical space is closed. See http://home.earthlink.net/~jhorne18 for "The General Theory and Method of Binary Logical Operations" and others on the subject.

As to the history of logic, there is always Aristotle A Priori Analytica and A Posteriori Analytica for your original set theory/quantifier logic.

Now, for some history on binary computers - Check out the April 1988 edition of Scientific American on some southeast Asian archipelago people using ropes as a binary computer for marine calculations some 600 or more years ago. I didn't check the accuracy of everything in the following, but it is intriguing:
http://wiki.answers.com/Q/What_is_the_history_of_binary_system
However, Leibniz was working with binary concepts; "Explication de l'Arithmétique Binaire" may be found at: http://ads.ccsd.cnrs.fr/docs/00/10/47/81/PDF/p85_89_vol3483m.pdf