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Re: The seven step series

by Bruno Marchal :: Rate this Message:

Hi Marty,

On 01 Jul 2009, at 18:57, m.a. wrote:

Hi Bruno,
I'm responding to the quiz (see below). What does "high non booleanity" mean in the context of para.2?

We will need more math for the details, but "boolean" refer to classical or even platonist logic, when applied in a frame where we can agree that propositions are either true or false.Like arithmetic: you surely agree that a postive integer is either even, or not even.

Classical logic is the one used implicitly or explicitly in most "scientific" discourses, especially in (classical) mathematics. "Boolean" comes from Boole who wrote a book "The laws of thought", and which plays some role in the birth of mathematical logic and computer science. But classical logic has a very long story, both in East and West.

>
> Example-exercise:
>
> 1°) Let A be the set {0, 1, 2, 3}. ("A" is said to be a local name for
> the set {0, 1, 2, 3}. And local means that such a name is used in a
> local context. One paragraph later "A" could designed another, so be
> careful). If "A" names {0, 1, 2, 3}, we will write "A = {0, 1, 2, 3}".
>
> OK, so with A = {0, 1, 2, 3}. Which of the following propositions are
> true
>
> 1) the number 2 is a member of A True

OK.

> 2) the number 12 is a member of A False

OK.

> 3) the number 12 is not a member of A True

OK.

> 4) (3 BELONGS-TO A) True: but you haven't told us whether the parenthesis cancels the locality of brackets.

OK.

You can suppress anything in the notations as far as you, and those who will read the text, can figure out what you mean. Here I could have simply

4) 3 BELONGS-TO A.

> 5) all members of A are numbers True

OK.

> 6) one element of A is not a number False: we've established that zero is a number.

OK. (I am not sure that we have established that, nor even what you mean by "established", but we surely welcome 0 in the numbers. Note that in the beginning 0 was rejected. And both the numbers 1 and 2 takes time to be accepted as number, for the reason that "number" means originally numerous. The "num" of "numerous" has the same origin as "number".

> 7) A can be defined in intension in the following way A = {x SUCH-THAT
> x is a positive integer little than 4} True...if zero is considered a positive integer.

OK.

>
> 2°) Same questions with the set A = {0, 1, 2, 3, ... , 61, 62, 63}
1. True
2. True
3. False
4. True: same question as 4 above.
5. True
6. False: zero is a number
7. False

OK.

Very good. A "sans faute" we would say in french. In cyclism, you would be suspected taking drug! Bravo!

next lesson: (but take your time)

Could you tell me if you understand and/or remember those definitions (where a and b denoting arbitrary sets):

(a INTERSECTION b) = {x SUCH-THAT (x BELONGS-TO a) and (x BELONGS-TO b)}

(a UNION b) = {x SUCH THAT (x BELONGS-TO a) or (x BELONGS-TO b)}

Can you compute

{1, 2, 7, 789} UNION {1, 2, 7, 5678} = ?

{1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = ?

Do you remember the empty set? Can you compute:

{1, 2} UNION { } = ?

{1} UNION { } = ?

{1, 2, 3} UNION {1, 2, 3} = ?

{ } UNION { } = ?

{1, 2} INTERSECTION { } = ?

{1} INTERSECTION { } = ?

{1, 2, 3} INTERSECTION {1, 2, 3} = ?

{ } INTERSECTION { } = ?

Now, an important distinction which will follow us through ... forever. I suggest you read attentively the next two paragraphs two times before breakfast, every day for one week. :), Really take all your time. It concerns the notion of operation, and relation.

INTERSECTION and UNION, are operations on sets, like addition (+, or PLUS) and multiplication (*, or TIMES) are operation on numbers. This means, typically, that, if x and y denote numbers, then x + y, and x * y, will denote, or are equal to, numbers. For example 3 + 4 is equal to 7.

Similarly, if x and y denotes, or are equal, to sets, then x INTERSECTION y denotes, or is equal to, some set. For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?

Operations are important, as you can guess, but relations are as well important. Operations lead to new elements, new objects. From the numbers 2 and 3, you get the element 5. Relations pertains or does not pertain, or equivalently, leads to true or false.

Example. The relation LESS-THAN, among the numbers. (x LESS-THAN y) is true if x is less than y. So (3 LESS-THAN 56) is true, and (56 LESS-THAN 3) is false. An important relation pertaining on sets is the relation of inclusion, or of being a subset of a set.

By definition a set x will be said included in y (or be said subset of y), when all the elements of x are among the elements of y. We will write (x INCLUDED-IN y) when the set x is included in the set y.

For example, the set {1, 2} is included in the set {3, 2, 1}, but is not included in the set {3, 1}.

Exercise: in the following, what is true or false?

45 LESS-THAN 67

0 LESS-THAN 1

999 LESS-THAN 4

{1, 2, 3} INCLUDED-IN {4, 1, 5, 2, 3, 8}

{1} INCLUDED-IN {1, 2}

oops, I must go. You are lucky ;)

Bruno

http://iridia.ulb.ac.be/~marchal/

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