Welcome to Gaia! ::

(Please note that this may in fact be a little more generalized logic than set logic, but it works for set logic). (I also hope this is along the lines of what this guild is for, else I have just spent forever typing something not needed).

Please ignore the striked portion for right now.
What needs to be first understood:
If I could, sucking out all your knowledge, teaching you this, then giving you all your knowledge back would be my plan of action, but I cannot. Instead, let me make a few statements.
First, you should understand that a class of objects is neither a set of objects or a list of objects. In fact, it is both and more.
Second, please understand nothing is allowed to be assumed, and even though we don't prove the theorems, you should not assume it until a theorem is states it. In other words, until we state a = a, you cannot assume a = a

Why set logic? Will I ever use it?

Set logic is logic involving sets, which are unordered groupings of elements. An element can be anything, and is only limited by what the set allows. So in fact, much of the logic used is set logic.
{a, b, c} is a set.
{b, c, a} is the same set, since sets are unordered. Please note this is different from list, which are ordered.
{} denotes a set.
[] denotes a list.

So {a, b} = {b, a}. But [a,b] != [b,a].

Now, let us plunge head first into set logic. First, we have 3 undefined symbols.


So, we have a class of objects C.

We have a second symbol, which is actually shown as (depending upon who you ask) either a upside down v or n with out the tail. For the time being, we will use the n to be this next symbol.

Finally, we have !,~,`, or ' depending upon who you speak to. For the time being, we will use ~.

There is a defined symbol, which looks like a c with a line in the middle (so a small curvy E if you will. for now, we will use c to be this symbol. This symbol represents the statement "is in" or "is an element of".

So a c {a, b, c}.

Ok, now, we have 3 undefined symbols, and one symbol which represents words. Next we have 7 undefined axioms, or assumptions we assume are true, but that cannot be proven. They are:

1. If a c C, then ~a, a n b c C.
2. If a, b c C, then a n b = b n a.
3. If a, b, d c C, then (a n b) n d = a n (b n d).

Up to this point, they shouldn't be too hard to understand. But now for the fun part.

4. If a, b, d c C, and a n ~b = d n ~d, then a n b = a.
5. If a, b, d c C, and a n b = a, then a n ~b = d n ~d.

So, these two are really one iff statement.

6. If a, b c C, and a = b, then ~a = ~b.
7. If a, b, d c C, and a = b, then a n d = b n d and d n a = d n b.

So, there they are. The unproven statements of logic. You may have to stare at them a wile to get them, I know I did. Now, for the even funner part, the 40 theorems of logic (which I will not prove, since that would be harmful to minors 3nodding ).

Some of these you know dearly, like 1, which is the basis of the reflexive property. Note, some of these seem to be restating what the 7 axioms did, but that is ok.

1. a = a.
2. If a = b, then b = a.
3. If a = b and b = c, then a = c.
4. a n b = b n a.
5. a n (b n d) = (a n b) n d.

Let us pause to give another symbol, though this one we can define. It is commonly shown as the u or v. So let us define it. a u b = ~(~a n ~b). This may seem a bit complicated, but don't worry about it.

Onwards, or ばんさい, as the Japanese like to yell.

6. a u b = b u a.
7. a u (b u d) = (a u b) u d.
8. a u a = a n a = a. (FYI, this is called the Idempotency of n and u.)
9. a n (b u d) = (a n b) u (a n d).
10. a u (b n d) = (a u b) n (a u d). (These two are the basis of the distributive property.
11. a u ~a = 1.
12. a n ~a = 0.

Hey look, new symbols. Though these are in fact numbers, they do not represent what we think of as number goes. Give a class C, 1 represents everything in it, and 0 represents nothing. Here, let us use a these new toys some more.

13. a n 1 = 1 u 0 = 1.
14. a u 1 = 1.
15. a n 0 = 0.
16. ~(~a) = a.
17. ~0 = 1, ~1 = 0.
18. ~(a u b) = ~a n ~b.
19. ~(a n b) = ~a u ~ b. (18/19 are called DeMorgans Law)
20. a u (a n b) = a n (a u b) = a.

Our friend c appears again.

21. a = b iff a c b and b c a.
22. a c b iff a n b = a.
23. a c b iff a u b = b.
24. a c b iff a - b = 0.

Hey a new symbol (again!). Let use define. a - b = a n ~b.

25. a c b iff ~a u b = 1.
26. a c a.
27. (a n b) c a c (a u b).
28. 0 c a c 1.

My notes are somewhat lacking here, so sorry if I do not have all of them. I will try to get them soon.

29.
30.
31. If a c b and b c d, then a c d.
32.
33. If a c b, then ~b c ~a.
34.
35.
36. (also note, some of these deal with a change in notation, we will not worry about them for now).

Ok, back on track. Let us introduce a new symbol, which should be a + with a dot on top, but we will just use the +. a + b = (a - b) u (b - a).

37. a + b = b + a; a + (b + d) = (a + b) + d.
38. a u (b + d) = (a u b) + (a u d); a n (b + d) = (a n b) + (a n d).
39. a + 0 = a.
40. a + a = 0.

Actually, they go past 40, but we will not care about those. Here, let me just show you 41., and you will understand why.

41. Let F be a boolean fucntion. The f(a) = (f(1) n a) u (f(0) n ~a).

When used correctly, 41 will make grown men cry. It is scary, and is where my teacher forgot he was the only one with a Ph.D.

So, once I get the missing few, we will have the first 40 (41) Theorems of logic, the base symbols, and the 7 axioms. Now, to make this whole thing make sense, let us name these properties so you can actually identify with them.

n is AND.
u is OR.
+ is XOR.
~ is NOT.

For those not familiar, I will give examples of AND, OR, and NOT it both boolean and set logic.

Boolean.

A/B/A AND B
1/1/ 1
1/0/ 0
0/1/ 0
0/0/ 0

A/B/A OR B
1/1/ 1
1/0/ 1
0/1/ 1
0/0/ 0

A/B/A XOR B
1/1/ 0
1/0/ 1
0/1/ 1
0/0/ 0

Let A = {a, b}
Let B = {b, c}

So:
A AND B = {b}
A OR B = {a, b, c}
A XOR B = {a, c}

Go back and read it again, it will make more sense (if you are familiar with these 4 terms).

You lost me here:




Keep in mind that most of us (well, maybe just me?) know jacksquat about logic. I know some elements of debate theory, but most of that was discussion of fallacies. It's the classic quandry of teaching: The more one understands something, the harder it is to understand someone who doesn't understand it.

EDIT: Actually, the more I look at this, the more I think it's useless to those who don't already understand set logic.

Jesus ******** cowboys.




Seriously? That made me want to cry.
I'll stick with my view of logic;
as in dissecting and cutting away.
Thank you, occam.

Oh, I see. Now it's starting to make sense.

It's mostly the introduction that doesn't make sense--it's badly organized. I think once I work that out, it'll get easier.

I'm honestly interested in this kind of thing, which is why I'm being so critical.

For fun, Occam is applied to this logic. Without it, this would be lethal.

So. Set logic looks like math.
Aaaand, that's as far as I got.
Seriously, I must be a moron.

That was so unbelieveably over my head that it made me want to end my own life. I.. confusion. One eye keeps closing before the other one.. You.. you broke me.

I'm going to stick with my logic.

My logic tells me that a=a can safely be assumed because people like you do the work for me. And I thank you for that. Truly, I thank you for that.

I understand the terminology now! Thankie heart

My graphing calculator has them in it...

EDIT: Could you please explain why #s 4 and 5 work out the way they do?