# Boolean algebra

The clever question is ontological and the answer cannot be provided in a way that is immediately clear to the intelligent beginner. And perhaps the answer cannot be provided clearly in the first explanation of itself. And perhaps the answer has to be indirectly noticed since the question is not itself clear. The ontological question is "What is that?". And two people look at the "that" which is pointed at and both perceive some different "thing". Perhaps that is one way to indirectly notice the answer. There are deep things like that, and there are not-deep things like that. So when I look at the "that" do I see what is deep or what is not-deep? And if I then were to explain didactically what "that" is, and provide an explanation of what is not-deep, perhaps that would be to disinform from what is deep. And if what is deep cannot be explained after the explanation of what is not-deep, and I have already explained that I only see what is not-deep, then maybe what is deep can no longer be explained ever. Which would be most unfortunate.

Or perhaps there are two ways of perceiving "that" which is not-deep and we have not yet decided which way that is not-deep we prefer. And so I jump in and say what "that" is and only explain one way of perceiving "that" and do not acknowledge that the other not-deep way of perceiving "that" even exists. Then perhaps any who had followed my explanation would think that "that" is only what I had said it to be.

They would be like people looking at a cube of three dimensions, told only to look at one side because "that" is a square, and then only have one corner explained so they miss every part of "that" except for a ninety degree angle. But maybe that is not even a cube, maybe it is a multidimensional hypercube capable of navigating the metacomplex for purposes quite unknown. Now in that case if I look at "that" and give an immediate explanation of it as a ninety degree angle which is two lines meeting at a point where one line is perpendicular to the other does that mean that it is false. No it is true. And it is still true even though some other person can look at the same thing and see a complete square including the corner I have described. And some person who sees deeper can look at the same thing and see a complete cube, including the square and corner. And some other person can look at the same thing and see a multidimensional hypercube.

My reason for the previous dialogue is to describe what is the problem with answering the question "What is that?". The question "What is that?" is only three words so seems so easy to answer in any particular case. But it is called the ontological question so it must be much deeper than it seems or else it would not be called the ontological question.

8QnMQ/x

8 Questions: Where and When, Who and What, How and Why, Which and Whether.

To the power of n: Where else and When else, Who else and What else, How else and Why else, Which other and Whether other.

Multiplied by MQ: Meta Questions directed towards answers to 8Qn in terms of structure, process, function, purpose, form, context, content.

Divided by x: A possibility versus probability categorisation in order to weight the values of the analytic.

a + 0 = a and a . 1 = a

a + 1 = 1 + a and a . 0 = 0

a . b = b . a and a + b = b + a

(a + b) + c = a + (b + c) and (a . b) . c = a . (b . c)

a + (b . c) = (a + b) . (a + c) and a . (b + c) = (a . b) + (a . c)

a + ~a = 0 and a . ~a = 0

a + a = a + a and a . a = a

~(a + b) = ~a + ~b and ~(a . b) = ~a . ~b

The above schematic is a prime pattern where further reduction is not possible without losing the information to which it refers. Addition of information to the above schematic serves no good purpose. If we classify the given prime pattern as the Boolean Prime then we can examine it according to a designed method. The first point is to distinguish between whether it is correct or incorrect and whether it is true or false. By correct or incorrect we are checking whether it is an accurate statement of the Boolean Prime. By true or false we are checking whether the Boolean Prime is itself true or false. If we discover that the given schematic is an accurate statement of the Boolean Prime and that the Boolean Prime is true, then we have a Prime Pattern that is cleared as sound. Prime Patterns of this sort are collectable items of significant value and the game then is to use the given Prime Pattern as a measure of the sort of thing to be collected. We could name the following as the Prime Pattern of P's and Q's.

p means "p is true" Affirmation.

~p means "p is false" Negation.

p v q means "either p is true, or q is true, or both" Or.

p ^ q means "both p and q are true" And.

p > q means "if p is true, then q is true". If - then.

p < q means "since q is true, then p is true. Since - then.

p <-> q means "p and q are either both true or both false" If and only if.

These are cardinal articulations that must be known previous to any more sophisticated information. The idea is to question in regard to the more sophisticated information what are the cardinal articulations that must be clearly stated at the previous level. And then to collect together only those patterns detailed as cardinal articulations and to exclude out any thing that is not specifically a cardinal articulation. Imaginatively it must be relatively easy to detail in a single volume what are the reduced prime patterns stated in terms of cardinal articulations. It is clear when the stated pattern is correct since we cannot find anything previous to itself. We could call it "The Book of Cardinal Articulations and Prime Patterns".

The prime pattern of p's and q's must exist before any truth tables can be detailed. P is true or p is not true is the decision that must be made before we can use the prime pattern of p's and q's. Given two different decision makers the same p at the same time can be both true and false since the nature of what truth is does not demand that two different decision makers assign the same value to p. This is provable. How do we decide whether p is true or p is false? A judgement as to the truth or falsity of p is an early decision that determines all the outcomes that are derived from p. The reason that two different decision makers can correctly assign contradictory value to p is because truth depends on the existence of three variables. We can provide an example of what is true in terms of 2+2=4 is correspondent, coherent and useful. By correspondent we mean it refers accurately to something outside itself. By coherent we mean it belongs to a larger system within which it makes sense. And by useful we mean it has a utility in regard to some purpose. Truth as hard as 2+2=4 is correspondent, coherent and useful. For any p its truth or falsity depends on correspondence, coherence and utility. And for that reason if two different decision makers are judging from two different systems with two different purposes and two different perspectives, then those two decision makers could assign two different values for any p at the same time, such that where one says p is true for the same p the other will say it is false. The decision as to whether p is true is a non-arbitrary choice. That means for any p that is said to be true based on given reasons, it is not possible to arbitrarily claim the contradictory, that p is false.

For any p that we state as true, we must be able to demonstrate these reasons: P is externally correspondent to some fact. P is internally coherent within a larger system. P is useful according to some pragmatic purpose.

For any p that we state as false, we must be able to demonstrate these reasons: P does not externally correspond to any known fact. P is entirely incoherent within a larger system. P has no use according to some pragmatic purpose.

It is not enough that some p fails to meet our criteria for truth that it is stated as false. For any p to be stated as false it must meet our criteria for falsity. And that necessitates the use of the universal operator which is to say: Every p is true, false, or neither true nor false. Since only those p which meet all of our criteria for falsity are called false, and only those p which meet all of our criteria for truth are called true, therefore some p must exist which are neither true nor false.

The following which begins with the Pythagoras code could be called the prime pattern of number. One plus two plus three plus four equals ten, proving the first four numbers are the sum of number. x squared equals x whether x equals one or x equals zero, proving one does not equal zero. Any number squared divided by two plus half the original number is equal to the number of times each of the consecutive integers in the original number combine with itself and each of the others once, as in dominoes. Pi squared and the square root of pi are both constant. A place holder exists that is neither one nor zero. What is the place holder?

It is useful to clarify in particular two peculiar understandings that seem to be evident from the analysis up to this point. One is that given our criteria for falsity and our criteria for truth that it is a necessity we accept some propositions exist that are neither true nor false, such that using the universal operator we must say for all propositions every p is true or false or neither true nor false. The second peculiar understanding is that given two decision makers as to the truth or falsity of p it is at least possible in many not all cases that the two decision makers can take a contradictory position.

For those two reasons we are forced to notice a difference between the truth of 2+2=4 and other propositions that meet our stated criteria for truth. Because unlike many of those other propositions the truth of 2+2=4 is not easily contradicted. That is, given two decision makers it is not clear how either decision maker could correctly claim the falsity of 2+2=4. That would suggest that we can add a fourth criteria that is met by this type of proposition. And that fourth criteria is that this type of proposition is capable of universal abstraction.

Given that, we can suppose a difference between any proposition that meets our three basic criteria for truth and those propositions that also meet the fourth criteria of being capable of universal abstraction. Understanding what is universal abstraction is a significant matter. Two plus two equals four and 2+2=4 are both late languages in sign label for something that is itself universally abstract. To understand this it is claimed that "equals means the power of making different things the same". So the sign label "equals =" is a functional power that combines what is different so that they seem the same. Then what is the universal abstraction?

It is the case that ** and ** is **** and that is not the same as @@ and @@ is @@@@. So because the universally abstract pattern is common to both we invent a sign label system which is above both. There is nothing in the sign label 2 which tells us anything about number. It is only that we have become so accustomed to thinking 2 and two are what they signify that we tend to not notice the complete lack of any content.

Now the third peculiar understanding is to clarify the difference between first propositions and the cardinal articulations of prime patterns. First propositions may nearly always be neither true nor false because of their hypothetical nature as first propositions. Meaning that they probably fail our truth criteria on grounds of not being coherent within a larger system since that larger system does not yet exist at the time of first propositions. We would tend to call these hypothetically true propositions and build our truth tables as if they were true unless we find a contradiction in which case that would count as incoherence so they would move to being hypothetically false. And if we do not find a contradiction we would forget that our first propositions are only hypothetically true.

The cardinal articulation of prime patterns is capable of universal abstraction. That is because the truth of the prime patterns is first in structure rather than first in time sequence. For this reason when it comes to hard truth comparable with the strength of 2+2=4 we do better to investigate the Boolean prime pattern, the prime pattern of p's and q's and the prime pattern of number. Given these three examples of what is meant by cardinal articulations then other prime patterns that meet all four criteria for truth can be collected together.

To re-iterate we can reinforce our earlier definition of the criteria for truth and falsity:

For any p that we state as true, we must be able to demonstrate these reasons: P is externally correspondent to some fact. P is internally coherent within a larger system. P is useful according to some pragmatic purpose. P is capable of universal abstraction.

For any p that we state as false, we must be able to demonstrate these reasons: P does not externally correspond to any known fact. P is entirely incoherent within a larger system. P has no use according to some pragmatic purpose. P is not capable of universal abstraction.

And why this is useful is because it means we can assess beforehand the hypothetical truth and falsity of our propositions that are neither true nor false. That is, where a proposition is neither true nor false, and we find that it is likely to be capable of universal abstraction then we are more confident about using it as hypothetically true even though it has not yet met all of our other criteria for truth. And where a proposition is neither true nor false, and we find that it is likely to never be capable of universal abstraction then we are more confident about using it as hypothetically false even though it has not yet met all of our other criteria for falsity.

a - 0 = a and a / 1 = a

a - 1 = -1 + a and a / 0 = 0

a / b = a / b and a - b = -b + a

0 - a = -a and 1 / a = 1 / a

1 - a = - a + 1 and 0 / a = 0

(a - b) - c = a - (b - c) and (a / b) / c = a / (b / c)

a - (b / c) = (a - b) / (a - c) and a / (b - c) = (a / b) - (a / c)

a - ~a = 1 and a / ~a = -1

a - a = 0 and a / a = 1

~(a - b) = ~a - ~b and ~(a / b) = ~a / ~b

The previous schematic is a continuation of the rules of the Boolean Prime. As stated it may be correct or incorrect. And when correctly stated it may be true or false. What is clear is that the prime pattern does lend itself to be correctly stated in its most true formulation. The difficulty is as to whether it is possible to check the correctness and truth of the patterns. It is not the case that a more correct true pattern can simply be offered as an alternative to the one given since we would be in the same position in regard to checking the more correct true pattern. And it is not the case that a different decision maker can affirm or negate any particular terms of the patterns, since any alteration to the given patterns would itself have to be demonstrated to be accurate and true. Therefore it is asked whether any person who has the stated patterns can follow a process by which to check their truth without having to refer to a different decision maker. And the reason that matters is because we may not want to agree with the other decision maker on any area of this that we cannot demonstrate in a universally abstract way.

The process then, that is best used, in order to check the correctness and truth of the patterns will now be explained. What makes the process in itself attractive is that it self-validates. And also that the process is organic and once understood can be individually used by different decision makers in the way they prefer, without the requirement to obtain confirmation from any other decision maker.

The first rule is to determine what is the simplest term from either pattern. And once that is determined to only include those terms which most closely cohere with the first term. The second rule is to not include any term that is contradictory to the first term. And the third rule is to bring in any not previously stated terms that clearly are coherent with the first term.

a = a and ~a = ~a and a =/= ~a

"a equals a", "not a equals not a", "a does not equal not a".

a + 0 = a and a - 0 = a and ~a + 0 = ~a and ~a - 0 = ~a

"a plus zero equals a" "a minus zero equals a" "not a plus zero equals not a" "not a minus zero equals not a"

a.1 = a and a/1 = a and ~a.1 = ~a and ~a/1 = ~a

"a multiplied by one equals a" "a divided by one equals a" "not a multiplied by one equals not a" "not a divided by one equals not a"

a.0 = 0 and a/0 = 0 and ~a.0 = 0 and ~a/0 = 0

"a multiplied by zero equals zero" "a divided by zero equals zero" "not a multiplied by zero equals zero" "not a divided by zero equals zero"

If we then clarify what we have decided is coherent up to this point without going any further it means we are then able to check our most simple terms on their own. If we can confirm these terms as accurate and correct then they are the measure of truth that we apply to the next group of terms that we select.

a = a and ~a = ~a and a =/= ~a

a + 0 = a and a - 0 = a and ~a + 0 = ~a and ~a - 0 = ~a

a.1 = a and a/1 = a and ~a.1 = ~a and ~a/1 = ~a

a.0 = 0 and a/0 = 0 and ~a.0 = 0 and ~a/0 = 0

All we do here before moving forward is confirm to ourselves that these are the simplest terms of the pattern and that from the given pattern we could not start with any more simple. If they are the simplest terms then all we need do is confirm our own agreement that as stated they are correct and that they are true because coherent without internal contradiction between themselves. We do not need do more at this stage, but cannot reasonably progress further until that point is accomplished.

Now crucially, given the patterns as stated in this document, whether they are correct as stated and whether in their most correct formulation they are true is no longer the matter. Because the method for checking them for correctness and truth is the way to organically formulate them in their most correct statement of themselves. And the next stage in that regard is to clarify a second small group that satisfy similar conditions to the first one stated. To do so it is correct to use any terms of the first group as required.

a + 0 = a and a - 0 = a and ~a + 0 = ~a and ~a - 0 = ~a

"a plus zero equals a" "a minus zero equals a" "not a plus zero equals not a" "not a minus zero equals not a"

a + 1 = 1 + a and a - 1 = -1 + a and ~a + 1 = 1 + ~a and ~a - 1 = -1 + ~a

"a plus one equals one plus a" "a minus one equals minus one plus a" "not a plus one equals one plus not a" "not a minus one equals minus one plus not a"

0 + a = a and 0 - a = -a and 0 + ~a = ~a and 0 - ~a = - ~a

"zero plus a equals a" "zero minus a equals minus a" "zero plus not a equals not a" "zero minus not a equals minus not a"

a + a = a + a and a . a = a and a - a = 0 and a / a = 1

"a plus a equals a plus a" "a multiplied by a equals a" "a minus a equals zero" "a divided by a equals one"

a + ~a = 0 and a . ~a = 0 and a - ~a = 1 and a / ~a = -1

"a plus not a equals zero" "a multiplied by not a equals zero" "a minus not a equals one" "a divided by not a equals minus one"

Isolating the second group of terms so that they can be themselves checked, the idea then is to determine whether we have maintained consistency throughout. That is, where the process of checking our terms has involved some correction of the original information, have we gone back to the original information and made any adjustment required by this process. In this way it does not matter as such what changes we make on the one condition that we correct towards coherency such that at any point in time the given pattern is more accurate and coherent than the previous statement of itself.

a + 0 = a and a - 0 = a and ~a + 0 = ~a and ~a - 0 = ~a

a + 1 = 1 + a and a - 1 = -1 + a and ~a + 1 = 1 + ~a and ~a - 1 = -1 + ~a

0 + a = a and 0 - a = -a and 0 + ~a = ~a and 0 - ~a = - ~a

a + a = a + a and a . a = a and a - a = 0 and a / a = 1

a + ~a = 0 and a . ~a = 0 and a - ~a = 1 and a / ~a = -1

And to facilitate this constant correction towards coherency of the stated patterns with less internal contradiction is the organic self-validation procedure that justifies the method that is described. Combining both our adjusted simplest possible patterns is the basis that we can now establish.

a = a and ~a = ~a and a =/= ~a

a + 0 = a and a - 0 = a and ~a + 0 = ~a and ~a - 0 = ~a

a.1 = a and a/1 = a and ~a.1 = ~a and ~a/1 = ~a

a.0 = 0 and a/0 = 0 and ~a.0 = 0 and ~a/0 = 0

a + 1 = 1 + a and a - 1 = -1 + a and ~a + 1 = 1 + ~a and ~a - 1 = -1 + ~a

0 + a = a and 0 - a = -a and 0 + ~a = ~a and 0 - ~a = - ~a

a + a = a + a and a . a = a and a - a = 0 and a / a = 1

a + ~a = 0 and a . ~a = 0 and a - ~a = 1 and a / ~a = -1

At this stage, which describes the pattern from its most simple possible terms, the only thing that I would do is clarify what seems to be a resolution of an internal contradiction so that if it is significant at any later stage, we know where to direct our attention towards. And that is:

a - a = 0 ^ a + ~a = 0 > a - ~a =/= 0

If a minus a equals zero and a plus not a equals zero, then a minus not a cannot equal zero.

a / a = 1 > a / ~a = -1

If a divided by a equals one then a divided by not a equals minus one.

Given those two areas are noticed, which are without internal contradiction as stated, then the coherence of the related terms and the absence of contradiction can be checked by detailing the associated terms:

a - a = 0 and a / a = 1 and a + a = a + a and a . a = a

a - ~a = 1 and a / ~a = -1 and a + ~a = 0 and a . ~a = 0

Then the point we arrive at here is that this pattern is a complete statement of the simplest possible terms of the Boolean Prime and to our present understanding it is correct, coherent and without internal contradiction. Since it is so tiny it is not the idea that we should be intent on some larger thing. Having that pattern in place, we can now consider the third main part of the original pattern which is to replace any number value with the letter b, such that we include terms based on a and b.

a . b = b . a and a + b = b + a

(a + b) + c = a + (b + c) and (a . b) . c = a . (b . c)

a + (b . c) = (a + b) . (a + c) and a . (b + c) = (a . b) + (a . c)

~(a + b) = ~a + ~b and ~(a . b) = ~a . ~b

a / b = a / b and a - b = -b + a

(a - b) - c = a - (b - c) and (a / b) / c = a / (b / c)

a - (b / c) = (a - b) / (a - c) and a / (b - c) = (a / b) - (a / c)

~(a - b) = ~a - ~b and ~(a / b) = ~a / ~b

Now whereas we are required to demonstrate the correctness of the a:b patterns it is clear from the shape of them as given that they are consistent with our previous argumentation and that as given there is no contradiction. That means that the demonstration of correctness is not the same process since our aim in this regard would be to challenge the rules that are being applied, in order to determine whether the rules when correctly applied are themselves appropriate. Which we are not in fact required to do. Therefore we can give the entire pattern in its most complete form up to this stage.

a = a and ~a = ~a and a =/= ~a

a + 0 = a and a - 0 = a and ~a + 0 = ~a and ~a - 0 = ~a

a.1 = a and a/1 = a and ~a.1 = ~a and ~a/1 = ~a

a.0 = 0 and a/0 = 0 and ~a.0 = 0 and ~a/0 = 0

a + 1 = 1 + a and a - 1 = -1 + a and ~a + 1 = 1 + ~a and ~a - 1 = -1 + ~a

0 + a = a and 0 - a = -a and 0 + ~a = ~a and 0 - ~a = - ~a

a + a = a + a and a . a = a and a - a = 0 and a / a = 1

a + ~a = 0 and a . ~a = 0 and a - ~a = 1 and a / ~a = -1

a . b = b . a and a + b = b + a

(a + b) + c = a + (b + c) and (a . b) . c = a . (b . c)

a + (b . c) = (a + b) . (a + c) and a . (b + c) = (a . b) + (a . c)

~(a + b) = ~a + ~b and ~(a . b) = ~a . ~b

a / b = a / b and a - b = -b + a

(a - b) - c = a - (b - c) and (a / b) / c = a / (b / c)

a - (b / c) = (a - b) / (a - c) and a / (b - c) = (a / b) - (a / c)

~(a - b) = ~a - ~b and ~(a / b) = ~a / ~b

The immediately previous pattern which begins with a equals a and completes with the bracketed term not a divided by b equals not a divided by not b is the smallest pattern of this sort. It is very easy to construct from the given pattern to variations on the same still without using the prime pattern of p's and q's. All one needs do is bracket any term and place a not in front of the bracket for example. And the same patterns can be extended into proofs by use of the pattern of p's and q's. When proving the important tool is to show what any term does not equal. For example, to say: If a minus a equals zero and a plus not a equals zero, then a minus not a cannot equal zero. In terms of the correctness of the pattern as detailed, there is no obligation on the part of any decision maker to maintain the pattern as stated. And as stated the pattern need not be held as the only possible statement of itself. All that is required is that if any one single term of the pattern is altered from its present statement, where there is no internal contradiction throughout the entire pattern, then the related terms would need to be altered in order to maintain the coherence throughout and to ensure that the altered pattern also has no contradiction.

Other areas can be noticed also. For example it is quite clear that a different decision maker need not adhere to our given criteria for what is truth and can choose to apply a completely different set of criteria not here detailed. All these sorts of different possibilities do, is satisfy our own property in the sense that we are defined by the systems we adhere to. As to the complex synergetic organic nature of the pattern as detailed, where as stated it is coherent throughout without contradiction, where it begins at the earliest possible point of a = a, and provides a complete set of terms without excess or deficiency and without any blanks, that complex synergy is organic in nature.

As to the difficulty of the complex synergy the secret is to take only the first term and confirm to oneself that it is understood. Then to take the second term and confirm to oneself that it is understood and also that is is coherent with the first term. In that way each small group of terms can be easily learned, even though when the several small groups of terms are combined the complex synergy is more difficult.

1."a equals a",

2."not a equals not a",

3."a does not equal not a".

4."a plus zero equals a"

5."a minus zero equals a"

6."not a plus zero equals not a"

7."not a minus zero equals not a"

8."a multiplied by one equals a"

9."a divided by one equals a"

10."not a multiplied by one equals not a"

11."not a divided by one equals not a"

12."a multiplied by zero equals zero"

13."a divided by zero equals zero"

14."not a multiplied by zero equals zero"

15."not a divided by zero equals zero"

16."a plus one equals one plus a"

17."a minus one equals minus one plus a"

18."not a plus one equals one plus not a"

19."not a minus one equals minus one plus not a"

20."zero plus a equals a"

21."zero minus a equals minus a"

22."zero plus not a equals not a"

23."zero minus not a equals minus not a"

24."a plus a equals a plus a"

25."a multiplied by a equals a"

26."a minus a equals zero"

27."a divided by a equals one"

28."a plus not a equals zero"

29."a multiplied by not a equals zero"

30."a minus not a equals one"

31."a divided by not a equals minus one"

The challenge as to the rules that are applied in the given pattern is best directed towards correspondence rather than as to their coherence. Where we have previously been checking the correctness of the statement of the patterns and the truth of the patterns when correctly stated the focus has been towards as to coherence. When we check the rules that are being used because of their position as first propositions they are already understood as neither true nor false. What we can demonstrate is that within the limits of the patterns as described they do satisfy correspondence, coherence and utility. And because we know they are neither true nor false, we can demonstrate that outside the very limited patterns they do not fail on coherence and will fail on grounds of correspondence.

The reason we would do such is so that we can determine what are the adjustments we are required to make in order to make use of the rules when outside the limited patterns as detailed. And before we can do that we must clarify the prime patterns of the rules in themselves as used within the limited patterns as detailed. As we construct the prime pattern of rules it will become evident what are the grounds of correspondence which will fail outside the limited given pattern.

First: a, b, 0, 1, +, -, ., /, =, =/=, ~, ().

Second: ++ = +, +- = +-, -- = -, +.+ = +, +/+ = +, -.- = +, -/- = +, +.- = -, +/- = -, -/+ = -

Third: + = + , - = -, . = ., / = /, ~ = ~, = = =, =/= = =/=, () = ()

Fourth: + =/= -, . =/= /, ~ =/= -, = =/= =/=

Fifth: =

First postulate is that the sign labels we use are only a, b, zero, one, plus, minus, multiplied by, divided by, equals, does not equal, not, brackets.

Second postulate is that the addition of positive terms gives a positive, the addition or subtraction of positive term with a negative term gives a positive or a negative, the subtraction of negative terms gives a negative, the multiplication of positive terms gives a positive, the division of positive terms gives a positive, the multiplication of negative terms gives a positive, the division of negative terms gives a positive, the multiplication of positive term with negative term gives a negative, the division of positive term by negative term gives a negative, the division of negative term by a positive term gives a negative.

Third postulate is that plus equals plus, minus equals minus, multiplied by equals multiplied by, divided by equals divided by, not equals not, equals equals equals, does not equal equals does not equal, brackets equals brackets.

Fourth postulate is that plus does not equal minus, multiplied by does not equal divided by, not does not equal minus, equals does not equal does not equal.

Fifth postulate is equals.

Where we will challenge the given system as to correspondence will focus primarily on the difference between object and process. And with a supposition that there is nothing in our given system that enables us to distinguish between when the sign labels refer to an object and when the same sign labels refer to a process. The second supposition will be that our given system does not enable us to signify when we move logical level in regard to genus and species. The third supposition will be that our given system does not specify the difference between ontologically existant and imaginatively existant. The fourth supposition will be that the multiplication of a positive and minus term gives a minus process.

In regard to the first supposition, two multiplied by three equals six, where if two is an object, then multiplied by three is not an object and is a process.

In regard to the second supposition, two apples plus three pears is five fruit, where apples and pears are species and fruit is genus.

In regard to the third supposition, positive integers can exist and negative integers cannot, where positive number can be object and negative number can only be process.

In regard to the fourth supposition, plus three multiplied by minus five equals minus twelve, where the difference between plus three and minus twelve is a total of fifteen units. The proposition that any positive number multiplied by minus one equals zero.

1: a, b, 0, 1, +, -, ., /, =, =/=, ~, ().

2: ++ = +, +- = +-, -- = -, +.+ = +, +/+ = +, -.- = +, -/- = +, +.- = -, +/- = -, -/+ = -

3: + = + , - = -, . = ., / = /, ~ = ~, = = =, =/= = =/=, () = ()

4: + =/= -, . =/= /, ~ =/= -, = =/= =/=

5: =

The prime pattern of rules is correspondent, coherent, useful and universally abstract. That means even though it is true, it is not in itself sufficient. The fifth rule details the matter. Essentially it raises the question "what is that?" in regard to equals. As all in the whole world know, the ontological question, what is that?, is not easy to answer. When the ontological question is directed towards equals we enter the field that studies being existant. Which is outside the domain of the study of the truth of prime patterns.

In the fourth rule where we state equals does not equal does not equal, we are noticing that there are two related sign labels which together have a contradictory function. The existance of the two contradictory functions enables us to satisfy what the difference is between the two. When we use equals we are saying some entity is the same as a different entity. When we use does not equal we are saying some entity is not the same as a different entity. By universal abstraction we can define this as to say being is being and the contradictory being is not being. That then explains the way in which we use equals, which is the mathematical equivalent of the word "is". And the way in which we use does not equal, which is the mathematical equivalent of the word phrase "is not".

Given that understanding, it becomes clear that if we attend to the matter "what is equals?", we will be forced to attend to the more abstract matter, what does "is" mean? The important knowledge here is to realise that we can not answer the second level question "what is equals?" unless we can answer the first level question "what does <is> mean?". And if we can answer the first level question, then answering the second level question becomes easy.

The concepts that are inter-relationally involved in this kind of study include being, doing, meaning, existence, non-existence, identity, non-identity and no other concepts. In mathematical terms we can say the concepts are one, some, more, less and none. Therefore, if we consider only those five mathematical concepts we can define some basic terms. One object is an existant identity. If the same identity is repeated that is some. Some is more than one. One is less than some. None is the non-existance of one and some. The concept of more is to do with addition. The concept of less is to do with subtraction. A repetitive addition is called multiplication. A repetitive subtraction is called division. Existence is an affirmation. Non-existance is a negation.

Any number squared divided by two plus half the original number is equal to the number of times each of the consecutive integers in the original number combine with itself and each of the others once, as in dominoes. In a standard nine bar set of dominoes the integers one to nine and zero are all represented in relation to each of the other integers. That is a total of ten units. Ten squared is one hundred. One hundred divided by two is fifty. Half the original number is five. Fifty plus five is fifty five. And in a standard nine bar set of dominoes there are fifty five counters. Given dominoes do not use late language sign label for number, and instead use a pattern of points to show the universal abstract that exists at the earlier level than sign label, they are useful in regard to demonstrating the ontological status of existant being of number.

At the level of dominoes, the sign labels and the concepts do not yet exist. Without the sign labels and without the concepts, still dominoes exist. Dominoes therefore are ontologically earlier than the sign labels and the concepts. And from the ontological existence of dominoes it is very easy to build the sign labels and concepts of number.

If we imagine we have a set of dominoes for only the units one and none, then that is two units. Two squared is four, four divided by two is two. Half the original number is one. Two plus one is three. Therefore we will have a set of dominoes with three counters. One of those counters will show none on both sides of the counter. Another will show one on both sides of the counter. Another will show none on one side and one on the other side of the counter. In that way without defining our concepts we are showing the concepts in their immanent form. And with the dominoe that has none on both sides we are saying none is the same as none or none equals none. And with the dominoe that has one on both sides we are saying one is the same as one or one equals one. And with the dominoe that has none on one side and one on the other side we are saying none is not the same as one or none does not equal one. And we know what we mean because we have three dominoes and we can see what we mean. And we think that each of the three dominoes is different to the other two. And if we thought of the connecting line between the two sides of any one dominoe as plus, then we would know that none plus none is none, one plus none is one, and one plus one is two. And if we look at the whole dominoe and thought of what happens when we take away one side that would be like minus, so we could say none minus none is none, one minus none is one, one minus one is none, two minus one is one.

All we need now do is notice what are the functions that are evident without specific sign label. And we can see that the difference between nothing and something is similar to the difference between the dominoe that is blank on both sides and the dominoe that has a blank on one side and a one on the other. And the difference between one and some is similar to the the difference between the dominoe that has a blank on one side and one on the other and the dominoe that has a one on both sides. And we can understand that some is more than one and one is less than some and none is the non-existance of one or some. So at this stage we have proved the first five concepts of number which are one, some, more, less, none. And we can apply those concepts as sign labels of functions such that we understand one is existant entity, some is a repetition of one, and none is the non-existance of entity. And we can apply the sign label functions of addition to some being more than one, and subtraction to one being less than some. Which means we have now invented our sign labels for the functions of addition and subtraction. And we know what we mean by "is" because "none <is> none" and "one <is> one" and we know what we mean by "is not" because "none <is not> one". Therefore we have demonstrated using three dominoes the function before sign label of the concepts of one, some, more, less, none, addition, subtraction. And now we simply invent the sign labels to specify those functions, which leads to the prime pattern of rules as detailed.

1: a, b, 0, 1, +, -, ., /, =, =/=, ~, ().

2: ++ = +, +- = +-, -- = -, +.+ = +, +/+ = +, -.- = +, -/- = +, +.- = -, +/- = -, -/+ = -

3: + = + , - = -, . = ., / = /, ~ = ~, = = =, =/= = =/=, () = ()

4: + =/= -, . =/= /, ~ =/= -, = =/= =/=

5: =

a = a and ~a = ~a and a =/= ~a

a + 0 = a and a - 0 = a and ~a + 0 = ~a and ~a - 0 = ~a

a.1 = a and a/1 = a and ~a.1 = ~a and ~a/1 = ~a

a.0 = 0 and a/0 = 0 and ~a.0 = 0 and ~a/0 = 0

a + 1 = 1 + a and a - 1 = -1 + a and ~a + 1 = 1 + ~a and ~a - 1 = -1 + ~a

0 + a = a and 0 - a = -a and 0 + ~a = ~a and 0 - ~a = - ~a

a + a = a + a and a . a = a and a - a = 0 and a / a = 1

a + ~a = 0 and a . ~a = 0 and a - ~a = 1 and a / ~a = -1

a . b = b . a and a + b = b + a

(a + b) + c = a + (b + c) and (a . b) . c = a . (b . c)

a + (b . c) = (a + b) . (a + c) and a . (b + c) = (a . b) + (a . c)

~(a + b) = ~a + ~b and ~(a . b) = ~a . ~b

a / b = a / b and a - b = -b + a

(a - b) - c = a - (b - c) and (a / b) / c = a / (b / c)

a - (b / c) = (a - b) / (a - c) and a / (b - c) = (a / b) - (a / c)

~(a - b) = ~a - ~b and ~(a / b) = ~a / ~b

When the prime pattern of p's and q's is applied to the prime pattern of number that we have established we obtain the prime pattern of sets in its most simple possible formulation. The prime pattern of sets is a universal abstraction of the prime pattern of number. It is more universal because it is not particular to number and can just as easily be applied to concepts. It is more abstract because its sign label system is common to both the functions of number and of language. What is strange is that the difference of sign label satisfies the thought of a more universally abstract system, such that it is not possible to follow the exact same idea shape of the prime pattern of number even though the formulaic terms may be similar. In that sense we are driven to better state our explanation of the meaning of the sign labels that we use, given that the first statement of their meaning is only adequate. Therefore in this pattern of sets, one of the continual processes is to clarify exactly how to better define the sign labels.

As with the prime pattern of number we will find that we can obtain the most simple reduction of the earliest pattern of sets, and that the complete pattern is itself quite tiny. And given that pattern the more complex extensions are indefinitely large. It is an old idea of invention that if a pattern that is known to be true can be exactly matched by a different pattern then the other pattern will be true.

a means "a is true" Affirmation. Or "a" means an entity in the universe of discourse that may be a genus or species, or may be a class or member of class, or may be a set or element of set.Or "a" means everything that is "a" in our universe of discourse.

~a means "a is false" Negation. Or "~a" means everything that is "not a" in the universe of discourse. Or "~a" means some entity that is an element of everything that is not "a" in the universe of discourse.

b means "b is true" Affirmation. Or some entity not a in the universe of discourse that may be defined in a similar manner to the way in which a is defined.

~b means "b is false" Negation. Or some entity not b in the universe of discourse that may be defined in relation to b in a similar manner to the way in which not a is defined in relation to a.

a v b means "either a is true, or b is true, or both" Either one or the other or both.

a ^ b means "both a and b are true" And. Both one and the other.

a => b means "if a is true, then b is true". If - then. If set of elements then any element of such a set.

a =/=> b means "if a is true, then b is false". If - then not . If set of elements then not any element of a different set.

a < b means "since b is true, then a is true." Since - then. Since the way this system uses the inclusion sign label may be inconsistent with the way other systems use the same sign label it requires further clarification. My definition is as follows: (a < b) ^ (b < c) => a < c. That is to say, if a contains b, and b contains c, then a contains c. I am using the sign label as an arrow to show that the species is part of the genus or that the element is contained by the set.

a /< b means "since b is true, then a is false." Since - not then. If member of a class, then not any different class.

a <=> b means "a and b are either both true or both false" If and only if. Only if both and not either one without the other.

a <=/=> b means "a is true and b is false, or a is false and b is true". Either one or the other not both.

ɸ means the empty set, or the class whose membership is none.

1 means the unity set, or the class whose membership is itself.

e means everything in the given universe of discourse.

! means not not. Or in regard to a single entity it means neither entity nor not entity.

Q means neither the unity set nor the empty set. Or an indefinite variable.

a => a and ~a => ~a and a =/=> ~a

a ^ ɸ => a and a ^ ~ɸ => a and ~a ^ ɸ => ~a and ~a ^ ~ɸ => ~a

a v 1 => a and a < 1 => a and ~a v 1 => ~a and ~a < 1 => ~a

a v ɸ => ɸ and a < ɸ => ɸ and ~a v ɸ => ɸ and ~a < ɸ => ɸ

a ^ 1 => 1 ^ a and a ^ ~1 => Q ^ a and ~a ^ 1 => 1 ^ ~a and ~a ^ ~1 => Q ^ ~a

ɸ ^ a => a and ɸ ^ ~a => ~a and ɸ ^ ~(~a) => !a

a ^ a => a ^ a and a v a => a and a ^ ~a => e and a < a => 1

a v ~a => Q and a ^ ~(~a) => a ^ !a and a < ~(~a) => a < !a

a v b => b v a and a ^ b => b ^ a

(a ^ b) ^ c => a ^ (b ^ c) and (a v b) v c => a v (b v c)

a ^ (b v c) => (a ^ b) v (a ^ c) and a v (b ^ c) => (a v b) ^ (a v c)

~(a ^ b) => ~a ^ ~b and ~(a v b) => ~a v ~b

a < b => a < b and a ^ ~b => ~b ^ a

(a ^ ~b) ^ ~c => a ^ (~b ^ ~c) and (a < b) < c => a < (b < c)

a ^ ~(b < c) => (a ^ ~b) < (a ^ ~c) and a < (b ^ ~c) => (a < b) ^ ~(a < c)

~(a ^ ~b) => ~a ^ ~(~b) and ~(a < b) => ~a < ~b

One of the understandings to develop is that the similar thought-idea can be expressed using different sign label languages. And that the difference in the larger system of the sign label languages alters the idea-shape of the thought-idea. Given that the prime pattern of number was detailed in the sign label of number and then the same thought-idea was detailed in the sign label of language, such that a = a means a equals a. In that sense the above pattern of sets is the exact same thought-idea given in the language of sets. Which we can then detail in the sign label of spoken language. When we detail the pattern of sets in the sign label of spoken language, the idea-shape will be different to when we detail the pattern of number in spoken language. That means a common thought-idea can be explained in two different universally abstract languages, that of the pattern of number and that of the pattern of sets, and when each of those different sign label systems are detailed in spoken language two different idea-shapes are detailed in one language for the same thought-idea.

What will be the case is that the familial relationship between the thought-idea, and the idea-shape when detailed in the two sign label systems, and the idea-shape when detailed in the spoken language will be very close. Meaning that it is natural that they are different to other decision maker systems in terms of internal coherence, correspondence and utility. As the earlier explanation clarified there is no requirement to satisfy coherence to some other decision maker's system, we do not have to satisfy correspondence from the other decision maker's perspective and we do not have to prove utility to the other decision maker's purpose.

At this earliest possible stage given that we have the common thought-idea in the idea-shape of different sign label systems that we have designed, it is easy to prove the terms of the pattern of sets. We shall do so in two ways. One is to explain each term in the spoken language. Second is to show each term next to the equivalent term in the sign label of number. This is important because at this stage it is easy and later it would not be possible. That is because the sign label system of sets can be used on its own without any requirement to refer to the prime pattern of number from which it is universally abstracted. And also, once the label system of the prime pattern of sets is understood it can be used without any reference to the prime pattern of sets in the same way that the sign label system of number can be used without any reference to the prime pattern of number.

In regards to the given spoken language explanation of the individual terms of the pattern of sets it has to be thought of as correct or incorrect in itself and in reference to terms that are correctly stated or not, and if correctly stated then true or not. For this reason it is permanently the possibility that a more correct spoken language explanation can be provided. And presumably to consider that the same universally abstract sign label pattern of sets could be explained in a variety of different spoken languages.

1. a => a If a, then a.

2. ~a => ~a If not a, then not a.

3. a =/=> ~a If a, then not not a.

4. a ^ ɸ => a If a and the empty set, then a.

5. a ^ ~ɸ => a If a and not the empty set, then a.

6. ~a ^ ɸ => ~a If not a and the empty set, then not a.

7. ~a ^ ~ɸ => ~a If not a and not the empty set, then not a.

8. a v 1 => a If a or the unity set, then a.

9. a < 1 => a If a includes the unity set, then a.

10. ~a v 1 => ~a If not a or the unity set, then not a.

11. ~a < 1 => ~a If not a includes the unity set, then not a.

12. a v ɸ => ɸ If a or the empty set, then the empty set.

13. a < ɸ => ɸ If a includes the empty set, then the empty set.

14. ~a v ɸ => ɸ If not a or the empty set, then the empty set.

15. ~a < ɸ => ɸ If not a includes the empty set, then the empty set.

16. a ^ 1 => 1 ^ a If a and the unity set, then the unity set and a.

17. a ^ ~1 => Q ^ a If a and not the unity set, then Q and a.

18. ~a ^ 1 => 1 ^ ~a If not a and the unity set, then the unity set and not a.

19. ~a ^ ~1 => Q ^ ~a If not a and not the unity set, then Q and not a.

20. ɸ ^ a => a If the empty set and a, then a.

21. ɸ ^ ~a => ~a If the empty set and not a, then not a.

22. ɸ ^ ~(~a) => !a If the empty set and not not a, then !a.

23. a ^ a => a ^ a If a and a, then a and a.

24. a v a => a If a or a, then a.

25. a ^ ~a => e If a and not a, then everything in the universe of discourse.

26. a < a => 1 If a includes a, then the unity set.

27. a v ~a => Q If a or not a, then Q.

28. a ^ ~(~a) => a ^ !a If a and not not a, then a and !a.

29. a < ~(~a) => a < !a If a includes not not a, then a includes entity neither a nor not a.

30. a v b => b v a If a or b, then b or a.

31. a ^ b => b ^ a If a and b, then b and a.

32. (a ^ b) ^ c => a ^ (b ^ c) If term a and b term and c, then a and term b and c.

33. (a v b) v c => a v (b v c) If term a or b term or c, then a or term b or c.

34. a ^ (b v c) => (a ^ b) v (a ^ c) If a and term b or c, then term a and b term or term a and c.

35. a v (b ^ c) => (a v b) ^ (a v c) If a or term b and c, then term a or b term and term a or c.

36. ~(a ^ b) => ~a ^ ~b If not term a and b, then not a and not b.

37. ~(a v b) => ~a v ~b If not term a or b, then not a or not b.

38. a < b => a < b If a includes b, then a includes b.

39. a ^ ~b => ~b ^ a If a and not b, then not b and a.

40. (a ^ ~b) ^ ~c => a ^ (~b ^ ~c) If term a and not b term and not c, then a and term not b and not c.

41. (a < b) < c => a < (b < c) If term a includes b term includes c, then a includes term b includes c.

42. a ^ ~(b < c) => (a ^ ~b) < (a ^ ~c) If a and not term b inc c, then term a and not b term inc term a and not c.

43. a < (b ^ ~c) => (a < b) ^ ~(a < c) If a inc term b and not c, then term a inc b term and not term a inc c.

44. ~(a ^ ~b) => ~a ^ ~(~b) If not term a and not b, then not a and not not b.

45. ~(a < b) => ~a < ~b In not term a inc b, then not a inc not b.

{{{Information contained in three brackets is a later note, not part of the original article.

Considering Idea 40 above, this seems to be the correct statement:(a ^ ~b) ^ ~c => a ^ (~b ^ ~c)

And this seems to be the false statement: (a ^ ~b) ^ ~c => a ^ ~(b ^ ~c)

The reason for the original statement is: (a - b) - c = a - (b - c)

Whereas the correct statement may be: (a - b) - c = a + (-b - c)

So the point I am drawing the attention towards is the double negative that is built in to the inference of the idea whereas the proposition does not contain a double negative.

Why this will matter later is because two different decision makers will be able to disagree as to the use of "and not".

The question will be does "and not" mean <and not> entity or does it mean <and> not entity.

It is similar to the question of does not equal and whether does not equal not entity makes a not not entity.

Both these questions are areas I have left unanswered in these articles, since the correct answer depends on the intention of the enquirer.}}}

Since the pattern of sets is more abstract than the pattern of number it has required the use of some sign labels that need not exist at a less abstract level. That is e, ! and Q. By universe of discourse we mean the restricted parameter bounded by the terms of meaning. It operates at a prime level in the sense that the permitted sign labels define the universe of discourse abstractly, and how they are used in particular applications define the universe of discourse specifically. In this sense, the prime number pattern is a particular application of the more abstract pattern of sets.

When the pattern of number is abstracted to provide the pattern of sets then using the sign label system of the pattern of sets our given terms can be forced to demonstrate a term that they can not answer with only the given terms. That is without ! we cannot show not not as a single entity, without the term e we cannot show everything in our given universe of discourse as a single entity, and without the term Q we cannot show not the empty set and not the unity set as a single entity. The basic rule in regard to the three terms Q ! e is that at any less abstract level they be replaced by any coherent non-contradictory term. They are required at the more abstract level so that in particular instances of the use of the pattern of sets the coherent non-contradictory term that is used in the same place is dependent on the particular instance.

The specific definition of the three terms in this system is:

a ^ ~a => e

~(~a) => !a

~1 ^ ~ɸ => Q

The equivalence function in regard to how I have maintained coherence in the two different sign label systems individually and correspondence between the two systems demands some explanation. The method used has not been to make a direct correspondence of "meaning" between the two systems. The method used has been to make a direct correspondence of "shape" between the two systems. The way I have done so is to make the easiest function to maintain as similar between both systems first. That is the decision to make "plus" in the number system mean "and" in the pattern of sets. Then the opposite of plus is minus, so I have made that to mean "and not" in the system of sets. Then the multiplication sign label in the number system is the "or" sign label in the system of sets. Since "or" is not to "and" what "multiplication" is to "plus" that prohibited my use of "or not" as the function against the number system function called "division". And since "if a contains b and b contains c then a contains c" is similar to the whole containing the part the inclusion sign label was used to meet the shape of the terms involving division in the number system. The next stage in the process is to maintain coherence between the terms. Such that the "and" sign label must give a different result to the "or" sign label where ever the two terms are otherwhise the same. "And" is considered opposite to "or" in this sense. Which then makes it inevitable that "and not" will give a different result to either since it ensures the two terms can not be otherwhise the same. Then given the inclusion sign label maintains the shape given by the division sign label the coherence of the pattern of sets is more likely. Finally the coherence of the pattern of sets does have to be checked on its own with no reference to the pattern of number. And all we are looking for is to make any adjustment necessary to increase internal coherence and non-contradiction.

The difference between minus and not requires clarification. Minus means subtraction of a specifically given entity. Minus is a negative value the exact shape of a positive entity. That is different to "not" which means everything that a given entity is not. "Not" in this sense means a positive statement of everything that is not a given entity. More specifically in regards to "~a" this means everything that is not a in our given universe of discourse. Or alternatively it means some entity that is an element of everything that is not a in our universe of discourse.

It is for that reason that we need the sign label "e". Because if a => a and ~a => ~a then a means everything that is a in the universe of discourse and not a means everything that is not a in the universe of discourse, the combination of a and not a being everything in the universe of discourse.

And it is for this reason that "not not" does not mean the simple positive affirmation. If we used the sign label "a" to signify "horse", then not not "a" means not not "horse", that being a unicorn. Since a unicorn is a horse but is not a horse, we can not say the positive affirmation. By the use of not not horse we are saying unicorn is neither a horse nor not a horse. Using an example given earlier in regard to "that" it is like saying that is neither a square nor not a square, but if I say that is a square I am missing that the square is part of a larger object called a cube. And it is like saying that is neither a cube nor not a cube, but if I say that is a cube I am missing that the cube is really a multi-dimensional hypercube capable of navigating the meta-complex for reasons unknown. We can prove "not not" does not equal the simple affirmative in this way. If not not does mean "neither entity nor not entity" then a third not would be the contradictory of "neither entity nor not entity". And that contradictory would be "either entity or not entity" which then is either the simple affirmation or the simple negation.

The sign label Q meets a similar condition in regard to the possible entity that is neither the empty set nor the unity set. Q => a v ~a v !a v ~ɸ v ~1 v b v ~b v !b. The rule is that in any particular term it is possible to use any sign label ~Q then that term must be used. And where ever the sign label Q is used it be replaced by a sign label ~Q where ever possible.

The common abstract condition that is met by both Q and ! is the concept "neither nor not".

1. a => a <=> a = a

2. ~a => ~a <=> ~a = ~a

3. a =/=> ~a <=> a =/= ~a

4. a ^ ɸ => a <=> a + 0 = a

5. a ^ ~ɸ => a <=> a - 0 = a

6. ~a ^ ɸ => ~a <=> ~a + 0 = ~a

7. ~a ^ ~ɸ => ~a <=> ~a - 0 = ~a

8. a v 1 => a <=> a.1 = a

9. a < 1 => a <=> a/1 = a

10. ~a v 1 => ~a <=> ~a.1 = ~a

11. ~a < 1 => ~a <=> ~a/1 = ~a

12. a v ɸ => ɸ <=> a.0 = 0

13. a < ɸ => ɸ <=> a/0 = 0

14. ~a v ɸ => ɸ <=> ~a.0 = 0

15. ~a < ɸ => ɸ <=> ~a/0 = 0

16. a ^ 1 => 1 ^ a <=> a + 1 = 1 + a

17. a ^ ~1 => Q ^ a <=> a - 1 = -1 + a

18. ~a ^ 1 => 1 ^ ~a <=> ~a + 1 = 1 + ~a

19. ~a ^ ~1 => Q ^ ~a <=> ~a - 1 = -1 + ~a

20. ɸ ^ a => a <=> 0 + a = a

21. ɸ ^ ~a => ~a <=> 0 - a = -a

22. ɸ ^ ~(~a) => !a <=> 0 - ~a = - ~a

23. a ^ a => a ^ a <=> a + a = a + a

24. a v a => a <=> a . a = a

25. a ^ ~a => e <=> a - a = 0

26. a < a => 1 <=> a / a = 1

27. a v ~a => Q <=> a . ~a = 0

28. a ^ ~(~a) => a ^ !a <=> a - ~a = 1

29. a < ~a => Q <=> a / ~a = -1

30. a v b => b v a <=> a . b = b . a

31. a ^ b => b ^ a <=> a + b = b + a

32. (a ^ b) ^ c => a ^ (b ^ c) <=> (a + b) + c = a + (b + c)

33. (a v b) v c => a v (b v c) <=> (a . b) . c = a . (b . c)

34. a ^ (b v c) => (a ^ b) v (a ^ c) <=> a + (b . c) = (a + b) . (a + c)

35. a v (b ^ c) => (a v b) ^ (a v c) <=> a . (b + c) = (a . b) + (a . c)

36. ~(a ^ b) => ~a ^ ~b <=> ~(a + b) = ~a + ~b

37. ~(a v b) => ~a v ~b <=> ~(a . b) = ~a . ~b

38. a < b => a < b <=> a / b = a / b

39. a ^ ~b => ~b ^ a <=> a - b = -b + a

40. (a ^ ~b) ^ ~c => a ^ (~b ^ ~c) <=> (a - b) - c = a (-b - c)

41. (a < b) < c => a < (b < c) <=> (a / b) / c = a / (b / c)

42. a ^ ~(b < c) => (a ^ ~b) < (a ^ ~c) <=> a - (b / c) = (a - b) / (a - c)

43. a < (b ^ ~c) => (a < b) ^ ~(a < c) <=> a / (b - c) = (a / b) - (a / c)

44. ~(a ^ ~b) => ~a ^ ~(~b) <=> ~(a - b) = ~a - ~b

45. ~(a < b) => ~a < ~b <=> ~(a / b) = ~a / ~b

The development of the prime pattern of sets has involved the use of sign labels from the prime pattern of the universal abstract. That is Q ! e. It is then possible to provide a simple clarification of other sign labels used in that sort of pattern and one of the possible definitions of the sign labels.

ʘ means the universal abstraction. ʘ => ʒ ^ e The universal abstraction is the universe of discourse and everything in it.

ψ means the universal operator. ψa => a ^ ~a ^ !a The universal operator is every and all of some possible entity.

Э means the existential operator. Эa => a v ~a v !a The existential operator is some possible entity of all such entity.

ʒ means the universe of discourse. ʒ => e The universe of discourse is a restricted parameter bounded by the terms of meaning.

ɸ means the empty set. ɸ => ʒ ^ ~e The empty set is the class whose membership is none or the set of no elements.

1 means the unity set. 1 => a < a The unity set is the class whose membership is itself or the set that contains itself as element.

e means everything in the universe of discourse. e => a ^ ~a Everything in the universe of discourse is only and all the sign labels used.

Q means not the empty set and not the unity set. Q => ~ɸ ^ ~1 Not the empty set and not the unity set is the indefinite variable or possible improbable.

! means not not entity. !a => ~(~a) Not not entity is neither entity nor not entity.

The sign labels of the universal abstract enable us to make a/an hypothetical proposition as to the ontological formula, that is:

ʘ => ʒ ^ e < (Q ^ !e) v [(ɸ ^ 1) ^ (ψ ^ Э)] is true or false or neither true nor false.

Given the ontological formula, then the universal abstract is given as:

ɸ => ʒ ^ ~e < [a ^ ɸ => a ; a ^ ~ɸ => a ; ~a ^ ɸ => ~a ; ~a ^ ~ɸ => ~a ; a v ɸ => ɸ ; a < ɸ => ɸ  ; ~a v ɸ => ɸ  ; ~a < ɸ => ɸ ;

ɸ ^ a => a ; ɸ ^ ~a => ~a ; ɸ ^ ~(~a) => !a]

1 => a < a < [a v 1 => a ; a < 1 => a ; ~a v 1 => ~a ; ~a < 1 => ~a ; a ^ 1 => 1 ^ a ; ~a ^ 1 => 1 ^ ~a ; a < a => 1]

Q => ~ɸ ^ ~1 < [a ^ ~1 => Q ; ~a ^ ~1 => Q ^ ~a ; a v ~a => Q]

e => a ^ ~a < [a ^ ~a => e]

ψ => Э < [a => a ; ~a => ~a ; a =/=> ~a ; a ^ a => a ^ a ; a v a => a ; a ^ ~(~a) => a ^ !a ; a < ~(~a) => a < !a

a v b => b v a ; a ^ b => b ^ a ; (a ^ b) ^ c => a ^ (b ^ c) ; (a v b) v c => a v (b v c) ;a ^ (b v c) => (a ^ b) v (a ^ c) ;

a v (b ^ c) => (a v b) ^ (a v c) ; ~(a ^ b) => ~a ^ ~b ; ~(a v b) => ~a v ~b ; a < b => a < b ; a ^ ~b => ~b ^ a ;

(a ^ ~b) ^ ~c => a ^ (~b ^ ~c) ; (a < b) < c => a < (b < c) ; a ^ ~(b < c) => (a ^ ~b) < (a ^ ~c) ;

a < (b ^ ~c) => (a < b) ^ ~(a < c) ;~(a ^ ~b) => ~a ^ ~(~b) ; ~(a < b) => ~a < ~b]

All of the various patterns previously detailed are either entirely correct or entirely incorrect or correct in part and incorrect in part, as given. If they are entirely correct then they do permit the possibility of a better formulation. If they are entirely incorrect then they do permit the possibility of their correct formulation. If they are correct in part and incorrect in part then they do permit the possibility of the correction of any part that is incorrect.

The criteria for truth, falsity and neither true nor false is given as:

For any p that we state as true, we must be able to demonstrate these reasons: P is externally correspondent to some fact. P is internally coherent within a larger system. P is useful according to some pragmatic purpose. P is capable of universal abstraction.

For any p that we state as false, we must be able to demonstrate these reasons: P does not externally correspond to any known fact. P is entirely incoherent within a larger system. P has no use according to some pragmatic purpose. P is not capable of universal abstraction.

In this sense, then, within the rules of the universe of discourse as given, the only condition for correcting the patterns as stated is that correction is only towards better coherency and non-contradiction. Any change in the directly opposite direction that increases internal contradiction and reduces internal coherency is not desirable. The patterns themselves do permit that they be adjusted individually, in part or in whole. As such the earlier proposition that they are an organic synergy is also held to be true. The patterns can be entirely written in a completely different sign label system. And the same sign label system could be used to write different patterns.

What are we missing? Since that is not the sort of question that can be answered in the totality of things, I will answer it in regard to a few important areas. One is in regard to "not not". A change that many will argue for is that "not not" should reduce to the simple positive affirmative. And that in fact is what we would tend to provide to children when they ask the same question. Because given the simple possible of a ^ ~a => e, that is much too early to provide children with the higher order studies. Many who study the early stages of this kind of system will meet the a and not a idea at the beginning so the only answer they are given is that it equals zero. And as well that not not reduces to the affirmation. And that neither the empty set nor the unity set being Q exists, so that if a choice is to be made and it is not the empty set it must be the unity set or the other way round. Therefore we provide a safe answer to people who learn the basics of this sign label, although the safe answer and the system it exists within cannot be proved true and remains without the required matter for the higher order system. On the other hand the stronger formulation can easily be permitted to intelligent children if they are potentially confident of understanding the higher order concepts.

As to the patterns previously stated I have made no adjustments to them since their final formulation. They seem correct as stated and accurate to what they should be saying. The difficulty is as to truth as strong as:

a = a and ~a = ~a and a =/= ~a

a + 0 = a and a - 0 = a and ~a + 0 = ~a and ~a - 0 = ~a

a.1 = a and a/1 = a and ~a.1 = ~a and ~a/1 = ~a

a.0 = 0 and a/0 = 0 and ~a.0 = 0 and ~a/0 = 0

Now since these were the first terms we noticed as the most simple possible from the entire number pattern what we can do at this stage is reflect again on these terms to remind ourselves of how we know if some terms are true. Since to obtain that level of confidence in our own judgement that is as sure as 2 + 2 = 4 we had to restrict ourselves to only this small group to start with, it is clear that we cannot immediately be sure as to the same definiteness of truth of the complete pattern of sets combined with the complete pattern of number. The complete aspect requires clarification. It is complete in the sense of most simple, not complete in the sense of exhaustive of all variations of possible true formulations. It is complete in the sense that from the simplest and tiniest pattern that is possible as given, any who works with the patterns can become skilled at generating other formulations of similar sort. For example 0 - 0 = 0 is a term that was not included in the original pattern and is clearly coherent with the patterns as given.

The better idea at this stage is to hard copy the entire discourse and to work at small units of it over a length of time. From an individual point of view any person should definitely hand write only those terms that they believe to be true. Even if that was only the a = a group it is a useful matter to begin with. That way any who choose to, can begin to keep a note book of what they are most sure of. And use the hard copy as an ongoing reference to see if they confirm or disagree with the formulation as presently stated. Since some of the sign labels used are not standard any individual may require that they replace with sign labels consistent with their own workstation. Maybe copy to desktop then send a copy to friends so that it can be worked with in a small team of players like a sort of game. Also that would mean no requirement to immediately keep up with what is necessarily the next area to consider, although any who worked through the patterns in order as stated up to this stage will easily be confident about this sort of area. Since the line through is very fast it can not mean that any who read all of it in one sitting can do all of it at one go. On the other hand any who chose to could easily hand write the entire combined pattern of number and sets exactly correctly even if they cannot at this immediate stage decide if it is true or not.

For a complete index to the various articles I have used to introduce these and related patterns, please follow the hyperlink:

http://en.wikiversity.org/wiki/Deductive_logic