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Inductive Hypothesis.

"What is the correct and proper statement of truth-holders?".

I am defining "truth-holders" in this context as: Those elements of first propositions that enable us to assert "p is true".

Whereas this article may be read on its own, it would make coherent sense only if understood in relation to the articles that describe the syllogism and algebraic deduction:

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

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

Additionally, it does not repeat information provided in the articles "Boolean Algebra" or "Truth-Value":

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

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

The truth-holders of inductive hypothesis are those elements that combine together in order to form any individual proposition. As such any individual proposition can be demonstrated to involve a specific combination of such truth-holders. The truth-holders themselves are given as the sixteen affirmations of inductive hypothesis, as follows:

The Sixteen Affirmations of Inductive Hypothesis.

1. Some X are Some Y

2. Some X are Not Y

3. Some Y are Some X

4. Some Y are Not X

5. Some Not X are Some Y

6. Some Not Y are Some X

7. Some Not X are Not Y

8. Some Not Y are Not X

9. No X are Some Y

10. No Y are Some X

11. No Y are Not X

12. No Not X are Some Y

13. No X are Not Y

14. No Not Y are Some X

15. No Not X are Not Y

16. No Not Y are Not X

As such the general affirmations are called "truth-holders" because by their nature they can be measured as true or false.

That is, the affirmation "Some X are Some Y" may be true or false.

And if the affirmation: "No X are Y" is true, then the affirmation "Some X are Some Y" must be false.

All of the general affirmations have this characteristic that they may be either true or false.

The Sixteen Affirmations in Algebraic Form.

1. ЭX < ЭY

2. ЭX < Э~Y

3. ЭY < ЭX

4. ЭY < Э~X

5. Э~X < ЭY

6. Э~Y < ЭX

7. Э~X < Э~Y

8. Э~Y < Э~X

9. [~]X < ЭY

10. [~]Y < ЭX

11. [~]Y < Э~X

12. [~]~X < ЭY

13. [~]X < Э~Y

14. [~]~Y < ЭX

15. [~]~X < ~Y

16. [~]~Y < ~X

Please note that in order to sign the term "no" as in "no x are y" I have used the square brackets around the "not" signlabel.

This is specifically to differentiate the term "no" from the term "not", since it should be clear that they mean different things. And also to ensure that in this context the use of both terms in no way leads towards the term "not not".

When combined together, the general affirmations provide the propositions used in deduction as follows:

The Propositions Used in Deduction.

A.

Proposition: All A is B. <=> Some B is All A <=> Some x contains all y <=> (ЭX < ψY)

Affirmation:

Some X are Y

Some X are Not Y

No Y are Not X

E.

Proposition: No A is B. <=> All B is Not A <=> All x contains not y <=> (ψX < ~Y)

Affirmation:

No X are Y

Some X are Not Y

Some Not X are Not Y

I.

Proposition: Some A is B. <=> Some B is Some A <=> Some x contains some y <=> (ЭX < ЭY)

Affirmation:

Some X are Y

Some X are Not Y

Some Y are X

Some Y are Not X

O.

Proposition: Some A is not B. <=> Not B is Some A <=> Not x contains some y <=> (~X < ЭY)

Affirmation:

Some Not X are Y

Some Not X are Not Y

Some Y are Not X

Some Y are X

My explanation to this point has involved a description of how we arrive at a proper statement of first propositions through the combination of specific general affirmations. I have presented this information without attending to any obscurity in the understanding so that the basic pattern of ideas can be attended towards.

However, it will be clear after some further consideration that we are presented with an ambiguity that it is useful to consider so that we have no vagueness in our comprehension.

Our rule of general affirmation is that they do not use the universal operator "all and every".

That is because the universal operator "All" involves two separate affirmations.

The first is "Some" as in "Some X are Y".

The second is "None" as in "No X are Not Y"

In combination "Some X are Y" and "No X are not Y" we obtain the proposition "All X are Y".

It must then be noted that the nature of the two component affirmations is not equivalent.

Since when we say: "Some X are Y" we are affirming what we have evidence for through experiment.

And when we say: "No X are Not Y" we are affirming what we have no evidence for.

That is, we are saying since we have never in all our experimentation discovered an x that is not y, we are extremely confident of making the claim that no x are not y. And since we know from our experimentation many x that are y, we are definite that some x are y. And for this reason we present the combined affirmations to make the true proposition that "All x are y".

However, the problem involves several elements on more that one level so it cannot be solved by only understanding the implications of the experimental aspect.

Because the meaning of "No X are Not Y" is actually that "All Not Y are Not X"

But that would mean we are saying the combined affirmations:

"Some Not Y are Some Not X" and "No Not Y are X"

Which as it happens may be okay, since the apparent circularity is not circular. All we are in fact doing is making the similar affirmation from the position of the other subject.

Now that seems to be the two basic elements of an apparent ambiguity as it exists in regard to the general affirmations of first propositions. Where as it may not describe every aspect of the perceived problem, it is probable that any variations of the problem are each some alternative similar to that which I have described.

The only real problem would be to recognise one aspect of the ambiguity and not know about the other since both aspects coincide at one point.

Just to clarify, the true proposition: "All X are Y" involves the strong experimental claim that "Some X are Y" and the weak experimental claim that "No X are Not Y" the combination of strong and weak claims giving a proposition we assert to be "true".

And the truth holder "No X are Not Y" can be stated as "All Not Y are Not X" which itself involves the combined affirmations "Some Not Y are Not X" and "No Not Y are X".

The next area to consider is the belief that: "Some X are Some Y" involves the necessary implication that "Some X are Not Y" and "Some Y are Not X". Now this is simply the necessary meaning of "Some". Because we remove the necessary implications through one of two methods. Either by changing "Some" to "All" or alternatively by combining the "Some" affirmation with a "No" affirmation applied to the contrary.

Therefore unless we state: "No X are Not Y" and "All X are Y" then the affirmation "Some X are Y" will necessarily involve the truth of the affirmations "Some X are Not Y" and "Some Y are Not X".

Furthermore, where a single proposition involves the combination of several affirmations it may often be the case that a necessary implication of the affirmations is not at all required to provide the given proposition. However the particular affirmation is still a necessary implication of the proposition as given.

For example the use of the proposition "No Y is X" necessarily implies the truth of the affirmation "Some Not X are Not Y" and "Some Not Y are Not X". Therefore we always assert the truth that "Some Not Y Not X exist" whenever we use the proposition "No Y is X". Even though the assertion that "Some Not Y Not X exist" is in no way necessary to provide the proposition "No Y is X". Using the xy/~x~y game we show:

E.

No A is B. <=> All B is Not A <=> All x contains not y <=> (ψX < ~Y)

A black counter on the top right corner and a white counter on the middle left line.

The black counter says: "No X are Y"

The white counter says: "Some X are Not Y" and "Some Not X are Not Y"

If some x are not y and no x are y then all x are not y.

This is the placement of counters on the xy grid to show the "No A is B" structure and it is the only placement that does so.

First Propositions.

Our general affirmations combine to provide the correct structure for the statement of the first propositions used in deductive argument, those propositions all according to the AEIO pattern as follows:

A. All A is B. <=> Some B is All A <=> Some x contains all y <=> (ЭX < ψY)

E. No A is B. <=> All B is Not A <=> All x contains not y <=> (ψX < ~Y)

I. Some A is B. <=> Some B is Some A <=> Some x contains some y <=> (ЭX < ЭY)

O. Some A is not B. <=> Not B is Some A <=> Not x contains some y <=> (~X < ЭY)

It may be thought that a more thorough consideration should be provided in order to demonstrate the proper hardness of truth obtained at the stage of the general affirmations of inductive hypothesis. And considering the matter that I outlined immediately previous to this note, that more attention should be dedicated to clarify how we can make definite assertions as to "All", "None", "Existant" and "Non-exitstant" at the point of general affirmations.

However, the belief is that the harder the evaluation of general affirmations and first propositions at the level of inductive hypothesis the more forcefully that evaluation demonstrates the soft nature of our knowledge at that level. My point is that a hard evaluation of inductive hypothesis can only prove the soft nature of its content.

Since the intention of this article is to offer ideas as to how we may obtain any hard basis for our development of knowledge it would not be reasonable to do that which is contrary to this purpose. That is not to say that the subject matter is not worth attention, only that it is not the focus of the matter of this article. My supposition being that the nature of what is essentially real before we have asserted hypothetically true first propositions is a different subject.

Now of course we will notice that such a subject exists when we find our analysis at the point of general affirmations and first propositions, but that does not require that we are confident of answering what ever the questions of that subject may be. In fact we must be permitted to consider the matter of inductive hypothesis without being required to answer the ontological question. Since all we find ourselves considering is the nature of all, nothing, existant, non-existant and how do we articulate the subject clearly in some way that furthers our comprehension usefully. All of this matter need not be demanded of people who choose to attend to inductive hypothesis. The only requirement is that they do not demand a hardness of truth be conferred on their content, when it is quite clear that the content at this level is by its nature soft.

The Positive Claim of Inductive Hypothesis:

The positive claim of inductive hypothesis is that we can assert hypothetically true first propositions that are coherent non-contradictory combinations of general affirmations. And that our general affirmations have the characteristic of being measured as true or false.

The Truth Hypothesis.

The "truth hypothesis" of this system of ideas is that truth only becomes hard as it is tested through a range of fields. That truth exists on a continuum of degrees of softness through to hardness. That hard truth is only obtained at the level of the cardinal articulation of prime patterns, the general characteristic of such patterns being a quality that we may call "constancy". That the cardinal articulation of a prime pattern will tend to demonstrate truth that meets conditions of universal abstraction, utility, coherence and correspondence. That the earliest sensible manifestation of truth is discovered at the level of general affirmations of inductive hypothesis, where such truth is by its nature entirely soft. The beauty of the general affirmations being that they enable the earliest articulation of some soft truth that may in the course of its development be discovered as part of a coherent system of such truths, the entirety of such being the content of what we call knowledge. Only as the general affirmations become first propositions, which in turn become valid deductive argument does their truth or falsity establish itself. Where we find some argument involves contradiction of what is already proved as true, then such argument has demonstrated itself as false. Where we find some argument coherent with what we already know as true, that strengthens what we already believe as the case, then that argument demonstrates itself as true on grounds of utility as well as correspondence and coherence. When we find an argument that is completely supportive of universal abstraction then such argument meets the conditions of prime patterns, and when properly stated will detail all cardinal articulations. Only at this stage do we obtain truth that is evaluated as hard.

Finally, the idea that the matter of inductive hypothesis suggests, that is very important to consider is that the "quantum indefinite" has a present nature existant at the point just before we say our first proposition. That is, when we consider some first proposition in terms of the general affirmations of which it is composed, it is at this level where we may discover the quantum indefinite to be operational.

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