Logic/Algebraic Deduction
Algebraic Deduction.
1. (ЭC < ψB) ^ (ЭB < ψA) => (ЭC < ψA)
2. (ψC < ~B) ^ (ЭB < ψA) => (ψC < ~A)
3. (ЭC < ψB) ^ (ЭB < ЭA) => (ЭC < ЭA)
4. (ψC < ~B) ^ (ЭB < ЭA) => (~C < ЭA)
5. (ψB < ~C) ^ (ЭB < ψA) => (ψC < ~A)
6. (ЭB < ψC) ^ (ψB < ~A) => (ψC < ~A)
7. (ψB < ~C) ^ (ЭB < ЭA) => (~C < ЭA)
8. (ЭB < ψC) ^ (~B < ЭA) => (~C < ЭA)
9. (ЭC < ψB) ^ (ЭA < ψB) => (ЭC < ЭA)
10. (ψC < ~B) ^ (ЭA < ψB) => (~C < ЭA)
11. (ЭC < ЭB) ^ (ЭA < ψB) => (ЭC < ЭA)
12. (ЭC < ψB) ^ (ЭA < ЭB) => (ЭC < ЭA)
13. (~C < ЭB) ^ (ЭA < ψB) => (~C < ЭA)
14. (ψC < ~B) ^ (ЭA < ЭB) => (~C < ЭA)
15.(ЭB < ψC) ^ (ЭA < ψB) => (ψC < ЭA)
16.(ЭB < ψC) ^ (ψA < ~B) => (~C < ψA)
17.(ЭB < ЭC) ^ (ЭA < ψB) => (ЭC < ЭA)
18.(ψB < ~C) ^ (ЭA < ψB) => (~C < ЭA)
19.(ψB < ~C) ^ (ЭA < ЭB) => (~C < ЭA)
The above notion shows nineteen patterns of algebra. Were one given the above patterns without any explanation as to their meaning it is possible that it would still show itself as worth attention. The approach of this article is to take a line similar to that circumstance combined with a supposition of available supporting information that would not in reality be available were that circumstance the case. For an in depth study of how we arrived at the above pattern, please see: http://en.wikiversity.org/wiki/Syllogism.
The concepts that are elements of the above patterns are only: " C B A ψ Э ~ < () ^ => ".
That is exactly ten sign labels specifically, and only ten sign labels for the entirety of the above pattern.
Three sign labels are the letters: C B A, where each letter signifies a different term, where the terms are in different combinations.
Three sign labels are the quantifiers: ψ Э ~, where each quantifier signifies a different existence value to any one term.
Four sign labels are the functional operators: < () ^ =>, where each functional operator signifies a mechanical process structure.
The ten concepts are the elements that form a total of "18" propositions, and those propositions are only:
CA: (ЭC < ψA) (ЭC < ЭA) (ψC < ~A) (~C < ЭA)
CB: (ЭC < ψB) (ЭC < ЭB) (ψC < ~B) (~C < ЭB)
BA: (ЭB < ψA) (ЭB < ЭA) (ψB < ~A) (~B < ЭA)
BC: (ЭB < ψC) (ЭB < ЭC) (ψB < ~C)
AB: (ЭA < ψB) (ЭA < ЭB) (ψA < ~B)
The eighteen propositions that form the entirety of the above patterns are themselves various forms of only four specific structural frames. The four structural frames are only:
1. Some x contains all y.
2. All x contains not y.
3. Some x contains some y.
4. Not x contains some y.
The totality of the nineteen algebraic patterns give only four possible conclusions, where the four possible conclusions are only the four structural frames detailed. The four possible conclusions are:
1. (ЭC < ψA) :: Some x contains all y.
2. (ψC < ~A) :: All x contains not y.
3. (ЭC < ЭA) :: Some x contains some y.
4. (~C < ЭA) :: Not x contains some y.
Each of the four conclusions may be given by several specific patterns of premisses, as follows:
1.
(ЭC < ψA)
<= (ЭC < ψB) ^ (ЭB < ψA)
2.
(ψC < ~A)
<= (ψC < ~B) ^ (ЭB < ψA)
v (ψB < ~C) ^ (ЭB < ψA)
v (ЭB < ψC) ^ (ψB < ~A)
3.
(ЭC < ЭA)
<= (ЭC < ψB) ^ (ЭB < ЭA)
v (ЭC < ψB) ^ (ЭA < ψB)
v (ЭC < ψB) ^ (ЭA < ЭB)
v (ЭC < ЭB) ^ (ЭA < ψB)
v (ЭB < ψC) ^ (ЭA < ψB)
v (ЭB < ЭC) ^ (ЭA < ψB)
4.
(~C < ЭA)
<= (ψC < ~B) ^ (ЭB < ЭA)
v (ψB < ~C) ^ (ЭB < ЭA)
v (ЭB < ψC) ^ (~B < ЭA)
v (ψC < ~B) ^ (ЭA < ψB)
v (~C < ЭB) ^ (ЭA < ψB)
v (ψC < ~B) ^ (ЭA < ЭB)
v (ЭB < ψC) ^ (ψA < ~B)
v (ψB < ~C) ^ (ЭA < ψB)
v (ψB < ~C) ^ (ЭA < ЭB)
That means the first conclusion structural frame is only given by one structure of premiss.
The second conclusion structural frame is given by three structure of premiss.
The third conclusion structural frame is given by six structure of premiss.
The fourth conclusion structural frame is given by nine structure of premiss.
Method of Proof By Analogy.
Using the method called proof by analogy we find that the model provided in the article "syllogism" gives "note III" as the structural patterns AEIO. Since the structural patterns AEIO are fundamentally tested we heuristically feel comfortable using them as a measure of correctness for any other similar pattern that may be discovered to exist. So then we notice that the algebraic patterns give four specific structural frames, which we have detailed as x and y patterns. When we then demonstrate that the xy patterns exactly fit the shape of our trusted AEIO patterns such that we can even provide them as an exact equivalence, we are suddenly more confident that our workings out are accurate and correct. Therefore we provide the analogy pattern showing equivalence of shape and meaning between our algebraic structural frames and 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)
Now because the AEIO patterns are considered fundamentally important to the nature of Silly Gisms, we are compelled to properly attend to the algebraic structural frames of x and y. And that makes sense because it requires we use Dodgson's Game, which we would expect since it is a necessary device at the deep structural level of individual premiss. This will be referred to as the "XY/~X~Y Game" in order to notify ourselves that it is only operational at the level of XY deep structure in regard to the component elements of individual proposition.
The XY/~X~Y Game.
The format which will be familiar to people who have read the syllogism article is:
Vertical line bisected by an horizontal line.
Vertical line where the upper point is called X, and the lower point is called ~X.
Horizontal line where the right point is called Y, and the left point is called ~Y.
Giving four boxes designated as XY, ~XY, ~Y~X, X~Y.
The use of a white counter designates existant, black counter designates not-existant.
Using this XY/~X~Y Game we now have to show the correct placement of counters on the grid to provide the structure of our propositions.
A.
All A is B. <=> Some B is All A <=> Some x contains all y <=> (ЭX < ψY)
A white counter on the upper line and a black counter in the bottom right corner box.
The white counter says: "Some X are Y" and "Some X are Not Y".
The black counter says: "No Y are Not X".
If some x are y and no y are not x then some x are all y.
This is placement of counters on the xy grid to show the "All A is B" structure and it is the only placement that so does.
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.
I.
Some A is B. <=> Some B is Some A <=> Some x contains some y <=> (ЭX < ЭY)
A white counter on the upper line and a white counter on the middle right line.
The white upper line counter says: "Some X are Y" and "Some X are Not Y"
The white middle right line counter says: "Some Y are X" and "Some Y are Not X"
If some x are y and some y are x then some xy exist.
This is the placement of counters on the xy grid to show the "Some A is B" structure and it is the only placement that does so.
O.
Some A is not B. <=> Not B is Some A <=> Not x contains some y <=> (~X < ЭY)
A white counter on the lower line and a white counter on the middle right line.
The white lower counter says: "Some Not X are Y" and "Some Not X are Not Y"
The white middle line counter says: "Some Y are Not X" and "Some Y are X"
If some not x are y and some y are not x then some notxy exist.
This is the placement of counters on the xy grid to show the "Some A is Not B" structure and it is the only placement that does so.
The above four xy/~x~y structural patterns have been given in a manner that categorically claims that as given they are correct and accurate. My reason for doing so, is because the correct statement must be given in that way. However, at this stage we must suppose that the patterns as given may be simply wrong, or at least partially inaccurate. And until we have checked our workings out we cannot suppose that as given the patterns are correct. Merely that the style of presentation has been as if they are. We are now required to challenge that matter in order to determine what if any are the alternative placement of counters, and is there a better placement of counters that would more correctly state our four propositions?
As the Japanese Zen Priest said: "Reading other people's work without doing any work of your own is like counting other people's money without having any money of your own."
For this reason we should bring in another part of the model for the syllogism in order to establish whether we can find any contradictions in the above placement of counters as given. If we find no contradiction then we may be sure that the patterns as given are correct, since we already think that they are. If one of the available alternative placement of counters seemed more correct then we would have used those instead, but they don't. If we find any contradiction then we would know to find what is the better placement of counters. The model we can use is as follows:
If A be true then: E is false, O false, I true.
If A be false then: E is unknown, O true, I unknown.
If E be true then: O is true, I false, A false.
If E be false then: O is unknown, I true, A unknown.
If O be true then: I is unknown, A false, E unknown.
If O be false then: I is true, A true, E false.
If I be true then: A is unknown, E false, O unknown.
If I be false then: A is false, E true, O true.
(A) All A is B, therefore No A is not-B.(E)
(E) No A is B, therefore All A is not-B.(A)
(I) Some A is B, therefore Some A is not B. (O)
(O) Some A is not B, therefore Some A is B. (I)
If we take the placement of counters as given, since to the best of our understanding at this stage, they are the correct placement for the particular four propositions that we are referring towards. Then a further way we can check our understanding is to clearly state what that placement of counters affirms as the case. If those clearly stated affirmations confirm what our original premisses are believed to mean, then we have our coherent model without contradiction.
The Affirmations of our Propositions.
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
The Sixteen Affirmations:
1. Some X are Y
2. Some X are Not Y
3. Some Y are X
4. Some Y are Not X
5. Some Not X are Y
6. Some Not Y are X
7. Some Not X are Not Y
8. Some Not Y are Not X
9. No X are Y
10. No Y are X
11. No Y are Not X
12. No Not X are Y
13. No X are Not Y
14. No Not Y are X
15. No Not X are Not Y
16. No Not Y are Not X
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".
Having detailed our analysis of algebraic deduction to the point of the Sixteen Affirmations we find ourselves at a different unknown. The different unknown is actually a commonplace understanding of Inductive Hypothesis. For this reason, within the study of deduction we are not required to investigate the nature of our "General Affirmations", a matter that will be properly considered in a different article related to the truth-holders of correct hypothesis within the field of induction: http://en.wikiversity.org/wiki/Inductive_Hypothesis.
Since that completes the entire analysis of the algebraic deduction patterns it cannot be thought too long. That is, whilst the given analysis is extensive and requiring of concentrated attention to follow, it is also of specifically limited size. Furthermore, it does not change with repeated study. Where there may be important error in the statement of some detail of any particular pattern, then that only means ongoing correction towards accuracy over time. Also, if some error is found that is not a mistake in statement but rather a mistake in analysis, then that also is simply a matter of making whatever the individual correction must involve. Any small error in a matter such as this must always be treated as basically important because it is the nature of the matter that it is minutely realised.
That being said, we are now at the point, given the two articles, "syllogism" and "algebraic deduction" where all of the parameters for this logic of inference are described. As such, it is very easy, not difficult. Where much information may be immediately acquired on first reading and soon assimilated by handwriting the basic patterns in a small notebook, a depth of knowledge will be the inevitable product of reviewing the essential points over a longer duration of time.
The only area that presumably remains to be considered is how an entirely abstract explanation of rational logic such as this can have any applicable utility. My first answer is that it need not. That simply as a discipline of mental development it will be discovered by any person who does work with it to be of value in itself.
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