Deductive logic
Deductive Logic.
This article provides the key mechanism of the deduction patterns described in the following:
http://en.wikiversity.org/wiki/Truth_value
http://en.wikiversity.org/wiki/Inductive_Hypothesis
http://en.wikiversity.org/wiki/Algebraic_Deduction
http://en.wikiversity.org/wiki/Syllogism
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.
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".
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
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)
(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 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.
The Premisses of Deduction.
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 Conclusions of Deduction.
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.
CA: (ЭC < ψA) (ЭC < ЭA) (ψC < ~A) (~C < ЭA)
The necessary inferences for each conclusion.
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)
The nineteen syllogisms in algebraic form.
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) => (ЭC < ЭA)
16.(ЭB < ψC) ^ (ψA < ~B) => (~C < ЭA)
17.(ЭB < ЭC) ^ (ЭA < ψB) => (ЭC < ЭA)
18.(ψB < ~C) ^ (ЭA < ψB) => (~C < ЭA)
19.(ψB < ~C) ^ (ЭA < ЭB) => (~C < ЭA)
The Four Figures of Deduction.
First Figure.
1. All B is C. All A is B. Therefore All A is C.
2. No B is C. All A is B. Therefore No A is C.
3. All B is C. Some A is B. Therefore Some A is C.
4. No B is C. Some A is B. Therefore Some A is not C.
Second Figure.
1. No C is B. All A is B. Therefore No A is C.
2. All C is B. No A is B. Therefore No A is C.
3. No C is B. Some A is B. Therefore Some A is not C.
4. All C is B. Some A is not B. Therefore Some A is not C.
Third Figure.
1. All B is C. All B is A. Therefore Some A is C.
2. No B is C. All B is A. Therefore Some A is not C.
3. Some B is C. All B is A. Therefore Some A is C.
4. All B is C. Some B is A. Therefore Some A is C.
5. Some B is not C. All B is A. Therefore Some A is not C.
6. No B is C. Some B is A. Therefore Some A is not C.
Fourth Figure.
1. All C is B. All B is A. Therefore Some A is C.
2. All C is B. No B is A. Therefore Some A is not C.
3. Some C is B. All B is A. Therefore Some A is C.
4. No C is B. All B is A. Therefore Some A is not C.
5. No C is B. Some B is A. Therefore Some A is not C.
The Rules of the Four Figures.
Figure 1: B-C, A-B, therefore A-C.
Figure 2: C-B, A-B, therefore A-C.
Figure 3: B-C, B-A, therefore A-C.
Figure 4: C-B, B-A, therefore A-C.
The First Figure is: When the middle term is subject in the major and predicate in the minor.
The Second Figure is: When the middle term is predicate in both premises.
The Third Figure is: When the middle term is subject in both premises.
The Fourth Figure is: When the middle term is predicate in the major premiss and subject in the minor.
Figure One: AAA. EAE. AII. EIO. AAI. EAO.
Figure Two: EAE. AEE. EIO. AOO. EAO. AEO.
Figure Three: AAI. IAI. AII. EAO. OAO. EIO.
Figure Four: AAI. AEE. IAI. EAO. EIO. AEO.
The Rules of the syllogism.
Deduction is two propositions, called the Premisses, and a third proposition known as the Conclusion, which is a necessary consequence of the two premisses.
Two terms are compared with one another by means of a third, which is called the Middle Term.
Let C be the major term; B the middle term; A the minor term.
In the premisses each of the two terms is compared separately with the middle term; and in the conclusion they are compared with one another.
Therefore every deduction consists of three terms, one of which occurs twice in the premisses and does not appear at all in the conclusion.
This term is called the Middle Term.
The predicate of the conclusion is called the Major Term and its subject the Minor Term.
The Major and Minor Terms are called the Extremes, as opposed to the Middle Term.
The premiss in which the Major Term is compared with the Middle is called the Major Premiss.
The premiss in which the Minor Term is compared with the Middle, is called the Minor Premiss.
This is a complete index to the various articles I have used to introduce these and other patterns.
1. http://en.wikiversity.org/wiki/Game_domain_model
2. http://en.wikiversity.org/wiki/Strategic_Context
3. http://en.wikiversity.org/wiki/Thinking_machines
4. http://en.wikiversity.org/wiki/Patterns
5. http://en.wikiversity.org/wiki/Intellect
6. http://en.wikiversity.org/wiki/Boolean_algebra
7. http://en.wikiversity.org/wiki/Zero_unity_and_infinity
8. http://en.wikiversity.org/wiki/Laws_of_Zero
9. http://en.wikiversity.org/wiki/Truth_value
10. http://en.wikiversity.org/wiki/Proving
11. http://en.wikiversity.org/wiki/Error_correction
12. wikiversity.org/wiki/Deductive_logic
13. http://en.wikiversity.org/wiki/Syllogism
14. http://en.wikiversity.org/wiki/Algebraic_Deduction
15. http://en.wikiversity.org/wiki/Inductive_Hypothesis
16. http://en.wikiversity.org/wiki/If_-_Then_Statements
17. http://en.wikiversity.org/wiki/Perfect_Syllogism
18. http://en.wikiversity.org/wiki/Algebraic_Perfect_Syllogism
19. http://en.wikiversity.org/wiki/Logic_patterns
20. http://en.wikiversity.org/wiki/Ontological_Questions
21. http://en.wikiversity.org/wiki/Geometria
22. http://en.wikiversity.org/wiki/Anomaly
Please feel free to copy above hyperlinks to any interested associates, thanks.
The twenty two articles above are basically complete 26th January Twenty Thirteen.
If you want a single complete transcription of the entirety of these twenty two chapters follow hyperlink below:
http://en.wikiversity.org/w/index.php?title=User_talk:Shadowjack&oldid=995340
As you will immediately notice the text has not been separated into paragraphs. However, it is an exact copy of the original articles on the date of 26.1.13.
I would suggest copy to word, change the font to your preferred style, centre everything and then add in the paragraphs.
In fact the work done as "the person mistakenly using the codename Shadowjack in this unfortunate misunderstanding" is now finished. I have generated a new codename "Glimmerguard" which I will use for all new articles. Meaning that all the Shadowjack articles are now entirely past tense.
The Glimmerguard articles are themselves complete and past tense on the date of 31.3.13
And if you want a complete transcription of the Thirty First March Twenty Thirteen articles, this can be found at:
http://en.wikiversity.org/w/index.php?title=User_talk:Glimmerguard&direction=prev&oldid=1022322
The various articles written under the codename Glimmerguard can be found at:
http://en.wikiversity.org/wiki/Quantum_Indefinite
General Disclaimer.
I would like to confirm that all of the twenty two articles referred to are the work of one individual person. And that to the best of my knowledge the work is original. Where I have directly copied some other source I have normally given some sign as to what that source is. Also that some or much of the information that I have provided is not available from any other source that I am aware of. And that where information given here is available from some other source it is often not reduced to the extent that I have done so.
That being said, I would be quite pleased if any other person would feel inclined to develop some of the ideas I have suggested, either within the logic department of wikiversity or elsewhere. If you do so, please do not ask permission for direct use of my work, since I do not have the time to say yes to any person who wants to. That is, do use my work directly if you want to, without permission.
If you do use my work directly, without permission, then I would ask that where possible, you include a hyperlink connection back to this page, so that people can go to the original source material. Where this may not be possible, for example in the sense of the game domain model and the advantage of secrecy, then I would ask that you do not present it as your own work.
Do take a copy and keep it, even if you cannot immediately invest the time in learning what is involved. The important theories are not topical and will be basically the same in a year or ten years time. If you do start working on some area, then have in mind two different timescales, one being what you can do easily in a couple of weeks, and one being the return you can obtain from working with the ideas over a couple of years.
