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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

If what you are after is the index to all the articles in the "Shadowjack" theme, then scroll down to the end of this article, thanks.

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.

12. wikiversity.org/wiki/Deductive_logic

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:

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